Fredrik Marinus Kruseman (1816-1882 ) was a Dutch painter.
The topics are
- Spot Market
- Riskless Portfolio Method
- Sensitivity Analysis
- Futures Market
- Option on a Dividend-Paying Stock
- Stock Price Volatility
We make throughout the following basic assumptions concerning market activities: trading takes place continuously in time, and unrestricted borrowing and lending of funds is possible at the same constant interest rate. Furthermore, the market is frictionless, meaning that there are no transaction costs or taxes, and no discrimination against the short sale. Finally, unless explicitly stated otherwise, we will assume that a stock which underlies an option does not pay dividends (at least during the option’s lifetime).
\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}
Spot Market.
Let us describe stochastic processes which model the prices of primary securities, a common stock and a risk-free bond. We take a geometric Brownian motion as a stochastic process which models the stock price. More specifically, the evolution of the stock price process \(S\) is assumed to be described by the following linear stochastic differential equation
\begin{equation}{\label{museq51}}\tag{1}
dS_{t}=\mu\cdot S_{t}dt+\sigma\cdot S_{t}dW_{t},
\end{equation}
where \(\mu\in \mathbb{R}\) is a constant appreciation rate of the stock price, \(\sigma >0\) is a constant volatility coefficient, and \(S_{0}\in \mathbb{R}_{+}\) is the initial stock price. Finally, \(W_{t}\) for \(t\in [0,T]\) stands for a one-dimensional standard Brownian motion defined on a filtered probability space \((\Omega ,\{{\cal F}_{t}\}_{t\in [0,T]},P)\). Let us emphasize that (\ref{museq51}) is merely a shorthand notation for the following Ito integral equation
\[S_{t}=S_{0}+\int_{0}^{t}\mu\cdot S_{s}ds+\int_{0}^{t}\sigma\cdot S_{s}dW_{s}\mbox{ for }t\in [0,T].\]
It is convenient to assume that the underlying filtration \(\{{\cal F}_{t}\}_{t\in [0,T]}\) is a standard augmentation of the natural filtration of the underlying Brownian motion, i.e., that the equality \({\cal F}_{t}={\cal F}_{t}^{W}\) holds for every \(t\in [0,T]\). This assumption is not essential when the aim is to value European options written on a stock \(S\). If this condition were not satisfied, the uniqueness of the martingale measure, and thus also the completeness of the market would fail to hold in general. However this would not affect the arbitrage valuation of standard options written on a stock \(S\).
It is elementary to check, using Ito’s formula, that the process which equals
\begin{equation}{\label{museq52}}\tag{2}
S_{t}=S_{0}\cdot\exp\left (\sigma\cdot W_{t}+\left (\mu-\frac{1}{2}\sigma^{2}\right )\cdot t\right )\mbox{ for all }t\in [0,T]
\end{equation}
is indeed a solution of (\ref{museq51}) starting from \(S_{0}\) at time \(0\). The uniqueness of a solution is an immediate consequence of a general result due to Ito, which states that a SDE with Lipschitz continuous coefficients has a unique solution. It is apparent from (\ref{museq52}) that the stock returns are lognormal, meaning that the random variable \(\ln (S_{t}/S_{s})\), under \(\mathbb{P}\), has a Gaussian probability distribution for any choice of dates \(s\leq t\leq T\). Since for any fixed \(t\), the random variable \(S_{t}=f(W_{t})\) for some invertible function \(f:\mathbb{R}\rightarrow\mathbb{R}_{+}\), it is clear that we have
\[{\cal F}_{t}^{W}=\sigma\{W_{s}:s\leq t\}=\sigma\{S_{s}:s\leq t\}={\cal F}_{t}^{S}.\]
Therefore, the filtration generated by the stock price coincides with the natural filtration of the underlying noise process \(W\), and thus \({\cal F}_{t}^{W}={\cal F}_{t}^{S}={\cal F}_{t}\). This means that the information structure of the model is based on observation of the stock price process only. Moreover, it is worthwhile to observe that the stock price \(S\) follows a time-homogeneous Markov process under \(\mathbb{P}\) with respect to the filtration \(\{{\cal F}_{t}\}_{t\in [0,T]}\). In particular, we have
\begin{align*}
\mathbb{E}_{\mathbb{P}}[S_{s}|{\cal F}_{t}] & =\mathbb{E}_{\mathbb{P}}[S_{s}|{\cal F}_{t}^{S}]\\
& =\mathbb{E}_{\mathbb{P}}[S_{s}|S_{t}]\mbox{(by Markov property with \({\cal F}_{t}^{S}=\sigma\{S_{s}:s\leq t\}\))}\\
& =S_{t}\cdot\exp\left (\mu (s-t)\right )
\end{align*}
for every \(t\leq s\leq T\). This follows from the fact that
\begin{equation}{\label{museq53}}\tag{3}
S_{s}=S_{t}\cdot\exp\left (\sigma\cdot (W_{s}-W_{t})+\left (\mu-\frac{1}{2}\sigma^{2}\right )\cdot (s-t)\right )
\end{equation}
and the increment \(W_{s}-W_{t}\) of the Borwnian motion \(W\) is independent of the \(\sigma\)-field \({\cal F}_{t}\) with the Gaussian law \(N(0,\sqrt{s-t})\).
The second security, whose price process is denoted by \(B\), represents in this model an accumulation factor corresponding to a savings account (known as a money market account). We assume throughout that the so-called short-term interest rate \(r\) is constant over the trading interval \([0,T]\). The risk-free security is assumed to continuously compound in value at the rate \(r\); that is,
\begin{equation}{\label{museq54}}\tag{4}
dB_{t}=r\cdot B_{t}dt, \mbox{ or equivalently }B_{t}=e^{rt}\mbox{ for all }t\in [0,T]
\end{equation}
as, by convention, we take \(B_{0}=1\).
Self-Financing Strategies.
By a trading strategy, we mean a pair \(\boldsymbol{\phi}=(\phi^{(1)},\phi^{(2)})\) of progressively measurable stochastic processes on the underlying probability space \((\Omega ,\{{\cal F}_{t}\}_{t\in [0,T]},\mathbb{P})\). The concept of a self-financing trading strategy in the Black-Scholes market is formally based on the notion of the Ito integral. Intuitively, such a choice of stochastic integral is supported by the fact that, in the case of the Ito integral (as opposed to the Fisk-Stratonovich integral), the underlying process is integrated in a predictable way, meaning that we take its values on the left-hand end of each (infinitesimal) time interval. Formally, we say that a trading strategy \(\boldsymbol{\phi}= (\phi^{(1)},\phi^{(2)})\) over the time interval \([0,T]\) is self-financing if its wealth process \(V(\boldsymbol{\phi})\), which is set to equal
\[V_{t}(\boldsymbol{\phi})=\phi^{(1)}_{t}\cdot S_{t}+\phi^{(2)}_{t}\cdot B_{t}\mbox{ for all }t\in [0,T],\]
satisfies the following condition
\begin{equation}{\label{museq56}}\tag{5}
V_{t}(\boldsymbol{\phi})=V_{0}(\boldsymbol{\phi})+\int_{0}^{t}\phi^{(1)}_{s}\cdot dS_{s}+\int_{0}^{t}\phi^{(2)}_{s}dB_{s}
\mbox{ for all }t\in [0,T],
\end{equation}
where the first integral is understood in the Ito sense. It is implicitly assumed that both integrals on the right-hand side of (\ref{museq56}) are well-defined. It is well-known that a sufficient condition for this is that
\begin{equation}{\label{museq57}}\tag{6}
\mathbb{P}\left\{\int_{0}^{T}(\phi^{(1)}_{s})^{2}ds<\infty\right\}=1\mbox{ and }
\mathbb{P}\left\{\int_{0}^{T}|\phi^{(2)}_{s}|ds<\infty\right\}=1.
\end{equation}
(Note that condition (\ref{museq57}) is invariant with respect to an equivalent change of a probability measure.) We denote by \(\boldsymbol{\Phi}\) the class of all self-financing trading strategies. There is an example shows that the arbitrage opportunities are not excluded a priori from the class of self-financing trading strategies.
Martingale Measure for the Spot Market.
A probability measure \(\bar{\mathbb{P}}\) on \((\Omega ,{\cal F}_{T})\), which is equivalent to \(\mathbb{P}\), is called a martingale measure for the discounted stock price process \(S^{*}\) when \(S^{*}\) is a local martingale under \(\bar{\mathbb{P}}\). Similarly, a probability measure \(\mathbb{P}^{*}\) is said to be a martingale measure for the spot market (or briefly, a spot martingale measure) when the discounted wealth of any self-financing trading strategy follows a local martingale under \(\mathbb{P}^{*}\). The following result shows that both notions coincide.
Proposition. A probability measure is a spot martingale measure if and only if it is a martingale measure for the discounted stock price \(S^{*}\).
Proof. The proof relies on the following equality, which easily follows from the Ito formula
\[V_{t}^{*}(\boldsymbol{\phi})=V_{0}^{*}(\boldsymbol{\phi})+\int_{0}^{t}\phi^{(1)}_{s}dS_{s}^{*}\mbox{ for all }t\in [0,T],\]
where \(V_{t}^{*}(\boldsymbol{\phi})=V_{t}(\boldsymbol{\phi})/B_{t}\) and \(\boldsymbol{\phi}\) is a self-financing strategy up to
time \(T\). It is now sufficient to make use of the local martingale property of the Ito stochastic integral. \(\blacksquare\)
Proposition. The unique martingale measure \(\bar{\mathbb{P}}\) for the discounted stock price process \(S^{*}\) is given by the Radon-Nikodym derivative
\begin{equation}{\label{museq58}}\tag{7}
\frac{d\bar{\mathbb{P}}}{d\mathbb{P}}=\exp\left (\frac{r-\mu}{\sigma}\cdot W_{T}-
\frac{1}{2}\cdot\frac{(r-\mu )^{2}}{\sigma^{2}}\cdot T\right )\mbox{ \(\mathbb{P}\)-a.s.}.
\end{equation}
Under the martingale measure \(\bar{\mathbb{P}}\), the discounted stock price \(S^{*}\) satisfies
\begin{equation}{\label{museq59}}\tag{8}
dS_{t}^{*}=\sigma\cdot S_{t}^{*}\cdot dW_{t}^{*},
\end{equation}
and the process \(W^{*}\) which equals
\begin{equation}{\label{musaeq259}}\tag{9}
W_{t}^{*}=W_{t}-\frac{r-\mu}{\sigma}\cdot t,
\end{equation}
for all \(t\in [0,T]\), follows a standard Brownian motion on a probability space under probability measure \(\bar{\mathbb{P}}\).
Proof. Essentially, all statements are direct consequence of Girsanov’s theorem. \(\blacksquare\)
Combining the above two propositions, we conclude that the unique spot martingale measure \(\mathbb{P}^{*}\) is given on \((\Omega ,{\cal F}_{T})\) by means of the Radon-Nikodym derivative
\[\frac{d\mathbb{P}^{*}}{d\mathbb{P}}=\exp\left (\frac{r-\mu}{\sigma}\cdot W_{T}-\frac{1}{2}\cdot\frac{(t-\mu )^{2}}{\sigma^{2}}\cdot T\right )\mbox{$\mathbb{P}$-a.s.}\]
The discounted stock price \(S^{*}\) follows, under \(\mathbb{P}^{*}\), a strictly positive martingale, since (clearly \(S_{0}=S_{0}^{*}\))
\begin{equation}{\label{museq511}}\tag{10}
S_{t}^{*}=S_{0}\cdot\exp\left (\sigma\cdot W_{t}^{*}-\frac{1}{2}\cdot\sigma^{2}\cdot t\right )
\end{equation}
for every \(t\in [0,T]\) (equation (\ref{museq511}) follows from equations (\ref{museq52}) and (\ref{museq59})). Notice also that in view of (\ref{musaeq259}), we have
\[\sigma dW_{t}=\sigma dW_{t}^{*}+(r-\mu )dt.\]
Then, the dynamics of the stock price \(S\) under \(\mathbb{P}^{*}\) are
\begin{equation}{\label{museq512}}\tag{11}
dS_{t}=r\cdot S_{t}dt+\sigma\cdot S_{t}dW_{t}^{*}\mbox{ for }S_{0}>0,
\end{equation}
and thus the stock price at time \(t\) equals
\begin{equation}{\label{museq513}}\tag{12}
S_{t}=S_{0}\cdot\exp\left (\sigma\cdot W_{t}^{*}+\left (r-\frac{1}{2}\cdot\sigma^{2}\right )\cdot t\right ).
\end{equation}
Finally, it is useful to observe that all filtrations involved in the model coincide; that is,
\[{\cal F}_{t}={\cal F}_{t}^{W}={\cal F}_{t}^{W^{*}}={\cal F}_{t}^{S}={\cal F}_{t}^{S^{*}}.\]
We are in a position to introduce the class of admissible trading strategies. Indeed, an unconstrained Black-Scholes market model would involve arbitrage opportunities, so that reliable valuation of derivative instruments would not be possible.
Definition. A trading strategy \(\boldsymbol{\phi}\in\boldsymbol{\Phi}\) is called \(\mathbb{P}^{*}\)-admissible when the discounted wealth process
\[V_{t}^{*}(\boldsymbol{\phi})=\frac{V_{t}(\boldsymbol{\phi})}{B_{t}}\mbox{ for all }t\in [0,T]\]
follows a martingale under \(\mathbb{P}^{*}\). We write \(\boldsymbol{\Phi}(\mathbb{P}^{*})\) to denote the class of all \(\mathbb{P}^{*}\)-admissible trading strategies. The triple \({\cal M}_{BS}=(S,B,\boldsymbol{\Phi}(\mathbb{P}^{*}))\) is called the arbitrage-free Black-Scholes model of a financial market, or briefly, the Black-Scholes market. \(\sharp\)
It is not hard to check that by restricting the attention to the class of \(\mathbb{P}^{*}\)-admissible strategies, we have guaranteed the absence of arbitrage opportunities in the Black-Scholes market. Consequently, given a contingent claim \(X\) which settles at time \(T\) and is attainable (i.e. it can be replicated by means of a \(\mathbb{P}^{*}\)-admissible strategy) we can uniquely define its arbitrage price \(\Pi_{t}(X)\) as the wealth \(V_{t}(\boldsymbol{\phi})\) at time \(t\) of any \(\mathbb{P}^{*}\)-admissible trading strategy \(\boldsymbol{\phi}\) which replicates \(X\); that is, satisfies \(V_{T}(\boldsymbol{\phi})=X\). If no replicating \(\mathbb{P}^{*}\)-admissible strategy exists, the arbitrage price of such a claim is not defined.
\begin{equation}{\label{musc511}}\tag{13}\mbox{}\end{equation}
Proposition \ref{musc511}. Let \(X\) be a \(\mathbb{P}^{*}\)-attainable European contingent claim which settles at time \(T\). Then, the arbitrage price \(\Pi_{t}(X)\) at time \(t\in [0,T]\) in \({\cal M}_{BS}\) is given by the risk-neutral valuation formula
\begin{equation}{\label{museq515}}\tag{14}
\Pi_{t}(X)=B_{t}\cdot E_{\mathbb{P}^{*}}\left [\left .\frac{X}{B_{T}}\right |{\cal F}_{t}\right ]\mbox{ for all }t\in [0,T].
\end{equation}
In particular, the price of \(X\) at time \(0\) equals
\[\Pi_{0}(X)=E_{\mathbb{P}^{*}}\left [\frac{X}{B_{T}}\right ]. \sharp\]
The Black-Scholes Option Valuation Formula.
Two alternative justifications of the option valuation formula are provided. The first approach is usually referred to as the risk-free portfolio method, and the second one is known as the equilibrium derivation of the Black-Scholes formula.
For concreteness, we shall first consider a European call option written on a stock \(S\) with expiry date \(T\) and strike price \(K\). Let the function \(c:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) be given by the formula
\begin{equation}{\label{museq516}}\tag{15}
c(s,t)=s\cdot N(d_{1}(s,t))-K\cdot e^{-rt}\cdot N(d_{2}(s,t)),
\end{equation}
where
\begin{equation}{\label{museq517}}\tag{16}
d_{1}(s,t)=\frac{\ln s-\ln K+(r+\frac{\sigma^{2}}{2})\cdot t}{\sigma\cdot\sqrt{t}}\mbox{ and }d_{2}(s,t)=d_{1}(s,t)-\sigma\cdot\sqrt{t}.
