Applications of Optimal Transport and Martingale Optimal Transport in Mathematical Finance
A central topic in mathematical finance is the pricing of financial derivatives—contracts that facilitate the exchange or allocation of resources across different time periods. We introduce the principle of arbitrage-free pricing in financial markets and focus on a fundamental class of financial derivatives known as options. Following the classical approach of model-based finance, we price options based on two different ideas: hedging and risk-neural pricing, which have essential underlying connections. Moving to an incomplete market, where the stock price dynamics are unknown, we motivate model-free finance, where the corresponding optimization problems for option pricing could be naturally interpreted in the context of optimal transport and martingale optimal transport.
Fundamentals of Mathematical Finance
We begin by restricting our discussions to the continuous-time setting, with the current time as \(0\) by default. Assume the market includes a risk-free asset (e.g., a bond), which we refer to as cash, with a constant risk-free interest rate \(r\), i.e., \(1\) dollor deterministically accumulates to \(e^{rt}\) dollars in time \(t\). In addition, there exists a risky asset, which we refer to as stock, whose price at time \(t\) is denoted by \(S_t\), a stochastic process on the filtered probability space \((\Omega,\{\mathscr{F_t}\}_{t\geq 0},\mathscr{F},\mathbb{P})\).
Arbitrage-Free Pricing
The fundamental principle of derivative pricing requires no arbitrage to exist in the market. Intuitively, arbitrage refers to the existence of a self-financing portfolio that requires no external endowment, yet guarantees a non-negative payoff almost surely, and yields a strictly positive profit with a positive probability. A mathematically rigorous definition of arbitrage relies on the notion of market viability, as discussed in (Karatzas and Kardaras 2021). Nevertheless, the concept of no-arbitrage pricing can be illustrated through the following simple example.
Consider a forward contract currently signed, which obligates the holder to receive one unit of stock at a future time \(T\) and to pay the forward price \(F(0,T)\) at time \(T\). A self-financing arbitrage portfolio can be constructed as follows:
Enter the forward contract at time \(0\).
Sell \(1\) unit of stock at time \(0\), receiving \(S_0\) cash.
Pay the forward price \(F(0,T)\) and receive \(1\) unit of stock at time \(T\).
Since the stock received at time \(T\) exactly offsets the short position, this strategy gives a risk-free portfolio. The net cash flow (amount paid by the portfolio holder) at time \(T\) is \(F(0,T)-S_0e^{rT}\), which shall be non-negative to prevent arbitrage, i.e., \(F(0,T)-S_0e^{rT}\geq 0\). Conversely, by reversing the strategy (buying the stock, shorting the forward contract), the no-arbitrage condition requires \(F(0,T)-S_0e^{rT}\leq 0\). Combining both inequalities yields the arbitrage-free price of the forward contract: \(F(0,T) = S_0e^{rT}\).
This example, though simple, illustrates the concept of hedging—reducing or eliminating risk by trading other assets. In the example above, the forward contract carries risk because the future stock price \(S_T\) is unknown at time \(0\). However, perfect hedging is achieved by holding one unit of stock into the future. Since the initial stock price \(S_0\) is observable at time \(0\), this strategy eliminates uncertainty, ensuring a risk-free position.
Options
In real financial markets, more complex financial derivatives are traded, one of which is called options. An option is a contract written on an underlying asset, signed at present but granting the holder the right, rather than the obligation, to execute it in the future. Unlike a forward contract, the option holder can choose whether to exercise the option based on its profitability at maturity. In our discussion, we assume the underlying asset to be the stock with price \(\{S_t\}\).
