Exotic Options

Charles Emile Hippolyte Lecomte-Vernet (1821-1900) was a French painter.

The topics are

The aim here is to study examples of more sophisticated option contracts. For convenience, we give the generic name exotic option to any option contract which is not a standard European or American option. Although the payoffs of exotic options are given by similar expressions for both spot and futures options, it is clear that the corresponding valuation formulas would not agree. Therefore, it should be made clear that we will restrict attention to the case of exotic spot options. We find it convenient to classify the large family of exotic options as follows:

Package options. Options that are equivalent to a portfolio of standard European options, cash and underlying asset (stock, say);

Forward-start options. Options that are paid for in the present but received by holders at a pre-specified future date;

Chooser options. Option contracts that are chosen by their holders to be call or put at a prescribed future date;

Compound options. Option contracts with other options playing the role of the underlying assets;

Digital (Binary) options. Contracts whose payoff is defined by means of some binary function;

Barrier options. Options whose payoff depends on whether the underlying asset price reaches some barrier during the option’s lifetime’

Lookback options. Options whose payoff depends on the minimum or maximum price of the underlying asset during options’ lifetimes;

Asian options. Options whose payoff depends on the average price of the underlying asset during a prespecified period;

Basket options. Options with a payoff depending on the average of prices of several assets;

$\alpha$-Quantile options. Options whose payoff depends on the percentage of time that the price of the underlying asset remains below some level;

Combined options on several assets. Including options on the minimum or maximum price of two risky assets;

Russian options. A “user friendly” variant of a standard American option.

\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}

Package Options.

An arbitrary financial contract whose terminal payoff is a piecewise linear function of the terminal price of the underlying asset may be seen as a package option; that is, a combination of standard options, cash and the underlying asset. As usual, we denote by $S$ the price process of the underlying asset; we shall refer to $S$ as a stock price. Unless explicitly stated otherwise, we shall place ourselves within the classic Black-Scholes framework.

Collars.

Let $K_{2}>K_{1}>0$ be fixed real numbers. The payoff at expiry date $T$ from the long position in a {\bf collar option} equals
\[{\bf CL}_{T}\equiv\min\{\max\{S_{T},K_{1}\},K_{2}\}.\]
It is easily seen that the payoff ${\bf CL}_{T}$ can be represented as follows
\[{\bf CL}_{T}=K_{1}+(S_{T}-K_{1})^{+}-(S_{T}-K_{2})^{+}\]
so that a collar option can be seen as a portfolio of cash and two standard call options. This implies that the arbitrage price of a collar option at any date $t$ before expiry equals
\[{\bf CL}_{t}=K_{1}\cdot e^{-r(T-t)}+C(S_{t},T-t,K_{1})-C(S_{t},T-t,K_{2}),\]
where $C(s,T-t,K)=C(s,T-t,K,r,\sigma )$ stands for the Black-Scholes call option price at time $t$, where the current level of the stock price is $s$, and the exercise price of the option equals $K$ (see formula (\ref{museq556})).

Break Forwards.

By a break forward, we mean a modification of a typical forward contract, in which the potential loss from the long position is limited by some prespecified number. More explicitly, the payoff from the long break forward is defined by the equality
\[{\bf BF}_{T}\equiv\max\{S_{T},F\}-K,\]
where $F=F_{S}(0,T)=S_{0}\cdot e^{rT}$ is the forward price of a stock for settlement at time $T$, and $K>F$ is some constant. The delivery price $K$ is set in such a manner that the break forward contract is worthless when it is entered into. Since
\[{\bf BF}_{T}=(S_{T}-F)^{+}+F-K,\]
it is clear that for every $t\in [0,T]$,
\[{\bf BF}_{t}=C(S_{t},T-t,F)+(F-K)\cdot e^{-r(T-t)}.\]
In particular, the right level of $K$, $K_{0}$ say, is given by the expression
\[K_{0}=e^{rT}\cdot\left (S_{0}+C(S_{0},T,S_{0}\cdot e^{rT})\right ).\]
Using the Black-Scholes valuation formula, we end up with the following equality
\[K_{0}=e^{rT}\cdot S_{0}\cdot\left (1+N(d_{1}(S_{0},T))-N(d_{2}(S_{0},T))\right ),\]
where $d_{1},d_{2}$ are given by (\ref{museq517}).

Range Forwards.

A range forward may be seen as a special case of a collar; one with zero initial cost. Its payoff at expiry is
\[{\bf RF}_{T}\equiv\max\{\min\{S_{T},K_{2}\},K_{1}\}-F=\max\{\min\{S_{T}-F,K_{2}-F\},K_{1}-F\},\]
where $K_{1}<F<K_{2}$, and as before $F=F_{S}(0,T)=S_{0}\cdot e^{rT}$. It appears convenient to decompose the payoff of a range forward in the following way
\[{\bf RF}_{T}=S_{T}-F+(K_{1}-S_{T})^{+}-(S_{T}-K_{2})^{+}.\]
Indeed, the above representation of the payoff implies directly that a range forward may be seen as a portfolio composed of a long forward contract, a long put option with strike price $K_{1}$, and finally a short call option with strike price $K_{2}$. Furthermore, its price at $t$ equals
\[{\bf RF}_{t}=S_{t}-S_{0}\cdot e^{rt}+P(S_{t},T-t,K_{1})-C(S_{t}-T-t,K_{2}).\]
As mentioned earlier, the levels $K_{1}$ and $K_{2}$ are chosen in such a way that the initial value of a range forward equals $0$.

\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}

Forward-start Options.

Let us consider two dates, say $T_{0}$ and $T$, with $T_{0}<T$. A forward-start option is a contract in which the holder receives, at time $T_{0}$ (at no additional cost), an option with expiry date $T$ and exercise price $K$ equal to $S_{T_{0}}$. On the other hand, the holder must pay at time $0$ an up-front fee, the price of a forward-start option. Let us consider the case of a forward-start call option, with terminal payoff
\[{\bf FS}_{T}\equiv (S_{T}-S_{T_{0}})^{+}.\]
To find the price at time $t\in [0,T_{0}]$ of such an option, it suffices to consider its value at the delivery date $T_{0}$, that is,
\[{\bf FS}_{T}=C(S_{T_{0}},T-T_{0},S_{T_{0}}).\]
Since we restrict the attention to the classic Black-Scholes model, it is easily seen that
\[C(S_{T_{0}},T-T_{0},S_{T_{0}})=S_{T_{0}}\cdot C(1,T-T_{0},1),\]
and thus the option’s value at time $0$ equals
\[{\bf FS}_{0}=S_{0}\cdot C(1,T-T_{0},1)=C(S_{0},T-T_{0},S_{0}).\]
If a stock continuously pays dividends at a constant rate $\lambda$, the above equality should be modified as follows
\[{\bf FS}_{0}^{\lambda}=e^{-\lambda T_{0}}\cdot C^{\lambda}(S_{0},T-T_{0},S_{0}),\]
where $C^{\lambda}$ stands for the call option price derived in Proposition \ref{musp621}. Similar formulas can be derived for the case of a forward-start put option.

\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}

Chooser Options.

As suggested by the name of the contract, a chooser option is an agreement in which one party has the right to choose at some future
date $T_{0}$ whether the option is to be a call or put option with a common exercise price $K$ and remaining time to expiry $T-T_{0}$. Therefore, the payoff at $T_{0}$ of a standard chooser option is
\[{\bf CH}_{T_{0}}\equiv\max\{C(S_{T_{0}},T-T_{0},K),P(S_{T_{0}},T-T_{0},K)\},\]
while its terminal payoff is given by the expression
\[{\bf CH}_{T_{0}}=(S_{T}-K)^{+}\cdot I_{A}+(K-S_{T})^{+}\cdot I_{A^{c}},\]
where $A$ stands for the following event, which belongs to the $\sigma$-field ${\cal F}_{T_{0}}$
\[A=\{\omega\in\Omega :C(S_{T_{0}},T-T_{0},K)>P(S_{T_{0}},T-T_{0},K)\}\]
and $A^{c}$ is the complement of $A$ in $\Omega$. Recall that the put-call parity implies that
\[P(S_{T_{0}},T-T_{0},K)=C(S_{T_{0}},T-T_{0},K)-S_{T_{0}}+K\cdot e^{-r(T-T_{0})},\]
and thus
\[{\bf CH}_{T_{0}}=\max\{C(S_{T_{0}},T-T_{0},K),C(S_{T_{0}},T-T_{0},K)-S_{T_{0}}+K\cdot e^{-r(T-T_{0})},\]
or finally
\[{\bf CH}_{T_{0}}=C(S_{T_{0}},T-T_{0},K)+(K\cdot e^{-r(T-T_{0})}-S_{T_{0}})^{+}.\]
The last equality implies immediately that the standard chooser option is equivalent to the portfolio composed of a long call option and a long put option (with different exercise prices and different expiry dates), so that its arbitrage price equals
\[{\bf CH}_{t}=C(S_{t},T-t,K)+P(S_{t},T_{0}-t,K\cdot e^{-t(T_{0}-T)})\]
for every $t\in [0,T_{0}]$. In particular, using the Black-Scholes formula, we get for $t=0$
\[{\bf CH}_{0}=S_{0}\cdot N(d_{1})-N(-\bar{d}_{1})+K\cdot e^{-rT}\cdot\left (N(-\bar{d}_{2}-N(d_{2})\right ),\]
where
\[d_{1,2}=\frac{\ln S_{0}-\ln K+(r\pm\frac{1}{2}\cdot\sigma^{2})T}{\sigma\cdot\sqrt{T}}\mbox{ and }
\bar{d}_{1,2}=\frac{\ln S_{0}-\ln K+rT\pm\frac{1}{2}\cdot\sigma^{2})\cdot T_{0}}{\sigma\cdot\sqrt{T_{0}}}.\]

