Charles Courtney Curran (1861–1942) was an American painter.
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We shall first review briefly the most important kinds of financial contracts, traded either on exchanges or over-the-counter (OTC), between financial institutions and their clients.
\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}
Financial Derivatives.
Options.
Options are examples of exchange-traded derivative securities in which the value of securities depends on the prices of other more basic securities, referred to as primary securities such as stocks or bonds. The stocks mean the common stocks, where the shares in the net asset value are not bearing fixed interest. They give the right to dividends according to profits. By contrast, the preferred stocks give some special rights to the stockholder, typically a guaranteed fixed dividend. A bond is a certificate issued by a government or a public company promising to repay borrowed money at a fixed rate of interest at a specified time. Basically, the call option and put option are the right to buy and sell, respectively, the option’s underlying asset at some future date for a pre-specified price.
It is worth noting that most of the traded options are of American style (or American options), which means that the holder has the right to exercise an option at any instant before the option’s expiry. When an option can be exercised only at its expiry date, it is known as an option of European style (or European options).
Futures Contracts and Options.
Another important class of exchange-traded derivative securities comprises futures contracts and options on futures contracts that is commonly known as futures options. Futures contracts apply to a wide range of commodities like sugar, wood, gold etc., and financial assets like currencies, bonds, stock indices etc. In what follows, we restrict the attention to financial assets. To make possible trading, the exchange specifies certain standard features of the contract. Futures prices are regularly reported in the financial press. They are determined on the floor in the same way as other prices by the law of supply and demand. If more investors want to go long position than to go short, the price goes up; if the reverse is true, the price falls. Positions in futures contracts are governed by a specific daily settlement procedure commonly referred to as marking to market. An investor’s initial deposit, known as the initial margin, is adjusted daily to reflect the gains or losses that are due to the futures price movements. Let us consider a party assuming a long position, i.e., the party who agreed to buy. When there is a decrease in the futures price, his/her margin account is reduced by an appropriate amount of money, his/her broker has to pay this sum to the exchange, and the exchange passes the money on to the broker of the party who assumes the short position. Similarly, when the futures price rises, brokers for parties with short positions pay money to the exchange, and brokers of parties with long positions receive money from the exchange. This way, the trade is marked to market at the close of each trading day. Finally, if the delivery period is reached and delivery is made by a party with a short position, the price received is generally the futures price at the time the contract was last marked to market.
In a futures option, the underlying asset is a futures contract. The futures contract normally matures shortly after the expiry of the option. When the holder of a call futures option exercises the option, he/she acquires from the writer a long position in the underlying futures contract plus a cash amount equal to the excess of the current futures price over the option’s strike price. Since futures contracts have zero value and can be closed out immediately, the payoff from a futures option is the same as the payoff from a stock option with the stock price replaced by the futures price.
Forward Contracts.
A forward contract is an agreement to buy or sell an asset at a certain future time for a certain price. One of the parties to a forward contract assumes a long position and agrees to buy the underlying asset on a certain specified future date for a delivery price. The other party assumes a short position and agrees to sell the asset on the same date for the same price. At the time the contract is entered into, the delivery price is determined so that the value of the forward contract to both parties is zero. It is clear that some features of forward contracts resemble those of futures contracts. However, unlike futures contracts, forward contracts do not trade on exchanges. Also, a forward contract is settled only once at the maturity date. The holder of the short position delivers the asset to the holder of the long position in return for a cash amount equal to the delivery price. The following lists summarize the main differences between forward and futures contracts.
1. Contract specification and delivery.
- Futures contracts. The contract precisely specifies the underlying instrument and price. Delivery dates and delivery procedures are standardized to a limited number of specific dates per year at approved locations. Delivery is not, however, the objective of the transaction, and less than \(2\%\) are delivered.
- Forward contracts. There is an almost unlimited range of instruments with individually negotiated prices. Delivery can take place on any individual negotiated date and location. Delivery is the object of the transaction with over \(90\%\) of forward contracts settled by delivery.
2. Prices.
- Futures contracts. The price is the same for all participants, regardless of transaction size. Typically, there is a daily limit. Trading is usually by open outcry auction on the trading floor of the exchange. Prices are disseminated publicly. Each transaction is conducted at the best price available at the time.
- Forward contracts. The price varies with the size of the transaction, the credit, risk, etc. There are no daily price limits. Trading takes place by telephone and telex between individual buyers and sellers. Prices are not disseminated publicly. Hence, there is no guarantee that the price is the best available.
3. Marketplace and trading hours.
- Futures contracts. Trading is centralized on the exchange floor with worldwide communications during hours fixed by the exchange.
- Forward contracts. Trading takes place by telephone and telex between individual buyers and sellers. Trading is over-the-counter worldwide, 24 hours per day, with telephone and telex access.
4. Security deposit and margin.
- Futures contracts. The exchange rules require an initial margin and the daily settlement of variation margins. A central clearing house is associated with each exchange to handle the daily revaluation of open positions, cash payments and delivery procedures. The clearing house assumes the credit risk.
- Forward contracts. The collateral level is negotiable with no adjustment for daily price fluctuations. There is no separate clearing house function. Therefore, the market participant bears the risk of the counter-party defaulting.
5. Volume and market liquidity.
- Futures contracts. Volume (and open interest) information is published. There is a very high liquidity and ease of offset with any other market participant due to standardized contracts.
- Forward contracts. Volume information is not available. The limited liquidity and offset is due to the variable contract terms. Offset is usually with the original counter-party.
\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}
Call and Put Spot Options.
We consider the so-called frictionless market meaning as follows
- All investors are price-takers.
- All parties have the same access to the relevant information.
- There are no transaction costs or commissions
- All assets are assumed to be perfectly divisible and liquid.
- There is no restriction whatsoever on the size of a bank credit, and the lending and borrowing rates are equal.
- Individuals are allowed to sell short any security and receive full use of the proceeds.
A European call option written on a common stock (we assume that the underlying stock pays no dividends during the option’s lifetime) is a financial security that gives its holder the right (but not the obligation) to buy the underlying stock on a pre-specified date and for a pre-specified price. The act of making this transaction is referred to as exercising the option. If an option is not exercised, we say that it is abandoned. Another class of options comprises the so-called American options. These may be exercised at any time on or before the pre-specified date. The pre-specified fixed price, denoted by \(K\), is termed the strike price or exercise price. The terminal date, denoted by \(T\), is called the expiry date or maturity. In order to purchase an option contract, an investor needs to pay an option’s price (or premium) to a second party at the initial date when the contract is entered into.