A European call option with time to maturity \(T\) and strike price \(K\) grants the holder the right to buy one unit of stock at time \(T\) at price \(K\). If \(S_T>K\), the holder exercises the option, receiving an immediate payoff of \(S_T - K\). If \(S_T\leq K\), the holder does not exercise the option, resulting in a payoff of \(0\). Written compactly, such a European call option has payoff \[ (S_T - K)_+ := \max\{S_T-K,0\}. \] Similarly, a European put option with time to maturity \(T\) and strike price \(K\) grants the holder the right to sell one unit of stock at time \(T\) at price \(K\). Its payoff function is given by: \[ (K-S_T)_+ := \max\{K-S_T,0\}. \] European options are the simplest ones, yet there exist various other types of options on the market. For example, American options allow the holder to exercise at any time before maturity. Path-dependent options have payoff functions of the form \(g(S_{[0,T]})\), which depend on the entire stock price trajectory over \([0,T]\). A common example is the lookback option, where the payoff is given by \[ g(S_{[0,T]}) = \sup_{t\in[0,T]} S_t. \] For a comprehensive introduction to continuous-time finance and option pricing, we refer the readers to (Björk 2009).
Option Pricing— A Model-Based Approach
We begin by introducing option pricing within the framework of model-based finance, which requires specifying an a priori model for the stock price dynamics \(\{S_t\}\). A well-known and widely used example is the following Black-Scholes (BS) Model: \[ dS_t = \mu S_t\,dt + \sigma S_t\,dW_t, \] where \(\mu\) denotes the expected return of the stock, \(\sigma>0\) denotes the volatility, and \(\{W_t\}\) is a standard Brownian motion under measure \(\mathbb{P}\).
To illustrate option pricing under the Black-Scholes model, we present two fundamental approaches for pricing a European call option:
The hedging approach.
The risk-neutral pricing (martingale measure) approach.
Those two approches are fundamentally equivalent, providing different perspectives illustrating the same underlying principle.
Pricing through Hedging
The first approach to pricing a European call option is based on hedging, which is motivated by the fundamental principle: “If two portfolios have the same payoff, they must have the same price to prevent arbitrage”. Therefore, if we can replicate the payoff of a European call option using a portfolio that consists of cash and stock, the option price is just the price of the replicating portfolio, which only requires trivial calculations. Replication is equivalent to perfect hedging, since selling the replicating portfolio while holding the option eliminates all risk.
Consider a replicating portfolio consisting of:
\(a_t\) units of stock at time \(t\).
\(b_t\) units of cash at time \(t\).
The portfolio value at time \(t\) is: \[ V_t = a_tS_t + b_te^{rt}. \] By Itô’s formula, \[ dV_t = a_t\,dS_t + S_t\,da_t + \,d\langle S,a\rangle_t + e^{rt}\,db_t + rb_te^{rt}\,dt, \] where \(\langle \cdot,\cdot\rangle_t\) denotes the quadratic variation up to time \(t\). The changes in the portfolio value are due to market movements \(a_t\,dS_t + rb_te^{rt}\,dt\) and investor adjustments \(S_t\,da_t + \,d\langle S,a\rangle_t + e^{rt}\,db_t\). Since no external endowments are allowed, the portfolio must be self-financing, meaning that the change in its value is only caused by market movements: \[ dV_t = a_t\,dS_t + rb_te^{rt}\,dt. \] To proceed, we assume the Markovian representation \(V_t = u(t,S_t)\), where \(u(t,s)\) denotes the portfolio value at time \(t\) on observing \(S_t = s\). Itô’s formula implies \[ dV_t = \partial_t u\,dt + \partial_s u\,dS_t + \frac{1}{2}\partial_{ss} u\,d\langle S,S\rangle_t. \] Substituting the dynamics of the Black-Scholes model yields \[ dV_t = \left(\partial_t u+ \mu S_t\partial_s u + \frac{\sigma^2}{2}S_t^2\partial_{ss}u\right)\,dt + \sigma S_t\partial_s u\,dW_t. \] By comparing the coefficients, we get \[ \begin{cases} a_t = \partial_s u(t,S_t)\\ b_t = \frac{1}{r}e^{-rt}(\partial_t u(t,S_t)+ \frac{\sigma^2}{2}S_t^2\partial_{ss}u(t,S_t)) \end{cases}. \] Recall that \(a_tS_t + b_te^{rt} = V_t = u(t,S_t)\). Substituting \(a_t,b_t\) yields the Black-Scholes PDE for \(u = u(t,s)\): \[ \partial_t u + rs\partial_s u + \frac{\sigma^2}{2}s^2\partial_{ss}u - ru = 0,\quad u(T,s) = (s - K)_+. \] Solving this PDE (a technical step omitted here) provides the celebrated Black-Scholes formula for the European call option price: \[ V_0 = u(0,S_0). \]
Risk-Neutral Pricing
The difficulty of pricing lies in the fact that under the physical measure \(\mathbb{P}\), the discounted expected payoff is not equal to the price, due to the existence of risk premium to compensate for the uncertainty taken by the holders of the derivative. That means, if a holder is risk-neutral, the discounted expected payoff is just the price, in which situation the pricing problem reduces to simple calculations of expectations.