\begin{equation}{\label{d}}\tag{D}\mbox{}\end{equation}

Compound Options.

A compound option is a standard option with another option being the underlying asset. One can distinguish four basic types of compound options: call on a call, put on a call, call on a put, and put on a put. Let us consider the case of a call on a cal compound option. For two future dates $T_{0}$ and $T$ with $T_{0}<T$, and two exercise prices $K_{0}$ and $K$, consider a call option with exercise price $K_{0}$ and expiry date $T_{0}$ on a call option with strike price $K$ and maturity $T$. It is clear that the payoff of the compound option at time $T_{0}$ is
\[{\bf CO}_{T_{0}}\equiv\left (C(S_{T_{0}},\tau ,K)-K_{0}\right )^{+},\]
where $C(S_{T_{0}},\tau ,K)$ stands for the value at time $T_{0}$ of a standard call option with strike price $K$ and expiry date $T=T_{0}+\tau$. In the Black-Scholes framework, we obtain the following equality
\[C(s,\tau ,K)=s\cdot N(d_{1}(s,\tau ,K))-K\cdot e^{-r\tau}\cdot N(d_{2}(s,\tau ,K)).\]
Moreover, since undr $\mathbb{P}^{*}$ we have
\[S_{T_{0}}=S_{0}\cdot\exp\left (\sigma\cdot\sqrt{T_{0}}\cdot\xi+\left (r-\frac{\sigma^{2}}{2}\right )\cdot T_{0}\right ),\]
where $\xi$ has a standard Gaussian probability law under $\mathbb{P}^{*}$, the price of the compound option at time $0$ equals
\[{\bf CO}_{0}=e^{-rT_{0}}\cdot\int_{x_{0}}^{\infty}(g(x)\cdot N(\hat{d}_{1})
-K\cdot e^{-r\tau}\cdot N(\hat{d}_{2})-K_{0})\cdot n(x)dx,\]
where $\hat{d}_{i}=d_{i}(g(x),\tau ,K)$ for $i=1,2$, the function $g:\mathbb{R}\rightarrow \mathbb{R}$ is given by the formula
\[g(x)=S_{0}\cdot\exp\left (\sigma\cdot\sqrt{T_{0}}\cdot x+\left (r-\frac{\sigma^{2}}{2}\right )\cdot T_{0}\right )\]
and, finally, the constant $x_{0}$ si defined implicitly by the equaltion
\[x_{0}=\inf\{x\in \mathbb{R}:C(g(x),\tau ,K)\geq K_{0}\}.\]
Straightforward calculations yield
\[d_{1}(g(x),\tau ,K)=\frac{\ln S_{0}-\ln K+\sigma\cdot\sqrt{T_{0}}\cdot x
+r\cdot T-\sigma^{2}\cdot T_{0}+\frac{1}{2}\cdot\sigma^{2}\cdot T}{\sigma\cdot\sqrt{T-T_{0}}}\]
and
\[d_{2}(g(x),\tau ,K)=\frac{\ln S_{0}-\ln K+\sigma\cdot\sqrt{T_{0}}\cdot x
+r\cdot T+\frac{1}{2}\cdot\sigma^{2}\cdot T}{\sigma\cdot\sqrt{T-T_{0}}}.\]

\begin{equation}{\label{e}}\tag{E}\mbox{}\end{equation}

Digital Options.

By a digital (or binary) option we mean a contract whose payoff depends in a discontinuous way on the terminal price of the underlying asset. The simplest examples of binary options are cash-or-nothing options and asset-or-nothing options. The payoffs at expiry of a cash-or-nothing call and put options are
\[{\bf BCC}_{T}\equiv X\cdot I_{\{S_{T}>K\}}\mbox{ and }{\bf BCP}_{T}\equiv X\cdot I_{\{S_{T}<K\}},\]
where in both cases $X$ stands for a prespecified amount of cash. Similarly, for the asset-or-nothing option we have
\[{\bf BAC}_{T}\equiv S_{T}\cdot I_{\{S_{T}>K\}}\mbox{ and }{\bf BAP}_{T}\equiv S_{T}\cdot I_{\{S_{T}<K\}}\]
for a call and put, respectively. All options introduced above may be easily priced by means of the risk-neutral valuation formula. Somewhat more complex binary options are the so-called gap options, whose payoff at expiry equals
\[{\bf GC}_{T}\equiv (S_{T}-X)\cdot I_{\{S_{T}>K\}}={\bf BAC}_{T}-{\bf BCC}_{T}\]
for the call option, and
\[{\bf GP}_{T}\equiv (X-S_{T})\cdot I_{\{S_{T}<K\}}={\bf BCP}_{T}-{\bf BAP}_{T}\]
for the corresponding put option. Once again, pricing these options involves no difficulties. As a last example of a binary option, let us mention a super-share, whose payoff is
\[{\bf SS}_{T}\equiv\frac{S_{T}}{K_{1}}\cdot I_{\{K_{1}<S_{T}<K_{2}\}}\]
for some positive constants $K_{1}<K_{2}$. The price of such an option at time $0$ is easily seen to equal
\[{\bf SS}_{0}=\frac{S_{0}}{K_{1}}\cdot\left (N(h_{1}(S_{0},T))-N(h_{2}(S_{0},T))\right ),\]
where
\[h_{i}(s,t)=\frac{\ln S_{0}-\ln K_{i}+(r+\frac{1}{2}\cdot\sigma^{2})\cdot t}{\sigma\cdot\sqrt{t}}.\]

\begin{equation}{\label{f}}\tag{F}\mbox{}\end{equation}

Barrier Options.

The generic term barrier options refers to the class of options whose payoff depends on whether or not the underlying prices hit a pre-specified barrier during the option’s lifetimes. To give the flavor of the mathematical techniques used when dealing with barrier options, we will examine a specific kind of currency barrier option, namely the down-and-out call option. The payoff at expiry of a down-and-out call option equals (in units of domestic currency)
\[C_{T}^{1}\equiv (Q_{T}-K)^{+}\cdot I_{\{\min_{0\leq t\leq T} Q_{t}\geq H\}},\]
where $K$ and $H$ are constants. It follows from the formula above that the down-and-out option becomes worthless (or is knocked out) if, at any time $t$ prior to the expiry date $T$, the current exchange rate $Q_{t}$ falls below a predetermined level $H$. It is thus evident that a down-and-out option is less valuable than a stabdard currency option. The aim is to find an explicit formula for a so-called knock-out discount.

Out-of-the-money knock-out option.