Let us denote by \(S_{T}\) the stock price at the terminal date \(T\). It is natural to assume that \(S_{T}\) is not known at time \(0\). Therefore, the price \(S_{T}\) gives rise to uncertainty in the model. We argue that, from the perspective of the option holder, the payoff \(g\) at expiry date \(T\) from a European call option is given by the expression
\[g(S_{T})=\left\{\begin{array}{ll} S_{T}-K & \mbox{if \(S_{T}>K\) (option is exercised)}\\ 0 & \mbox{if \(S_{T}\leq K\) (option is abandoned)}. \end{array}\right .\]
In other words, we have
\[g(S_{T})=(S_{T}-K)^{+}\equiv\max\{S_{T}-K,0\},\]
In fact, when, at the expiry date \(T\), the stock price is lower than the strike price, the holder of the call option can purchase an underlying stock directly on a spot market by paying less than \(K\). On the other hand, when, at the expiry date, the stock price is greater than \(K\), an investor should exercise his/her right to buy the underlying stock at the strike price \(K\). Indeed, by selling the stock immediately at the spot market, the holder of the call option is able to realize an instantaneous net profit \(S_{T}-K\), where the transaction costs and/or commissions are ignored here.
In contrast to a call option, a put option gives its holder the right to sell the underlying asset by a certain date for a pre-specified price. Using the same notation as above, we arrive at the following expression for the payoff at maturity \(T\) from a European put option
\[h(S_{T})=\left\{\begin{array}{ll} K-S_{T} & \mbox{if \(S_{T}<K\) (option is exercised)}\\ 0 & \mbox{if \(S_{T}\geq K\) (option is abandoned)}.\end{array}\right .\]
In other words, we have
\[h(S_{T})=(K-S_{T})^{+}\equiv\max\{K-S_{T},0\},\]
It follows immediately that the payoffs of call and put options satisfy the following simple but useful equality
\begin{equation}{\label{museq13}}\tag{1}g(S_{T})-h(S_{T})=(S_{T}-K)^{+}-(K-S_{T})^{+}=S_{T}-K.\end{equation}
The above equality can be used to derive the so-called put-call parity relationship for options prices. Basically, put-call parity means that the price of a European put option is determined by the price of a European call option with the same strike and expiry date, the current price of the underlying asset, and the properly discounted value of the strike price.
One-period Spot Market.
Let us start by considering an elementary example of an option contract.
\begin{equation}{\label{muse141}}\tag{2}\mbox{}\end{equation}
Example \ref{muse141}. Assume that the current stock price is \(\$280\). After three months, the stock price may either rise to \(\$320\) or decline to \(\$260\). We shall find the rational price of a \(3\)-month European call option with strike price \(K=\$280\) provided that the simple risk-free interest rate \(r\) for \(3\)-month deposits and loans is \(r=5\%\). We shall usually assume that the borrowing and lending rates are equal. \(\sharp\)
Suppose that the subjective probability of the price rise is \(0.2\), and that of the fall is \(0.8\). Theses assumptions correspond to a so-called bear market. Note that the word “subjective” means that we take the point of view of a particular individual. Generally speaking, the two parties involved in an option contract may have differing assessments of these probabilities. To model a bull market, one may assume that the first probability is \(0.8\) so that the second is \(0.2\).
Let us focus first on the bear market case. The terminal stock price \(S_{T}\) may be seen as a random variable on a probability space \(\Omega=\{\omega_{1},\omega_{2}\}\) with a probability measure \(\mathbb{P}\) given by
\[\mathbb{P}\{\omega_{1}\}=0.2=1-\mathbb{P}\{\omega_{2}\}.\]
By referring to Example \ref{muse141}, \(S_{T}\) can be treated as a function \(S_{T}:\Omega\rightarrow \mathbb{R}_{+}\) given by the following expression
\[S_{T}(\omega )=\left\{\begin{array}{ll}S_{u}=320, & \mbox{if \(\omega =\omega_{1}\)};\\S_{d}=260, & \mbox{if \(\omega =\omega_{2}\)}.\end{array}\right .\]
Consequently, the terminal option’s payoff \(X=C_{T}=(S_{T}-K)^{+}\) satisfies
\[C_{T}(\omega )=\left\{\begin{array}{ll}C_{u}=320-280=40, & \mbox{$\omega =\omega_{1}$};\\ C_{d}=0, & \mbox{$\omega =\omega_{2}$}. \end{array}\right .\]
The expected value under \(\mathbb{P}\) of the discounted option’s payoff equals
\[\mathbb{E}_{\mathbb{P}}\left [\frac{C_{T}}{1+r}\right ]=\frac{0.2\cdot 40}{1.05}=7.62.\]
It is clear that the above expectation depends on the choice of the probability measure \(\mathbb{P}\), which depends on the investor’s assessment of the market. For a call option, the expectation corresponding to the case of a bull market would be greater than that which assumes a bear market. In this example, the expected value of the discounted payoff from the option under the bull market hypothesis equals \(30.81\). To construct a reliable model of a financial market, one has to guarantee the uniqueness of the price of any derivative security. This can be done by applying the concept of the so-called replicating portfolios.
Replicating Portfolios.