Risk-neutral pricing follows this main idea and aims to view the current world through the lens of a risk-neutral (martingale) measure \(\mathbb{Q}\), under which all investors are risk-neutral, i.e., \(\{e^{-rt}S_t\}\) is a martingale. The European call price is nothing but \[ e^{-rT}\mathbb{E}_{\mathbb{Q}}(S_T - K)_+. \] When it comes to the calculation, one needs to guarantee the existence of \(\mathbb{Q}\) and the knowledge on the distribution of \(S_T\) under \(\mathbb{Q}\). Followed from the fact that \(\{e^{-rt}S_t\}\) is a martingale, we rewrite the stock price dynamics: \[ dS_t = rS_t\,dt + \sigma S_t\left(\frac{\mu - r}{\sigma}\,dt+dW_t\right), \] denoting \(dW_t^\mathbb{Q}:= \frac{\mu - r}{\sigma}\,dt+dW_t\) as the BM under \(\mathbb{Q}\). An application of Girsanov theorem guarantees the existence of \(\mathbb{Q}\) and provides the Radon-Nikodym derivative between the physical measure and the risk-neutral measure. Solving for the SDE yields \[ S_t = e^{(r-\frac{\sigma^2}{2})t + \sigma W_t^{\mathbb{Q}}}, \] which tells the explicit distribution of \(S_T\) under \(\mathbb{Q}\). After performing calculations, one yields exactly the same Black-Scholes formula for the European call option price as the one derived from hedging.
Market Completeness and Super-Replication
Martingale measures serve as a crucial component in mathematical finance and are known to be associated with several fundamental theorems. To name a few, the First Fundamental Theorem of Asset Pricing states that the market is viable (arbitrage-free) if and only if at least one martingale measure exists. The Second Fundamental Theorem of Asset Pricing claims that the market is complete iff the martingale measure exists and is unique. Without even realizing it, the market completness greatly simplifies the problem of option pricing in a Black-Scholes market.
A viable market is called complete if any European payoffs at any time of maturity \(T\) can be perfectly replicated. It turns out that the perfect replicability of European call and put payoffs implies the perfect replicability of any European payoff functions (Carr-Madan formula). That means, the European call and put payoffs serve as the “basis” for all European payoffs. Option pricing in a complete market (e.g., the Black-Scholes market) is therefore easy, but unfortunately is almost never the case in real life.
One of the common examples of an incomplete market is the stochastic volatility model, where the volatility \(\{\sigma_t\}\) is also a stochastic process with its dynamics given by an SDE under a different Brownian motion. There is only one risky asset but two sources of randomness (i.e., one Brownian motion for \(S\) and one for \(\sigma\)), which causes the failure of perfect replication. Intuitively understanding through an analogue to the solvability of linear systems, the market is complete (a linear system has a unique solution) if and only if the number of different risky assets (number of equations) equals the number of different sources of randomness (number of unknowns).
As we would naturally expect, pricing becomes much harder in an incomplete market, since the arbitrage-free price is an interval, rather than a single number. On the side of hedging, perfectly hedging becomes impossible. On the side of risk-neutral pricing, there exist infinitely many martingale measures. A new pricing criterion is given by super-replication instead, i.e., ensuring the payoff of the replicating portfolio is larger than the payoff of the option. The minimal cost of super-replication for an option writer, denoted by \(C_{\text{sell}}\), is the highest arbitrage-free price on the market. Conversely, taking the perspective of an option buyer, one can define \(C_{\text{buy}}\) as the maximal cost of sub-replication, which is the lowest arbitrage-free price on the market. Therefore, the interval of arbitrage-free price is provided by \([C_{\text{buy}},C_{\text{sell}}]\).
