Martingale optimal transport and mathematical finance

In the financial market, an option is a contract signed at present that gives the holder some nonnegative payoff in the future, which depends on the price of one or more assets. One needs to pay for holding an option in order to get the payoff in the future, and this leads to the problem of option pricing. In the following, we will first introduce some fundamental results regarding option pricing, and then discuss how an option pricing problem can be reformulated as an optimal transport problem possibly with martingale condition.

Backgrounds For Option Pricing

Suppose $(\Omega ,{\mathcal {F}},\mathbb {P} )$ is the underlying probability space with a filtration $({\mathcal {F}}_{t})_{t\geq 0}$ . For simplicity, we only consider stocks and money as our assets. The price of an asset at time $t$ , denoted by $S_{t}$ , is a random variable on this probability space. An option is called vanilla if the payoff is given at a fixed time $T$ in the future, called maturity, and is called European style if the payoff only depends on the asset price only at the maturity $T$ . A vanilla European call option on a stock has the payoff $(S_{T}-K)^{+}\equiv \max(S_{T}-K,0)$ , where $K$ is a constant called strike determined in the contract. An option with payoff $(\max _{0\leq t\leq T}S_{t})$ is not European style because the payoff depends on the whole path of stock price before time $T$ .

A basic assumption for option pricing is that the market is free of arbitrage opportunity, meaning we cannot generate a positive return starting with nothing. A fair price of an option should not give an arbitrage. No-arbitrage condition is equivalent with the existence of the equivalent martingale measure, or risk-neutral measure.

Definition 1. Equivalent martingale measure (EMM)

A probability measure $\mathbb {Q}$ is an equivalent martingale measure, if:

(i) $\mathbb {Q}$ is equivalent with $\mathbb {P} :\mathbb {Q} \sim \mathbb {P}$ , meaning $\mathbb {Q} (A)=0\Leftrightarrow \mathbb {P} (A)=0$ , and

(ii) the discounted price is a martingale under $\mathbb {Q}$ for all assets: $\mathbb {E} ^{\mathbb {Q} }[e^{-rt}S_{t}|{\mathcal {F}}_{s}]=e^{-rs}S_{s}$ , $\mathbb {Q}$ - $a.s.$ , where $r$ is the risk-free rate, i.e., the growth rate of money.

A well-known result in option pricing is that the expected value of the discounted payoff under an EMM gives a no-arbitrage price. This result also holds for path-dependent payoffs, but we use an European style payoff as an example. According to this result, a no-arbitrage price at time $t$ of an option with payoff $h(S_{T})$ is given by $\mathbb {E} ^{\mathbb {Q} }[e^{-r(T-t)}h(S_{T})|{\mathcal {F}}_{t}]$ for some EMM $\mathbb {Q}$ . Denote the set of EMMs to be ${\mathcal {M}}$ , the set of no-arbitrage prices for this option is:

{\mathcal {P}}_{t}=\{\mathbb {E} ^{\mathbb {Q} }[e^{-r(T-t)}h(S_{T})|{\mathcal {F}}_{t}]:\mathbb {Q} \in {\mathcal {M}}\}

Another useful result is that the prices of all European call options with different strikes $K$ give the marginal distribution of the stock at time $T$ .

Theorem 1. Call option gives marginal distribution^[1]

Let $C(0,T;x,K)$ be the price of a European call option at time $0$ , with maturity $T$ , strike $K$ , and $S_{0}=x$ . Then the marginal distribution of the stock price $S_{T}$ is given by

f(S_{T}=K|S_{0}=x)={\frac {\partial ^{2}C}{\partial K^{2}}}

We assume that the European options of stocks are traded very frequently so that their prices for each strike are available, which gives us the marginal distribution of the stock prices.

Option Pricing as an Optimal Transport Problem

For simplicity we assume the risk-free rate to be $0$ . We introduce the following two option pricing problems.^[2]

Problem 1. Option with payoff $f(S_{T}^{1},S_{T}^{2})$ .

Suppose the payoff $f$ depends on two assets but only the price at maturity $T$ : $f(S_{T}^{1},S_{T}^{2})$ . Suppose we have all the prices of call options for both stocks $S^{1}$ and $S^{2}$ , then Theorem 1 gives that we have the marginal distribution of $S_{T}^{1}$ and $S_{T}^{2}$ , denoted by $\mu _{1}$ and $\mu _{2}$ . Denote the set of probability measures on $\mathbb {R} ^{2}$ that have the right marginals as

\Gamma (\mu _{1},\mu _{2})\equiv \{\gamma :(\pi ^{1})_{\#}\gamma =\mu _{1},(\pi ^{2})_{\#}\gamma =\mu _{2}\}

The upper and lower bound of the no-arbitrage price is given by the supremum and infimum of

\mathbb {E} ^{\mathbb {Q} }[f(S_{T}^{1},S_{T}^{2})]=\int f(x_{1},x_{2})d\mu _{1}(x_{1})d\mu _{2}(x_{2})

over $\gamma \in \Gamma (\mu _{1},\mu _{2})$ , where $\gamma$ is the joint distribution of $(S_{T}^{1},S_{T}^{2})$ under $\mathbb {Q}$ . This is exactly an optimal transport problem with the cost function being $f(x_{1},x_{2})$ . We point out that $\mathbb {Q}$ needs to be an EMM, but here we drop this restriction because the payoff only depends on the stock prices at maturity, so the evolution of the stock price before $T$ does not effect the problem.

However, in the following problem, the martingale condition cannot be avoided.

Problem 2. Option with path-dependent payoff.

Consider a payoff function $f$ depending on one stock, but the whole path $(S_{t})_{0\leq t\leq T}$ . For simplicity, we only consider finitely many time points before the maturity and the payoff function is denoted by $f(S_{0},S_{1},\cdots ,S_{N})$ . Similar as Problem 1, we obtain the marginal distributions $\mu _{1},\cdots ,\mu _{N}$ of stock prices $S_{1},\cdots ,S_{N}$ . To estimate the no-arbitrage price of this option, we consider the following quantity:

\mathbb {E} ^{\mathbb {Q} }[f(S_{0},S_{1},\cdots ,S_{N})]=\int f(x_{1},x_{2},\cdots ,x_{N})d\gamma

where $\gamma =(S_{0},\cdots ,S_{N})_{\#}\mathbb {Q}$ so $(\pi ^{i})_{\#}\gamma =\mu _{i}$ for each $i=0,\cdots ,N$ . In addition, the stock price should be an martingale under $\mathbb {Q}$ . Therefore, we need to add conditions that $\mathbb {E} ^{\mathbb {Q} }[S_{i+1}|{\mathcal {F}}_{i}]=S_{i}$ , meaning