The idea is to construct a portfolio at time \(0\) which replicates exactly the option’s terminal payoff at time \(T\). Let
\[\phi=\phi_{0}=(\alpha_{0},\beta_{0})\in \mathbb{R}^{2}\]
denote a portfolio of an investor with a short position in one call option. More precisely, let \(\alpha_{0}\) stand for the number of shares of stock held at time \(0\), and \(\beta_{0}\) be the amount of money deposited on a bank account or borrowed from a bank. We denote by \(V_{t}(\phi)\) the wealth of this portfolio at dates \(t=0\) and \(t=T\). In other words, the payoff from the portfolio \(\phi\) at given dates. Therefore, we have
\[V_{0}(\phi)=\alpha_{0}\cdot S_{0}+\beta_{0}\mbox{ and }V_{T}(\phi)=\alpha_{0}\cdot S_{T}+\beta_{0}\cdot (1+r).\]
We say that a portfolio \(\phi\) replicates the option’s terminal payoff whenever \(V_{T}(\phi)=C_{T}\), i.e.,
\[V_{T}(\phi)(\omega )=\left\{\begin{array}{ll}V_{u}(\phi)=\alpha_{0}\cdot S_{u}+(1+r)\cdot\beta_{0}=C_{u}, & \mbox{if \(\omega =\omega_{1}\)};\\V_{d}(\phi)=\alpha_{0}\cdot S_{d}+(1+r)\cdot\beta_{0}=C_{d}, & \mbox{if \(\omega =\omega_{2}\)}. \end{array}\right .\]
For the data of Example \ref{muse141}, the portfolio \(\phi\) is determined by the following system of equations
\[\left\{\begin{array}{l} 320\alpha_{0}+1.05\beta_{0}=40\\ 260\alpha_{0}+1.05\beta_{0}=0 \end{array}\right .\]
with unique solution \(\alpha_{0}=2/3\) and \(\beta_{0}=-165.8\). It is natural to define the manufacturing cost \(C_{0}\) of a call option as the initial investment needed to construct a replicating portfolio, i.e.,
\[C_{0}=V_{0}(\phi)=\alpha_{0}\cdot S_{0}+\beta_{0}=(2/3)\cdot 280-165.08=21.59.\]
It should be stressed that, in order to determine the manufacturing cost of a call, we are not necessarily to know the probability of the rise or fall of the stock price. In other words, it appears that the manufacturing cost is invariant with respect to individual assessments of market behavior. In particular, it is identical under the bull and bear market hypothesis. The investor’s transactions and the corresponding cash flows may be summarized by the following two expressions
\[\mbox{at time \(t=0\) }\left\{\begin{array}{ll}\mbox{one written call option} & C_{0}\\ \mbox{$\alpha_{0}$ shares purchased} & -\alpha_{0}\cdot S_{0}\\ \mbox{amount of cash borrowed} & \beta_{0} \end{array}\right .\]
and
\[\mbox{at time \(t=T\) }\left\{\begin{array}{ll} \mbox{payoff from the call option} & -C_{T}\\ \mbox{$\alpha_{0}$ shares sold} & \alpha_{0}\cdot S_{T}\\ \mbox{loan paid back} & -(1+r)\cdot\beta_{0}. \end{array}\right .\]
Observe that no net initial investment is needed to establish the above portfolio, i.e., the portfolio is costless. It is easy to verify that if the call were not priced at \(\$21.59\), it would be possible for a sure profit to be gained, either by the option’s writer (if the option’s price were greater than its manufacturing cost) or by its buyer (in the opposite case). The manufacturing cost cannot be seen as a fair price of a claim \(X\) unless the market model is arbitrage-free. Indeed, it may happen that the manufacturing cost of a nonnegative claim is a strictly negative number. Such a phenomenon contradicts the usual assumption that it is not possible to make riskless profits.
Martingale Measure for a Spot Market.
Although, as shown above, subjective probabilities are useless when pricing an option, probabilistic methods play an important role in the valuation of contingent claims. They rely on the notion of a martingale, which is a probabilistic model of a fair game. In order to apply the so-called martingale method of derivative pricing, one has to find a probability measure \(\mathbb{P}^{*}\) that is equivalent to \(\mathbb{P}\) such that the discounted (or relative) stock price process \(S^{*}\), which is defined by the formula
\[S_{0}^{*}=S_{0}\mbox{ and }S_{T}^{*}=S_{T}/(1+r)\]
follows a \(\mathbb{P}^{*}\)-martingale. In other word, the equality \(S_{0}^{*}=\mathbb{E}_{\mathbb{P}^{*}}[S_{T}^{*}]\) holds. Such a probability measure is called a martingale measure for the discounted stock price process \(S^{*}\). In the case of a two-state model, the probability measure \(\mathbb{P}^{*}\) is easily seen to be uniquely determined by the following linear equation
\[S_{0}=\frac{p^{*}\cdot S_{u}+(1-p^{*})\cdot S_{d}}{1+r},\]
where \(p^{*}=\mathbb{P}^{*}\{\omega_{1}\}\) and \(1-p^{*}=\mathbb{P}^{*}\{\omega_{2}\}\). Solving this equation for \(p^{*}\), we obtain
\[\mathbb{P}^{*}\{\omega_{1}\}=\frac{(1+r)\cdot S_{0}-S_{d}}{S_{u}-S_{d}}\mbox{ and }
\mathbb{P}^{*}\{\omega_{2}\}=\frac{S_{u}-(1+r)\cdot S_{0}}{S_{u}-S_{d}}.\]
Let us now check that the price \(C_{0}\) coincides with \(C_{0}^{*}\), where we write \(C_{0}^{*}\) to denote the expected value under \(\mathbb{P}^{*}\) of an option’s discounted terminal payoff; that is
\[C_{0}^{*}\equiv\mathbb{E}_{\mathbb{P}^{*}}\left [\frac{C_{T}}{1+r}\right ]=\mathbb{E}_{\mathbb{P}^{*}}\left [
\frac{(S_{T}-K)^{+}}{1+r}\right ].\]
Indeed, using the data of Example \ref{muse141}, we find \(p^{*}=17/30\) so that
\[C_{0}^{*}=\frac{p^{*}\cdot C_{u}+(1-p^{*})\cdot C_{d}}{1+r}=21.59=C_{0}.\]
Since the process \(S^{*}\) follows a \(\mathbb{P}^{*}\)-martingale, we may say that the discounted stock price process may be seen as a fair game model in a risk-neutral economy. That is to say, in the stochastic economy in which the probabilities of future stock price fluctuations are determined by the martingale measure \(\mathbb{P}^{*}\). It should be stressed that the fundamental idea of arbitrage pricing is based exclusively on the existence of a portfolio that hedges perfectly the risk exposure related to uncertain future prices of risky securities. Therefore, the probabilistic properties of the model are not essential. In particular, we do not assume that the real world economy is actually risk-neutral. On the contrary, the notion of a risk-neutral economy should be seen rather as a technical tool. The aim of introducing the martingale measure is twofold.
- It simplifies the explicit evaluation of arbitrage prices of derivative securities.
- It describes the arbitrage-free property of a given pricing model for primary securities in terms of the behavior of relative prices. This approach is frequently referred to as the partial equilibrium approach, as opposed to the general equilibrium approach.
Let us stress that, in the later theory, the investors’ preferences, usually described in stochastic models by means of their (expected) utility functions, play an important role.
To summarize, the notion of an arbitrage price for a derivative security does not depend on the choice of a probability measure in a particular pricing model for primary securities. More precisely, using standard probabilistic terminology, this means that the arbitrage price depends on the support of a subjective probability measure \(\mathbb{P}\), but is invariant with respect to the choice of a particular probability measure from the class of mutually equivalent probability measures. In financial terminology, this can be restated as follows. All investors agree on the range of future price fluctuations of primary securities. However, they may have different assessments of the corresponding subjective probabilities.