Suppose first that the inequalities $H<K$ and $H<Q_{0}$ are satisfied. From the general features of a down-and-out call, it is clear that the option is knocked out when it is out-of-the-money. Recall that under the domestic martingale measure $\mathbb{P}^{*}$ we have (ref. (\ref{museq73}))
\[Q_{t}=Q_{0}\cdot\exp\left (\sigma_{Q}\cdot W_{t}^{*}+\lambda\cdot t\right )=Q_{0}\cdot e^{X_{t}},\]
where $X_{t}=\sigma_{Q}\cdot W_{t}^{*}+\lambda\cdot t$ for $t\in [0,T]$, and $\lambda =r_{d}-r_{f}-\frac{1}{2}\cdot\sigma_{Q}^{2}$. Therefore,
\[\left\{\omega\in\Omega :\min_{0\leq t\leq T}Q_{t}\geq H\right\}=
\left\{\omega\in\Omega :m_{T}\geq\ln H-\ln Q_{0}\right\},\]
where $m_{T}=\min_{0\leq t\leq T}X_{t}$, and thus
\[C_{T}^{1}=(Q_{T}-K)\cdot I_{\{Q_{T}\geq K,\min_{0\leq t\leq T}Q_{t}\geq H\}}=Q_{0}\cdot e^{X_{T}}\cdot I_{D}-K\cdot I_{D},\]
where $D$ stands for the set
\[D=\left\{\omega\in\Omega :X_{T}\geq\ln K-\ln Q_{0},m_{T}\geq\ln K-\ln Q_{0}\right\}.\]
We conclude that the price at time $0$ of a down-and-out call option admits the following representation
\[C_{0}^{1}=e^{-r_{d}T}\cdot Q_{0}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [e^{X_{T}}\cdot
I_{D}\right ]-e^{-r_{d}T}\cdot K\cdot \mathbb{P}^{*}\{D\},\]
where $\mathbb{P}^{*}$ is the martingale measure of the domestic market. In order to directly evaluate $C_{0}^{1}$ by means of integration, we need to find first the joint probability distribution of random variables $X_{T}$ and $m_{T}$. On can show that for all $x,y$ such that $y\leq 0$ and $y\leq x$, we have
\[\mathbb{P}^{*}\left\{X_{T}\geq x,m_{T}\geq y\right\}=N\left (\frac{-x+\lambda T}{\sigma\cdot\sqrt{T}}\right )-
e^{2\lambda y/\sigma^{2}}\cdot N\left (\frac{-x+2y+\lambda T}{\sigma\cdot\sqrt{T}}\right ),\]
where, for the sake of notational convenience, we write $\sigma$ in place of $\sigma_{Q}$. Consequently, the probability density function of $(X_{T},m_{T})$ equals
\[f(x,y)=\frac{-2(2y-x)}{\sigma^{3}\cdot T^{3/2}}\cdot e^{2\lambda y/\sigma^{2}}\cdot n\left (\frac{-x+2y+\lambda T}{\sigma\cdot
\sqrt{T}}\right )\]
for $y\leq 0, y\leq x$, where $n$ stands for the standard Gaussian density function. From the above, it follows that
\[\mathbb{P}^{*}\{D\}=N\left (\frac{\ln Q_{0}-\ln K+\lambda T}{\sigma\cdot\sqrt{T}}
\right )-\left (\frac{H}{Q_{0}}\right )^{2\lambda /\sigma^{2}}\cdot N\left (\frac{\ln H^{2}-\ln (Q_{0}\cdot K)+\lambda T}
{\sigma\cdot\sqrt{T}}\right ).\]
To find the expectation
\[I_{1}\equiv \mathbb{E}_{\mathbb{P}^{*}}\left [e^{X_{T}}\cdot I_{D}\right ]=\mathbb{E}_{\mathbb{P}^{*}}\left [
e^{X_{T}}\cdot I_{\{X_{T}\geq\ln K-\ln Q_{0},m_{T}\geq\ln H-\ln Q_{0}\}}\right ],\]
we need to evaluate the double integral
\[\int\int_{A} e^{x}\cdot f(x,y)dxdy,\]
where
\[A=\{(x,y):x\geq\ln K-\ln Q_{0},y\geq\ln H-\ln Q_{0},y\leq 0,y\leq x\}.\]
Straightforward integration leads to the following result
\[I_{1}=e^{r_{d}-r_{f})T}\cdot\left (N(h_{1}(Q_{0},T))-\left (\frac{H}{Q_{0}}
\right )^{2+2\lambda/\sigma^{2}}\cdot N(c_{1}(Q_{0},T))\right ),\]
where
\[h_{1,2}(q,t)=\frac{\ln q-\ln K+(r_{d}-r_{f}\pm\frac{1}{2}\cdot\sigma^{2}) \cdot t}{\sigma\cdot\sqrt{t}}\]
and
\[c_{1,2}(q,t)=\frac{\ln H^{2}-\ln (qK)+(r_{d}-r_{f}\pm\frac{1}{2}\cdot\sigma^{2})\cdot t}{\sigma\cdot\sqrt{t}}.\]
By collecting and rearranging the formulas above, we conclude that the price at initiation of the knock-out option admits the following
representation (recall that we write $\sigma =\sigma_{Q}$)
\begin{equation}{\label{museq91}}\tag{1}
C_{0}^{1}=C_{0}^{Q}-J_{0}=\mbox{Standard Call Price}-\mbox{Knockout Discount}.
\end{equation}
where (ref. Proposition \ref{musp722})
\[C_{0}^{Q}=Q_{0}\cdot e^{-r_{f}T}\cdot N(h_{1})-K\cdot e^{-r_{d}T}\cdot N(h_{2})\]
and
\[J_{0}=Q_{0}\cdot e^{-r_{f}T}\cdot\left (\frac{H}{Q_{0}}\right )^{2+2\lambda /\sigma^{2}}\cdot N(c_{1})-K\cdot e^{-r_{d}T}\cdot\left (\frac{H}{Q_{0}}\right )^{2\lambda /\sigma^{2}}\cdot N(c_{2}),\]
where $h_{1,2}=h_{1,2}(Q_{0},T)$ nd $c_{1,2}=c_{1,2}(Q_{0},T)$. Notice that the proof of this formula can be substantially simplified by an application of Girsanov’s theorem. We define an auxiliary probability measure $\tilde{\mathbb{P}}$ by setting
\[\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}^{*}}=\exp\left (\sigma\cdot W_{T}^{*}-\frac{1}{2}
\cdot\sigma^{2}\cdot T\right )=\eta_{T}\mbox{ $\mathbb{P}^{*}$-a.s.}\]
It follows from the Girsanov theorem that the process $\bar{W}=W_{t}^{*}-\sigma\cdot t$ follows a standard Brownian motion under the probability measure $\bar{\mathbb{P}}$. Moreover, taking into account the definition o $X$, we obtain
\[\mathbb{E}_{\mathbb{P}^{*}}\left [e^{X_{T}}\cdot I_{D}\right ]=e^{(r_{d}-r_{f})T}\cdot
\mathbb{E}_{\mathbb{P}^{*}}\left [\eta_{T}\cdot I_{D}\right ]\]
and thus
\[I_{1}=e^{(r_{d}-r_{f})T}\cdot\bar{P}\{D\}=e^{(r_{d}-r_{f})T}\cdot\bar{P}
\left\{X_{T}\geq\ln K-\ln Q_{0},m_{T}\geq\ln H-\ln Q_{0}\right\}.\]
Finally, the semimartingale decomposition of the process $X$ under $\tilde{\mathbb{P}}$ is
\[X_{t}=\sigma\cdot\bar{W}_{t}+\left (r_{d}-r_{f}+\frac{\sigma^{2}}{2}\right )\cdot t\mbox{ for all }t\in [0,T],\]
hence for every $y\leq 0,y\leq x$, we have
\[\bar{P}\{D\}=N(h_{1}(Q_{0},T))-\left (\frac{H}{Q_{0}}\right )^{2+2\lambda /\sigma^{2}}\cdot N(c_{1}(Q_{0},T)).\]
Representation (\ref{museq91}) of the option’s price now follows easily.

In-the-money knock-out option.