\end{equation}
Furthermore, \(N\) stands for the standard Gaussian cumulative distribution function
\[N(x)=\frac{1}{\sqrt{\pi}}\cdot\int_{-\infty}^{x}e^{-z^{2}/2}dz\mbox{ for all }x\in \mathbb{R}.\]
We adopt the following notational convention
\[d_{1,2}(s,t)=\frac{\ln s-\ln K+(r\pm\frac{\sigma^{2}}{2})\cdot t}{\sigma\cdot\sqrt{t}}.\]
Let us denote by \(C_{t}\) the arbitrage price of a European call option at time \(t\) in the Black-Scholes market.
\begin{equation}{\label{must511}}\tag{17}\mbox{}\end{equation}
Theorem \ref{must511}. The arbitrage price at time \(t\in [0,T]\) of the European call option with expiry date \(T\) and strike price \(K\) in the Black-Scholes market is given by the formula
\[C_{t}=c(S_{t},T-t)\mbox{ for all }t\in [0,T],\]
where the function \(c:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) is given by (\ref{museq516}) and (\ref{museq517}). Moreover, the unique \(\mathbb{P}^{*}\)-admissible replicating strategy \(\boldsymbol{\phi}\) of the call option satisfies
\[\phi^{(1)}_{t}=\frac{\partial c}{\partial s}(S_{t},T-t)\mbox{ and }
\phi_{t}^{(2)}=e^{-rt}\cdot\left (c(S_{t},T-t)-\phi_{t}^{(1)}\cdot S_{t}\right )\]
for every \(t\in [0,T]\).
Proof. We provide two alternative proofs of the Black-Scholes result. The first method relies on the direct determination of the replicating strategy. Therefore, it gives not only the valuation formula (this requires solving the Black-Scholes PDE (\ref{museq526})), but also explicit formulas for the replicating strategy. The second method makes direct use of the risk-neutral valuation formula (\ref{museq515}). It focuses on the explicit computation of the arbitrage price of the option, rather than on the derivation of the hedging strategy.
First method. We start by assuming that the option price \(C_{t}\) satisfies the equality \(C_{t}=v(S_{t},t)\) for some function \(v(s,t):\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\). We may assume that the replicating strategy \(\boldsymbol{\phi}\) we are looking for has the following form
\[\boldsymbol{\phi}_{t}=(\phi^{(1)}_{t},\phi^{(2)}_{t})=(g(S_{t},t),h(S_{t},t))\]
for \(t\in [0,T]\), where \(g,h:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) are unknown functions. Since \(\boldsymbol{\phi}\) is assumed to be self-financing, the wealth process \(V(\boldsymbol{\phi})\), which equals
\begin{equation}{\label{museq522}}\tag{18}
V_{t}(\boldsymbol{\phi})=g(S_{t},t)\cdot S_{t}+h(S_{t},t)\cdot B_{t}=v(S_{t},t),
\end{equation}
needs to satisfy the following equality
\begin{equation}{\label{museq2031}}\tag{19}
dV_{t}(\boldsymbol{\phi})=g(S_{t},t)\cdot dS_{t}+h(S_{t},t)\cdot dB_{t}.
\end{equation}
From (\ref{museq522}), we have
\begin{equation}{\label{museq2032}}\tag{20}
h(S_{t},t)=\frac{v(S_{t},t)-g(S_{t},t)\cdot S_{t}}{B_{t}}.
\end{equation}
From (\ref{museq51}), (\ref{museq54}) and (\ref{museq2032}), the equality (\ref{museq2031}) can be given the following form
\begin{equation}{\label{museq523}}\tag{21}
dV_{t}(\boldsymbol{\phi})=(\mu-r)\cdot S_{t}\cdot g(S_{t},t)dt+\sigma\cdot S_{t}\cdot g(S_{t},t)dW_{t}+r\cdot v(S_{t},t)dt.
\end{equation}
We shall searach for the wealth function \(v\) in the class of smooth functions on the open domain \({\cal D}=(0,+\infty )\times (0,T)\); more exactly, we assume that \(v\in C^{2,1}({\cal D})\). An application of Ito’s formula yields
\begin{equation}{\label{museq2033}}\tag{22}
dv(S_{t},t)=\left (v_{t}(S_{t},t)+\mu\cdot S_{t}\cdot v_{s}(S_{t},t)+
\frac{1}{2}\cdot\sigma^{2}\cdot S_{t}^{2}\cdot v_{ss}(S_{t},t)\right )dt+\sigma\cdot S_{t}\cdot v_{s}(S_{t},t)dW_{t}.
\end{equation}
Let \(Y_{t}=v(S_{t},t)-V_{t}(\boldsymbol{\phi})\). Combining (\ref{museq2033}) with (\ref{museq523}), we get
\begin{align}
dY_{t} & =dv(S_{t},t)-dV_{t}(\boldsymbol{\phi})\nonimber\\
& =\left (v_{t}(S_{t},t)+\mu\cdot S_{t}\cdot v_{s}(S_{t},t)+
\frac{1}{2}\cdot\sigma^{2}\cdot S_{t}^{2}\cdot v_{ss}(S_{t},t)\right )dt+\sigma\cdot S_{t}\cdot v_{s}(S_{t},t)dW_{t}\nonumber\\
& +(r-\mu)\cdot S_{t}\cdot g(S_{t},t)dt-\sigma\cdot S_{t}\cdot g(S_{t},t)dW_{t}-r\cdot v(S_{t},t)dt.\label{museq2034}\tag{23}
\end{align}
On the other hand, in view of (\ref{museq522}), \(Y\) vanishes identically, thus \(dY_{t}=0\). By virtue of the uniqueness of canonical decomposition of continuous semimartingales, the diffusion term in the above decomposition of \(Y\) vanishes. In our case, this means that for every \(t\in [0,T]\), we have
\begin{equation}{\label{museq524}}\tag{24}
\int_{0}^{t}\sigma\cdot S_{s}\cdot\left (g(S_{s},s)-v_{s}(S_{s},s)\right )
dW_{s}=0,\mbox{ or equivalently }\int_{0}^{t}S_{s}^{2}\cdot\left (g(S_{s},s)-v_{s}(S_{s},s)\right )^{2}ds=0.
\end{equation}
For (\ref{museq524}) to hold, it is sufficient and necessary that the function \(g\) satisfies
\begin{equation}{\label{museq525}}\tag{25}
g(s,t)=v_{s}(s,t)\mbox{ for all }(s,t)\in \mathbb{R}_{+}\times [0,T].
\end{equation}
We shall assume from now on that (\ref{museq525}) holds. Using (\ref{museq525}), from (\ref{museq2034}), we can get the representation for \(Y\) as follows (the \(dW_{t}\) will disappear)
\[Y_{t}=\int_{0}^{t}\left (v_{t}(S_{s},s)+\frac{1}{2}\cdot\sigma^{2}\cdot
S_{s}^{2}\cdot v_{ss}(S_{s},s)+r\cdot S_{s}\cdot v_{s}(S_{s},s)-r\cdot v(S_{s},s)\right )ds.\]
Therefore, it is apparent that \(Y\) vanishes whenever \(v\) satisfies the following partial differential equation, referred to as the Black-Scholes PDE
\begin{equation}{\label{museq526}}\tag{26}
v_{t}(s,t)+\frac{1}{2}\cdot\sigma^{2}\cdot s^{2}\cdot v_{ss}(s,t)+r\cdot s\cdot v_{s}(s,t)-r\cdot v(s,t)=0.
\end{equation}
Since \(C_{T}=v(S_{T},T)=(S_{T}-K)^{+}\), we need to impose also the terminal condition \(v(s,T)=(s-K)^{+}\) for \(s\in \mathbb{R}_{+}\). It is not hard to check that by direct computation that the function \(v(s,t)=c(s,T-t)\), where \(c\) is given by (\ref{museq516}) and (\ref{museq517}), actually solves this problem. It thus remains to check that the replicating strategy
$\boldsymbol{\phi}$, which equals
\[\phi^{(1)}_{t}=g(S_{t},t)=v_{s}(S_{t},t)\mbox{ and }\phi^{(2)}_{t}=h(S_{t},t)=\frac{v(S_{t},t)-g(S_{t},t)\cdot S_{t}}{B_{t}},\] is
$\mathbb{P}^{*}$-admissible. Let us first check that \(\boldsymbol{\phi}\) is indeed self-financing. We need to check
\[dV_{t}(\boldsymbol{\phi})=\phi^{(1)}_{t}\cdot dS_{t}+\phi^{(2)}_{t}\cdot dB_{t}.\]
Since \(V_{t}(\boldsymbol{\phi})=\phi^{(1)}_{t}\cdot S_{t}+\phi^{(2)}_{t}\cdot B_{t}=v(S_{t},t)\), by applying Ito’s formula, we get
\[dV_{t}(\boldsymbol{\phi})=v_{s}(S_{t},t)dS_{t}+\frac{1}{2}\cdot
\sigma^{2}\cdot S_{t}^{2}\cdot v_{ss}(S_{t},t)dt+v_{t}(S_{t},t)dt.\]
In view of (\ref{museq526}), the last equality can also be given the following form
\[dV_{t}(\boldsymbol{\phi})=v_{s}(S_{t},t)dS_{t}+r\cdot v(S_{t},t)dt-r\cdot S_{t}\cdot v_{s}(S_{t},t)dt,\]
and thus
\[dV_{t}(\boldsymbol{\phi})=\phi^{(1)}_{t}\cdot dS_{t}+r\cdot B_{t}\cdot\frac{v(S_{t},t)-S_{t}\cdot\phi^{(1)}_{t}}{B_{t}}dt=
\phi^{(1)}_{t}dS_{t}+\phi^{(2)}_{t}dB_{t}.\]
This ends the verification of the self-financing property. In view of the definition of admissibility of trading strategies, we need to verify that the discounted wealth process \(V^{*}(\boldsymbol{\phi})\), which satisfies
\[V_{t}^{*}(\boldsymbol{\phi})=V_{0}^{*}(\boldsymbol{\phi})+\int_{0}^{t}v_{s}(S_{s},s)dS_{s}^{*}\]
follows a martingale under the martingale measure \(\mathbb{P}^{*}\). By direct computation, we obtain \(v_{s}(s,t)=N(d_{1}(s,T-t))\) for every \((s,t)\in\mathbb{R}_{+}\times [0,T]\), and thus, using also (\ref{museq59}), we find
\[V_{t}^{*}(\boldsymbol{\phi})=V_{0}^{*}(\boldsymbol{\phi})+\int_{0}^{t}\sigma\cdot S_{s}\cdot N(d_{1}(S_{s},T-s))dW_{s}^{*}=
V_{0}^{*}(\boldsymbol{\phi})+\int_{0}^{t}\eta_{s}dW_{s}^{*},\]
where \(\eta_{s}=\sigma\cdot S_{s}\cdot N(d_{1}(S_{s},T-s))\). The existence of the stochastic integral is an immediate consequence of the sample path continuity of the process \(\eta\). From the general properties of the Ito stochastic integral, it is thus clear that the discounted wealth \(V^{*}(\boldsymbol{\phi})\) follows a local martingale under \(\mathbb{P}^{*}\). To show that \(V^{*}(\boldsymbol{\phi})\) is a genuine martingale, it is enough to observe that
\[\mathbb{E}_{\mathbb{P}^{*}}\left [\left (\int_{0}^{T}\eta_{s}dW_{s}^{*}\right )^{2}\right ]=
\mathbb{E}_{\mathbb{P}^{*}}\left [\int_{0}^{T}\eta_{s}^{2}ds\right ]\leq\sigma^{2}\cdot\int_{0}^{T}\mathbb{E}_{\mathbb{P}^{*}}[S_{s}^{2}]ds<\infty,\]
where the second inequality follows easily from the existence of the exponential moments of a Gaussian random variable.
Second method. The second method of the proof puts more emphasis on the explicit evaluation of the price function \(c\). The form of the replicating strategy will not be examined here. Since we wish to apply Proposition \ref{musc511}, we need to ckeck first that the contingent claim \(X=(S_{T}-K)^{+}\) is attainable in the Black-Scholes market model. This follows easily from the general results; more specifically, from the predictable representation property (see Proposition \ref{mustb13}) combined with the square-integrability of the random variable \(X^{*}=(S_{T}-K)^{+}/B_{T}\) under the martingale measure \(\mathbb{P}^{*}\). We conclude that there exists a predictable
process \(\theta\) such that the stochastic integral
\[V_{t}^{*}=V_{0}^{*}+\int_{0}^{t}\theta_{s}dW_{s}^{*}\mbox{ for all }t\in [0,T]\]
follows a (square-integrable) continuous martingale under \(\mathbb{P}^{*}\), and
\[X^{*}=\frac{(S_{T}-K)^{+}}{B_{T}}=E_{\mathbb{P}^{*}}[X^{*}]+\int_{0}^{T}
\theta_{s}dW_{s}^{*}=E_{\mathbb{P}^{*}}[X^{*}]+\int_{0}^{T}h_{s}dS_{s}^{*},\]
where we have put \(h_{t}=\theta_{t}/(\sigma\cdot S_{t}^{*})\). Let us consider a trading strategy \(\boldsymbol{\phi}\) that is given by
\[\phi^{(1)}_{t}=h_{t}\mbox{ and }\phi^{(2)}_{t}=V_{t}^{*}-h_{t}\cdot S_{t}^{*}=\frac{V_{t}-h_{t}\cdot S_{t}}{B_{t}},\]
where \(V_{t}=B_{t}\cdot V_{t}^{*}\). Now from (\ref{museq59}) and (\ref{museq512}), we get
\begin{equation}{\label{museq2037}}\tag{27}
dS_{t}=r\cdot S_{t}dt+B_{t}dS_{t}^{*}.
\end{equation}
Let us check first that the strategy \(\boldsymbol{\phi}\) is self-financing. Observe that the wealth process \(V(\boldsymbol{\phi})\) agrees with \(V\), and thus
\begin{align*}
dV_{t}(\boldsymbol{\phi}) & =d(B_{t}\cdot V_{t}^{*})=B_{t}dV_{t}^{*}+r\cdot V_{t}^{*}\cdot B_{t}dt\\
& =B_{t}\cdot h_{t}dS_{t}^{*}+r\cdot V_{t}dt\\
& =h_{t}\cdot (dS_{t}-r\cdot S_{t}dt)+r\cdot V_{t}dt\mbox{ (from (\ref{museq2037}))}\\
& =h_{t}dS_{t}+r\cdot (V_{t}-h_{t}\cdot S_{t})dt\\
& = phi_{t}^{(1)}dS_{t}+\phi^{(2)}_{t}dB_{t}
\end{align*}
as expected. Finally, it is clear that \(V_{T}(\boldsymbol{\phi})= V_{T}=B_{T}\cdot V_{T}^{*}=B_{T}\cdot X^{*}=(S_{T}-K)^{+}\) so that \(\boldsymbol{\phi}\) is in fact a \(\mathbb{P}^{*}\)-admissible replicating strategy for \(X\). So far, we have shown that the call option is represented by a contingent claim that is attainable in the Black-Scholes market \({\cal M}_{BS}\). The goal is now to evaluate the arbitrage price of \(X\) using the risk-neutral valuation formula. Since \({\cal F}_{t}^{W}={\cal F}_{t}^{S}\) for every \(t\in [0,T]\), the risk-neutral valuation formula (\ref{museq515}) can be rewritten as follows
\begin{equation}{\label{museq531}}\tag{28}
C_{t}=B_{t}\cdot E_{\mathbb{P}^{*}}\left [\left .\frac{(S_{T}-K)^{+}}{B_{T}}\right |{\cal F}_{t}^{S}\right ]=c(S_{t},T-t)
\end{equation}
for some function \(c:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\). The second equality in (\ref{museq531}) can be inferred from the Markovian property of \(S\) (it is easily seen that \(S\) follows a time-homogeneous Markov process under \(\mathbb{P}^{*}\)). Alternatively, we can make use of equality (\ref{museq53}). The increment \(W_{T}^{*}-W_{t}^{*}\) of the Brownian motion is independent of the \(\sigma\)-field \({\cal F}_{t}^{W}= {\cal F}_{t}^{W^{*}}\); on the other hand, the stock price \(S_{t}\) is manifestly \({\cal F}_{t}^{W}\)-measurabel. From (\ref{museq513}), we have
\begin{equation}{\label{museq2039}}\tag{29}
S_{T}=S_{t}\cdot\exp\left (\sigma\cdot (W_{T}^{*}-W_{t}^{*})+\left (r-\frac{\sigma^{2}}{2}\right )\cdot (T-t)\right ).