Absence of Arbitrage.
Let us consider a simple two-state, one-period, two-security market model defined on a probability space \(\Omega =\{\omega_{1},\omega_{2}\}\) equipped with the \(\sigma\)-fields \({\cal F}_{0}=\{\emptyset ,\Omega\}\), \({\cal F}_{T}=2^{\Omega}\) (i.e., \({\cal F}_{T}\) contains all subsets of \(\Omega\)), and a probability measure on \((\Omega ,{\cal F}_{T})\) such that \(\mathbb{P}\{\omega_{1}\}\) and \(\mathbb{P}\{\omega_{2}\}\) are strictly positive numbers. The first security is a stock whose price process is modeled as a strictly positive discrete-time process \(S=\{S_{t}\}_{t\in\{0,T\}}\). We assume that the process \(S\) is \({\cal F}_{t}\)-adapted. This means that \(S_{0}\) is a real number, and
\[S_{T}(\omega )=\left\{\begin{array}{ll} S_{u} & \mbox{if \(\omega =\omega_{1}\)},\\ S_{d} & \mbox{if \(\omega =\omega_{2}\)}, \end{array}\right .\]
where, without loss of generality, \(S_{u}>S_{d}\). The second security is a riskless bond whose price is \(p_{0}=1\) and \(p_{T}=1+r\) for some real \(r\geq 0\). Let \(\Phi\) stand for the linear space of all stock-bond portfolios \(\phi=\phi_{0}=(\alpha_{0},\beta_{0})\), where \(\alpha_{0}\) and \(\beta_{0}\) are real numbers. Clearly, the class \(\Phi\) may be identified with \(\mathbb{R}^{2}\). We shall consider the pricing of contingent claims in a security market model \({\cal M}=(S,p,\Phi)\). We shall now check that an arbitrary contingent claim \(X\) which settles at time \(T\) (i.e., any \({\cal F}_{T}\)-measurable real-valued random variable) admits a unique replicating portfolio in the market model \({\cal M}\). Indeed, if
\[X(\omega )=\left\{\begin{array}{ll}X_{u} & \mbox{if \(\omega =\omega_{1}\)},\\ X_{d} & \mbox{if \(\omega =\omega_{2}\)},\end{array}\right .\]
then the replicating portfolio \(\phi\) is determined by a linear system of two equations in two unknowns, i.e.,
\[\left\{\begin{array}{l}\alpha_{0}\cdot S_{u}+(1+r)\cdot\beta_{0}=X_{u}\\\alpha_{0}\cdot S_{d}+(1+r)\cdot\beta_{0}=X_{d}, \end{array}\right .\]
which admits a unique solution
\[\alpha_{0}=\frac{X_{u}-X_{d}}{S_{u}-S_{d}}\mbox{ and }\beta_{0}=\frac{X_{d}\cdot S_{u}-X_{u}\cdot S_{d}}{(1+r)(S_{u}-S_{d}}.\]
As already mentioned, the manufacturing cost of a strictly positive contingent claim may appear to be a negative number in general. If this were the case, there would be a profitable riskless trading strategy (so-called arbitrage opportunity) involving only the stock and riskless borrowing and lending. To exclude such situations, we have to impose further essential restrictions on the simple market model.
Definition. We say that a security pricing model \({\cal M}\) is arbitrage-free when there is no portfolio \(\phi\in\Phi\) for which
\begin{equation}{\label{museq110}}\tag{3}V_{0}(\phi)=0, V_{T}(\phi)\geq 0\mbox{ and }\mathbb{P}\left\{V_{T}(\phi)>0\right\}>0. \end{equation}
A portfolio \(\phi\) for which the set (\ref{museq110}) of conditions is satisfied is called an arbitrage opportunity. A strong arbitrage opportunity is a portfolio \(\phi\) for which
\begin{equation}{\label{museq111}}\tag{4}V_{0}(\phi)<0\mbox{ and }V_{T}(\phi)\geq 0. \sharp \end{equation}
It is customary to take either (\ref{museq110}) or (\ref{museq111}) as the definition of an arbitrage opportunity. Note that both notions are not necessarily equivalent. Suppose that the security market \({\cal M}\) is arbitrage-free. Then, the manufacturing cost \(\Pi_{0}(X)\) is called the arbitrage price of \(X\) at time \(0\) in security market \({\cal M}\). The following result says that the arbitrage opportunity exists when the no-arbitrage condition \(H_{0}=\Pi_{0}(X)\) is violated.
Proposition. Suppose that the spot market model \({\cal M}=(S,p,\Phi)\) is arbitrage-free. Let \(H\) stand for the rational price process of some attainable contingent claim \(X\). More precisely, we have \(H_{0}\in \mathbb{R}\) and \(H_{T}=X\). Let \(\Phi_{H}\) denote the class of all portfolios in stock, bond and derivative security \(H\). The extended market model \((S,p,H,\Phi_{H})\) is arbitrage-free if and only if \(H_{0}=\Pi_{0}(X)\). \(\sharp\)
Optimality of Replication.
Let us show that replication is an optimal way of hedging. Firstly, we say that a portfolio \(\phi\) perfectly hedges against \(X\) when \(V_{T}(\phi)\geq X\), that is,
\begin{equation}{\label{museq112}}\tag{5}\left\{\begin{array}{l}\alpha_{0}\cdot S_{u}+(1+r)\cdot\beta_{0}\geq X_{u}\\ \alpha_{0}\cdot S_{d}+(1+r)\cdot\beta_{0}\geq X_{d},\end{array}\right .\end{equation}
The minimal initial cost of a perfect hedging portfolio against \(X\) is called the seller’s price of \(X\), and it is denoted by \(\Pi_{0}^{s}(X)\). Let us check \(\Pi_{0}^{s}(X)=\Pi_{0}(X)\). By denoting \(c=V_{0}(\phi)=\alpha_{0}\cdot S_{0}+\beta_{0}\), we may rewrite (\ref{museq112}) as follows
\begin{equation}{\label{museq113}}\tag{6}\left\{\begin{array}{l}\alpha_{0}\cdot (S_{u}-S_{0}(1+r)))+c(1+r)\geq X_{u}\\ \alpha_{0}\cdot (S_{d}-S_{0}(1+r)))+c(1+r)\geq X_{d},\end{array}\right .\end{equation}
It is trivial to check that the minimal \(c\in \mathbb{R}\) for which (\ref{museq113}) holds is actually that value of \(c\) for which inequalities in (\ref{museq113}) becomes equalities. This means that the replication appears to be the least expensive way of perfect hedging for the seller of \(X\).