The super-replication approach sounds natural and attractive from the hedging perspective, but what can we say about martingale measures? Does that mean the hedging perspective would be superior over the martingale measure approach in an incomplete market? Surprisingly, the super-replication duality (Henry-Labordère 2017) holds, which can be proved using the Fenchel-Rockafeller Duality, stating that the minimum cost of super-replication is equal to the maximum martingale price. This result implies that hedging strategies and martingale measures are two sides of the same coin, in the sense that they should be understood as dual optimization variables of each other.
Option Pricing— A Model-Free Approach
Motivated by the super-replication that goes beyond the requirement of market completeness, we introduce option pricing in the model-free finance context. The basic settings of model-free option pricing are:
No a priori knowledge on the stock price dynamics is assumed.
The only information available are the empirical observations of derivative prices traded on the market.
The market is incomplete.
By the Breeden-Litzenberger Formula, empirical observations of European call prices for a continuum of strikes \(K\in\mathbb{R}_+\) with maturity \(T\) recovers the marginal distribution of the stock price \(S_T\) under the martingale measure. Therefore, within the discussions below, we always assume the knowledge of the marginal distribution of \(S_T\).
Although it is practically impossible to observe European call prices for a continuum of strikes \(K\), several approaches can be taken to fix this problem:
Adding constraints to the optimization problems stated below (Henry-Labordère 2017). The constraints reflect the fact that European call prices are observed for only finitely many strikes \(K\).
Interpolate or approximate the implied volatility surface to predict European call prices at unknown strikes \(K\).
In the following context, we always assume \(r=0\) without loss of generality, and all optimization problems are stated based on the perspective of an option writer. In other words, the solutions to those optimization problems provide the maximum arbitrage-free prices of the options. Similar formulations can be adopted from the perspective of option buyers that provide the minimum arbitrage-free prices of the options.
Pricing Options with Joint Payoffs
Assume there are two stocks on the market with stock prices \(\{S^1_t\}\) and \(\{S^2_t\}\). We wish to price a European option with maturity \(T\) and a given payoff function \(C\), which is a joint function in both stock prices, i.e. \(c(S^1_T,S^2_T)\). Denote by \(\mathbb{P}^1\) and \(\mathbb{P}^2\) the laws of \(S^1_T\) and \(S^2_T\) under the martingale measure (known). The seller’s price is the minimum cost of super-replication: \[ \text{MK}_2:=\inf_{(\lambda_1,\lambda_2)\in \mathscr{P}^*(\mathbb{P}^1,\mathbb{P}^2)} \mathbb{E}_{\mathbb{P}^1}\lambda_1(S^1_T) + \mathbb{E}_{\mathbb{P}^2}\lambda_2(S^2_T), \] where \[ \mathscr{P}^*(\mathbb{P}^1,\mathbb{P}^2) := \{(\lambda_1,\lambda_2):\lambda_1(s_1) + \lambda_2(s_2)\geq c(s_1,s_2)\}. \] The objective function is the cost of replication with European payoffs \(\lambda_1\) and \(\lambda_2\), while \(\mathscr{P}^*(\mathbb{P}^1,\mathbb{P}^2)\) is the collection of payoff functions that admits a super-replication.
Through the Kantorovich Duality, it is clear that \[ \text{MK}_2=\sup_{\mathbb{P}\in\mathscr{P}(\mathbb{P}^1,\mathbb{P}^2)}\mathbb{E}_{\mathbb{P}}[c(S^1_T,S^2_T)], \] where \(\mathscr{P}(\mathbb{P}^1,\mathbb{P}^2)\) is the collection of probability measures, under which \(S^1_T\sim \mathbb{P}^1\) and \(S^2_T\sim \mathbb{P}^2\).