\end{equation}
Therefore we get
\begin{equation}{\label{museq2040}}\tag{30}
E_{\mathbb{P}^{*}}\left [\left .(S_{T}-K)^{+}\right |{\cal F}_{t}^{S}\right ]=E_{\mathbb{P}^{*}}\left [(S-K)^{+}\right ].
\end{equation}
Now we consider case \(t=0\). Note that \(B_{0}=1\). It is enough to find the unconditional expectation
\begin{equation}{\label{museq2038}}\tag{31}
E_{\mathbb{P}^{*}}\left [\frac{(S_{T}-K)^{+}}{B_{T}}\right ]=E_{\mathbb{P}^{*}}\left [
\frac{S_{T}}{B_{T}}\cdot I_{D}\right ]-E_{\mathbb{P}^{*}}\left [\frac{K}{B_{T}}\cdot I_{D}\right ]=J_{1}-J_{2},
\end{equation}
where \(D\) stands for the set \(\{S_{T}>K\}\). Since \(B_{T}=e^{rT}\) and the fact that \(\xi =-W_{T}^{*}/\sqrt{T}\) has a standard normal distribution \(N(0,1)\) under the martingale measure \(\mathbb{P}^{*}\) (since \(W_{T}^{*}\) is \(N(0,T)\) implies that \(-W_{T}^{*}\) is also \(N(0,T)\)), for \(J_{2}\), we have
\begin{align*}
J_{2} & =e^{-rT}\cdot K\cdot \mathbb{P}^{*}\{S_{T}>K\}\\
& =e^{-rT}\cdot K\cdot \mathbb{P}^{*}\left\{S_{0}\exp\left (\sigma\cdot W_{T}^{*}+\left (r-\frac{1}{2}\sigma^{2}\right )\cdot T\right )>K\right\}\mbox{ (from (\ref{museq513}))}\\
& =e^{-rT}\cdot K\cdot \mathbb{P}^{*}\left\{-\sigma\cdot W_{T}^{*}<\ln S_{0}-\ln K+\left (r-\frac{1}{2}\sigma^{2}\right )\cdot T\right\}\\
& =e^{-rT}\cdot K\cdot \mathbb{P}^{*}\left\{\xi <\frac{\ln S_{0}-\ln K+(r-\frac{1}{2}\sigma^{2})\cdot T}{\sigma\cdot\sqrt{T}}\right\}\\
& =e^{-rT}\cdot K\cdot N(d_{2}(S_{0},T)).
\end{align*}
For \(J_{1}\), note first that
\begin{equation}{\label{museq533}}\tag{32}
J_{1}=\mathbb{E}_{\mathbb{P}^{*}}\left [\frac{S_{T}}{B_{T}}\cdot I_{D}\right ]=\mathbb{E}_{\mathbb{P}^{*}}[S^{*}_{T}\cdot I_{D}].
\end{equation}
It is convenient to introduce an auxiliary probability measure \(\bar{\mathbb{P}}\) on \((\Omega ,{\cal F}_{T})\) by setting
\begin{equation}{\label{apeq1}}\tag{33}
\frac{d\bar{\mathbb{P}}}{d\mathbb{P}^{*}}=\exp\left (\sigma\cdot W_{T}^{*}-\frac{1}{2}
\sigma^{2}\cdot T\right )\mbox{ \(\mathbb{P}^{*}\)-a.s.}
\end{equation}
(this can refer to equation (\ref{museq58}) by replacing \(\frac{r-\mu} {\sigma}\) by \(\sigma\)). By virtue of the Girsanov’s theorem, the process \(\bar{W}_{t}=W_{t}^{*}- \sigma\cdot t\) follows a standard Brownian motion on the space \((\Omega ,{\bf F},\bar{\mathbb{P}})\). Moreover, using (\ref{museq511}) we obtain
\begin{equation}{\label{museq534}}\tag{34}
S_{T}^{*}=S_{0}\cdot\exp\left (\sigma\cdot (\bar{W}_{T}+\sigma\cdot T)-\frac{1}{2}\cdot\sigma^{2}\cdot T\right )
=S_{0}\cdot\exp\left (\sigma\cdot\bar{W}_{T}+\frac{1}{2}\sigma^{2}\cdot T\right ).
\end{equation}
Combining (\ref{museq533}) and (\ref{museq534}), we find that
\begin{align*}
J_{1} & =\int_{\{S_{T}^{*}>K/B_{T}\}}S_{T}^{*}d\mathbb{P}^{*}\\
& =\int_{\{S_{T}^{*}>K/B_{T}\}}S_{T}^{*}\cdot\exp\left (-\sigma
\cdot W_{T}^{*}+\frac{1}{2}\sigma^{2}\cdot T\right )d\bar{\mathbb{P}}\\
& =\int_{\{S_{T}^{*}>K/B_{T}\}}S_{0}\cdot\exp\left (\sigma\cdot\bar{W}_{T}
+\frac{1}{2}\sigma^{2}\cdot T\right )\cdot\exp\left (-\sigma\cdot (
\bar{W}_{T}+\sigma\cdot T)+\frac{1}{2}\sigma^{2}\cdot T\right )d\bar{\mathbb{P}}\\
& =S_{0}\cdot\bar{\mathbb{P}}\left\{S_{T}^{*}>K/B_{T}\right\}\\
& =S_{0}\cdot\bar{\mathbb{P}}\left\{S_{0}\cdot\exp\left (\sigma\cdot\bar{W}_{T}
+\frac{1}{2}\sigma^{2}\cdot T\right )>Ke^{-rT}\right\}\\
& =S_{0}\cdot\bar{\mathbb{P}}\left\{-\sigma\cdot\bar{W}_{T}<\ln S_{0}-\ln K+
\left (r+\frac{1}{2}\sigma^{2}\right )\cdot T\right\}.
\end{align*}
Using similar arguments as for \(J_{2}\), we find that \(J_{1}=S_{0}\cdot N(d_{1}(S_{0},T))\). Summarizing, we have shown that the price at time \(0\) of a call option equals (from (\ref{museq2038}) and the fact the \(B_{0}=1\))
\[C_{0}=c(S_{0},T)=S_{0}\cdot N(d_{1}(S_{0},T))-K\cdot e^{-rT}\cdot N(d_{2}(S_{0},T)),\]
where
\[d_{1,2}(S_{0},T)=\frac{\ln S_{0}-\ln K+(r\pm\frac{1}{2}\sigma^{2})\cdot T}{\sigma\cdot\sqrt{T}}\]
This ends the proof for the special case of \(t=0\). The valuation formula for \(t>0\) can be easily deduced from (\ref{museq2039}) and (\ref{museq2040}). This completes the proof. \(\nlacksquare\)
It can be checked that the probability measure \(\bar{\mathbb{P}}\) introduced in (\ref{apeq1}) is the martingale measure corresponding to the choice of the stock price as the numeraire asset, that is, the unique probability measure, equivalent to \(\mathbb{P}\), under which the process \(B^{*}=B/S\) follows a martingale. Notice that we have shown that
\[C_{0}=S_{0}\cdot\bar{\mathbb{P}}\left\{S_{T}>K\right\}-e^{-rT}\cdot K\cdot\mathbb{P}^{*}\left\{S_{T}>K\right\}.\] Undoubtedly, the most striking feature of the Black-Scholes result is the fact that the appreciation rate \(\mu\) does not enter the valuation formula. This is not surprising, however, as expression (\ref{museq513}), which describes the evolution of the stock price under the martingale measure \(\mathbb{P}^{*}\), does not involve the stock appreciation rate \(\mu\). More generally, we could have assumed that the appreciation rate is not constant, but is varying in time, or even follows a stochastic process (adapted to the underlying filtration). Assume, for instance, that the stock price process is determined by the SDE (it is implicitly assumed that SDE (\ref{museq536}) below admits a unique strong solution \(S\), which follows a continuous, Strictly positive process)
\begin{equation}{\label{museq536}}\tag{35}
dS_{t}=\mu (S_{t},t)\cdot S_{t}dt+\sigma (t)\cdot S_{t}dW_{t}\mbox{ for }S_{0}>0,
\end{equation}
where \(\mu :\mathbb{R}\times [0,T]\rightarrow \mathbb{R}\) is a deterministic function satsifying certain regularity conditions, and \(\sigma :[0,T]\rightarrow \mathbb{R}\) is also deterministic with \(\sigma (t)>\epsilon >0\) for some constant \(\epsilon\). We introduce the accumulation factor \(B\) by setting
\begin{equation}{\label{museq537}}\tag{36}
B_{t}=\exp\left (\int_{0}^{t} r(s)ds\right )\mbox{ for all }t\in [0,T]
\end{equation}
for a deterministic function \(r:[0,T]\rightarrow \mathbb{R}_{+}\). In view of (\ref{museq537}), we have
\[dB_{t}=r(t)\cdot B_{t}dt\mbox{ with }B_{0}=1,\]
so that it is clear that \(r(t)\) represents the instantaneous, continuously compounded interest rate prevailing at the market at time \(t\). It is easily seen that, under the present hypothesis, the martingale measure \(\mathbb{P}^{*}\) is unique, and the risk-neutral valuation formula (\ref{museq515}) is valid. In particular, the price of a European call option equals
\begin{equation}{\label{museq538}}\tag{37}
C_{t}=\exp\left (-\int_{t}^{T}r(s)ds\right )\cdot E_{\mathbb{P}^{*}}\left [(S_{T}-K)^{+}|{\cal F}_{t}\right ]\mbox{ for every }t\in [0,T].
\end{equation}
Notice that under the martingale measure \(\mathbb{P}^{*}\) we have
\begin{equation}{\label{museq539}}\tag{38}
dS_{t}=r(t)\cdot S_{t}dt+\sigma (t)\cdot S_{t}dW_{t}^{*}.
\end{equation}
If \(r\) and \(\sigma\) are continuous functions, the unique solution to (\ref{museq539}) is known to be
\[S_{t}=S_{0}\cdot\exp\left (\int_{0}^{t}\sigma (s)dW_{s}^{*}+\int_{0}^{t}\left (r(s)-\frac{1}{2}\sigma^{2}(s)\right )ds\right ).\]
It is now an easy task to derive a suitable generalization of the Black-Scholes formula using (\ref{museq538}). Indeed, it appears that it is enough to substitute the quantities \(r(T-t)\) and \(\sigma^{2}(T-t)\) in the standard Black-Scholes formula by
\[\int_{t}^{T} r(s)ds\mbox{ and }\int_{t}^{T}\sigma^{2}(s)ds,\]
respectively. The function obtained in such a way solves the Black-Scholes PDE with time-dependent coefficients.
The Put-Call Parity for Spot Options.
\begin{equation}{\label{musp511}}\tag{39}\mbox{}\end{equation}
Proposition \ref{musp511}. The arbitrage prices of European call and put options with the same expiry date \(T\) and strike price \(K\) satisfy the put-call parity relationship
\begin{equation}{\label{museq540}}\tag{40}
C_{t}-P_{t}=S_{t}-K\cdot e^{-r(T-t)}
\end{equation}
for every \(t\in [0,T]\).
Proof. It is sufficient to observe that the payoffs of the call and put options at expiry satisfy the equality
\[(S_{T}-K)^{+}-(K-S_{T})^{+}=S_{T}-K.\]
Relationship (\ref{museq540}) now follows from the risk-neutral valuation formula. Alternatively, one may derive (\ref{museq540}) using simple no-arbitrage arguments. \(\blacksquare\)
The put-call parity can be used to derive a closed-form expression for the arbitrage price of a European put option. Let us denote by \(p:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) the function
\begin{equation}{\label{museq541}}\tag{41}
p(s,t)=K\cdot e^{-rt}\cdot N(-d_{2}(s,t))-s\cdot N(-d_{1}(s,t))
\end{equation}
with \(d_{1}(s,t)\) and \(d_{2}(s,t)\) given by (\ref{museq517}).
Proposition. The Black-Scholes price at time \(t\in [0,T]\) of a European put option with strike price \(K\) equals \(P_{t}=p(S_{t},T-t)\), where the function \(\mathbb{P}\) is given by \((\ref{museq541})\).
Proof. The result follows immediately from Proposition \ref{musp511} and Theorem \ref{must511}.
In particular, the price at time \(0\) of a European put option equals
\[P_{0}=K\cdot e^{-rT}\cdot N(-d_{2}(S_{0},T))-S_{0}\cdot N(-d_{1}(S_{0},T)).\]
Since in typical situations it is not difficult to find a proper form of the put-call parity, we shall usually restrict the attention to the case of a call option. In some circumstances, it will be convenient to explicitly account for the dependence of the option’s price on its strike price \(K\), as well as on the parameters \(r\) and \(\sigma\) of the model. For this reason, we shall sometimes write
\[C_{t}=c(S_{t},T-t,K,r,\sigma )\mbox{ and }P_{t}=p(S_{t},T-t,K,r,\sigma )\]
in what follows.
The Black-Scholes PDE.
The purpose is now to extend the valuation procedure of Theorem \ref{must511} to any contingent claim attainable in \({\cal M}_{BS}\), and whose values depend only on the terminal value of the stock price. Random payoffs of such a simple form are termed path-independent claims, as opposed to path-dependent contingent claims; that is, payoffs which depend on the fluctuation of the stock price over a pre-specified period of time before the settlement date. Path-dependent payoffs correspond to the various kinds of OTC options, known under the generic name of exotic options. We start with the auxiliary result, which deals with a special case of the classic Feynman-Kac formula. Bascially, the Feynman-Kac formula expresses the solution of a parabolic PDE as the expected value of a certain functional of a Brownian motion.
\begin{equation}{\label{musl513}}\tag{42}\mbox{}\end{equation}
Proposition \ref{musl513}. Let \(W\) be the one-dimensional Brownian motion defined on a filtered probability space \((\Omega ,\{{\cal F}_{t}\}_{t\in [0,T]},\mathbb{P})\). For a Borel-measurable function \(h:\mathbb{R}\rightarrow \mathbb{R}\), we define the function \(u:\mathbb{R}\times [0,T]\rightarrow \mathbb{R}\) by setting
\[u(x,t)=E_{\mathbb{P}}\left [\left .e^{-r(T-t)}\cdot h(W_{T})\right |W_{t}=x\right ]\mbox{ for all }(x,t)\in \mathbb{R}\times [0,T].\]
Suppose that
\[\int_{-\infty}^{+\infty} e^{-ax^{2}}\cdot |h(x)|dx<\infty\]
for some \(a>0\). Then the function \(u\) is defined for \(0<T-t<1/2a\) and \(x\in \mathbb{R}\), and has derivatives of all orders. In particular, it belongs to the class \(C^{2,1}(\mathbb{R}\times (0,T))\) and satisfies the followng PDE
\[-\frac{\partial u}{\partial t}(x,t)=\frac{1}{2}\cdot\frac{\partial^{2}u}
{\partial x^{2}}(x,t)-r\cdot u(x,t)\mbox{ for }(x,t)\in \mathbb{R}\times (0,T)\]
with the terminal condition \(u(x,T)=h(x)\) for \(x\in \mathbb{R}\). \(\sharp\)
Suppose we are given a Borel-measurable function \(g:\mathbb{R}\rightarrow\mathbb{R}\). Then we have the following result, which generalizes Theorem \ref{must511}.
Corollary. Let \(g:\mathbb{R}\rightarrow \mathbb{R}\) be a Borel-measurable function such that the random variable \(X=g(S_{T})\) is integrable under \(\mathbb{P}^{*}\). Then, the arbitrage price in \({\cal M}_{BS}\) of the claim \(X\) which settles at time \(T\) is given by the equality \(\Pi_{t}(X)=v(S_{t},t)\), where the function \(v:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) solves the Black-Scholes PDE
\begin{equation}{\label{museq545}}\tag{43}
\frac{\partial v}{\partial t}+\frac{1}{2}\cdot\sigma^{2}\cdot s^{2}\cdot \frac{\partial^{2}v}{\partial s^{2}}+r\cdot s\cdot\frac{\partial v}{\partial s}-r\cdot v=0\mbox{ for all }(s,t)\in (0,\infty )\times (0,T)
\end{equation}
subject to the terminal condition \(v(s,T)=g(s)\).