When $K\leq H$ and $H<Q_{0}$, the option is knocked out when it is in-the-money. In thos case, we have
\[D=\{\omega\in\Omega :m_{T}\geq\ln K-\ln Q_{0}\}.\]
since $\{m_{T}\geq\ln H-\ln Q_{0}\}\subset\{X_{T}\geq\ln K-\ln Q_{0}\}$. It is well known that for every $y\leq 0$
\[\mathbb{P}^{*}\{m_{T}\geq y\}=N\left (\frac{-y+\lambda T}{\sigma\cdot\sqrt{T}}
\right )-e^{2\lambda y/\sigma^{2}}\cdot N\left (\frac{y+\lambda T}{\sigma\cdot\sqrt{T}}\right )\]
and thus
\[\mathbb{P}^{*}\{D\}=N\left (\frac{\ln Q_{0}-\ln H+\lambda T}{\sigma\cdot\sqrt{T}}
\right )-\left (\frac{H}{Q_{0}}\right )^{2\lambda /\sigma^{2}}\cdot N\left (
\frac{-\ln Q_{0}-\ln H+\lambda\cdot T}{\sigma\cdot\sqrt{T}}\right ).\]
On the other hand,
\[I_{1}=\mathbb{E}_{\mathbb{P}^{*}}\left [e^{X_{T}}\cdot I_{D}\right ]=e^{(r_{d}-r_{f})T}\cdot
\bar{P}\left\{m_{T}\geq\ln H-\ln Q_{0}\right\}\]
so that
\[I_{1}=e^{(r_{d}-r_{f})T}\cdot\left (N(\hat{h}_{1}(Q_{0},T))-\left (
\frac{H}{Q_{0}}\right )^{2+2\lambda /\sigma^{2}}\cdot N(\hat{c}_{1}(Q_{0},T))\right ),\]
where
\[\hat{h}_{1,2}=\frac{\ln q-\ln H+(r_{d}-r_{f}\pm\frac{1}{2}\cdot\sigma^{2})\cdot t}{\sigma\cdot\sqrt{t}}\]
and
\[\hat{c}_{1,2}=\frac{\ln H-\ln q+(r_{d}-r_{f}\pm\frac{1}{2}\cdot\sigma^{2})\cdot t}{\sigma\cdot\sqrt{t}}.\]
Consequently, the option price at time $0$ equals
\[C_{0}^{1}=\hat{C}_{0}-\hat{J}_{0},\]
where
\[\hat{C}_{0}=Q_{0}\cdot e^{-r_{f}T}\cdot N(\hat{h}_{1}(Q_{0},T))-K\cdot e^{-r_{d}T}\cdot N(\hat{h}_{2}(Q_{0},T))\]
is the price of the standard currency cal with strike $H$, and denoting $\hat{c}_{1,2}=\hat{c}_{1,2}(Q_{0},T)$, we get
\[\hat{J}_{0}=Q_{0}\cdot e^{-r_{f}T}\cdot\left (\frac{H}{Q_{0}}
\right )^{2+2\lambda /\sigma^{2}}\cdot N(\hat{c}_{1})-K\cdot e^{-r_{d}T}\cdot
\left (\frac{H}{Q_{0}}\right )^{2\lambda /\sigma^{2}}\cdot N(\hat{c}_{2}).\]

\begin{equation}{\label{g}}\tag{G}\mbox{}\end{equation}

Lookback Options.

Lookback options are another example of path-dependent options; i.e., option contracts whose payoff at expiry depends not only on the terminal prices of the underlyng assets, but also on asset price fluctuations during the options’ lifetimes. We will examine the follwoing two cases: that of a standard lookback call option, with payoff at expiry
\[{\bf LC}_{T}\equiv (S_{T}-m_{T}^{S})^{+}=S_{T}-m_{T}^{S},\]
where $m_{T}^{S}=\min_{t\in [0,T]}S_{t}$; and that of a standard lookback put option, whose terminal payoff equals
\[{\bf LP}_{T}\equiv (M_{T}^{S}-S_{T})^{+}=M_{T}^{S}-S_{T},\]
where $M_{T}^{S}=\max_{t\in [0,T]}S_{t}$. Note that a lookback option is not a genuiune option contract since the (European) lookback option is always exercised by its holder at its expiry date. It is clear that the arbitrage prices of a lookback options are
\[{\bf LC}_{t}=e^{-r(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}[S_{T}|{\cal F}_{t}]-
e^{-r(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .m_{T}^{S}\right |{\cal F}_{t}\right ]=I_{1}-I_{2}\]
and
\[{\bf LP}_{t}=e^{-r(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .M_{T}^{S}\right |
{\cal F}_{t}\right ]-e^{-r(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}[S_{T}|{\cal F}_{t}=J_{1}-J_{2}\]
for the lookback call and put, respectively.

Proposition. The price at time $t\in [0,T]$ of a European lookback call option equals
\begin{align*}
{\bf LC}_{t} & =s\cdot N\left (\frac{\ln s-\ln m+r_{1}\cdot\tau}
{\sigma\cdot\sqrt{\tau}}\right )-m\cdot e^{-r\tau}\cdot N\left (\frac{\ln s-\ln m+r_{2}\cdot\tau}{\sigma\cdot\sqrt{\tau}}\right )\\
& -\frac{s\cdot\sigma^{2}}{2r}\cdot N\left (\frac{\ln s-\ln m-r_{1}\cdot\tau}{\sigma\cdot\sqrt{\tau}}\right )+e^{-r\tau}\cdot
\frac{s\cdot\sigma^{2}}{2r}\cdot N\left (\frac{\ln s-\ln m+r_{2}\cdot\tau}{\sigma\cdot\sqrt{\tau}}\right ),
\end{align*}
where $s=S_{t}, m=m_{t}^{S}, \tau =T-t$, and $r_{1,2}=r\pm\frac{1}{2}\cdot \sigma^{2}$. Equivalently,
\begin{align*}
{\bf LC}_{t} & =s\cdot N(\tilde{d})-m\cdot e^{-t\tau}\cdot N(\tilde{d}-\sigma\cdot\sqrt{\tau})-\frac{s\cdot\sigma^{2}}{2r}\cdot
N(-\tilde{d})\\
& +e^{-r\tau}\cdot\frac{s\cdot\sigma^{2}}{2r}\cdot\left (\frac{m}{s}
\right )^{2r/\sigma^{2}}\cdot N\left (-\tilde{d}+\frac{2r\cdot\sqrt{\tau}}{\sigma}\right ),
\end{align*}
where
\[\tilde{d}=\frac{\ln s-\ln m+r_{1}\cdot\tau}{\sigma\cdot\sqrt{\tau}}.\]
In particular, if $s=S_{t}=m=m_{t}^{S}$, then by setting $d=r_{1}\cdot \sqrt{\tau}/\sigma$, we get
\[{\bf LC}_{t}=s\cdot\left (N(d)-e^{-r\tau}\cdot N(d-\sigma\cdot\sqrt{\tau})
-\frac{\sigma^{2}}{2r}\cdot N(-d)+e^{-r\tau}\cdot\frac{\sigma^{2}}{2r}\cdot N(d-\sigma\cdot\sqrt{\tau})\right ).\]

