From Fair Games to Fair Prices: Martingale Theory in Modern Finance

Motivation: The idea of a "fair game" is central to both probability theory and finance. In the realm of finance, fairness translates to the absence of arbitrage opportunities. Martingale theory provides a mathematical framework to describe this principle. Investment Management and Stochastic Proccesses were two particularly interesting courses I have taken @ Penn. I thought I'd connect some dots of some of the material I have learned and show a very basic introduction to the relationship between these topics.

Martingales 101

Martingales are stochastic processes where the expected value of the next observation is equal to the current value, given all past observations. Intuitively, it represents a "fair game" where future winnings are unpredictable given the current state.

Conditions for Martingales

For a stochastic process $X_t$ to be a martingale, the following conditions must be satisfied:

1. $\mathbb{E}[|X_t|]<\mathrm{\infty }$ for all $t\ge 0$ (finite expectation).
2. $\mathbb{E}[X_{t+1}\mid {\mathcal{F}}_t]=X_t$ (martingale property).
3. $X_t$ is ${\mathcal{F}}_t$-adapted (depends only on past and present, not future).

Simple Example: $M_t=S_t^2-\frac{{\sigma }^{2}t}{4}$

We now consider the process $M_t=S_t^2-\frac{{\sigma }^{2}t}{4}$, where $S_t$ represents a discrete random walk or a simple Brownian motion approximation. Let us verify the martingale property by computing $\mathbb{E}[M_{t+1}\mid {\mathcal{F}}_t]$.

1. Step 1: Substitute $M_{t+1}$ in terms of $S_{t+1}$: $$M_{t+1}=S_{t+1}^2-\frac{{\sigma }^2(t+1)}{4}.$$ Using $S_{t+1}=S_t+X_{t+1}$, where $X_{t+1}$ is an independent random increment with $\mathbb{E}[X_{t+1}\mid {\mathcal{F}}_t]=0$: $$M_{t+1}=(S_t+X_{t+1}{)}^2-\frac{{\sigma }^2(t+1)}{4}.$$
2. Step 2: Expand $(S_t+X_{t+1}{)}^2$: $$M_{t+1}=S_t^2+2S_t{X}_{t+1}+X_{t+1}^2-\frac{{\sigma }^2(t+1)}{4}.$$ Taking the conditional expectation given ${\mathcal{F}}_t$: $$\mathbb{E}[M_{t+1}\mid {\mathcal{F}}_t]=\mathbb{E}[S_t^2+2S_t{X}_{t+1}+X_{t+1}^2\mid {\mathcal{F}}_t]-\frac{{\sigma }^2(t+1)}{4}.$$
3. Step 3: Simplify the expectation: Since $\mathbb{E}[X_{t+1}\mid {\mathcal{F}}_t]=0$: $$\mathbb{E}[M_{t+1}\mid {\mathcal{F}}_t]=S_t^2+\mathbb{E}[X_{t+1}^2\mid {\mathcal{F}}_t]-\frac{{\sigma }^2(t+1)}{4}.$$ The variance of $X_{t+1}$ is ${\sigma }^2$, so $\mathbb{E}[X_{t+1}^2\mid {\mathcal{F}}_t]={\sigma }^2$: $$\mathbb{E}[M_{t+1}\mid {\mathcal{F}}_t]=S_t^2+{\sigma }^2-\frac{{\sigma }^2(t+1)}{4}.$$
4. Step 4: Simplify further: $$\mathbb{E}[M_{t+1}\mid {\mathcal{F}}_t]=S_t^2-\frac{{\sigma }^{2}t}{4}.$$ This equals $M_t$, confirming the martingale property.

By substituting the definitions of $S_{t+1}$ and applying the properties of conditional expectation, we verified that $M_t=S_t^2-\frac{{\sigma }^{2}t}{4}$ satisfies the martingale property.

Mini Intro to Put-Call Parity: An Arbitrage Example

Setup

• Stock Price ($S$) = $10
• Strike Price ($K$) = $10
• Risk-Free Rate ($r$) = 0% (for simplicity)
• Time to Expiration ($T$) = 1 year
• No dividends on the stock

Put-Call Parity (with $r=0$) states:

$$C-P=S-K$$ Substituting values: $$C-P=10-10=0⟹C=P$$

This says the call price ($C$) should equal the put price ($P$).

The Mismatch

Suppose, in the market, the call is trading at $3 and the put is trading at $1. Clearly:

$$C-P=3-1=2\ne 0$$

The put-call parity says it should be 0. This discrepancy indicates a potential arbitrage.

Constructing the Arbitrage

We outline a classic arbitrage strategy to profit immediately and lock in a risk-free payoff at expiration:

• Buy 1 share of the stock at $10
• Buy 1 put at $1
• Sell 1 call at $3

Net Cash Flow at the Start

• Outflow for stock = $-\mathrm{\$}10$
• Outflow for put = $-\mathrm{\$}1$
• Inflow from selling call = $+\mathrm{\$}3$

Total Initial Outlay:

$$-10-1+3=-8$$

You pay $8 net today to set this up.

What Happens at Expiration?

1) If Stock Price $S_{\text{exp}}>\mathrm{\$}10$:

• The call you sold will be exercised by the buyer. You must deliver 1 share at $10.
• You already own 1 share (bought at $10), so you simply hand it over and receive $10 from the call buyer.
• The put you bought is worthless (since no one will exercise the right to sell at $10 if the stock is trading above $10).