Proof. We shall focus on the straightforward derivation of (\ref{museq545}) from the risk-neutral valuation formula. By reasoning along the same lines as in the second proof of Theorem \ref{must511}, we find that the price \(\Pi_{t}(X)\) satisfies
\[\Pi_{t}(X)=E_{\mathbb{P}^{*}}\left [\left .e^{-r(T-t)}\cdot g(S_{T})\right |{\cal F}_{t}^{S}\right ]=v(S_{t},t)\]
for some function \(v:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\). Furthermore,
\begin{equation}{\label{museq2041}}\tag{44}
\Pi_{t}(X)=E_{\mathbb{P}^{*}}\left [\left .e^{-r(T-t)}\cdot g(f(W_{T}^{*},T))\right |{\cal F}_{t}^{W^{*}}\right ]=v(S_{t},t),
\end{equation}
where \(f:\mathbb{R}\times [0,T]\rightarrow \mathbb{R}\) is a strictly positive function given by the formula
\begin{equation}{\label{museq547}}\tag{45}
f(x,t)=S_{0}\exp\left (\sigma\cdot x+\left (r-\frac{\sigma^{2}}{2}\right )\cdot t\right )\mbox{ for all }x\in \mathbb{R}
\end{equation}
from (\ref{museq513}); that is,
\begin{equation}{\label{museq2042}}\tag{46}
S_{t}=f(W_{t}^{*},t).
\end{equation}
Let us denote
\begin{equation}{\label{museq2043}}\tag{47}
u(x,t)=E_{\mathbb{P}^{*}}\left [\left .e^{-r(T-t)}\cdot g(f(W_{T}^{*},T))\right |W_{t}^{*}=x\right ]
\end{equation}
By virtue of Proposition \ref{musl513}, the function \(u(x,t)\) satisfies
\begin{equation}{\label{museq548}}\tag{48}
-u_{t}(x,t)=\frac{1}{2}\cdot u_{xx}(x,t)-r\cdot u(x,t)\mbox{ for all }(x,t)\in \mathbb{R}\times (0,T)
\end{equation}
subject to the terminal condition \(u(x,T)=g(f(x,T))\). A comparison of (\ref{museq2041}) and (\ref{museq2043}) using (\ref{museq2042}) yields the following relationship between \(v(s,t)\) and \(u(s,t)\)
\begin{equation}{\label{museq2044}}\tag{49}
u(x,t)=v(f(x,t),t)\mbox{ for all }(x,t)\in \mathbb{R}\times [0,T].
\end{equation}
We denote by \(s=f(x,t)\) so that \(s\in (0,+\infty )\). From (\ref{museq2044}), we obtain
\[u_{t}(x,t)=v_{s}(s,t)\cdot f_{t}(x,t)+v_{t}(s,t)\mbox{ and }u_{x}(x,t)=v_{s}(s,t)\cdot f_{x}(x,t),\]
and thus
\[u_{xx}(x,t)=v_{ss}(s,t)\cdot f_{x}^{2}(x,t)+v_{s}(s,t)\cdot f_{xx}(x,t).\]
On the other hand, it follows from (\ref{museq547}) that
\[f_{x}(x,t)=\sigma\cdot f(x,t), f_{t}(x,t)=\left (r-\frac{\sigma^{2}}{2}
\right )\cdot f(x,t),\mbox{ and }f_{xx}(x,t)=\sigma^{2}\cdot f(x,t).\]
We conclude that
\[u_{t}(x,t)=s\cdot\left (r-\frac{\sigma^{2}}{2}\right )\cdot v_{s}(s,t)+
v_{t}(s,t)\mbox{ and }u_{xx}(x,t)=\sigma^{2}\cdot s^{2}\cdot v_{ss}(s,t)+\sigma^{2}\cdot s\cdot v_{s}(s,t).\]
Substitution into (\ref{museq548}) gives
\[s\cdot\left (\frac{\sigma^{2}}{2}-r\right )\cdot v_{s}(s,t)-v_{t}(s,t)=
\frac{\sigma^{2}}{2}\cdot s^{2}\cdot v_{ss}(s,t)+\frac{\sigma^{2}}{2}\cdot s\cdot v_{s}(s,t)-r\cdot v(s,t).\]
Simplified, this yields (\ref{museq545}). Both the terminal and boundary conditions are also evident. This completes the proof. \(\blacksquare\)
\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}
Riskless Portfolio Method.
A risk-free portfolio is a trading strategy created by taking positions in an option and a stock in such a way that the portfolio’s wealth follows a process of finite variation. It is an easy matter to show that the wealth of a risk-free portfolio should appreciate at the risk-free rate (otherwise, it would be possible to create profitable risk-free strategies). Let \(v=v(s,t)\) be a smooth function \(v:\mathbb{R}_{+}\times [0,T]\rightarrow\mathbb{R}\) such that \(v(s,T)=(s-K)^{+}\). As before, we assume a priori that the arbitrage price of a call option at time \(t\) equals \(C_{t}=v(S_{t},t)\). We shall examine a specific trading strategy which involves a short position in one call option, and a long position in the underlying stock. Formally, we consider a dynamic portfolio \(\boldsymbol{\phi}_{t}=(\phi^{(1)}_{t},\phi^{(2)}_{t})\), where \(\phi^{(1)}_{t}\) and \(\phi^{(2)}_{t}\) stand for the number of shares of stock and the number of call options held at instant \(t\), respectively. More specifically, we assume that for every \(t\in [0,T]\),
\begin{equation}{\label{museq552}}\tag{50}
\boldsymbol{\phi}_{t}=(v_{s}(S_{t},t),-1).
\end{equation}
The wealth at time \(t\) of this strategy equals
\[V_{t}(\boldsymbol{\phi})=\phi^{(1)}_{t}\cdot S_{t}+\phi_{t}^{(2)}\cdot C_{t}=v_{s}(S_{t},t)\cdot S_{t}-v(S_{t},t).\]
Suppose that the trading strategy \(\boldsymbol{\phi}\) is self-financing. Then \(V\) satisfies
\begin{equation}{\label{museq554}}\tag{51}
dV_{t}=\phi_{t}^{(1)}dS_{t}+\phi_{t}^{(2)}dC_{t}=v_{s}(S_{t},t)dS_{t}-dv(S_{t},t).
\end{equation}
The aim is to derive the Black-Scholes PDE by taking (\ref{museq554}) as a starting point. Notice that Ito’s formula gives
\[dv(S_{t},t)=\left (\mu\cdot S_{t}\cdot v_{s}(S_{t},t)+\frac{\sigma^{2}}{2}
\cdot S_{t}^{2}\cdot v_{ss}(S_{t},t)+v_{t}(S_{t},t)\right )dt+\sigma\cdot S_{t}\cdot v_{s}(S_{t},t)dW_{t}.\]
Substituting this into (\ref{museq554}), we obtain
\begin{equation}{\label{museq555}}\tag{52}
dV_{t}=-\left (v_{t}(S_{t},t)+\frac{\sigma^{2}}{2}\cdot S_{t}^{2}\cdot v_{ss}(S_{t},t)\right )dt
\end{equation}
so that the strategy \(\boldsymbol{\phi}\) appears to be instataneously risk-free, meaning that its wealth process is a continuous, adapted process of finite variation. Since under the martingale measure the discounted wealth process \(V_{t}^{*}=V_{t}/B_{t}\) of any self-financing strategy is known to follow a local martingale, the equality (\ref{museq555}) implies that the continuous process of finite variation \(V_{t}^{*}\) is a constant (It is well known that all (local) martingales with respect to the natural filtration of a Brownian motion have simple paths that are almost all of infinite variation, unless constant). Put another way, we have shown that \(dV_{t}^{*}=0\), or equivalently, that \(dV_{t}=rV_{t}dt\). On the other hand, in view of (\ref{museq555}), the differential \(dV_{t}^{*}\) satisfies also
\[dV_{t}^{*}=B^{-1}_{t}dV_{t}-r\cdot B^{-1}_{t}\cdot V_{t}dt,\]
or more explicitly
\[dV_{t}^{*}=-B^{-1}_{t}\cdot\left (v_{t}(S_{t},t)+r\cdot S_{t}\cdot
v_{s}(S_{t},t)+\frac{\sigma^{2}}{2}\cdot S_{t}^{2}\cdot v_{ss}(S_{t},t)-r\cdot v(S_{t},t)\right )dt.\]
Therefore, the equality \(dV_{t}^{*}=0\) is satisfied if and only if, for any \(t\in [0,T]\), we have
\[\int_{0}^{t}\left (v_{t}(S_{u},u)+r\cdot S_{u}\cdot v_{s}(S_{u},u)+
\frac{\sigma^{2}}{2}\cdot S_{u}^{2}\cdot v_{ss}(S_{u},u)-r\cdot v(S_{u},u)\right )du=0.\]
This in turn holds if and only if the function \(v\) solves the Black-Scholes PDE. This shows that the Black-Scholes PDE, and thus also the Black-Scholes option valuation result, can be obtained via the risk-free portfolio approach.
So far we have implicitly assumed that the portfolio \(\boldsymbol{\phi}\) given by (\ref{museq552}) is self-financing. However, to completely justify the above proof of the Black-Scholes formula, we still need to verify whether the strategy \(\boldsymbol{\phi}\) is self-financing when \(v\) solves the Black-Scholes PDE. Unfortunately, this is not the case, as Proposition \ref{musp521} shows below. Therefore, the rather frequent derivation of the Black-Scholes formula in existing financial literature through the risk-free portfolio approach is mathematically unsatisfactory. A point worth stressing is that the risk-free portfolio approach is unquestionable in a discrete-time setting.
\begin{equation}{\label{musp521}}\tag{53}\mbox{}\end{equation}
Proposition \ref{musp521}. Suppose that the trading strategy \(\boldsymbol{\phi}\) is given by (\ref{museq552}) with the function \(v(s,t)=c(s,T-t)\), where \(c\) is given by the Black-Scholes formuls. Then the trading strategy is not self-financing; that is, condition \((\ref{museq554})\) fails to hold. \(\sharp\)
\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}
Sensitivity Analysis.
We say that at a given instant \(t\) before or at expiry, a call option is in-the-money and out-of-the-money if \(S_{t}>K\) and \(S_{t}<K\), respectively. Similarly, a put option is said to be in-the-money and out-of-the-money at time \(t\) when \(S_{t}<K\) and \(S_{t}>K\), respectively. Finally, when \(S_{t}=K\) both options are said to be at-the-money. The intrinsic values of a call and a put options are defined by the formulas
\[I_{t}^{C}=(S_{t}-K)^{+}\mbox{ and }I_{t}^{P}=(K-S_{t})^{+},\]
respectively, and the time values equal
\[J_{t}^{C}=C_{t}-(S_{t}-K)^{+}\mbox{ and }J_{t}^{P}=P_{t}-(K-S_{t})^{+}\]
for \(t\in [0,T]\). It is thus evident that an option is in-the-money if and only if its intrinsic value is strictly positive. A short position in a call option is referred to as a {\bf covered call} if the writer of the option hedges his or her risk exposure by holding the underlying stock; in the opposite case, the position is known as a {\bf naked call}. Writing covered call options is a popular practice among portfolio managers, as such a strategy seems to offer obvious advantages. Let us consider a call option with the strike price \(K\) above the current stock price \(S_{t}\). If the stock appreciates and is called away, the portfolio still gains the option price and the difference \(K-S_{t}\) (also, it receives dividends for the period before exercise). If the stock price rises but the option is out-of-the-money at expiry, the porfolio will gain the option price and the stock appreciation up to the strike price; that is, \(S_{T}-S_{t}\). Finally, if the stock price declines in price, the loss \(S_{t}-S_{T}\) is buffered by the option price, so that the total loss from the position will be less than if no option were written. Summarizing, writing covered call options significantly reduces the risk exposure of stockholder, traditionally measured by means of the variance of the return distribution. It should be observed that most of the drop in variance occurs in that part of the return distribution where high variance is desirable; that is, on the right-hand side. Still, it is possible for a covered call strategy to give a higher expected return and a lower variance than the underlying stock.
When an investor who holds a stock also purchases a put option on this stock as a protection against stock price decline, the position is referred to as a protective put. While writing covered calls truncates, roughly speaking, the right-hand side of the return distribution and simultaneously shifts it to the right, buying protective puts truncates the left-hand side of the return distribution and at the same time shifts the distribution to the left. The last effect is due to the fact that the cost of a put increases the initial investment of a portfolio.
To measure quantitatively the influence of an option’s position on a given portfolio of financial assets, we will now examine the dependence of its price on the fluctuations of the current stock price, time to expiry, strike price, and other relevant parameters. For a fixed expiry date \(T\) and arbitrary \(t\leq T\), we denote by \(\tau =T-t\) the time to expiry. We write \(p(S_{t},\tau ,K,r,\sigma )\) and \(c(S_{t},\tau ,K,r,\sigma )\) to denote the price of a call and a put option, respectively. The functions \(c\) and \(\mathbb{P}\) are thus given by the formulas
\begin{equation}{\label{museq556}}\tag{54}
c(s,\tau ,K,r,\sigma )=s\cdot N(d_{1})-K\cdot e^{-r\cdot\tau}\cdot N(d_{2})
\end{equation}
and
\[p(s,\tau ,K,r,\sigma )=K\cdot e^{-r\cdot\tau}\cdot N(-d_{2})-s\cdot N(-d_{1}),\]
where
\[d_{1,2}=d_{1,2}(s,\tau ,K,r,\sigma )=\frac{\ln s-\ln K+(r\pm\frac{\sigma^{2}}{2})\cdot\tau}{\sigma\cdot\sqrt{\tau}}.\]
Recall that at any time \(t\in [0,T]\), the replicating portfolio of a call option involves \(\alpha_{t}\) shares of stock and \(\beta_{t}\) units of borrowed funds, where
\[\alpha_{t}=c_{s}(S_{t},\tau )=N(d_{1}(S_{t},\tau ))\mbox{ and }\beta_{t}=c(S_{t},\tau )-\alpha_{t}\cdot S_{t}.\]
The strictly positive number \(\alpha_{t}\), which determines the number of shares in the replicating portfolio, is commonly referred to as the hedge ratio or, briefly, the delta of the option. It is not hard to verify by straightforward calculations that
\begin{align*}
c_{s} & =N(d_{1})=\delta >0;\\
c_{ss} & =\frac{n(d_{1})}{s\cdot\sigma\sqrt{\tau}}=\gamma >0;\\
c_{\tau} & =\frac{s\cdot\sigma}{2\sqrt{\tau}}\cdot n(d_{1})+K\cdot r\cdot e^{-r\tau}\cdot N(d_{2})=\theta >0;\\
c_{\sigma} & =s\cdot\sqrt{\tau}\cdot n(d_{1})=\lambda >0;\\
c_{r} & =\tau\cdot K\cdot e^{-r\tau}\cdot N(d_{2})=\rho >0;\\
c_{K} & = e^{-r\tau}\cdot N(d_{2})<0,
\end{align*}
where \(n\) stands for the standard Gaussian probability density function; that is
\[n(x)=\frac{1}{\sqrt{2\pi}}\cdot e^{-x^{2}/2}\mbox{ for all }x\in \mathbb{R}.\]
Similarly, in the case of a put option we get
\begin{align*}
p_{s} & =N(d_{1})-1=-N(-d_{1})=\delta <0;\\
p_{ss} & = frac{n(d_{1})}{s\cdot\sigma\sqrt{\tau}}=\gamma >0;\\
p_{\tau} & =\frac{s\cdot\sigma}{2\sqrt{\tau}}\cdot n(d_{1})+K\cdot r\cdot e^{-r\tau}\cdot (N(d_{2}-1)=\theta ;\\
p_{\sigma} & =s\cdot\sqrt{\tau}\cdot n(d_{1})=\lambda >0;\\
p_{r} & =\tau\cdot K\cdot e^{-r\tau}\cdot (N(d_{2}-1)=\rho <0;\\
p_{K} & =-e^{-r\tau}\cdot (1-N(d_{2}))>0.