Let us now consider the other party of the contract, i.e., the buyer of \(X\). Since the buyer of \(X\) can be seen as the seller of \(-X\), the associated problem is to minimize \(c\in \mathbb{R}\) subject to the following constraints
\[\left\{\begin{array}{l}
\alpha_{0}\cdot (S_{u}-S_{0}(1+r)))+c(1+r)\geq -X^{u}\\
\alpha_{0}\cdot (S_{d}-S_{0}(1+r)))+c(1+r)\geq -X^{d},
\end{array}\right .\]
It is clear that the solution to this problem is
\begin{equation}{\label{museq2010}}\tag{7}\Pi_{o}^{s}(-X)=-\Pi_{0}(X)=\Pi_{0}(-X)\end{equation}
so that replication appears to be optimal for the buyer also. We conclude that the least price the seller is ready to accept for \(X\) equals the maximal amount the buyer is ready to pay for it. If we define the buyer’s price of \(X\), denoted by \(\Pi_{0}^{b}(X)\), by setting \(\Pi_{0}^{b}(X)=-\Pi_{0}^{s}(-X)\), then, from (\ref{museq2010}), we have
\[\Pi_{0}^{b}(X)=-\Pi_{0}^{s}(-X)=-(-\Pi_{0}(X))=\Pi_{0}(X)=\Pi_{0}^{s}(X);\]
that is, all price coincide. This shows that in a two-state, arbitrage-free model, the arbitrage price of any contingent claim can be defined using the optimality criterion. It appears that such an approach to arbitrage pricing can be extended to other models. However, we prefer to define the arbitrage price as that value of the price which excludes arbitrage opportunities.
The next result explains the role of the so-called risk-neutral economy in arbitrage pricing of derivative securities. Notice that the use of a martingale measure \(\mathbb{P}^{*}\) in arbitrage pricing corresponds to the assumption that all investors are risk-neutral, meaning that they do not differentiate between all riskless and risky investments with the same expected rate of return. The arbitrage valuation of derivative securities is done as if an economy actually were risk-neutral. Formula (\ref{museq114a}) below shows that the arbitrage price of a contingent claim \(X\) can be found by first modifying the model so that the stock earns at the riskless rate, and then computing the expected value of the discounted claim.
\begin{equation}{\label{musp142}}\tag{8}\mbox{}\end{equation}
Proposition \ref{musp142}. The spot market \({\cal M}=(S,p,\Phi)\) is arbitrage-free if and only if the discounted stock price \(S^{*}\) admits a martingale measure \(\mathbb{P}^{*}\) equivalent to \(\mathbb{P}\). In this case, the arbitrage price at time \(0\) of any contingent claim \(X\) which settles at time \(T\) is given by the risk-neutral valuation formula
\begin{equation}{\label{museq114a}}\tag{9}\Pi_{0}(X)=E_{P^{*}}\left [\frac{X}{1+r}\right ],\end{equation}
or explicitly
\begin{equation}{\label{museq115a}}\tag{10}\Pi_{0}(X)=\frac{S_{0}(1+r)-S_{d}}{S_{u}-S_{d}}\cdot\frac{X_{u}}{1+r}+\frac{S_{u}-S_{0}(1+r)}{S_{u}-S_{d}}\cdot\frac{X_{d}}{1+r}.\end{equation}
The choice of the bond price process as a discount factor is not essential. Suppose, on the contrary, that we have chosen the stock price \(S\) as a numeraire. In other words, we now consider the bond price \(p\) discounted by the stock price \(S\)
\[p_{t}^{*}=p_{t}/S_{t}\]
for \(t\in\{0,T\}\). The martingale measure \(\bar{\mathbb{P}}\) for the process \(p^{*}\) is determined by the equality \(p_{0}=\mathbb{E}_{\bar{\mathbb{P}}}[p_{T}^{*}]\), or explicitly
\[\frac{\bar{p}\cdot (1+r)}{S_{u}}+\frac{(1-\bar{p})(1+r)}{S_{d}}=\frac{1}{S_{0}}.\]
One finds
\[\bar{\mathbb{P}}(\omega_{1})=\bar{p}=\left (\frac{1}{S_{d}}-\frac{1}
{S_{0}(1+r)}\right )\cdot\frac{S_{u}\cdot S_{d}}{S_{u}-S_{d}}\] and
\[\bar{\mathbb{P}}(\omega_{2})=1-\bar{p}=\left (\frac{1}{S_{U}}-\frac{1}
{S_{0}(1+r)}\right )\cdot\frac{S_{u}\cdot S_{d}}{S_{d}-S_{u}}.\]
It is easy to show that the properly modified version of the risk-neutral valuation formula has the following form
\begin{equation}{\label{museq119}}\tag{11}\Pi_{0}(X)=S_{0}\cdot\mathbb{E}_{\bar{\mathbb{P}}}\left [\frac{X}{S_{T}}\right ]\end{equation}
where \(X\) is a contingent claim which settles at time \(T\). It appears that, in some circumstances, the choice of the stock price as a numeraire is more convenient than that of the savings account.
Let us apply this approach to the call options of Example \ref{muse141}. One finds \(\bar{p}=0.62\). Therefore, formula (\ref{museq119}) gives
\[\bar{C}_{0}=S_{0}\cdot E_{\bar{\mathbb{P}}}\left [\frac{(S_{T}-K)^{+}}{S_{T}}\right ]=21.59=C_{0}\]
as expected.
Put Option.