Proof. By the martinagle property of the discounted stock price $S^{*}$ under the martingale measure $\mathbb{P}^{*}$, it is clear that
\[I_{1}=e^{-r(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}[S_{T}|{\cal F}_{t}]=e^{rT}\cdot
\mathbb{E}_{\mathbb{P}^{*}}[S_{T}^{*}|{\cal F}_{t}]=S_{t}.\]
To evaluate $I_{2}$, observe first that for every $u\in [t,T]$, we have
\[S_{u}=S_{t}\cdot\exp\left (\sigma\cdot (W_{s}^{*}-W_{t}^{*})+r_{2}(u-t)\right )=S_{t}\cdot e^{-(X_{u}-X_{t})},\]
where $X_{t}=-\sigma\cdot W_{t}^{*}+\nu\cdot t$ with $\nu =\frac{1}{2}\cdot\sigma^{2}-r$, and $W^{*}$ follows the standard Brownian motion under $\mathbb{P}^{*}$. Hence
\[m_{t,T}^{S}\equiv\min_{u\in [t,T]}S_{u}=S_{t}\cdot e^{-M_{t,T}^{X}},\]
where $X_{t,T}^{X}=\max_{u\in [t,T]}(X_{u}-X_{t})$. From the poperties of the Brownian motion it is clear that the random variable
$M_{t,T}^{X}$ under $\mathbb{P}^{*}$ coincides with the law under $\mathbb{P}^{*}$ of $M_{\tau}^{X}$, where $M_{\tau}^{X}=\max_{u\in [0,\tau ]}X_{u}$ and $\tau =T-t$. Furthermore,
\[m_{T}^{S}=\min\left\{m_{t}^{S},m_{t,T}^{S}\right\}=\min\left\{m_{t}^{S},S_{t}\cdot e^{-M_{t,T}^{X}}\right\},\]
where both $m_{t}^{S}$ nd $S_{t}$ are ${\cal F}_{t}$-measurable random variables, and the random variable $M_{t,T}^{X}$ is independent of the $\sigma$-field ${\cal F}_{t}$. Theorefore, it is sufficient to evaluate the expectation
\[L(s,m)\equiv \mathbb{E}_{\mathbb{P}^{*}}\left [\min\{m,s\cdot e^{-M_{t,T}^{X}}\}\right ]=
\mathbb{E}_{\mathbb{P}^{*}}\left [\min\{m,s\cdot e^{-M_{\tau}^{X}}\}\right ]\]
for fixed real numbers $s\geq m>0$. Indeed, we have
\[I_{2}=e^{-r(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .m_{T}^{S}\right |{\cal F}_{t}
\right ]=e^{-r\tau}\cdot L(S_{t},m_{t}^{S}).\]
To find explicitly $L(s,m)$, note first that
\[L(s.m)-m=\mathbb{E}_{\mathbb{P}^{*}}\left [\min\{m,s\cdot e^{-M_{\tau}^{X}}\}\right ]-m=
\mathbb{E}_{\mathbb{P}^{*}}\left [(s\cdot e^{-M_{\tau}^{X}}-m)\cdot I_{\{M_{\tau}^{X}\geq z\}}\right ],\]
where $z=-\ln m+\ln s$. Consequently,
\[L(s,m)-m=s\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left (e^{-M_{\tau}^{X}}-e^{-z}\right )
\cdot I_{\{M_{\tau}^{X}}\}\right ]=-s\cdot\int_{z}^{\infty}e^{-y}\cdot\mathbb{P}^{*}\left\{M_{\tau}^{X}\geq y\right\}dy.\]
By virtue of equality (\ref{museqb34}), we have
\[\mathbb{P}^{*}\left\{M_{\tau}^{X}\geq y\right\}=\mathbb{P}^{*}\left\{X_{\tau}\geq y
\right\}+e^{2\nu y/\sigma^{2}}\cdot \mathbb{P}^{*}\left\{X_{\tau}\geq y+2\nu\tau\right\}.\]
Therefore, using a trivial equality $2\nu /\sigma^{2}-1=-2r/\sigma^{2}$,we obtain
\begin{align*}
L(s,m)-m & =-s\cdot\int_{z}^{\infty} e^{-y}\cdot \mathbb{P}^{*}\left\{
X_{\tau}\geq y\right\}dy-s\cdot\int_{z}^{\infty}e^{-2ry/\sigma^{2}}\cdot
\mathbb{P}^{*}\left\{X_{\tau}\geq y+2\nu\tau\right\}dy\\
& =L_{1}(s,m)+L_{2}(s,m).
\end{align*}
The first integral can be represented in the following way
\[L_{1}(s,m)=-s\cdot\int_{z}^{\infty}e^{-y}\cdot \mathbb{P}^{*}\left\{X_{\tau}\geq y
\right\}dy=s\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [(e^{-X_{\tau}}-e^{-z})\cdot I_{\{X_{\tau}\geq z\}}\right ]\]
and thus
\[L_{1}(s,m)=s\cdot e^{r\tau}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\exp (\sigma\cdot
W_{\tau}^{*}-\sigma^{2}\cdot\tau /2)\cdot I_{\{X_{\tau}\geq z\}}\right ]-m\cdot \mathbb{P}^{*}\left\{X_{\tau}\geq z\right\}.\]
equivalently, we have
\[L_{1}(s,m)=s\cdot e^{r\tau}\cdot Q\left\{X_{\tau}\geq z\right\}-
m\cdot \mathbb{P}^{*}\left\{X_{\tau}\geq z\right\},\]
where the probability measure $Q$ satisfies on $(\Omega ,{\cal F}_{T})$
\[\frac{dQ}{d\mathbb{P}^{*}}=\exp\left (\sigma\cdot W_{\tau}^{*}-\frac{1}{2}\cdot
\sigma^{2}\cdot\tau\right )\mbox{ $\mathbb{P}^{*}$-a.s.}\]
It is now easily seen that
\[L_{1}(s,m)=s\cdot e^{r\tau}\cdot N\left (\frac{\ln m-\ln s-r_{1}\cdot
\tau}{\sigma\cdot\sqrt{\tau}}\right )-m\cdot N\left (\frac{\ln m-\ln s-
r_{2}\cdot\tau}{\sigma\cdot\sqrt{\tau}}\right ).\]
For $L_{2}(s,m)$, observe that
\[L_{2}(s,m)=-s\cdot\int_{z}^{\infty}e^{-2ry/\sigma^{2}}\cdot
\mathbb{P}^{*}\left\{X_{\tau}\geq y+2\nu\tau\right\}dy,\]
or equivalently
\begin{align*}
L_{2}(s,m) & =\frac{s\cdot\sigma^{2}}{2r}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left (
\exp\left ((-2r/\sigma^{2})(X_{\tau}-2\nu\tau )\right )-\exp\left (
-2rz/\sigma^{2}\right )\right )\cdot I_{\{X_{\tau}\geq z+\nu\tau\}}\right ]\\
& =\frac{s\cdot\sigma^{2}}{2r}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\exp\left ((-2r/\sigma^{2})(X_{\tau}-2\nu\tau )\right )
\cdot I_{\{X_{\tau}\geq z+\nu\tau\}}\right ]-e^{-2rz/\sigma^{2}}\cdot
\frac{s\cdot\sigma^{2}}{2r}\cdot \mathbb{P}^{*}\left\{X_{\tau}\geq z+2\nu\tau\right\}.
\end{align*}
Since $X_{\tau}=-\sigma\cdot W_{\tau}^{*}+\nu\cdot\tau$, we get
\[\exp\left ((-2r/\sigma^{2})(X_{\tau}-2\nu\tau )\right )=
e^{r\tau}\cdot\exp\left (2rW_{\tau}^{*}/\sigma -2r^{2}\tau /\sigma^{2}\right ).\]
Let us define the probability measure $\tilde{Q}$ by setting
\[\frac{d\tilde{Q}}{d\mathbb{P}^{*}}=\exp\left (2rW_{\tau}^{*}/\sigma
-2r^{2}\tau /\sigma^{2}\right )\mbox{ $\mathbb{P}^{*}$-a.s.}\]
Notice that the process $\tilde{W}_{t}=W_{t}^{*}-2rt/\sigma$ follows a Brownian motion under $\tilde{Q}$. Furthermore
\[\frac{s\cdot\sigma^{2}}{2r}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\exp\left (
(-2r/\sigma^{2})(X_{\tau}-2\nu\tau )\right )\cdot I_{\{X_{\tau}\geq z+
2\nu\tau\}}\right ]=e^{r\tau}\cdot\frac{s\cdot\sigma^{2}}{2r}\cdot\tilde{Q}\left\{X_{\tau}\geq z+2\nu\tau\right\},\]
and finally
\[L_{2}(s,m)=e^{r\tau}\cdot\frac{s\cdot\sigma^{2}}{2r}\cdot N\left (\frac{-z-r_{1}\cdot\tau}{\sigma\cdot\sqrt{\tau}}\right )-
e^{-2rz/\sigma^{2}}\cdot\frac{s\cdot\sigma^{2}}{2r}\cdot N\left (\frac{-z+r_{2}\cdot\tau}{\sigma\cdot\sqrt{\tau}}\right ),\]
where $z=-\ln m+\ln s$. the asserted formula is now immediate consequence of the relationship
\[{\bf LC}_{t}=s-e^{-r\tau}\cdot (L_{1}(s,m)+L_{2}(s,m)+m)\]
with $s=S_{t}$ and $m=m_{t}^{S}$. $\blacksquare$