\end{align*}
Consequently, the delta of a long position in a put option is a strictly negative number (equivalently, the price of a put option is a strictly decreasing function of a stock price). Generally speaking, the price of a put moves in the same direction as a short position in the asset. In
particular, in oreder to hedge a written put option, an investor needs to short a certain number of shares of the underlying stock. Another useful coefficient which measures the relative change of an option’s price as the stock price moves is the {\bf elasticity}. For any date \(t\leq T\), the elasticity of a call option is given by the equality
\[\eta_{t}^{c}=\frac{c_{s}(S_{t},\tau )\cdot S_{t}}{C_{t}}=\frac{N(d_{1}(S_{t},t))\cdot S_{t}}{C_{t}},\]
and for a put option it equals
\[\eta_{t}^{p}=\frac{p_{s}(S_{t},\tau )\cdot S_{t}}{P_{t}}=\frac{-N(-d_{1}(S_{t},t))\cdot S_{t}}{P_{t}},\]
Let us check that the elasticity of a call option price is always greater than \(1\). Indeed, for every \(t\in [0,T]\), we have
\[\eta_{t}^{c}=1+\frac{e^{-r\tau}\cdot K\cdot N(d_{2}(S_{t},\tau ))}{C_{t}}>1.\]
This implies aslo that \(C_{t}-c_{s}(S_{t},\tau )\cdot S_{t}<0\) so that the replicating portfolio of a call option always involves the borrowing of funds. Similarly, the elasticity of a put option satisfies
\[\eta_{t}^{0}=1-\frac{e^{-r\tau}\cdot K\cdot N(-d_{2}(S_{t},\tau ))}{P_{t}}<1.\]
This in turn implies that \(P_{t}-S_{t}\cdot p_{s}(S_{t},\tau )>0\) and thus the replicating portfolio of a short put option generates funds which are invested in risk-free bonds. It is instructive to determine the dynamics of the option price \(C\). Using Ito’s formula, one easily finds that under the martingale measure \(\mathbb{P}^{*}\) we have
\[dC_{t}=r\cdot C_{t}dt+\sigma\cdot C_{t}\cdot\eta_{t}^{c}dW_{t}^{*}.\]
This shows that the appreciation rate of the option price in a risk-neutral economy equals the risk-free rate \(r\); however, the volatility coefficient equals \(\sigma\cdot\eta_{t}^{c}\), so that, in contrast to the stock price volatility, the volatility of the option price follows a stochastic process (note also that \(\sigma\cdot\eta_{t}^{c}>\sigma\)). This feature makes the influence of an option’s position on the performance of a portfolio of financial assets rather difficult to quantify.
The position delta is obtained by multiplying the face value of the option position by its delta (The face value equals the number of
underlying assets, e.g., the face value of an option on a lot of \(100\) shares of stock equals \(100\)). Clearly, the position delta of a long call option (or a short put option) is positive; on the contrary, the position delta of a short call option (and of a long put option) is a negative number. The position delta of a portfolio is obtained by summing up the position deltas of its components. In this context, let us make the trivial observation that the position delta of a long stock equals \(1\), and that of a short stock is \(-1\). It is clear that the option’s position delta measures only the market exposure at the current price level of underlying assets. More precisely, it gives the first oreder approximation of the change in option price, which is sufficiently accurate only for a small move in the underlying asset price. To measure the change in the option delta as the underlying asset price moves, one should use the second derivative with respect to \(s\) of the option’s price; that is, the option’s gamma. The gamma effects means that position deltas also move as asset prices fluctuate, so that predictions of revaluation profit and loss based on position deltas are not sufficiently accurate, except for small moves. It is easily seen that bought options have positive gammas, while sold options have negative gammas. A portfolio’s gamma is the weighted sum of its option’s gammas, and the resulting gamma is determined by the dominant options in the portfolio. In this regard, options close to the money with a short time to expiry have a dominant influence on the portfolio’s gammas. Generally speaking, a portfolio with a positive gamma is more attractive than negative gamma portfolio. Recall that by theta we have denoted the derivative of the option price with respect to time to expiry. Generally, a portfolio dominated by bought options will have a negative theta, meaning that the portfolio will lose value as time passes. In contrast, short options generally have positive thetas. Finally, options close to the money will have more influence on the position theta than options far from the money. The derivative of the option price with respect to volatility is known as the vega of an option. A positive vega position will result in profits from increases in volatility; similarly, a negative vega means a strategy will profit from falling volatility. To create a positive vega, a trader needs to dominate the portfolio with bought options, bearing in mind that the vega will be dominated by those options that are close to the money and have significant remaining time to expiry.
\begin{equation}{\label{muse531}}\tag{55}\mbox{}\end{equation}
Example \ref{muse531}. Consider a call option on a stock \(S\) with strike price \(\$30\) and with \(3\) months to expiry. Suppose, in adition, that the current stock price equals \(\$31\), the stock price volatility is \(\sigma =10\%\) per annum, and the risk-free interest rate is \(r=5\%\) per annum with continuous compounding. We may assume, without loss of generality, that \(t=0\) and \(T=0.25\). Using (\ref{museq517}), we obtain (approximately) \(d_{1}(S_{0},T)=0.93\), and thus \(d_{2}(S_{0},T)=d_{1}(S_{0},T)-\sigma\sqrt{T}=0.88\). Consequently, using formula (\ref{museq516}) and the following values of the standard Gaussain probability distribution function: \(N(0.93)=0.8238\) and \(N(0.88)=0.8106\), we find that (approximately) \(C_{0}=1.52\), \(\phi_{0}^{1}=0.82\) and \(\phi_{0}^{2}=-23.9\). This means that to hedge a short position in the call option, which was sold at the arbitrage price \(C_{0}=\$1.52\), an investor needs to purchase at time \(0\) the number \(\delta =0.82\) shares of stock (this transaction requires an adidtional borrowing of \(23.9\) units of cash). The elasticity at time \(0\) of the call option price with respect to the stock price equals
\[\eta_{0}^{c}=\frac{N(d_{1}(S_{0},T)\cdot S_{0}}{C_{0}}=16.72.\]
Suppose that the stock price rises immediately from \(\$31\) to \(\$31.2\), yielding a return rate of \(0.65\%\) flat. Then the option price will move by approximately \(16.5\) cents from \(\$1.52\) to \(\$1.685\), giving a return rate of \(10.76\%\) flat. Roughly speaking, the option has nearly \(17\) times the return rate of the stock; of course, this also means that it will drop \(17\) times as fast. If an investor’s portfolio involves \(5\) long call options (each on a round lot of \(100\) shares of stock), the position delta equals \(500\times 0.82=410\), so that it is the same as for a portfolio involving \(410\) shares of the underlying stock. Let us now assume that an option is a put. the price of a put option at time \(0\) equals (alternatively, \(P_{0}\) can be found from the put-call parity (\ref{museq540}))
\[P_{0}=30\cdot e^{-0.05/4}\cdot N(-0.88)-31\cdot N(-0.93)=0.15.\]
The hedge ratio corresponding to a short position in the put option equals approximately \(\delta =-0.18\) (since \(N(-0.93)=0.18\)), therefore to hedge the exposure, using the Black-Scholes recipe, an investor needs to short \(0.18\) shares of stock for one out option. The proceeds from the option and share-selling transactions, which amount to \(\$5.73\), should be invested in risk-free bonds. Notice that the elasticity of the put option is several times larger than the elasticity of the call option. If the stock price rises immediately from \(\$31\) to \(\$31.2\), the price of the put option will drop to less than \(12\) cents. \(\sharp\)
\begin{equation}{\label{d}}\tag{D}\mbox{}\end{equation}
Futures Market.
Let \(f_{S}(t,T)\) for \(t\in [0,T]\) stand for the futures price of a certain stock \(S\) for the date \(T\). The evolution of futures prices \(f_{t}=f_{S}(t,T)\) is given by the familiar expression
\begin{equation}{\label{museq61}}\tag{56}
df_{t}=\mu_{f}\cdot f_{t}dt+\sigma_{f}\cdot f_{t}dW_{t}\mbox{ for }f_{0}>0,
\end{equation}
where \(\mu_{f}\) and \(\sigma_{f}>0\) are real numners, and \(W_{t}\) stands for a one-dimensional standard Brownian motion, defined on a filtered probability space \((\Omega ,\{{\cal F}_{t}^{W}\}_{t\in [0,T]},\mathbb{P})\). The unique solution of SDE (\ref{museq61}) equals (ref. (\ref{museq52}))
\[f_{t}=f_{0}\cdot\exp\left (\sigma_{f}\cdot W_{t}+\left (\mu_{f}-\frac{\sigma_{f}^{2}}{2}\right )\cdot t\right )\mbox{ for all }t\in [0,T].\]
The price of the second security, a risk-free bond, is given as before (\ref{museq54}). In the Black-Scholes setting, the futures price dynamics of a stock \(S\) can be found by combining (\ref{museq51}) with the following chain of equalities (ref. Proposition \ref{musp332})
\begin{equation}{\label{museq62}}\tag{57}
f_{t}=f_{S}(t,T)=F_{S}(t,T)=S_{t}\cdot e^{r(T-t)}\mbox{ for all }t\in [0,T],
\end{equation}
where, as usual, we write \(F_{S}(t,T)\) to denote the forward price of the stock for the settlement date \(T\). The equality (\ref{museq62}) can be easily derived from the absence of arbitrage in the spot/forward market (also referring to Hull \cite[p.55-56]{hul}); the second is a consequence of the assumption that the interest rate is deterministic. If the dynamics of the stock price \(S\) are given by the SDE (\ref{museq51}), the Ito’s formula yields (by considering (\ref{museq62}))
\[df_{t}=(\mu -r)\cdot f_{t}dt+\sigma\cdot f_{t}dW_{t},\]
with \(f_{0}=S_{0}\cdot e^{rT}\), so that \(f\) satisfies (\ref{museq61}) with \(\mu_{f}=\mu -r\) and \(\sigma_{f}=\sigma\). Since futures contracts are not necessarily associated with a physical underlying security, such as a stock or bond, we prefer to study the case of futures options in an abstract way. This means that we consider (\ref{museq61}) as the exogenously given the dynamics of the futures price \(f\). However, for the sake of notational simplicity, we write \(\mu =\mu_{f}\) and \(\sigma =\sigma_{f}\) in what follows. It follows from (\ref{museq62}) that
\begin{equation}{\label{museq63}}\tag{58}
f_{S}(t,T)=F_{S}(t,T)=E_{P^{*}}[S_{T}|{\cal F}_{t}]\mbox{ for all }t\in [0,T],
\end{equation}
but also
\begin{equation}{\label{museq64}}\tag{59}
f_{S}(t,T)=F_{S}(t,T)=\frac{S_{t}}{p(t,T)}\mbox{ for all }t\in [0,T],
\end{equation}
where \(p(t,T)\) stands for the price at time \(t\) of the zero-coupon bond that matures at \(T\) (also referring to Proposition \ref{musp3331}). It appears that under uncertainty of interest rates, the right-hand sides of (\ref{museq63}) and (\ref{museq64}) characterize the futures price and the forward price of \(S\), respectively.
Self-Financing Strategies.
We consider a European contingent claim \(X\) which settles at time \(T\). By a {\bf futures strategy} we mean a pair \(\boldsymbol{\phi}_{t}= (\phi_{t}^{(1)},\phi_{t}^{(2)})\) of real-valued adapted stochastic processes defined on the filtered probability space \((\Omega ,\{{\cal F}_{t}\}_{t\in [0,T]},\mathbb{P})\). Since it costs nothing to take a long or short positions in a futures contract, the wealth process \(V^{(f)}(\boldsymbol{\phi})\) of a futures strategy \(\boldsymbol{\phi}\) equals
\begin{equation}{\label{museq65}}\tag{60}
V_{t}^{(f)}(\boldsymbol{\phi})=\phi_{t}^{(2)}\cdot B_{t}\mbox{ for all }t\in [0,T]
\end{equation}
(also referring to (\ref{museq311})). We say that a futures strategy \(\boldsymbol{\phi}=(\phi^{(1)},\phi^{(2)})\) is self-financing if its wealth process \(V^{(f)}(\boldsymbol{\phi})\) satisfies for every \(t\in [0,T]\)
\[V_{t}^{(f)}(\boldsymbol{\phi})=V_{0}^{(f)}(\boldsymbol{\phi})+
\int_{0}^{t}\phi_{s}^{(1)}df_{s}+\int_{0}^{t}\phi_{s}^{(2)}dB_{s},\]
or equivalently,
\begin{equation}{\label{museq2047}}\tag{61}
dV_{t}^{(f)}(\boldsymbol{\phi})=\phi_{t}^{(1)}df_{t}+\phi_{t}^{(2)}dB_{t}
\end{equation}
We write \(\boldsymbol{\Phi}^{(f)}\) to denote the class of all self-financing futures strategies.
Martingale Measure for the Futures Market.
A probability measure \(\bar{\mathbb{P}}\) equivalent to \(\mathbb{P}\) is called the futures martingale measure when the discounted wealth \(\bar{V}^{(f)}(\boldsymbol{\phi})\) of any strategy \(\boldsymbol{\phi}\in\boldsymbol{\Phi}^{(f)}\), which equals \(\bar{V}^{(f)}(\boldsymbol{\phi})=V^{(f)}(\boldsymbol{\phi})/B\), follows a local martingale under \(\bar{\mathbb{P}}\).
Proposition. Let \(\bar{\mathbb{P}}\) be a probability measure on \((\Omega ,{\cal F}_{T})\) equivalent to \(\mathbb{P}\). Then \(\bar{\mathbb{P}}\) is a futures martingale measure if and only if the futures price \(f\) follows a local martingale under \(\bar{\mathbb{P}}\).
Proof. The discounted wealth \(\bar{V}^{(f)}(\boldsymbol{\phi})\) for any trading strategy \(\boldsymbol{\phi}\in\boldsymbol{\Phi}^{(f)}\) satisfies
\begin{align*}
d\bar{V}_{t}^{(f)}(\boldsymbol{\phi}) & =B^{-1}_{t}dV_{t}^{(f)}(\boldsymbol{\phi})+V_{t}^{(f)}(\boldsymbol{\phi})dB^{-1}_{t}\\
& =B_{t}^{-1}\left (\phi_{t}^{(1)}df_{t}+\phi_{t}^{(2)}dB_{t}\right )-
r\cdot B_{t}^{-1}V_{t}^{(f)}(\boldsymbol{\phi})dt\mbox{ (using (\ref{museq2047}) and \(B^{-1}_{t}=e^{-rt}\))}\\
& =\phi_{t}^{(1)}\cdot B^{-1}_{t}df_{t}
\end{align*}
since (\ref{museq65}) yields the equality
\begin{align*}
B^{-1}_{t}(\phi_{t}^{(2)}dB_{t}-r\cdot V_{t}^{(f)}(\boldsymbol{\phi})dt)
& =B_{t}^{-1}(B_{t}^{-1}\cdot V_{t}^{(f)}(\boldsymbol{\phi})dB_{t}-r\cdot V_{t}^{(f)}(\boldsymbol{\phi})dt)\\
& =B_{t}^{-1}(B_{t}^{-1}\cdot V_{t}^{(f)}(\boldsymbol{\phi})r\cdot B_{t}dt-r\cdot V_{t}^{(f)}(\boldsymbol{\phi})dt)=0.
\end{align*}
Then the result now easily follows (since \(d\bar{V}_{t}^{(f)}(\boldsymbol{\phi})\) and \(df_{t}\) have zero drifts). This completes the proof. \(\blacksquare\).
The next result is an immediate consequence of Girsanov’s theorem
Proposition. The unique martingale measure \(\bar{\mathbb{P}}\) for the process \(f\) is given by the Radon-Nikodym derivative
\[\frac{d\bar{\mathbb{P}}}{d\mathbb{P}}=\exp\left (-\frac{\mu}{\sigma}\cdot W_{T}-
\frac{1}{2}\cdot\frac{\mu^{2}}{\sigma^{2}}\cdot T\right )\mbox{$\mathbb{P}$-a.s.}\]
The dynamics of the futures price \(f\) under \(\bar{\mathbb{P}}\) are
\begin{equation}{\label{museq66}}\tag{62}
df_{t}=\sigma\cdot f_{t}d\bar{W}_{t},
\end{equation}
and the process \(\bar{W}_{t}=W_{t}+\mu\cdot t/\sigma\) follows a standard Brownian motion under the probability measure \(\bar{\mathbb{P}}\).