We refer once again to Example \ref{muse141}. However, we shall now focus on a European put option instead of a call option. Since the buyer of a put option has the right to sell a stock at a given date \(T\), the terminal payoff from the option is now \(P_{T}=(K-S_{T})^{+}\), i.e.,
\[P_{T}(\omega )=\left\{\begin{array}{ll}P_{u}=0 & \mbox{if \(\omega =\omega_{1}\)},\\ P_{d}=20 & \mbox{if \(\omega =\omega_{2}\)},\end{array}\right .\]
where we have taken, as before, \(K=\$280\). The portfolio \(\phi=(\alpha_{0},\beta_{0})\) which replicates the European put option is determined by the following system of linear equations
\[\left\{\begin{array}{l}320\alpha_{0}+1.05\beta_{0}=0\\ 260\alpha_{0}+1.05\beta_{0}=20\end{array}\right .\]
so that \(\alpha_{0}=-1/3\) and \(\beta_{0}=101.59\). Consequently, the arbitrage price \(P_{0}\) of the European put option equals
\[P_{0}=-(1/3)\cdot 280+101.59=8.25.\]
Notice that the number of shares in a replicating portfolio is negative. This means that an option writer who wishes to hedge risk exposure should sell short at time \(0\) the number \(-\alpha_{0}=1/3\) shares of stock for each sold put option. The proceeds from the short-selling of shares, as well as the option’s premium, are invested in an interest-earning account. To find the arbitrage price of the put option, we may alternatively apply Proposition \ref{musp142}. By virtue of (\ref{museq114a}) with \(X=P_{T}\), we get
\[P_{0}=\mathbb{E}_{\mathbb{P}^{*}}\left [\frac{P_{T}}{1+r}\right ]=8.25.\]
Finally, the put option value can also be found by applying the following relationship between the prices of call and put options.
\begin{equation}{\label{musc141}}\tag{12}\mbox{}\end{equation}
Proposition \ref{musc141}. The following put-call parity relationship is valid
\begin{equation}{\label{museq120}}\tag{13}C_{0}-P_{0}=S_{0}-\frac{K}{1+r}.\end{equation}
Proof. The formula is an immediate consequence of equality (\ref{museq13}) and the pricing formula (\ref{museq115a}) applied to the claim \(S_{T}-K\). \(\blacksquare\)
It is worthwhile to mention that relationship (\ref{museq120}) is universal. In other words, it does not depend on the choice of the model. Using the put-call parity, we can calculate once again the arbitrage price of the put option. Formula (\ref{museq120}) yields immediately
\[P_{0}=C_{0}-S_{0}+\frac{K}{1+r}=8.25.\]
For ease of further reference, we shall write down explicit formulas for the call and put price in the one-period, two-state model. We assume that \(S_{u}>K>S_{d}\). Then, we have
\[C_{0}=\frac{S_{0}\cdot (1+r)-S_{d}}{S_{u}-S_{d}}\cdot\frac{S_{u}-K}{1+r}\] and
\[P_{0}=\frac{S_{u}-S_{0}\cdot (1+r)}{S_{u}-S_{d}}\cdot\frac{K-S_{d}}{1+r}.\]
\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}
Futures Call and Put Options.
We will focus mainly on the arbitrage pricing of European call and put options. Instead of the spot price of the underlying asset, we shall now consider its futures price. The model of futures prices we adopt is quite similar to the one used to describe spot prices. Due to the specific features of futures contracts used to set up a replicating strategy, one has to modify significantly the way in which the payoff from a portfolio is defined.
Futures Contracts and Futures Prices.
A futures contract is an agreement to buy or sell an asset at a certain date in the future for a certain price. The important feature of these contracts is that they are traded on exchanges. Consequently, the authorities need to define precisely all the characteristics of each futures contract in order to make trading possible. More importantly, the futures price, the price at which a given futures contract is entered into, is determined on a given futures exchange by the usual law of demand and supply. Therefore, futures prices are settled daily and the quotations are reported in the financial press. A futures contract is referred to by its delivery month, however an exchange specifies the period within that month when delivery must be made. The exchange specifies the amount of the asset to be delivered for one contract, as well as some additional details when necessary, e.g., the quality of the given commodity or the maturity of a bond. From our perspective, the most fundamental feature of a futures contract is the way the contract is settled. The procedure of daily settlement of futures contracts is called marking to market. A futures contract is worth zero when it is entered into. However, each investor is required to deposit funds into a margin account. The amount that should be deposited when the contract is entered into is known as the initial margin. At the end of each trading day, the balance of the investor’s margin account is adjusted in a way that reflects daily movements of futures prices. To be more specific, if an investor assumes a long position, and on a given day the futures price rises, the balance of the margin account will also increase. Conversely, the futures contract will be properly reduced. Intuitively, it is possible to argue that futures contracts are actually closed out after each trading day, and then start afresh the next trading day. Obviously, to offset a position in a futures contract, an investor enters into the opposite trade to the original one. Finally, if the delivery period is reached, the delivery is made by the party with a short position.
One-Period Futures Market.
It will be convenient to start a simple example which, in fact, is a straightforward modification of Example \ref{muse141} to the case of a futures market
\begin{equation}{\label{muse151}}\tag{14}\mbox{}\end{equation}
Example \ref{muse151}. Let \(f_{t}=f_{S}(t,T^{*})\) be a one-period process which models the futures price of a certain asset \(S\) for the settlement date \(T^{*}\geq T\). We assume \(f_{0}=280\) and
\[f_{T}(\omega )=\left\{\begin{array}{ll}f_{u}=320 & \mbox{if \(\omega =\omega_{1}\)},\\f_{d}=260 & \mbox{if \(\omega =\omega_{2}\)},\end{array}\right .\]
where \(T=3\) months. In the present context, the knowledge of the settlement date \(T^{*}\) of a futures contract is not essential. It is implicitly assumed \(T^{*}\geq T\). We consider a \(3\)-month European futures call option with strike price \(K=\$280\). As before, we assume that the simple risk-free interest rate for \(3\)-month deposits and loans is \(r=5\%\). \(\sharp\)
The payoff from the futures call option \(C_{T}^{f}=(f_{T}-K)^{+}\) equals