Proposition. The price of a European lookback put option at time $t$ equals
\begin{align*}
{\bf LP}_{t} & =-s\cdot N\left (-\frac{\ln s-\ln M+r_{1}\cdot\tau}{\sigma\cdot\sqrt{\tau}}\right )
+M\cdot e^{-r\tau}\cdot N\left (-\frac{\ln s-\ln M+r_{2}\cdot\tau}{\sigma\cdot\sqrt{\tau}}\right )\\
& +\frac{s\cdot\sigma^{2}}{2r}\cdot N\left (\frac{\ln s-\ln M+r_{1}\cdot\tau}{\sigma\cdot\sqrt{\tau}}\right )
-e^{-r\tau}\cdot\frac{s\cdot\sigma^{2}}{2r}\cdot\left (\frac{M}{s}\right)^{2r/\sigma^{2}}\cdot
N\left (\frac{\ln s-\ln M-r_{2}\cdot\tau}{\sigma\cdot\sqrt{\tau}}\right ),
\end{align*}
where $s=S_{t}$, $M=M_{t}^{S}$, $\tau =T-t$, and $r_{1,2}=r\pm\frac{1}{2}\sigma^{2}$. Equivalently,
\begin{align*}
{\bf LP}_{t} & =-s\cdot N(-\hat{d})+M\cdot e^{-r\tau}\cdot
N(-\hat{d}+\sigma\cdot\sqrt{\tau})+s\cdot\frac{\sigma^{2}}{2r}\cdot N(\hat{d})\\
& -e^{-r\tau}\cdot\frac{s\cdot\sigma^{2}}{2r}\cdot\left (\frac{M}{s}
\right )^{2r/\sigma^{2}}\cdot N(\hat{d}-2r\cdot\sqrt{\tau}/\sigma ),
\end{align*}
where
\[\hat{d}=\frac{\ln s-\ln M+(r+\frac{1}{2}\cdot\sigma^{2})\cdot\tau}{\sigma\cdot\sqrt{\tau}}.\]
In particular, if $s=S_{t}=M=M_{t}^{S}$, then denoting $d=r_{1}\cdot\sqrt{\tau}/\sigma$, we obtain
\[{\bf LP}_{t}=s\cdot\left (-N(-d)+e^{-r\tau}\cdot N(-d+\sigma\cdot\sqrt{\tau})+\frac{\sigma^{2}}{2r}\cdot N(d)-e^{-r\tau}
\cdot\frac{\sigma^{2}}{2r}\cdot N(-d+\sigma\cdot\sqrt{\tau})\right ).\]
This completes the proof. $\blacksquare$

Notice that an American lookback call option is equivalent to its European counterpart. Indeed, the process $Z_{t}=e^{-rt}\cdot (S_{t}-m_{t}^{S})=S_{t}^{*}-A_{t}$ is a submartingale, since the process $A$ has nonincreasing sample paths with probability $1$. American and European lookback put options are not equivalent, however. The following bounds for the price ${\bf LP}_{t}^{a}$ of an American lookback put option can be established
\[{\bf LP}_{t}\leq {\bf LP}_{t}^{a}\leq e^{r\tau}\cdot {\bf LP}_{t}+S_{t}\cdot (e^{r\tau}-1).\]

\begin{equation}{\label{h}}\tag{H}\mbox{}\end{equation}

Asian Options.

An Asian option (or an average option) is a generic name for the class of options (of European or American style) whose terminal payoff is based on average asset values during some period within the options’ lifetimes. Let $T$ be the exercise date, and let $0\leq T_{0}<T$ stand for the beginning date of the averaging period. Then the payoff at expiry of an Asian call option equals
\[C_{T}^{A}\equiv (A_{S}(T_{0},T)-K)^{+},\]
where
\[A_{S}(T_{0},T)=\frac{1}{T-T_{0}}\cdot\int_{T_{0}}^{T}S_{u}du\]
is the arithmetic average of the asset price over the time interval $[T_{0},T]$, $K$ is the fixed strike price, and the price $S$ of the stock is assumed to follow a geometric Borwnian motion. The main difficulty in pricing and hedging Asian option is due to the fact that the random variable $A_{S}(T_{0},T)$ does not have a lognormal distribution. This feature makes the task of finding an explicit formula for the price of an Asian option surprisingly involved. For this reason, early studies of Asian options were based either on approximations or on the direct application of the Monte Carlo method. The numerical approach to the valuation of Asian options, proposed by Vorts \cite{vor}, is based on the approximation of the arithmetic average using the geometric average. Note first that it is natural to substitute the continuous-time average $A_{S}(T_{0},T)$ with its discrete-time counterpart
\begin{equation}{\label{museq96}}\tag{2}
A_{S}^{n}(T_{0},T)=\frac{1}{n}\cdot\sum_{i=0}^{n-1}S_{T_{i}},
\end{equation}
where $T_{i}=T_{0}+i\cdot (T-T_{0})/n$. The arithmetic average $A_{S}^{n}(T_{0},T)$ can be replaced with the geometric average, denoted by $G_{S}^{n}(T_{0},T)$ in what follows. Recall that the random variables $S_{T_{i}}$ for $i=1,\cdots ,n$ are given explicitly by the expression
\[S_{T_{i}}=S_{T_{0}}\cdot\exp\left (\sigma\cdot (W_{T_{i}}^{*}-
W_{T_{0}}^{*})+\left (r-\frac{\sigma^{2}}{2}\right )(T_{i}-T_{0})\right ),\]
where $W^{*}$ is a standard Brownian motion under the martingale measure $\mathbb{P}^{*}$. Therefore, the geometric average admits the following representation
\[G_{S}^{n}(T_{0},T)=\left (\prod_{i=0}^{n-1}S_{T_{i}}\right )^{1/n}=
c\cdot S_{T_{0}}\cdot\exp\left (\frac{\sigma}{n}\cdot\sum_{i=0}^{n-1}
(n-i-1)\cdot (W_{T_{i+1}}^{*}-W_{T_{i}}^{*})\right )\]
for a strictly positive constant $c$. In view of the independent of incerments of the Brownian motion, the last formula makes clear that the geometric average $G_{S}^{n}(T_{0},T)$ has a lognormal distribution under $\mathbb{P}^{*}$. The approximate Black-Scholes-like formula for the price of an Asian call option can thus be easily found by the direct evaluation of the conditional expectation
\[\tilde{C}_{T_{0}}^{n}=e^{-r(T-T_{0})}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .
(G_{S}^{n}(T_{0},T)-K)^{+}\right |{\cal F}_{T_{0}}\right ].\]
It appears however that such an approach significantly underprices Asian call option. To overcome this deficiency, one may directly approximate the true distribution of the arithmetic average using an approximate distribution, typically a lognormal law with the appropriate parameters (see Turnbull and Wakeman \cite{tur}, Levy \cite{lev} and Bousziz et al. \cite{bou}). Another approach, initiated by Carverhill and Clewlow \cite{car}, relies on the use of the fast Fourier transform to evaluate the density of the sum of the random variables as the convolution of individual densities. The second step in this method involves numerical integration of the option’s payoff function with respect to this density function. Kemma and Vorst \cite{kem} apply the Monte Carlo simulation with variance reduction to price Asian options. They replace $A_{S}(T_{0},T)$ with the arithmetic average (\ref{museq96}), so that the approximate value of an Asian call option is given by the formula
\[\bar{C}_{T_{0}}^{n}=e^{-r(T-T_{0})}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .\left (
\frac{1}{n}\sum_{i=0}^{n-1}S_{T_{i}}-K\right )^{+}\right |{\cal F}_{T_{0}}\right ].\]
Since the random variables $S_{T_{i}}$ for $i=1,\cdots ,n$ are given by an explicit formula, they can easily be generated using any standard procedure.

The most efficient analytical tools which lead to quasi-explicit pricing formulas were developed by Geman and Yor \cite{gem}. Consider first the special case of an Asian option which is already known to be in-the-money, i.e., assume that $t<T$ belongs to the average period, and the past values of stock price are satisfying
\begin{equation}{\label{museq97}}\tag{3}
A_{S}(T_{0},T)=\frac{1}{T-T_{0}}\int_{T_{0}}^{T}S_{u}du>\frac{1}{T-T_{0}}\int_{T_{0}}^{t}S_{u}du\geq K.
\end{equation}
In this case, the value at time $t$ od the Asian option equals
\[C_{t}^{A}=\frac{S_{t}(1-e^{-r(T-t)}}{r(T-T_{0})}-e^{-r(T-t)}\cdot\left (
K-\frac{1}{T-T_{0}}\int_{T_{0}}^{t}S_{u}du\right ).\]
Indeed, under (\ref{museq97}), the price of the option satisfies
\[C_{t}^{A}=e^{-r(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .\frac{1}{T-T_{0}}
\int_{T_{0}}^{T}S_{u}du-K\right |{\cal F}_{t}\right ],\]
or equivalently
\[C_{t}^{A}=\frac{e^{-r(T-t)}}{T-T_{0}}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .
\int_{t}^{T}S_{u}du\right |{\cal F}_{t}\right ]+e^{-r(T-t)}\cdot\left (
\frac{1}{T-T_{0}}\int_{T_{0}}^{t}S_{u}du-K\right ).\]
Furthermore, we have
\[\mathbb{E}_{\mathbb{P}^{*}}\left [\left .\int_{t}^{T}S_{u}du\right |{\cal F}_{t}\right ]=
\frac{S_{t}\cdot (e^{r(T-t)}-1)}{r}\]
since (recall that $S_{t}^{*}=e^{-rt}\cdot S_{t}$)
\[S_{t}\cdot (e^{r(T-t)}-1)=e^{rt}\cdot S_{T}^{*}-e^{rt}\cdot S_{t}^{*}=
\int_{t}^{T}d(r^{ru}\cdot S_{u}^{*})=\int_{t}^{T}r\cdot S_{u}du+\int_{t}^{T}e^{ru}dS_{u}^{*}.\]
For an Asian option which is not known at time $t$ to be in-the-money at time $T$, and explicit valuation formula is not available. For the proof of the following result, we refer to the original paper by Geman and Yor \cite{gem}. Let us only mention that the proof of the quasi-explicit formula (\ref{museq99}) below is based on a connection between an exponential Brownian motion and time-changed Bessel processes.