Proof. Using Musiela and Rutkowski \cite[p.466, Theorem B.2.1]{mus} by taking \(\gamma_{s}=-\mu /\sigma\), we have
\[df_{t}=\mu f_{t}dt+\sigma f_{t}dW_{t}=\mu f_{t}dt+\sigma f_{t}(d\bar{W}_{t}-(\mu /\sigma )dt)=\sigma\cdot f_{t}d\bar{W}_{t}\]
since \(dW_{t}=d\bar{W}_{t}-(\mu /\sigma )dt\). \(\blacksquare\)
It is clear from (\ref{museq66}) that
\begin{equation}{\label{museq67}}\tag{63}
f_{t}=f_{0}\cdot\exp\left (\sigma\cdot\bar{W}_{t}-\frac{\sigma^{2}}{2}\cdot t\right )\mbox{ for all }t\in [0,T]
\end{equation}
so that \(f\) follows a strictly positive martingale under \(\bar{\mathbb{P}}\). As expected, we say that a futures strategy \(\boldsymbol{\phi}\in \boldsymbol{\Phi}^{(f)}\) is \(\bar{\mathbb{P}}\)-{\bf admissible} if the discounted wealth \(\bar{V}^{(f)}(\boldsymbol{\phi})\) follows a martingale under \(\bar{\mathbb{P}}\). We shall study an arbitrage-free futures market \({\cal M}^{(f)}=(f,B,\boldsymbol{\Phi}^{(f)}(\bar{\mathbb{P}}))\), where \(\boldsymbol{\Phi}^{(f)}(\bar{\mathbb{P}})\) is the class of all \(\bar{\mathbb{P}}\)-admissible futures trading strategies. The futures market \({\cal M}^{(f)}\) is referred to as the Black futures market in what follows. The notion of an arbitrage price is defined in a similar way to the case of the Black-Scholes market.
The Black Futures Option Formula.
Let the function \(c^{(f)}:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) be given by Black’s futures formula
\begin{equation}{\label{museq68}}\tag{64}
c^{(f)}(f,t)=e^{-rt}\cdot\left (f\cdot N(\bar{d}_{1}(f,t))-K\cdot N(\bar{d}_{2}(f,t))\right ),
\end{equation}
where
\begin{equation}{\label{museq69}}\tag{65}
\bar{d}_{1,2}(f,t)=\frac{\ln f-\ln K\pm\frac{\sigma^{2}}{2}\cdot t}{\sigma\cdot\sqrt{t}}
\end{equation}
and \(N\) denotes the standard Gaussian cumulative distribution function. The futures option valuation result (\ref{museq68}) and (\ref{museq69}) can be found directly from the Black-Scholes formula by setting \(S_{t}=f_{t}\cdot e^{-r(T-t)}\) (referring to (\ref{museq62}); this also applies to the replicating strategy). Intuitively, this follows from the simple observation that in this case we have \(S_{T}=f_{T}\) at the option’s expiry, and thus the payoffs from both options agree. In practice, the expiry date of a futures option usually precedes the settlement date of the underlying futures contract. In such a case we have
\[C_{t}^{(f)}=\left (f_{S}(t,T)-K\right )^{+}=e^{r(T-t)}\cdot\left (S_{t}-K\cdot e^{-r(T-t)}\right )^{+},\]
and we may still value the futures option as if we were the spot option. Such considerations rely on the equality \(f_{S}(t,T)=F_{S}(t,T)\) (ref. Proposition \ref{musp332}), which in turn hinges on the assumption that the interest rate is a deterministic function. They thus cannot be easily extended to the case of stochastic interest rates.
\begin{equation}{\label{must611}}\tag{66}\mbox{}\end{equation}
Theorem \ref{must611}. \(C^{(f)}\) in the arbitrage-free futures market \({\cal M}^{(f)}\) of a European futures call option with expiry date \(T\) and strike price \(K\), is given by the equality \(C_{t}^{(f)}=c^{(f)}(f_{t},T-t)\). The futures strategy \(\boldsymbol{\phi}\in \boldsymbol{\Phi}^{(f)}(\bar{\mathbb{P}})\) that replicates a European futures call option is given by
\begin{equation}{\label{museq610}}\tag{67}
\phi_{t}^{(1)}=\frac{\partial c^{(f)}}{\partial f}(f_{t},T-t)\mbox{ and }\phi_{t}^{(2)}=e^{-rt}\cdot c^{(f)}(f_{t},T-t),
\end{equation}
for every \(t\in [0,T]\).
Proof. We will follow rather closely the proof of Theorem \ref{must511}. Some technical details, such as integrability of random variables or admissibility of trading portfolios, are left aside.
First method. Assume that the price process \(C_{t}^{(f)}\) is of the form \(C_{t}^{(f)}=v(f_{t},t)\) for some function \(v:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\), and consider a function strategy \(\boldsymbol{\phi}\in\boldsymbol{\Phi}^{(f)}\) of the form \(\boldsymbol{\phi}_{t}=(g(f_{t},t),h(f_{t},t))\) for some functions \(g,h:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\). Since replicating portfolio \(\boldsymbol{\phi}\) is assumed to be self-financing, the wealth process \(V^{(f)}(\boldsymbol{\phi})\), which equals
\begin{equation}{\label{museq611}}\tag{68}
V_{t}^{(f)}(\boldsymbol{\phi})=h(f_{t},t)\cdot B_{t}=v(f_{t},t)
\end{equation}
satisfies
\[dV_{t}^{(f)}(\boldsymbol{\phi})=g(f_{t},t)df_{t}+h(f_{t},t)dB_{t},\]
or more explicitly from (\ref{museq61}), (\ref{museq611}) and the fact \(dB_{t}=r\cdot B_{t}dt\)
\begin{equation}{\label{museq612}}\tag{69}
dV_{t}^{(f)}(\boldsymbol{\phi})=f_{t}\cdot\mu\cdot g(f_{t},t)dt+f_{t}\cdot\sigma\cdot g(f_{t},t)dW_{t}+r\cdot v(f_{t},t)dt.
\end{equation}
On the other hand, assuming that the function \(v\) is sufficiently smooth, we obtain
\[dv(f_{t},t)=\left (v_{t}(f_{t},t)+\mu\cdot f_{t}\cdot v_{f}(f_{t},t)+
\frac{\sigma^{2}}{2}\cdot f_{t}^{2}\cdot v_{ff}(f_{t},t)\right )dt+\sigma\cdot f_{t}\cdot v_{f}(f_{t},t)dW_{t}.\]
Combining the last equality with (\ref{museq612}), we get the following expression for the Ito differential of the process \(Y_{t}=v(f_{t},t)-V_{t}^{(f)}(\boldsymbol{\phi})\)
\begin{align*}
dY_{t} & =\left (v_{t}(f_{t},t)+\mu\cdot f_{t}\cdot v_{f}(f_{t},t)+
\frac{\sigma^{2}}{2}\cdot f_{t}^{2}\cdot v_{ff}(f_{t},t)\right )dt+\sigma\cdot f_{t}\cdot v_{f}(f_{t},t)dW_{t}\\
& -\mu\cdot f_{t}\cdot g(f_{t},t)dt-\sigma\cdot f_{t}\cdot g(f_{t},t)dW_{t}-r\cdot v(f_{t},t)dt=0.
\end{align*}
Arguing along the same lines as in the proof of Theorem \ref{must511}, we infer that
\begin{equation}{\label{museq613}}\tag{70}
g(f,t)=v_{f}(f,t)\mbox{ for all }(f,t)\in \mathbb{R}_{+}\times [0,T],
\end{equation}
and thus also
\[Y_{t}=\int_{0}^{t}\left (v_{t}(f_{s},s)+\frac{\sigma^{2}}{2}\cdot f_{s}^{2}\cdot v_{ff}(f_{s},s)-r\cdot v(f_{s},s)\right )ds=0,\]
where the last equality follows from the definition of \(Y\). To guarantee the last equality we assume that \(v\) satisfies the following PDE (referred to as the Black PDE)
\[v_{t}+\frac{\sigma^{2}}{2}\cdot f^{2}\cdot v_{ff}-r\cdot v=0\]
on \((0,\infty )\times (0,T)\) with the terminal condition \(v(f,T)=(f-K)^{+}\). Since the function \(v(f,t)=c^{(f)}(f,T-t)\), where \(c^{(f)}\) is given by (\ref{museq68})-(\ref{museq610}), is easily seen to solve this problem, to complete the proof it is sufficient to note that, by virtue of (\ref{museq613}) and (\ref{museq611}), the unique \(\bar{\mathbb{P}}\)-admissible strategy \(\boldsymbol{\phi}\) that replicates the option satisfies
\[\phi_{t}^{(1)}=g(f_{t},t)=v_{f}(f_{t},t)\mbox{ and }\phi_{t}^{(2)}=h(f_{t},t)=\frac{v_{f}(f_{t},t)}{B_{t}}.\]
Details are left aside.
Second method. Since the random variable \(X^{*}=(f_{T}-K)^{+}/B_{T}\) is integrable with respect to the martingale measure \(\bar{\mathbb{P}}\), it is enough to evaluate the conditional expectation
\[C_{t}^{(f)}=B_{t}\cdot E_{\bar{\mathbb{P}}}\left [\left .\frac{(f_{T}-K)^{+}}
{B_{T}}\right |{\cal F}_{t}^{(f)}\right ]=B_{t}\cdot E_{\bar{\mathbb{P}}}\left [
\left .\frac{(f_{T}-K)^{+}}{B_{T}}\right |f_{t}\right ].\]
This means that, in particular for \(t=0\), we need to find the expectation
\[E_{\bar{\mathbb{P}}}\left [\frac{f_{T}-K)^{+}}{B_{T}}\right ]=E_{\bar{\mathbb{P}}}\left [\frac{f_{T}}{B_{T}}\cdot I_{D}\right ]-
E_{\bar{\mathbb{P}}}\left [\frac{K}{B_{T}}\cdot I_{D}\right ]=I_{1}-I_{2},\]
where \(D\) denotes the set \(\{f_{T}>K\}\). For \(I_{2}\), we have, from (\ref{museq67}),
\[I_{2}=e^{-rT}\cdot K\cdot\bar{\mathbb{P}}\{f_{T}>K\}=e^{-rT}\cdot K\cdot
\bar{\mathbb{P}}\left\{f_{0}\cdot\exp\left (\sigma\cdot\bar{W}_{t}-\frac{\sigma^{2}}{2}\cdot T\right )>K\right\},\]
and thus
\begin{align*}
I_{2} & =e^{-rT}\cdot K\cdot\bar{\mathbb{P}}\left\{-\sigma\cdot\bar{W}_{t}<\ln f_{0}-\ln K-\frac{\sigma^{2}}{2}\cdot T\right\}\\
& =e^{-rT}\cdot K\cdot\bar{\mathbb{P}}\left\{\xi <\frac{\ln f_{0}-\ln K-\frac{\sigma^{2}}{2}}{\sigma\cdot\sqrt{T}}\right\}\\
& e^{-rT}\cdot K\cdot N(\bar{d}_{2}(f_{0},T))
\end{align*}
since the random variable \(\xi =-\bar{W}_{t}/\sqrt{T}\) has under \(\bar{\mathbb{P}}\) the standard Gaussian law. To evaluate \(I_{1}\), we define an auxiliary probability measure \(\hat{P}\) on \((\Omega,{\cal F}_{T})\) by setting
\[\frac{d\hat{P}}{d\bar{\mathbb{P}}}=\exp\left (\sigma\cdot\bar{W}_{t}-\frac{\sigma^{2}}{2}\cdot T\right )\mbox{ \(P\)-a.s.}\]
and thus (ref. (\ref{museq67}))
\[I_{1}=\bar{\mathbb{P}}\left\{\frac{f_{T}}{B_{T}}\cdot I_{D}\right\}=e^{-rT}\cdot f_{0}\cdot\hat{P}\{f_{T}>K\}.\]
Moreover, the process \(\widehat{W}_{t}=\bar{W}_{t}-\sigma\cdot t\) follows a standard Brownian motion on the filtered probability space \((\Omega ,{\bf F},\hat{P})\), and
\[f_{T}=f_{0}\cdot\exp\left (\sigma\cdot\widehat{W}_{T}+\frac{\sigma^{2}}{2}\cdot T\right ).\]
Consequently,
\begin{align*}
I_{1} & =e^{-rT}\cdot f_{0}\cdot\hat{P}\{f_{T}>K\}\\
& =e^{-rT}\cdot f_{0}\cdot\hat{P}\left\{f_{0}\cdot\exp\left (\sigma\cdot\widehat{W}_{T}+\frac{\sigma^{2}}{2}\cdot T\right )>K\right\}\\
& =e^{-rT}\cdot f_{0}\cdot\hat{P}\left\{-\sigma\cdot\widehat{W}_{T}<\ln f_{0}-\ln K+\frac{\sigma^{2}}{2}\cdot T\right\}\\
& =e^{-rT}\cdot f_{0}\cdot N(\bar{d}_{1}(f_{0},T)).
\end{align*}
The general valuation result for any date \(t\) is a consequence of the Markov property of \(f\). \(\blacksquare\)
The method of arbitrage pricing in the Black futures market can be easily extended to any path-independent claim contingent on the futures price. In fact, the following result follows easily from the first proof of Theorem \ref{must611}. As already mentioned, in financial literature, the partial differential equation (\ref{museq614}) below is commonly referred to as the Black PDE.
Corollary. The arbitrage price in \({\cal M}^{(f)}\) of any attainable contingent claim \(X=g(f_{T})\) which settles at time \(T\) is given by \(\Pi_{t}^{(f)}(X)=v(f_{t},t)\), where the function \(v:\mathbb{R}_{+}\times [0,T]\rightarrow\mathbb{R}\) is a function of the following partial differential equation
\begin{equation}{\label{museq614}}\tag{71}
\frac{\partial v}{\partial t}+\frac{\sigma^{2}}{2}\cdot f^{2}\cdot
\frac{\partial^{2}v}{\partial f^{2}}-r\cdot v=0\mbox{ for all }(f,t)\in (0,\infty )\times (0,T)
\end{equation}
subject to the terminal condition \(v(f,T)=g(f)\). \(\sharp\)
Let us denote by \(P_{t}^{(f)}=p^{(f)}(f_{t},T-t)\) the price of a put futures option with strike price \(K\) and \(T-t\) to its expiry date, provided that the current futures price is \(f_{t}\). To find the price of a futures put option, we can use the following result.
Corollary. The following relationship, known as the put-call parity for futures options, holds for every \(t\in [0,T]\)
\begin{equation}{\label{museq615}}\tag{72}
C_{t}^{(f)}-P_{t}^{(f)}=c^{(f)}(f_{t},T-t)-p^{(f)}(f_{t},T-t)=e^{-r(T-t)}\cdot (f_{t}-K).
\end{equation}
Consequently,
\[p^{(f)}(f_{t},T-t)=e^{-r(T-t)}\cdot\left (K\cdot N(-\bar{d}_{2}(f_{t},T-t))-f_{t}\cdot N(-\bar{d}_{1}(f_{t},T-t))\right ),\]
where \(\bar{d}_{1}(f,t)\) and \(\bar{d}_{2}(f,t)\) are given by (\ref{museq69}). \(\sharp\)
Example. Suppose that the call option considered in Example \ref{muse531} is a futures option. This means that the price is now interpreted as the futures price. Using (\ref{museq68}), one finds that the arbitrage price of a futures call option equals (approximately) \(C_{0}^{(f)}=1.22\). Moreover, the portfolio that replicates the option is composed at time \(0\) of \(\phi_{0}^{1}\) futures contracts and \(\phi_{0}^{2}\) invested in risk-free bonds, where \(\phi_{0}^{1}=0.75\) and \(\phi_{0}^{2}=1.22\). Since the number \(\phi_{0}^{1}\) is positive, it is clear that an investor who assumes a short option position needs to enter \(\phi_{0}^{1}\) (long) futures contracts. Such a position, commonly referred to as the long hedge, is also a generally accepted practical strategy for a party who expects to purchase a given asset at some future date. To find the arbitrage price of the corresponding put futures option, we make use of the put-call parity relationship (\ref{museq615}). We find that \(P_{0}^{(f)}=0.23\); moreover, for the replicating portfolio of the put option we have \(\phi_{0}^{1}=-0.25\) and \(\phi_{0}^{2}=0.23\). Since now \(\phi_{0}^{1}<0\), we deal here with the {\bf short hedge}; a strategy typical for an investor who expects to sell a given asset at some future date. \(\sharp\)
Options on Forward Contracts.