\[C_{T}^{f}(\omega )=\left\{\begin{array}{ll}C_{u}^{f}=320-280=40 & \mbox{if \(\omega =\omega_{1}\)},\\C_{d}^{f}=0 & \mbox{if \(\omega =\omega_{2}\)},\end{array}\right .\]
A portfolio \(\phi\) which replicates the option is composed of \(\alpha_{0}\) futures contracts and \(\beta_{0}\) units of cash invested in riskless bond (or borrowed). The wealth process \(V_{t}^{f}(\phi)\) for \(t\in\{0,T\}\) of this portfolio equals to \(V_{0}^{f}(\phi)=\beta_{0}\) since futures contract are worthless when they are first entered into. Furthermore, the terminal wealth of \(\phi\) is
\begin{equation}{\label{museq123}}\tag{15}V_{T}^{f}(\phi)=\alpha_{0}\cdot (f_{T}-f_{0})+\beta_{0}(1+r),\end{equation}
where the first term on the right-hand side represents gains or losses from the futures contract, and the second corresponds to a savings account (or loan). Note that (\ref{museq123}) reflects the fact that futures contracts are marked to market daily (that is, after each period in the model). A portfolio \(\phi=(\alpha_{0},\beta_{0})\) is said to replicate the option when \(V_{T}^{f}=C_{T}^{f}\), or more explicitly, when the equalities
\[V_{T}^{f}(\omega )=\left\{\begin{array}{ll}\alpha_{0}\cdot (f_{u}-f_{0})+(1+r)\beta_{0}=C_{u}^{f} & \mbox{if \(\omega=\omega_{1}\)},\\ \alpha_{0}\cdot (f_{d}-f_{0})+(1+r)\beta_{0}=C_{d}^{f} & \mbox{if \(\omega=\omega_{2}\)} \end{array}\right .\]
are satisfied. For Example \ref{muse151}, this gives the following system of linear equations
\[\left\{\begin{array}{l}40\alpha_{0}+1.05\beta_{0}=40\\ -20\alpha_{0}+1.05\beta_{0}=20\end{array}\right .\]
so that \(\alpha_{0}=-1/3\) and \(\beta_{0}=12.70\) in this case. Consequently, the manufacturing costs of call and put futures options are equal in this example. As we shall see soon, this is not a pure coincidence. As a matter of fact, by virtue of formula (\ref{museq129a}) below, the prices of call and put futures options are equal when the option’s strike price coincides with the initial futures price of the underlying asset. The above considerations may summarized by means of the following exhibits (note that \(\beta_{0}\) is a positive number)
\[\mbox{at time \(t=0\) }\left\{\begin{array}{ll}\mbox{one sold futures option} & C_{0}^{f}\\\mbox{futures contracts} & 0\\ \mbox{cash deposited in a bank} & -\beta_{0}=-C_{0}^{f}\end{array}\right .\]
and
\[\mbox{at time \(t=T\) }\left\{\begin{array}{ll}\mbox{option’s payoff} & -C_{T}^{f}\\\mbox{profits or losses from futures} & \alpha_{0}\cdot (f_{T}-f_{0})\\ \mbox{cash withdraw} & (1+r)\beta_{0}\end{array}\right .\]
Martingale Measure for a Futures Market.
Now, we are looking for a probability measure \(\tilde{\mathbb{P}}\) which makes the futures price process (with no discounting) follow a \(\tilde{\mathbb{P}}\)-martingale. A probability \(\tilde{\mathbb{P}}\), if it exists, is determined by the equality
\[f_{0}=\mathbb{E}_{\tilde{\mathbb{P}}}[f_{T}]=\tilde{p}\cdot f_{u}+(1-\tilde{p})\cdot f_{d}.\]
It is easily seen that
\[\tilde{\mathbb{P}}\{\omega_{1}\}=\tilde{p}=\frac{f_{0}-f_{d}}{f_{u}-f_{d}}\mbox{ and }\tilde{\mathbb{P}}\{\omega_{2}\}=1-\tilde{p}=\frac{f_{u}-f_{0}}{f_{u}-f_{d}}.\]
Using the data of Example \ref{muse151}, one finds that \(\tilde{p}=1/3\). Consequently, the expected value under the probability \(\tilde{\mathbb{P}}\) of the discounted payoff from the futures call option equals
\[\tilde{C}_{0}^{f}=\mathbb{E}_{\tilde{\mathbb{P}}}\left [\frac{(f_{T}-K)^{+}}{1+r}\right ]=12.70=C_{0}^{f}.\]
This illustrates the fact that the martingale approach may be used also in the case of futures markets with a suitable modification of the notion of a martingale measure.
Using the traditional terminology of mathematical finance, we may conclude that the risk-neutral futures economy is characterized by the fair-game property of the process of futures price. Remember that the risk-neutral spot economy is the one in which discounted stock price models a fair game.
Absence of Arbitrage.
We shall study a general two-state, one-period model of a futures price. The futures price of a certain asset for the fixed settlement date
$T^{*}\geq T$ is an adapted and strictly positive process \(f_{t}=f_{S}(t,T^{*})\) for \(t\in\{0,T\}\). More specifically, \(f_{0}\) is assumed to be a real number, and \(f_{T}\) is the following random variable
\[f_{T}(\omega )=\left\{\begin{array}{ll}f_{u} & \mbox{if \(\omega =\omega_{1}\)},\\f_{d} & \mbox{if \(\omega =\omega_{2}\)},\end{array}\right .\]
where, by convention, \(f_{u}>f_{d}\). The second security is, as in the case of a spot market, a riskless bond whose price process is \(p_{0}=1\) and \(p_{T}=1+r\) for some real \(r\geq 0\). Let \(\Phi^{f}\) stand for the linear space of all futures contracts-bonds portfolios \(\phi=\phi_{0}=(\alpha_{0},\beta_{0})\). It may be identified with the linear space \(\mathbb{R}^{2}\). The wealth process \(V^{f}(\phi)\) of any portfolio equals
\[V_{0}^{f}(\phi)=\beta_{0}\mbox{ and }V_{T}^{f}(\phi)=\alpha_{0}\cdot (f_{T}-f_{0})+(1+r)\beta_{0}.\]
We shall study the valuation of derivatives in the futures market model \({\cal M}^{f}=(f,p,\Phi^{f})\). It is easily seen that an arbitrary contingent claim \(X\) which settles at time \(T\) admits a unique replicating portfolio \(\phi\in\Phi^{f}\). In other words, all contingent claims which settle at time \(T\) are attainable in the market model \({\cal M}^{f}\). In fact, if \(X\) is given by the formula
\[X(\omega )=\left\{\begin{array}{ll}X_{u} & \mbox{if \(\omega =\omega_{1}\)},\\X_{d} & \mbox{if \(\omega =\omega_{2}\)},\end{array}\right .\]
then its replicating portfolio \(\phi\in\Phi^{f}\) may be found by solving the following system of linear equations
\begin{equation}{\label{museq127}}\tag{16}\left\{\begin{array}{l}\alpha_{0}\cdot (f_{u}-f_{0})+(1+r)\beta_{0}=X_{u}\\\alpha_{0}\cdot (f_{0}-f_{d})+(1+r)\beta_{0}=X_{d}.\end{array}\right .\end{equation}
The unique solution of (\ref{museq127}) is
\[\alpha_{0}=\frac{X_{u}-X_{d}}{f_{u}-f_{d}}\mbox{ and }\beta_{0}=\frac{X_{u}\cdot (f_{0}-f_{d})+X_{d}(f_{u}-f_{0})}{(1+r)\cdot (f_{u}-f_{d}}.\]
Consequently, the manufacturing cost \(\Pi_{0}^{f}(X)\) in \({\cal M}^{f}\) equals
\begin{equation}{\label{museq129a}}\tag{17}\Pi_{0}^{f}(X)\equiv V_{0}^{f}(\phi)=\beta_{0}=\frac{X_{u}\cdot (f_{0}-f_{d})+X_{d}(f_{u}-f_{0})}{(1+r)\cdot (f_{u}-f_{d}}.\end{equation}
We say that a model \({\cal M}^{f}\) of the futures market is arbitrage-free when there are no arbitrage opportunities in the class \(\Phi^{f}\) of trading strategies.