Proposition. The price of an Asian call option admits the representation
\begin{equation}{\label{museq99}}\tag{4}
C_{t}^{A}=\frac{4e^{-r(T-t)}\cdot S_{t}}{\sigma^{2}\cdot (T-T_{0})}\cdot C^{\nu}(h,q),
\end{equation}
where
\[\nu =\frac{2r}{\sigma^{2}}-1, h=\frac{\sigma^{2}}{4}\cdot (T-t),
q=\frac{\sigma^{2}}{4S_{t}}\cdot\left (K\cdot (T-T_{0})-\int_{T_{0}}^{t}S_{u}du\right ),\]
and the Laplace transform of $C^{\nu}(h,q)$ with respect to $h$ equals
\[\int_{0}^{\infty}e^{-\lambda h}\cdot C^{\nu}(h,q)dh=d\cdot\int_{0}^{1/2q}
e^{-x}\cdot x^{\gamma -2}\cdot (1-2qx)^{\gamma +1}dx\equiv g(\lambda ),\]
where
\[\mu =\sqrt{2\lambda +\nu^{2}},\gamma =\frac{1}{2}(\mu -\nu ),\mbox{ and }
d=\frac{1}{\lambda\gamma (\lambda -2-2\nu )(\gamma -1)}. \sharp\]

In order to apply the last result to price the option, one needs to find the inverse Laplace transform of the function $g(\lambda )$. for this purpose, let us denote $f(h)=C^{\nu}(h,q)$ and introduce an auxiliary function $\tilde{f}(h)=e^{-\alpha h}\cdot f(h)$ with $\alpha >2\nu +2$. Then the Laplace transform of $\tilde{f}$ equals $g(\lambda +\alpha )=\tilde{g}(\lambda )$. Moreover, since the function $\tilde{g}$ is regular for $\lambda\geq 0$, we are in a position to make use of the inverse Fourier transform; that is,
\[\tilde{f}(h)=\frac{1}{2\pi i}\int_{0-i\infty}^{0+i\infty} e^{-i\lambda h}\cdot\tilde{g}(\lambda )d\lambda .\]
This can be done numerically, using the fast Fourier transform; a detailed description of this approach (as well as numerical examples) is provided in Geman and Eydeland \cite{gem95}.

\begin{equation}{\label{i}}\tag{I}\mbox{}\end{equation}

Basket Options.

A basket option, as suggested by its name, is a kind of option contract which serves to hedge against the risk exposure of a basket of assets; that is, a prespecified portfolio of assets. Generally speaking, a basket option is more cost-effective than a portfolio of single options, as the latter over-hedges the exposure, and costs more than a basket option. An intuitive explanation for this feature is that basket option takes into account the correlation between different risk factors. For instance, in the case of a strong negative correlation between two or more underlying assets, the total risk exposure may almost vanish, and this nice feature is not reflected in payoffs and prices of single options. Let us observe that from the analytical viewpoint, there is a close analogy between basket options and Asian options. Let us denote by $S^{i}$, $i=1,\cdots ,k$ the price processes of $k$ underlying assets, which will be referred to as stocks in what follows. In this case, it seems natural to refer to such a basket option as the stock index option (in market practice, options on a basket of currencies are also quite common). The payoff at expiry of a basket call option is defined in the following way
\begin{equation}{\label{museq910}}\tag{5}
C_{T}^{B}\equiv\left (\sum_{i=1}^{k}w_{i}\cdot S_{T}^{i}-K\right )^{+}=(A_{T}-K)^{+},
\end{equation}
where $w_{i}\geq 0$ is the weight of the $i$th asset, so that $\sum_{i=1}^{k}w_{i}=1$. Note that by $A_{T}$ we denote here the weighted arithmetic average
\[A_{T}=\sum_{i=1}^{k}w{i}\cdot S_{T}^{i}.\]
We assume that each stock price $S^{i}$ follows a geometric Brownaian motion. More explicitly, under the martingale measure $\mathbb{P}^{*}$ we have
\begin{equation}{\label{museq911}}\tag{6}
dS_{t}^{i}=S_{t}^{i}\cdot (rdt+\hat{\boldsymbol{\sigma}}\cdot d{\bf W}_{t}^{*})
\end{equation}
for some nonzero vectors $\hat{\boldsymbol{\sigma}}\in \mathbb{R}^{k}$, where ${\bf W}^{*}=(W^{1*},\cdots ,W^{k*})$ stands for a $k$-dimensional Brownian motion under $\mathbb{P}^{*}$, and $r$ is the risk-free interest rate. Observe that for any fixed $i$, we can find a standard one-dimensional Brownian motion $\tilde{W}^{i}$ satisfying
\begin{equation}{\label{museq912}}\tag{7}
dS_{t}^{i}=S_{t}^{i}\cdot (rdt+\sigma_{i}d\tilde{W}_{t}^{i})
\end{equation}
and $\sigma_{i}=\parallel\hat{\boldsymbol{\sigma}}\parallel$. Let us denote by $\rho_{ij}$ the instantaneous correlation coefficient
\[\rho_{ij}=\frac{\hat{\boldsymbol{\sigma}}_{i}\cdot \hat{\boldsymbol{\sigma}}_{j}}{\sigma_{i}\cdot\sigma_{j}}=
\frac{\hat{\boldsymbol{\sigma}}_{i}\cdot \hat{\boldsymbol{\sigma}}_{j}}{\parallel
\hat{\boldsymbol{\sigma}}_{i}\parallel\cdot\parallel\hat{\boldsymbol{\sigma}}_{j}\parallel}.\]
We may thus alternatively assume that the dynamics of price processes $S^{i}$ are given by (\ref{museq912}), where $\tilde{W}_{i}$, $i=1,\cdots ,k$, are one-dimensional Brownian motions, whose cross-variations satisfy $\langle\tilde{W}^{i},\tilde{W}_{j}\rangl\mathbb{E}_{t}=\rho_{ij}\cdot t$ for every $i,j=1,\cdots ,k$. Gentle \cite{gen} proposed valuation of a basket using an approximation of the weighted arithmetic mean in the form of its geometric counterpart (this follows the approach of Vorst \cite{vor} to Asian options). For a fixed $t\leq T$, let us denote by $\hat{w}_{i}$ the modified weights
\begin{equation}{\label{museq913}}\tag{8}
\hat{w}_{i}=\frac{w_{i}\cdot S_{t}^{i}}{\sum_{j=1}^{k}w_{j}\cdot
S_{t}^{j}}=\frac{w_{i}\cdot F_{S^{i}}(t,T)}{\sum_{j=1}^{k}F_{S^{j}}(t,T)},
\end{equation}
where $F_{S^{i}}(t,T)$ is the forward price at time $t$ of the $i$th asset for the settlement date $T$. We may rewrite (\ref{museq910}) as follows
\[C_{T}^{B}=\left (\sum_{j=1}^{k}w_{j}\cdot F_{S^{j}}(t,T)\right )\cdot
\left (\sum_{i=1}^{k}\hat{w}_{i}\cdot\tilde{S}_{T}^{i}-\tilde{K}\right )^{+}
=\left (\sum_{j=1}^{k}w_{j}\cdot F_{S^{j}}(t,T)\right )\cdot\left (\tilde{A}_{T}-\tilde{K}\right )^{+},\]
where $\tilde{S}_{T}^{i}=S_{T}^{i}/F_{S^{i}}(t,T)$, $\tilde{A}_{T}=\sum_{i=1}^{k}\hat{w}_{i}\cdot\tilde{S}_{T}^{i}$, and
\begin{equation}{\label{museq914}}\tag{9}
\tilde{K}=\frac{K}{\sum_{j=1}^{k}w_{j}\cdot F_{S^{j}}(t,T)}=\frac
{e^{-r(T-t)}\cdot K}{\sum_{j=1}^{k}w_{j}\cdot S_{t}^{j}}.
\end{equation}
The arbitrage price at time $t$ of a basket call option thus equals
\[C_{t}^{B}=e^{-r(T-t)}\cdot\left (\sum_{j=1}^{k}w_{j}\cdot
F_{S^{j}}(t,T)\right )\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .(\tilde{A}_{T}-\tilde{K})^{+}\right |{\cal F}_{t}\right ],\]
or equivalently
\[C_{t}^{B}=\left (\sum_{j=1}^{k}w_{j}\cdot S_{t}^{j}\right )\cdot
\mathbb{E}_{\mathbb{P}^{*}}\left [\left .(\tilde{A}_{T}-\tilde{K})^{+}\right |{\cal F}_{t}\right ].\]
The next step relies on an approximation of the weighted arithmetic mean $\sum_{j=1}^{k}\hat{w}_{i}\cdot\tilde{S}_{T}^{i}$ using a similarly weighted geometric mean. More specifically, we approximate the price $C_{0}^{B}$ of the basket option using $\hat{C}_{0}^{B}$, which is given by the formula (for the sake of notational simplicity, we put $t=0$ in what follows)
\begin{equation}{\label{museq916}}\tag{10}
\hat{C}_{0}^{B}=\left (\sum_{j=1}^{k}w_{j}\cdot S_{0}^{j}\right )\cdot
\mathbb{E}_{\mathbb{P}^{*}}\left [(\tilde{G}_{T}-\hat{K})^{+}\right ],
\end{equation}
where
\[\tilde{G}_{T}=\prod_{i=1}^{k}(\tilde{S}_{T}^{i})^{\hat{w}_{i}}\]
and
\begin{equation}{\label{museq917}}\tag{11}
\hat{K}=\tilde{K}+\mathbb{E}_{\mathbb{P}^{*}}\left [\tilde{G}_{T}-\tilde{A}_{T}\right ].
\end{equation}
In view of (\ref{museq911}), we have (recall that $F_{S^{i}}(0,T)=e^{rT}\cdot S_{0}^{i}$)
\[\tilde{S}_{T}^{i}=\frac{S_{T}^{i}}{F_{S^{i}}(0,T)}=\exp\left (
\hat{\boldsymbol{\sigma}}_{i}\cdot {\bf W}_{T}^{*}-\frac{\sigma^{2}_{i}\cdot T}{2}\right ),\]
and thus the weighted geometric average $\tilde{G}_{T}$ equals
\begin{equation}{\label{museq918}}\tag{12}
\tilde{G}_{T}=\exp\left (\eta_{T}-\frac{c_{2}\cdot T}{2}\right ),
\end{equation}
where $\eta_{T}=c_{1}\cdot {\bf W}_{T}^{*}$, $c_{1}=\sum_{j=1}^{k}\hat{w}_{i}\cdot\hat{\boldsymbol{\sigma}}_{i}$ and $c_{2}+\sum_{i=1}^{k}\hat{w}_{i}\cdot\sigma_{i}^{2}$. We conclude that the random variable $\tilde{G}_{T}$ is lognormally distributed under $\mathbb{P}^{*}$. More precisely, the random variable $\eta_{T}$ in (\ref{museq918}) has Gaussian law with zero mean and the variance (We use here the equality $\mathbb{E}_{\mathbb{P}^{*}}\left [W_{T}^{l*}\cdot W_{T}^{m*}\right ] =\delta_{lm}\cdot T$, where $\delta_{lm}$ stands for Kronecker’s delta; that is, $\delta_{lm}=1$ if $l=m$, and zero otherwise.)