We will consider a forward contract with delivery date \(T>0\) written on a non-dividend-paying stock \(S\). Recall that the forward price at time \(t\) of a stock \(S\) for the settlement date \(T\) equals
\[F_{S}(t,T)=S_{t}\cdot e^{r(T-t)}\mbox{ for all }t\in [0,T].\]
This means that the forward contract, established at time \(t\), in which the delivery price is set to be equal to \(F_{S}(t,T)\) is worthless at time \(t\). Of course, the value of such a contract at time \(s\in [t,T]\) is no longer zero in general. It is intuitively clear that the value \(V^{(f)}(t,s,T)\) of such a contract at time \(s\) equals to the discounted value of the difference between the current forward price of \(S\) at time \(s\) and its value at time \(t\), that is,
\[V^{(f)}(t,s,T)=e^{-r(T-s)}\cdot\left (S_{s}\cdot e^{r(T-s)}-S_{t}\cdot e^{r(T-t)}\right )=S_{s}-S_{t}\cdot e^{r(s-t)}\]
for every \(s\in [t,T]\). The last equality can also be derived by applying directly the risk-neutral valuation formula to the claim \(X=S_{T}-F_{S}(t,T)\), which settles at time \(T\). Indeed, we have
\begin{align*}
V^{(f)}(t,s,T) & =B_{s}\cdot E_{P^{*}}\left [\left .\frac{S_{T}}{B_{T}}-\frac{S_{t}\cdot e^{r(T-t)}}{B_{T}}\right |{\cal F}_{s}\right ]\\
& =B_{s}\cdot E_{P^{*}}[S^{*}_{T}|{\cal F}_{s}]-S_{t}\cdot e^{r(T-t)}\cdot e^{-r(T-s)}\\
& =S_{s}-S_{t}\cdot e^{r(s-t)}=S_{s}-F_{S}(t,s)
\end{align*}
since the random variable \(S_{t}\) is \({\cal F}_{s}\)-measurable. It is worthwhile to observe that \(V^{(f)}(t,s,T)\) is in fact independent of the settlement date \(T\), therefore we may and do write \(V^{(f)}(t,s,T)= V^{(f)}(t,s)\) in what follows.
By definition, a call option written at time \(t\) on a forward contract with the expiry date \(t<T\) is simply a call option, with zero strike price, which is written on the value of the underlying forward contract. The terminal option’s payoff thus equals
\[C_{T}^{(f)}=(V^{(f)}(t,T))^{+}=(S_{T}-S_{t}\cdot e^{r(T-t)})^{+}.\]
It is clear that the call option on the forward contract purchased at time \(t\) gives the right to enter at time \(T\) into the forward contract on the stock \(S\) with delivery date \(T^{*}\) and delivery price \(F_{S}(t,T^{*})\). If the forward price at time \(T\) is less than it was at time \(t\), the option is abandoned. In the opposite case, the holder exercises the option, and either enters, at no additional cost, into a forward contract under more favorable conditions than those prevailing at time \(T\), or simply takes the payoff of the option. Assume now that the option was written at time \(0\) so that \(C_{T}^{(f)}=(V^{(f)}(0,T))^{+}=(S_{T}-S_{0}\cdot e^{rT})^{+}\). To value such an option at time \(t\leq T\), we can make use of the Black-Scholes formula with the (fixed) strike price \(K=S_{0}\cdot e^{rT}\). After simple manipulations, we find that the option’s value at time \(t\) is
\begin{equation}{\label{museq616}}\tag{73}
C_{t}^{(f)}=S_{t}\cdot N(d_{1}(S_{t},t))-S_{0}\cdot e^{rt}\cdot N(d_{2}(S_{t},t)),
\end{equation}
where
\[d_{1,2}(S_{t},t)=\frac{\ln S_{t}-\ln (S_{0}\cdot e^{rt})\pm\frac{\sigma^{2}}{2}\cdot (T-t)}{\sigma\cdot\sqrt{T-t}}.\]
Alternatively, we can make use of Black’s futures formula. Since the futures price \(f_{S}(T,T)\) equals \(S_{T}\), we have \(C_{T}^{(f)}=(f_{S}(T,T)-S_{0}\cdot e^{rT})^{+}\). An application of Black’s formula yields
\begin{equation}{\label{museq617}}\tag{74}
C_{t}^{(f)}=e^{-r(T-t)}\cdot\left (f_{t}\cdot N(\bar{d}_{1}(f_{t},t))-S_{0}\cdot e^{rt}\cdot N(\bar{d}_{2}(f_{t},t))\right ),
\end{equation}
where \(f_{t}=f_{S}(t,T)\) and
\[\bar{d}_{1,2}(f_{t},t)=\frac{\ln f_{t}-\ln (S_{0}\cdot e^{rT})\pm\frac{\sigma^{2}}{2}\cdot (T-t)}{\sigma\cdot\sqrt{T-t}}.\]
Since in the Black-Scholes setting the relationship \(f_{S}(t,T)=S_{t}\cdot e^{r(T-t)}\) is satisfied, it is apparent that expressions (\ref{museq616}) and (\ref{museq617}) are equivalent.
\begin{equation}{\label{e}}\tag{E}\mbox{}\end{equation}
Option on a Dividend-Paying Stock.
Up to now, we have studied arbitrage pricing within the Black-Scholes framework under the assumption that the stock upon which an option is written pays no dividends during option’s lifetime. Now we assume that the dividends (or the dividend rate) that will be paid to the shareholders during an option’s lifetime can be predicted with certainty.
Case of a Constant Dividend Yield.
We assume that the stock \(S\) continuously pays dividends at some fixed rate \(\lambda\). We also assume that the effective dividend rate is proportional to the level of the stock price. Although this is rather impractical as a realistic dividend policy associated with a particular stock, this model fits the case of a stock index option reasonably well. The dividend payments should be used in full, either to purchase additional shares of stock, or to invest un risk-free bonds (however, inter-temporal consumption or infusion of funds is not allowed). Consequently, a trading strategy \(\boldsymbol{\phi}=(\phi^{(1)},\phi^{(2)})\) is said to be self-financing when its wealth process
\[V_{t}(\boldsymbol{\phi})=\phi_{t}^{(1)}\cdot S_{t}+\phi_{t}^{(2)}\cdot B_{t}\]
satisfies
\[dV_{t}(\boldsymbol{\phi})=\phi_{t}^{(1)}dS_{t}+\lambda\cdot\phi_{t}^{(1)}\cdot S_{t}dt+\phi_{t}^{(2)}dB_{t},\]
or equivalently
\[dV_{t}(\boldsymbol{\phi})=\phi_{t}^{(1)}\cdot (\mu +\lambda )\cdot
S_{t}dt+\phi_{t}^{(1)}\sigma\cdot S_{t}dW_{t}+\phi_{t}^{(2)}dB_{t}.\]
We find it convenient to introduce an auxiliary process \(\tilde{S}_{t}=r^{\lambda t}\cdot S_{t}\), whose dynamics are given by the SDE
\begin{align*}
d\tilde{S}_{t} & =\lambda\cdot e^{\lambda t}dS_{t}+e^{\lambda t}dS_{t}\\
& =\lambda\cdot e^{\lambda t}dS_{t}+e^{\lambda t}\cdot (\mu\cdot S_{t}dt+\sigma\cdot S_{t}dW_{t})\\
& =(\mu +\lambda )\cdot\tilde{S}_{t}dt+\sigma\cdot\tilde{S}_{t}dW_{t}.
\end{align*}
In terms of this process, we have
\[V_{t}(\boldsymbol{\phi})=\phi_{t}^{(1)}\cdot e^{-\lambda t}\cdot\tilde{S}_{t}+\phi_{t}^{(2)}\cdot B_{t}\mbox{ and }
dV_{t}(\boldsymbol{\phi})=\phi_{t}^{(1)}\cdot e^{-\lambda t}d\tilde{S}_{t}+\phi_{t}^{(2)}dB_{t}.\]
Also, it is not difficult to check that the discounted wealth \(V^{*}(\boldsymbol{\phi})\) satisfies
\[dV_{t}^{*}(\boldsymbol{\phi})=\phi_{t}^{(1)}\cdot e^{-\lambda t}dS\tilde{S}_{t}^{*},\]
where \(\tilde{S}_{t}^{*}=\tilde{S}_{t}/B_{t}\). Put another way, we have
\[dV_{t}^{*}(\boldsymbol{\phi})=(\mu +\lambda -r)\cdot\phi_{t}^{(1)}
\cdot\tilde{S}_{t}^{*}+\sigma\cdot\phi_{t}^{(1)}\cdot\tilde{S}_{t}^{*}dW_{t}.\]
In view of the last equality, the unique martingale measure \(Q\) for our model is given by (\ref{museq58}) with \(\mu\) replaced by \(\mu +\lambda\). The dynamics of \(V^{*}(\boldsymbol{\phi})\) under \(Q\) are given by the expression
\[dV_{t}^{*}(\boldsymbol{\phi})=\sigma\cdot\phi_{t}^{(1)}\cdot\tilde{S}_{t}^{*}d\bar{W}_{t},\]
while those of \(\tilde{S}^{*}\) are
\begin{equation}{\label{museq618}}\tag{75}
d\tilde{S}_{t}^{*}=\sigma\cdot\tilde{S}_{t}^{*}d\bar{W}_{t},
\end{equation}
and the process \(\bar{W}_{t}=W_{t}-(r-\mu -\lambda )t/\sigma\) follows a standard Brownian motion on the probability space \((\Omega ,{\bf F},P)\). It is thus possible to construct, by defining in a standard way the class of admissible trading strategies, an arbitrage-free market in which a risk-free bond and a dividend-paying stock are primary securities. Assuming that this is done, the valuation of stock-dependent contingent claim is now standard.
\begin{equation}{\label{musp621}}\tag{76}\mbox{}\end{equation}
Proposition \ref{musp621}. The arbitrage price at time \(t\leq T\) of a call option on a stock which pays dividends at a constant rate \(\lambda\) during the option’s lifetime is given by the risk-neutral formula
\begin{equation}{\label{museq619}}\tag{77}
C_{t}^{\lambda}=B_{t}\cdot E_{Q}\left [\left .\frac{(S_{T}-K)^{+}}{B_{T}}\right |{\cal F}_{t}\right ]\mbox{ for all }t\in [0,T],
\end{equation}
or explicitly
\begin{equation}{\label{museq620}}\tag{78}
C_{t}^{\lambda}=\bar{S}_{t}\cdot N(d_{1}(\bar{S}_{t},T-t))-K\cdot e^{-r(T-t)}\cdot N(d_{2}(\bar{S}_{t},T-t)),
\end{equation}
where \(\bar{S}_{t}=S_{t}\cdot e^{-\lambda (T-t)}\), and \(d_{1},d_{2}\) are given by (\ref{museq517}) and (\ref{museq518}). Equivalently,
\[C_{t}^{\lambda}=e^{-\lambda (T-t)}\cdot\left (S_{t}\cdot
N(\hat{d}_{1}(S_{t},T-t))-K\cdot e^{-(r-\lambda )(T-t)}\cdot N(\hat{d}_{2}(S_{t},T-t))\right ),\]
where
\begin{equation}{\label{museq621}}\tag{79}
\hat{d}_{1,2}(s,t)=\frac{\ln s-\ln K+(r-\lambda\pm\frac{\sigma^{2}}{2})\cdot t}{\sigma\cdot\sqrt{t}}.
\end{equation}
Proof. The first equality is obvious. For the second, note first that we may rewrite (\ref{museq619}) as follows
\[C_{t}^{\lambda}=e^{-r(T-t)}\cdot \mathbb{E}_{Q}\left [\left .(S_{T}-K)^{+}
\right |{\cal F}_{t}\right ]=e^{-\lambda T}\cdot e^{-r(T-t)}\cdot
\mathbb{E}_{Q}\left [\left .(\tilde{S}_{T}-e^{\lambda T}\cdot K)^{+}\right |{\cal F}_{t}\right ].\]
Using (\ref{museq618}), and proceeding along the same lines as in the proof of Theorem \ref{must511}, we find that
\[C_{t}^{\lambda}=e^{-\lambda T}\cdot c(\tilde{S}_{t},T-t,e^{\lambda T}\cdot K),\]
where \(c\) is a standard Black-Scholes call option valuation formula. Put another way, \(C_{t}^{\lambda}=c^{\lambda}(S_{t},T-t)\), where
\[c^{\lambda}(s,t)=s\cdot e^{-\lambda t}\cdot N(\hat{d}_{1}(s,t))-K\cdot e^{-rt}\cdot N(\hat{d}_{2}(s,t))\]
and \(\hat{d}_{1},\hat{d}_{2}\) are given by (\ref{museq621}). \(\blacksquare\)
Alternatively, to derive the valuation formula for a call option (or for any European claim of the form \(X=g(S_{T})\)), we may first show that its arbitrage price equals \(v(S_{t},t)\), where \(v\) solves the following PDE
\[\frac{\partial v}{\partial t}+\frac{\sigma^{2}}{2}\cdot s^{2}\cdot
\frac{\partial^{2}v}{\partial s^{2}}+(r-\lambda )\cdot s\cdot\frac{\partial v}{\partial s}-r\cdot v=0\]
on \((0,\infty )\times (0,T)\) subject to the standard terminal condition \(v(s,T)=g(s)\). Under the assumptions of Proposition \ref{musp621}, one can show also that the value at time \(t\) of the forward contract with expiry date \(T\) and delivery price \(K\) is given by the equality
\[V_{t}(K)=e^{-\lambda (T-t)}\cdot S_{t}-e^{-r(T-t)}\cdot K.\]
Consequently, the forward price at time \(t\leq T\) of the stock \(S\), for settlement at date \(T\), equals
\begin{equation}{\label{museq622}}\tag{80}
F_{S}^{\lambda}(t,T)=e^{(r-\lambda )(T-t)}\cdot S_{t}.
\end{equation}
It is not difficult to check that the following version of the put-call parity relationship is valid
\[c^{\lambda}(S_{t},T-t)-p^{\lambda}(S_{t},T-t)=e^{-\lambda (T-t)}\cdot S_{t}-e^{-r(T-t)}\cdot K,\]
where \(p^{\lambda}(S_{t},T-t)\) stands for the arbitrage price at time \(t\) of the European put option with maturity date \(T\) and strike price \(K\). In particular, if the exercise price equals the forward price of the underlying stock, then
\[c^{\lambda}(S_{t},T-t)-p^{\lambda}(S_{t},T-t)=0.\]
As already mentioned, formula (\ref{museq620}) is commonly used by market practioners when valuing stock index options. For this purpose, one needs to assume that the stock index follows a lognormal process; actually, a geometric Brownian motion. The dividend yield \(\lambda\), which can be estimated from the historical data, slowly varies on a monthly (or quarterly) basis. Therefore, for options with relatively short maturity, it is reasonable to assume that the dividend yield is constant.
Case of Known Dividends.
We will now focus on the valuation of European options on a stock with declared dividens during the option’s lifetime. We assume that the amount of dividends to be paid before option’s expiry, as well as the dates of their payments, are known in advance. The aim is to show that the Black-Scholes formula remains in force, provided that the current stock price is reduced by the discounted value of all the dividends to be paid during the life of the option. As usual, it is sufficient to consider the case \(t=0\). Assume that the dividends \(\lambda_{1},\cdots ,\lambda_{m}\) are to be paid at the dates \(0<T_{1}<\cdots <T_{m}<T\). We assume, in addition, that dividend payments are known in advance; that is, \(\lambda_{1},\cdots ,\lambda_{m}\) are real numbers. The present value of all future dividend payments equals
\begin{equation}{\label{museq623}}\tag{81}
\tilde{I}_{t}=\sum_{j=1}^{m}\lambda_{j}\cdot e^{-r(T_{j}-t)}\cdot I_{[t,T]}(T_{j})\mbox{ for all }t\in [0,T],
\end{equation}
and the value of all the dividends paid after time \(t\) and compounded at the risk-free rate to the option’s expiry date \(T\) is given by the expression
\begin{equation}{\label{museq624}}\tag{82}
I_{t}=\sum_{j=1}^{m}\lambda_{j}\cdot e^{r(T-T_{j})}\cdot I_{[t,T]}(T_{j})\mbox{ for all }t\in [0,T].
