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(i.e., anything that can be traded). 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. We equip this probability space with a filtration $({\mathcal {F}}_{t})_{t\geq 0}$, which is an increasing sequence of $\sigma$-algebras indexed by time $t$. That is, for every pair $s<t$, we have ${\mathcal {F}}_{s}\subset {\mathcal {F}}_{t}\subset {\mathcal {F}}$. A $\sigma$-algebra can be viewed as a collection of information, so the filtration represents the information accumulating as time evolves.

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 the underlying probability space. The collection of random variable indexed by time $(S_{t})_{t\geq 0}$ is a stochastic process and assumed to be adapted to the filtration $({\mathcal {F}}_{t})_{t\geq 0}$, meaning the random variable $S_{t}$ is measurable with ${\mathcal {F}}_{t}$. The intuition of this condition is that the price of the stock at time $t$ can be observed in the market, so it should be part of the information we have up to time $t$.

The payoff of an option is a function of the stock price process $(S_{t})_{t\geq 0}$. 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 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. The rigorous definition is the following.

Definition 1. Arbitrage opportunity

A portfolio $V$(i.e. a linear combination of different assets with weights possibly changing by time) gives an arbitrage opportunity if (i) $\mathbb {P} (V_{0}=0)=1$, (ii) $\mathbb {P} (V_{T}\geq 0)=1$, and (iii) $\mathbb {P} (V_{T}>0)>0$, for some time $T>0$.

In other words, a portfolio generates an arbitrage if it starts with nothing, guarantees no loss, and has a positive probability to obtain a positive return at the end. A fair price of an option should exclude arbitrage opportunities. No-arbitrage condition is equivalent with the existence of the equivalent martingale measure, or risk-neutral measure. The definition involves the notion of martingales.

Definition 3. Martingale

A stochastic process $(X_{t})_{t\geq 0}$ is called a martingale adapted to the filtration $({\mathcal {F}}_{t})_{t\geq 0}$ if (i) $X_{t}$ is measurable with ${\mathcal {F}}_{t}$ for every $t\geq 0$, (ii) $X_{t}$ is integrable for every $t\geq 0$, and (iii) for every pair $t>s$, $\mathbb {E} [X_{t}|{\mathcal {F}}_{s}]=X_{s},\mathbb {P}$-a.e.

The notation of conditional expectation given a $\sigma$-algebra can be viewed as an estimation of the random variable given the information represented by the $\sigma$-algebra. By taking expectation on both sides of (iii) and using tower property of conditional expectations($\mathbb {E} [\mathbb {E} [X|{\mathcal {G}}]]=\mathbb {E} [X]$), we obtain that a martingale has a constant expectation over the whole time horizon, so it can be treated as an analogue of constant function in the stochastic world. Now we are ready to state the definition of equivalent martingale measure.

Definition 4. 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.

The existence of EMM excludes arbitrage. To see this, we assume $r=0$ for simplicity and $\mathbb {Q}$ is an EMM. Assume that a portfolio $V$ satisfies $\mathbb {P} (V_{0}=0)=1$ and $\mathbb {P} (V_{T}\geq 0)=1$, so by equivalence of $\mathbb {P}$ and $\mathbb {Q}$ we have $\mathbb {Q} (V_{0}=0)=1$ and $\mathbb {Q} (V_{T}\geq 0)=1$ as well. The portfolio $V$ as a linear combination of assets, should also be a martingale under the EMM $\mathbb {Q}$ by linearity of conditional expectations. Therefore, $\mathbb {E} ^{\mathbb {Q} }[V_{T}]=\mathbb {E} ^{\mathbb {Q} }[V_{0}]=0$ which implies $\mathbb {Q} (V_{T}=0)=1$. Applying the equivalence of $\mathbb {P}$ and $\mathbb {Q}$ again gives $\mathbb {P} (V_{T}=0)=1$ so it cannot be the case that $\mathbb {P} (V_{T}>0)>0$. Thus, the existence of EMM leads to an arbitrage-free market. Actually, the converse argument is also correct so we have the equivalent of two conditions: the existence of EMM and no-arbitrage.

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 a European style payoff as an example.

Theorem 1. No arbitrage price

Suppose $\mathbb {Q}$ is an EMM and the payoff of an option is $h(S_{T})$. Then the expected discounted payoff under $\mathbb {Q}$

$P_{t}\equiv \mathbb {E} ^{\mathbb {Q} }[e^{-r(T-t)}h(S_{T})|{\mathcal {F}}_{t}]$

gives a no-arbitrage price of this option at time $t\in [0,T]$.

Proof: Since an option can be traded, it is an asset, thus its discounted process $(e^{-rt}P_{t})_{t\geq 0}$ is a martingale under the EMM $\mathbb {Q}$. Therefore, for $t\in [0,T]$, $e^{-rt}P_{t}=\mathbb {E} ^{\mathbb {Q} }[e^{-rT}P_{T}|{\mathcal {F}}_{t}]=\mathbb {E} ^{\mathbb {Q} }[e^{-rT}h(S_{T})|{\mathcal {F}}_{t}]$, since the price of the option at time $T$ is exactly the payoff at time $T$. Multiply $e^{rt}$ on both sides we get the desired result.

Denote the set of EMMs to be ${\mathcal {M}}$, according to this result, the set of no-arbitrage prices for this option is:

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

Sometimes the EMM is not unique, and it is the market that choose the EMM by giving assets proper prices through trading.

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 2. Call option gives marginal distribution

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

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

where $f$ is the probability density function of $S_{T}$ given that $S_{0}=x$. We assume that the European options of stocks are traded very frequently so that the 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 $0$, and denote the EMM by $\mathbb {Q}$. We introduce the following two option pricing problems. By Theorem 1, we always price the option under EMMs, so in the following discussion we consider the problem under EMMs.

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 2 gives that we have the marginal distribution of $S_{T}^{1}$ and $S_{T}^{2}$ under $\mathbb {Q}$, denoted by $\mu _{1}$ and $\mu _{2}$. By Theorem 1, the price of the option is:

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

where $\gamma$ is the joint distribution of $S_{T}^{1}$ and $S_{T}^{2}$ under the EMM $Q$. The upper and lower bound of the no-arbitrage price is given by the supremum and infimum of the above quantity over all the possible probability measures on $\mathbb {R} ^{2}$ that have the right marginals, and denote the set of these probability measures as:

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

The problem then becomes:

$\min _{\gamma \in \Gamma (\mu _{1},\mu _{2})}/\max _{\gamma \in \Gamma (\mu _{1},\mu _{2})}\int f(x_{1},x_{2})d\gamma (x_{1},x_{2})$

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 supremum and infimum of 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 $\mathbb {E} ^{\mathbb {Q} }[S_{i+1}|{\mathcal {F}}_{i}]=S_{i}$, $\mathbb {Q}$-a.e., meaning

$\int x_{i+1}\phi (x_{0},\cdots ,x_{i})d\gamma =\int x_{i}\phi (x_{0},\cdots ,x_{i})d\gamma$

for any Borel measurable function $\phi :\mathbb {R} ^{i}\rightarrow \mathbb {R}$.