\begin{align*}
Var (\eta_{T}) & =\mathbb{E}_{\mathbb{P}^{*}}\left [\left (\sum_{i,l=1}^{k}\hat{w}_{i}
\cdot\hat{\sigma}_{il}\cdot W_{T}^{l*}\right )\cdot\left (\sum_{j,m=1}^{k}
\hat{w}_{j}\cdot\hat{\sigma}_{jm}\cdot W_{T}^{m*}\right )\right ]\\
& =\sum_{i,j,l,m=1}^{k}\hat{w}_{i}\cdot\hat{w}_{j}\cdot
\hat{\sigma}_{il}\cdot\hat{\sigma}_{jm}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [W_{T}^{l*}\cdot W_{T}^{m*}\right ]\\
& =\sum_{i,j=1}^{k}\hat{w}_{i}\cdot\hat{w}_{j}\cdot\hat{\sigma}_{i}\cdot\hat{\sigma}_{j}\cdot T=v^{2}\cdot T,
\end{align*}
where
\[v^{2}=\sum_{i,j=1}^{k}\rho_{ij}\cdot\hat{w}_{i}\cdot\hat{w}_{j}\cdot\sigma_{i}\cdot\sigma_{j}.\]
Notice also that the last term on the right-hand side of (\ref{museq917}) equals $1$ since
\[\mathbb{E}_{\mathbb{P}^{*}}[\tilde{A}_{T}]=\sum_{j=1}^{k}\hat{w}_{j}\cdot S_{0}^{j}\cdot
\mathbb{E}_{\mathbb{P}^{*}}\left [e^{-rt}\cdot S_{T}^{i}\right ]=\sum_{j=1}^{k}\hat{w}_{j}=1,\]
and the expected value $\mathbb{E}_{\mathbb{P}^{*}}[\tilde{G}_{T}]$ equals
\[\mathbb{E}_{\mathbb{P}^{*}}[\tilde{G}_{T}]=\exp\left (\frac{(v^{2}-c_{2})T}{2}\right )
\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [e^{\eta_{T}-\frac{1}{2}\cdot Var(\eta_{T})}\right ]
=\exp\left (\frac{(v^{2}-c_{2})T}{2}\right )\equiv c.\]
We conclude that $\hat{K}=\tilde{K}+c-1$. The expectation in (\ref{museq916}) can now be evaluated explicitly using the following result (ref. Proposition \ref{musl751}).

\begin{equation}{\label{museql991}}\tag{13}\mbox{}\end{equation}

Proposition \ref{museql991}. Let $\xi$ be a Gaussian random variable on $(\Omega ,{\cal F},P)$ with zero mean and the variance $\sigma^{2}>0$. For any strictly positive real numbers $a$ and $b$, we have
\[\mathbb{E}_{P}\left [(a\cdot e^{\xi -\frac{1}{2}\sigma^{2}}-b)^{+}\right ]=a\cdot N(h)-b\cdot N(h-\sigma ),\]
where
\[h=\frac{\ln a-\ln b}{\sigma}+\frac{\sigma}{2}.\sharp\]

We have
\[\mathbb{E}_{\mathbb{P}^{*}}\left [\tilde{G}_{T}-\hat{K})^{+}\right ]=\mathbb{E}_{\mathbb{P}^{*}}\left [\left (
c\cdot e^{\eta_{T}-\frac{1}{2}\cdot Var(\eta_{T})}-(\tilde{K}+c-1)\right )^{+}\right ],\]
so that $\xi =\eta_{T}$, $a=c$ and $b=\tilde{K}+c-1$. In view of Proposition \ref{musl991}, the following result is straightforward.

Proposition. The approximate value $\hat{C}_{t}^{B}$ of the price $C_{t}^{B}$ of a basket call option with strike price $K$ and expiry date $T$ equals
\begin{equation}{\label{museq920}}\tag{14}
\hat{C}_{t}^{B}=\left (\sum_{j=1}^{k}w_{j}\cdot S_{t}^{j}\right )\cdot
\left (c\cdot N(l_{1}(T-t))-(\tilde{K}+c-1)\cdot N(l_{2}(T-t))\right ),
\end{equation}
where
\[c=\exp\left [\left (\frac{1}{2}\sum_{i,j=1}^{k}\rho_{ij}\cdot\hat{w}_{i}
\cdot\hat{w}_{j}\cdot\sigma_{i}\cdot\sigma_{j}-\sum_{j=1}^{k}\hat{w}_{j}\cdot\sigma_{j}^{2}\right )\cdot (T-t)\right ],\]
and where the modified weights $\hat{w}_{i}$ are given by $(\ref{museq913})$, $\tilde{K}$ is given by $(\ref{museq914})$, and
\[l_{1,2}(t)=\frac{\ln c-\ln (\tilde{K}+c-1)\pm\frac{1}{2}\cdot v^{2}\cdot t}{v\cdot\sqrt{t}}. \sharp\]

Suppose that $k=1$. In this case, $w_{1}=1$, and the arithmetic average agrees with the geometric one. Consequently, $c=1$ and (\ref{museq920}) reduces to the standard Black-Scholes formula.