\end{equation}
In modeling the stock price we will separate the capital gains process from the impact of dividend payments. The capital gains process \(G\) is assumed to follow the usual SDE
\begin{equation}{\label{museq625}}\tag{83}
dG_{t}=\mu\cdot G_{t}dt+\sigma\cdot G_{t}dW_{t}\mbox{ for }G_{0}>0,
\end{equation}
hence \(G\) follows a geometric Brownian motion. The process \(S\), representing the price of a stock which pays dividends \(\lambda_{1},\cdots ,\lambda_{m}\) at times \(T_{1},\cdots ,T_{m}\), may be introduced by setting
\begin{equation}{\label{museq626}}\tag{84}
S_{t}=G_{t}-\sum_{j=1}^{m}\lambda_{j}\cdot e^{r(t-T_{j})}\cdot I_{[T_{j},T]}(t)=G_{t}-D_{t}\mbox{ for }t\in [0,T],
\end{equation}
where the process \(D\) given by the equality
\[D_{t}=\sum_{j=1}^{m}\lambda_{j}\cdot e^{r(t-T_{j})}\cdot I_{[T_{j},T]}(t)\]
account for the impact of ex-dividend stock price decline. Alternatively, one may put
\begin{equation}{\label{museq627}}\tag{85}
S_{t}=G_{t}+\sum_{j=1}^{m}\lambda_{j}\cdot e^{-r(T_{j}-t)}\cdot I_{[0,T_{j}]}(t)=G_{t}+\bar{d}_{t}\mbox{ for }t\in [0,T],
\end{equation}
where \(\bar{d}\) now statisfies
\[\bar{d}_{t}=\sum_{j=1}^{m}\lambda_{j}\cdot e^{-r(T_{j}-t)}\cdot I_{[0,T_{j}]}(t).\]
Notice that in the first case we have
\begin{equation}{\label{museq2050}}\tag{86}
G_{0}=S_{0}\mbox{ and }G_{T}=S_{T}+D_{T}=S_{T}+I_{0},
\end{equation}
while in the second case
\[G_{0}=S_{0}-\bar{d}_{0}=S_{0}-\tilde{I}_{0}\mbox{ and }G_{T}=S_{T}.\]
Not surprisingly, the option valuation formulas corresponding to various specifications of the stock price (i.e., to (\ref{museq626}) and (\ref{museq627})).
As usual, the first step in option valuation is to introduce the notion of a self-financing trading strategy. Under the present assumptions, it is natural to assume that if \(\boldsymbol{\phi}=(\phi^{(1)},\phi^{(2)})\) is a self-financing trading strategy, then all dividends received
during the option’s lifetime are immediately reinvested in stocks or bonds. Therefore, the wealth process of any trading strategy \(\boldsymbol{\phi}\), which equals
\[V_{t}(\boldsymbol{\phi})=\phi_{t}^{(1)}\cdot S_{t}+\phi_{t}^{(2)}\cdot B_{t},\]
satisfies, as usual,
\begin{equation}{\label{museq628}}\tag{87}
dV_{t}(\boldsymbol{\phi})=\phi_{t}^{(1)}dG_{t}+\phi_{t}^{(2)}dB_{t}.
\end{equation}
Intuitively, since the decline of the stock price equals the dividend, the wealth of a portfolio is not influenced by dividend payments. It turns
out that the resulting option valuation formula will not agree with the standard Black-Scholes result, however. In view of (\ref{museq628}), the martingale measure \(P^{*}\) for the security market model corresponds to the unique martingale measure for the discounted capital gains process \(G_{t}^{*}=G_{t}/B_{t}\). Hence, \(P^{*}\) can be found in exactly the same way as in the proof of Theorem \ref{must511}. We can also price options using the standard risk-neutral valuation approach.
\begin{equation}{\label{musp622}}\tag{88}\mbox{}\end{equation}
Proposition \ref{musp622}. Consider a European call option with strike price \(K\) and expiry date \(T\), written on a stock \(S\) which pays deterministic dividends \(\lambda_{1},\cdots ,\lambda_{m}\) at times \(T_{1},\cdots ,T_{m}\). Assume that thestock price \(S\) satisfies \((\ref{museq625})\) and \((\ref{museq626})\). Then, the arbitrage price at time \(t\) of this option equals
\[C_{t}^{\lambda}=S_{t}\cdot N(h_{1}(S_{t},T-t))-e^{-r(T-t)}\cdot (K+I_{t})\cdot N(h_{2}(S_{t},T-t)),\]
where \(I_{t}\) is given by the expression \((\ref{museq624})\), and
\[h_{1,2}(S_{t},T-t)=\frac{\ln S_{t}-\ln (K+I_{t})+(r\pm\frac{\sigma^{2}}{2})\cdot (T-t)}{\sigma\cdot\sqrt{T-t}}.\]
Proof. It is sufficient to consider the case of \(t=0\). An application of the risk-neutral valuation formula yields
\[C_{0}^{\lambda}=e^{-rT}\cdot E_{P^{*}}\left [(S_{T}-K)^{+}\right ]=
E_{P^{*}}\left [(G_{T}^{*}-e^{-rT}\cdot (I_{0}+K))^{+}\right ]\]
from (\ref{museq2050}) and the fact \(e^{-rT}=B^{-1}_{T}\). The dynamics of the capital gains process \(G\) under \(P^{*}\) are
\[dG_{t}=r\cdot G_{t}dt+\sigma\cdot G_{t}dW_{t}\mbox{ for }G_{0}=S_{0}>0.\]
We conclude that the price of the European call option is given by the standard Black-Scholes formula with strike price \(K\) replaced by \(I_{0}+K\). \(\blacksquare\)
By virtue of Proposition \ref{musp622}, if the stock price behavior is described by (\ref{museq625}) and (\ref{museq626}), then the pricing of European options corresponds to use of the Black-Scholes formula with the strike price increased by the value of the dividends compounded to time \(T\) at the risk-free rate. The next result corresponds to the second specification of the stock price, given by (\ref{museq627}).
Proposition. Consider a European call option with strike price \(K\) and expiry date \(T\), written on a stock \(S\) which pays deterministic dividends \(\lambda_{1},\cdots ,\lambda_{m}\) at times \(T_{1},\cdots ,T_{m}\). Assume that the stock price \(S\) satisfies (\ref{museq625})–(\ref{museq627}). Then, the arbitrage price at time \(t\) of this option equals
\[\tilde{C}_{t}^{\lambda}=(S_{t}-\tilde{I}_{t})\cdot N(d_{1}(S_{t}-\tilde{I}_{t}))-e^{-r(T-t)}\cdot K\cdot N(d_{2}(S_{t}-\tilde{I}_{t},T-t)),\]
where \(\tilde{I}_{t}\) is given by equality \((\ref{museq623})\), and the functions \(d_{1},d_{2}\) are given by (\ref{museq517}) and
(\ref{museq518}).
Proof. Once again we consider the case of \(t=0\), We need to find
\[\tilde{C}_{t}^{\lambda}=e^{-rT}\cdot E_{P^{*}}\left [(S_{T}-K)^{+}\right ]=e^{-rT}\cdot E_{P^{*}}\left [(G_{T}-K)^{+}\right ].\]
But now under \(P^{*}\), we have
\[dG_{t}=G_{t}\cdot (rdt+\sigma dW_{t})\mbox{ for }G_{0}=S_{0}-\tilde{I}_{0}>0.\]
The assertion is now obvious. \(\blacksquare\)
\begin{equation}{\label{f}}\tag{F}\mbox{}\end{equation}
Stock Price Volatility.
Apart from its theoretical appeal, the Black-Scholes model has been largely adopted by market practitioners, either in its original form or after suitable modifications. Let us summarize once again the basic assumptions of the Black-Scholes model, which allowed us to value European options by arbitrage reasoning:
- the market for the stock, the option and cash is perfect;
- present and future interest rates are known with certainty;
- the lending and borrowing interest rates are equal;
- the stock price has a known variance in rate of return;
- there are no transaction costs or taxes;
- there are no margin requirements.
Let us comment on the last assumption. Market regulations usually impose restrictions upon the amount of funds one can borrow to purchase securities. For instance, if there is a \(50\%\) margin requirement, \(50\%\) at most of the stock’s value can be borrowed at the time of purchase. If the stock price declines after initiation, the borrowings may rise by up to \(75\%\) of the stock’s value. On the other hand, when a party shorts a stock, a margin account must be established with a balance of at least \(50\%\) of stock’s initial value. If the stock price rises, it remains at least \(30\%\) of the subsequent stock’s value. The proceeds from the short sale are held by the broker and usually don not earn interest (on the contrary, the margin account earns interest). When purchasing naked calls, no margin is required, but when selling a naked call, a margin must be maintained, just as if it were s short sale. Alternatively, the underlying security can be placed with the broker, resulting in a covered call. When the fifth assumption is relaxed, the frequency of transactions may mean that the exact formulas are very sensitive to the imposition of even small transaction costs. In essence, positive transaction costs impose some risk on neutral hedgers who must adopt finite holding periods.
Historical Volatility.
All potential practical applications of the Black-Scholes formula hinge on knowledge of the volatility parameter of the return of stock prices. Indeed, of the five variables necessary to specify the model, all are directly observable except for the stock price volatility. The most natural approach uses an estimate of the standard deviation based upon an ex-post series of returns from the underlying stock. The stock volatilities can be estimated from daily data over the year preceding each option price observation. Although the estimation of stock price
volatility from historical data is a fairly straightforward procedure, some important points should be mentioned. Firstly, to reduce the estimation risk arising from the sampling error, it seems natural to increase the sample size, e.g., by using a longer series of historical observations or by increasing the frequency of obervations. Unfortunately, there is evidence to suggest that the variance is non-stationary, so that extending the observation period may make matters even worse. Furthermore, in many cases only daily data are available, so that there is a limit on the number of observations available within a given period. Finally, since the option pricing formula is nonlinear in the standard deviation, an unbiased estimate of the standard deviation does not produce an unbiased estimate of option price. To summarize, since the volatility is usually unstable through time, historical precedent is a poor guide for estimating future volatility. Moreover, estimates of option prices based on historical volatilities are systematically biased.
Implied Volatility.
Alternatively, one can infer the investment community’s consensus outlook as to the volatility of a given asset by examining the prices at which options on that asset trade. Since an option price appears to be an increasing function of the underlying stock volatility, and all other factors determining the option price are known with certainty, one can infer the volatility that is implied in the observed market price of an option. More specifically, the {\bf implied volatility} \(\sigma_{imp}\) is derived from the nonlinear equation
\[C_{t}=s\cdot N(d_{1}(s,T-t))-K\cdot e^{-r(T-t)}\cdot N(d_{2}(s,T-t)),\]
where the only unknown parameter is \(\sigma\) since, for \(C_{t}\), we take the current market price of the call option. In other words, the implied volatility is the value of the standard deviation of the stock returns that, when put in the Black-Scholes formula, results in a model price equal to the current market price. The actual value of the implied volatility \(\sigma_{imp}\) determined in this way depends on an option’s contractual features; that is, on the value \(K\) of the strike price, as well as on the time \(T-t\) to maturity. A properly weighted average of these implied standard deviations is used as a measure of the market forecasts of return variability.
Volatility Mis-specification.
In view of the discussion above, an important issue is the dependence of an option’s price on the level of stock price volatility. The answer is well known when the actual volatility is assumed to be a deterministic constant. On the other hand, relatively little is known if the actual volatility is assumed to be random; more specifically, if it follows a stochastic process.
The first step is to examine the dependence of an option’s price on the level of the volatility. It appears that the answer depends essentially on specific features of a volatility process. Generally speaking, we restrict the attention to those volatility process \(\sigma_{t}\) which are of the form \(\sigma_{t}=\sigma (S_{t},t)\), where \(\sigma :\mathbb{R}_{+} \times [0,T]\rightarrow \mathbb{R}_{+}\) is a dterministic function. We thus have the following dynamics of the stock price under the martingale measure
\[dS_{t}=r(t)\cdot S_{t}dt+\sigma (S_{t},t)\cdot S_{t}dW_{t},\]
where \(r:[0,T]\rightarrow \mathbb{R}_{+}\) is a deterministic function. In this case, the answer is positive; that is, the option’s price is monotone: if \(\tilde{\sigma}\geq\sigma\), then the price assuming \(\tilde{\sigma}\) is never less than the price evaluated for \(\sigma\). Somewhat surprisingly, if the path-dependent volatilities are also allowed, it is possible to construct an example in which this property is violated.
Stochastic Volatility Models.
In a continuous-time framework, volatility \(\sigma_{t}\) is assumed to follow a diffusion process. Let the stock price \(S\) is given by the expression
\begin{equation}{\label{museq629}}\tag{89}
dS_{t}=\mu (S_{t},t)dt+\sigma_{t}\cdot S_{t}dW_{t},
\end{equation}
with the volatility \(\sigma\) satisfying
\begin{equation}{\label{museq630}}\tag{90}
d\sigma_{t}=a(\sigma_{t},t)dt+b(\sigma_{t},t)d\bar{W}_{t},
\end{equation}
where \(W\) and \(\tilde{W}\) are standard one-dimensional Brownian motions defined on some filtered probability space \((\Omega ,{\bf F},P)\) with the cross-variation that equals \(d\langle W,\tilde{W}\rangle_{t}=\rho dt\) for some constant \(\rho\) (processes \(W\) and \(\tilde{W}\) are mutually independent if and only if \(\rho =0\)). Under suitable regularity condition, a unique solution \((S,\sigma )\) to SDEs (\ref{museq629}) and (\ref{museq630}) is known to follow a two-dimensional diffusion process. Generally speaking, a generalization of the Black-Scholes price model described by SDEs (\ref{museq629}) and (\ref{museq630}) is referred to as a stochastic volatility model.
A special case of a stochastic volatility model with \(W=\tilde{W}\) can be obtained by asuming that the stock price is described by the expression
\[dS_{t}=\mu (S_{t},t)dt+g(S_{t})dW_{t}\]
for some strictly increasing (or decreasing) smooth function \(g:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\). To check this, note that by virtue of Ito’s formula, we have
\[d\sigma_{t}=g'(S_{t})dS_{t}+\frac{1}{2}g”(S_{t})d\langle S_{t}\rangle_{t}\]
and thus
\[d\sigma_{t}=\left (g'(h(\sigma_{t}))\cdot\mu (h(\sigma_{t}),t)+\frac{1}{2}
g”(h(\sigma_{t}))\cdot\sigma^{2}\right )dt+g'(h(\sigma_{t}))\cdot\sigma_{t}dW_{t},\]
where \(h\) stands for the inverse function of \(g\). The process \(\sigma\) is thus specified by a particular case of equation (\ref{museq630}). Generally speaking, stochastic volatility models are not complete, hence typical contingent claims (such as European options) cannot be priced by arbitrage. Still, it is possible to derive, under additional hypotheses, the partial differential equation satisfied by the value of a contingent claim. To derive this PDE, which generalizes the Black-Scholes PDE, one needs first to specify the so-called market price for risk, which reflects the expected excess return per unit risk over the risk-free rate. Intuitively, the market price for risk represents the return-to-risk trade-off demanded by investors for bearing the volatility risk of the stock (Mathematically speaking, the market price for risk is associated with the Girsanov transformation of the underlying probability measure leading to a particular martingale measure. The necessity of specifying the market prices for risk is related to the incompleteness of most stochastic volatility models.). It is thus clear that pricing of contingent claims using the market price for risk is not preference-free in general (typically, one assumes that the representative investor is risk-averse and has a constant relative risk-aversion utility function). Assume that the dynamics of two-dimensional diffusion process \((S,\sigma )\) are given by (\ref{museq629}) and (\ref{museq630}) with mutually independent Brownian motions \(W\) and \(\tilde{W}\). Then the price function \(v=v(s,\sigma ,t)\) of a European contingent claim can be shown to satisfy the following PDE
\[v_{t}+\frac{\sigma^{2}}{2}\cdot s^{2}\cdot v_{ss}+r\cdot s\cdot v_{s}-
r\cdot v+\frac{b^{2}}{2}\cdot v_{\sigma\sigma}+(a+\eta\cdot b)\cdot v_{\sigma}=0,\]
where \(\eta =\eta (t,\sigma )\) represents the market price for volatility risk, which needs to be exogenously specified.
Discretization of a diffusion-type stochastic volatility model leads to the so-called autoregessive random variance models, the ARV models for short. For instance, the evolution of a discrete-time, two-dimensional process \((S_{t},\sigma_{t})\) may be described by the following recurrence relation
\[\ln S_{t}=\ln S_{t-1}+\mu +\sigma_{t-1}\cdot\xi_{t}\mbox{ and }
\ln\sigma_{t}=\nu -\lambda\cdot (\nu -\ln\sigma_{t-1})+\theta\cdot\zeta_{t},\]
where \((\xi_{t},\zeta_{t})\) for \(t\in {\bf N}\) are independent identically distributed random variables with Gaussian law (ref. Duffie \cite{duf}). Still another approach to the modelling of stochastic volatility in a discrete-time framework is based on so-called ARCH (or GARCH) models; that is, models with the property of general autoregressive conditional heteroskedasticity.