Proposition. The futures market \({\cal M}^{f}=(f,p,\Phi^{f})\) is arbitrage-free if and only if the process \(f\) that models the futures price admits a unique martingale measure \(\tilde{\mathbb{P}}\) equivalent to \(\mathbb{P}\). In this case, the arbitrage price at time \(0\) of any contingent claim \(X\) which settles at time \(T\) equals
\begin{equation}{\label{museq130}}\tag{18}\Pi_{0}^{f}(X)=E_{\tilde{P}}\left [\frac{X}{1+r}\right ],\end{equation}
or explicitly,
\[\Pi_{0}^{f}(X)=\frac{f_{0}-f_{d}}{f_{u}-f_{d}}\cdot\frac{X_{u}}{1+r}+
\frac{f_{u}-f_{0}}{f_{u}-f_{d}}\cdot\frac{X_{d}}{1+r}. \sharp\]
When the price of the futures call option is already known, in order to find the price of the corresponding put option, one may use the following relation, which is an immediate consequence of equality (\ref{museq13}) and the pricing formula (\ref{museq130})
\begin{equation}{\label{museq132}}\tag{19}C_{0}^{f}-P_{0}^{f}=\frac{f_{0}-K}{1+r}.\end{equation}
It is now obvious that the equality \(C_{0}^{f}=P_{0}^{f}\) is valid if and only if \(f_{0}=K\); that is, when the current futures price and the strike price of the option are equal. Equality (\ref{museq132}) is referred to as the put-call parity relationship for futures option.
One-Period Spot/Futures Market.
Consider an arbitrage-free, one-period spot market \((S,p,\phi)\) described above. Moreover, let \(f_{t}=f_{S}(t,T)\) for \(t\in\{0,T\}\) be the process of futures prices with the underlying asset \(S\) and for the maturity date \(T\). In order to preserve consistency with the financial interpretation of the futures price, we have to assume \(f_{T}=S_{T}\). The aim is to find the right value \(f_{0}\) of the futures price at time \(0\). In such a market, trading in stocks, bonds, as well as entering into futures contracts is allowed.
\begin{equation}{\label{musc151}}\tag{20}\mbox{}\end{equation}
Proposition \ref{musc151}. The futures price at time \(0\) for the delivery date \(T\) of the underlying asset \(S\) which makes the spot/futures market arbitrage-free equals \(f_{0}=(1+r)S_{0}\).
Proof. Suppose an investor enters at time \(0\) into one futures contract. The payoff of his position at time \(T\) corresponds to a time \(T\) contingent claim \(X=f_{T}-f_{0}=S_{T}-f_{0}\). Since it costs nothing to enter a futures contract we should have
\[\Pi_{0}(X)=\Pi_{0}(S_{T}-f_{0})=0,\]
or equivalently, from (\ref{museq115a}) (or the proof of Proposition \ref{musc141}),
\[\Pi_{0}(X)=S_{0}-\frac{f_{0}}{1+r}=0.\]
This proves the asserted formula. Alternatively, one can check that if the futures price \(f_{0}\) were different from \((1+r)S_{0}\), this would lead to arbitrage opportunities in the spot/futures market. \(\blacksquare\)
\begin{equation}{\label{d}}\tag{D}\mbox{}\end{equation}
Forward Contracts.
A forward contract is an agreement, signed at the initial date \(0\), to buy or sell an asset at a certain time \(T\) for a pre-specified price \(K\). In contrast to stock options and futures contracts, forward contracts are not traded on exchanges. By convention, the party who agrees to buy the underlying asset at time \(T\) for the delivery price \(K\) is said to assume a long position in a given contract. Consequently, the other party, who is obliged to sell the asset at the same date for the price \(K\), is said to assume a short position. Since a forward contract is settled at maturity and a party in a long position is obliged to buy an asset worth \(S_{T}\) at maturity for \(K\), it is clear that the payoff from the long position (resp. from the short position) in a given forward contract with a stock \(S\) being the underlying asset corresponds to the time \(T\) contingent claim \(X\) (resp. \(-X\)), where
\begin{equation}{\label{museq133}}\tag{21}X=S_{T}-K.\end{equation}
Let us emphasize that there is no cash flow at the time the forward contract is entered into. In other words, the price of a forward contract at its initiation is zero. However, for \(t>0\), the value of a forward contract may be negative or positive. As we shall see now, a forward contract is worthless at time \(0\) provided that a judicious choice if the delivery price \(K\) is made.
Forward Price.
We shall find the rational delivery price for a forward contract. Recall that there is no cash flow at the initiation of a forward contract.
Definition. The delivery price \(K\) that makes a forward contract worthless at initiation is called the forward price of an underlying financial asset \(S\) for the settlement date \(T\). \(\sharp\)
Proposition. Assume that the one-period, two-state security market model \((S,p,\Phi)\)is arbitrage-free. Then, the forward price at time \(0\) for the settlement date \(T\) of one share of stock \(S\) equals \(F_{S}(0,T)=S_{0}(1+r)\).
Proof. We shall apply the martingale method of Proposition \ref{musp142}. By applying formulas (\ref{museq114a}) and (\ref{museq133}), we get
\[\Pi_{0}(X)=E_{P^{*}}\left [\frac{X}{1+r}\right ]=E_{P^{*}}[S_{T}^{*}]-\frac{K}{1+r}=S_{0}-\frac{K}{1+r}=0.\]
This completes the proof. \(\blacksquare\)
By combining Proposition \ref{musc151} with the above proposition, we conclude that in a one-period model of a spot market, the futures and forward prices of financial assets for the same settlement date are equal.
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\end{thebibliography}