We remark that, the primal and dual optimization problems align with those in optimal transport. However, those optimization problems no longer have the physical interpretation as transport problems, but are interpreted in the sense of hedging. Nevertheless, this optimization problem can be numerically solved through techniques that have been developed for optimal transport, e.g., Sinkhorn’s Algorithm.
Pricing Options with Path-dependent Payoffs
Assume there is a single stock considered at two different time points \(t_1<t_2\). For simplicity, we denote \(S_1:= S_{t_1}\) and \(S_2:= S_{t_2}\). The path-dependent option payoff has the form \(c(S_1,S_2)\). Denote by \(\mathbb{P}^1\) and \(\mathbb{P}^2\) the laws of \(S^1\) and \(S^2\) under the martingale measure (known).
The seller’s price is the minimum cost of super-replication: \[ \widetilde{\text{MK}}_2:=\inf_{(\lambda_1,\lambda_2,H)\in \mathscr{M}^*(\mathbb{P}^1,\mathbb{P}^2)} \mathbb{E}_{\mathbb{P}^1}\lambda_1(S_1) + \mathbb{E}_{\mathbb{P}^2}\lambda_2(S_2), \] where \[ \mathscr{M}^*(\mathbb{P}^1,\mathbb{P}^2) := \{(\lambda_1,\lambda_2,H):\lambda_1(s_1) + \lambda_2(s_2) + H(s_1)(s_2-s_1)\geq c(s_1,s_2)\}. \] The objective function is the cost of replication with European payoffs \(\lambda_1\) and \(\lambda_2\), while \(\mathscr{M}^*(\mathbb{P}^1,\mathbb{P}^2)\) is the collection of payoff functions and the static trading strategy \(H\) that admits a super-replication. Note that due to the presence of two different time points, one can trade the stock at time \(0,t_1,t_2\), which results in the extra \(H\) variable in the optimization problem.
Through Fenchel-Rockafeller Duality, one can prove that \[ \widetilde{\text{MK}}_2=\sup_{\mathbb{P}\in\mathscr{M}(\mathbb{P}^1,\mathbb{P}^2)}\mathbb{E}_{\mathbb{P}}[c(S_1,S_2)], \] where \(\mathscr{M}(\mathbb{P}^1,\mathbb{P}^2)\) is the collection of probability measures, under which \(S^1_T\sim \mathbb{P}^1\) and \(S^2_T\sim \mathbb{P}^2\) and \(\mathbb{E}_{\mathbb{P}}(S_2|S_1) = S_1\). Clearly, the presence of the constraint \(\mathbb{E}_{\mathbb{P}}(S_2|S_1) = S_1\) in the dual problem is the direct consequence of the extra \(H\) variable in the primal problem, which is actually a martingale condition restricted to time points \(t_1,t_2\).
This problem is called martingale optimal transport due to the extra martingale constraint, which is reasonable to appear due to \(\{S_t\}\) being a martingale under the martingale measure when \(r=0\). In other words, the transport trajectory from \(\mathbb{P}^1\) to \(\mathbb{P}^2\) cannot be arbitrary due to the introduction of two different time points.
Note that unlike \(\mathscr{P}^*(\mathbb{P}^1,\mathbb{P}^2)\) which always contains the trivial coupling, extra conditions are required to ensure that \(\mathscr{M}(\mathbb{P}^1,\mathbb{P}^2)\) is not empty. An if and only if condition is given by the convex order between two measures, denoted \(\mathbb{P}^1\leq \mathbb{P}^2\), which is defined as \[ \mathbb{E}_{\mathbb{P}^1}(S_1-K)_+\leq \mathbb{E}_{\mathbb{P}^2}(S_2-K)_+,\ \forall K. \] Through the Black-Scholes formula, one can easily check that if \(\{S_t\}\) denotes the stock price in a Black-Scholes market with a single stock, then \(\mathbb{P}^1\leq \mathbb{P}^2\) for \(\forall 0\leq t_1<t_2\).