Net Cash Flow:

$$\mathrm{\$}10-\mathrm{\$}8=\mathrm{\$}2\text{ (risk-free profit)}$$

2) If Stock Price $S_{\text{exp}}<\mathrm{\$}10$:

• The call you sold expires worthless (no one will exercise a call to buy at $10 if the stock is below $10).
• The put you bought is in the money. You can exercise it to sell your share at $10.

Net Cash Flow:

$$\mathrm{\$}10-\mathrm{\$}8=\mathrm{\$}2\text{ (risk-free profit)}$$

Why This Is Arbitrage

In both outcomes, you end up with a $+\mathrm{\$}2$ profit. You never lose money and are not exposed to where the stock finally trades. This is the classic definition of risk-free arbitrage.

Recall the Execution Payoffs of Calls and Puts

Call Option Payoff = $\text{Max}(0,S_t-K)$, where $S_t$ is the strike price at some time $t$.

Put Option Payoff = $\text{Max}(0,K-S_t)$

Expanding the Call Option Formula

Start with the given payoff formula:

$$C_0=e^{-rT}{\mathbb{E}}_Q[\text{Max}(S_T-K,0)]$$

Here:

• $C_0$: The current price of the European call option.
• $S_T$: The stock price at maturity $T$.
• $K$: The strike price of the call.
• $e^{-rT}$: The discount factor for the risk-free rate $r$ over time $T$.
• ${\mathbb{E}}_Q[\cdot ]$: Expectation under the equivalent martingale measure $Q$.

The term $\text{Max}(S_T-K,0)$ represents the call option payoff at maturity, i.e., the value $S_T-K$ if $S_T>K$, or 0 otherwise.

Expanding the Max Function

Recall that the max function can be written using the indicator function $1\{S_T>K\}$, which is 1 when $S_T>K$ and 0 otherwise:

$$\text{Max}(S_T-K,0)=(S_T-K)\cdot 1\{S_T>K\}$$

Substituting this into the formula for $C_0$:

$$C_0=e^{-rT}{\mathbb{E}}_Q[(S_T-K)\cdot 1\{S_T>K\}]$$

Separating the Expectation

Using the linearity of expectation, split the expression:

$${\mathbb{E}}_Q[(S_T-K)\cdot 1\{S_T>K\}]={\mathbb{E}}_Q[S_T\cdot 1\{S_T>K\}]-{\mathbb{E}}_Q[K\cdot 1\{S_T>K\}]$$

Substituting back, we get:

$$C_0=e^{-rT}{\mathbb{E}}_Q[S_T\cdot 1\{S_T>K\}]-e^{-rT}K\cdot Q(S_T>K)$$

Interpretation

The term ${\mathbb{E}}_Q[S_T\cdot 1\{S_T>K\}]$ represents the expected value of $S_T$ under $Q$, conditioned on $S_T>K$, multiplied by the probability of $S_T>K$:

$${\mathbb{E}}_Q[S_T\cdot 1\{S_T>K\}]={\int }_K^{\mathrm{\infty }}{S}_T{f}_Q(S_T)d{S}_T$$

Where $f_Q(S_T)$ is the probability density of $S_T$ under $Q$.

Expressing Call and Put Prices Under the Risk-Neutral Measure

From the proposition, the call price $C_0$ is given by:

$$C_0=e^{-rT}{\mathbb{E}}_Q[S_T\cdot 1\{S_T>K\}]-e^{-rT}KQ(S_T>K)$$

Similarly, the put price $P_0$ is given by:

$$P_0=e^{-rT}{\mathbb{E}}_Q[K\cdot 1\{S_T\le K\}]-e^{-rT}{\mathbb{E}}_Q[S_T\cdot 1\{S_T\le K\}]$$

Substitute Into the Put-Call Parity Formula

Substituting the expressions for $C_0$ and $P_0$ into the parity formula:

$$(e^{-rT}{\mathbb{E}}_Q[S_T\cdot 1\{S_T>K\}]-e^{-rT}KQ(S_T>K))$$ $$-(e^{-rT}{\mathbb{E}}_Q[K\cdot 1\{S_T\le K\}]-e^{-rT}{\mathbb{E}}_Q[S_T\cdot 1\{S_T\le K\}])=S_0-K{e}^{-rT}$$

Simplify Expectations

After expanding and simplifying, you observe that the terms involving $S_T$ and $K$ under the probabilities $1\{S_T>K\}$ and $1\{S_T\le K\}$ will combine into the full expectation:

$$e^{-rT}{\mathbb{E}}_Q[S_T]-e^{-rT}K=S_0-K{e}^{-rT}$$

Rearrange and Focus on the Martingale Property

The equality implies:

$$S_0=e^{-rT}{\mathbb{E}}_Q[S_T]$$

This is exactly the definition of the martingale property: The current stock price $S_0$ equals the discounted expected future stock price $e^{-rT}{\mathbb{E}}_Q[S_T]$, showing that the discounted stock price $e^{-rt}{S}_t$ is a martingale under the risk-neutral measure $Q$.

Final Remarks: Martingales, (Sub)Supermartingales, and Arbitrage

Putting everything together, the condition $$S_0=e^{-rT}{\mathbb{E}}_Q[S_T]$$ shows that the discounted stock price $e^{-rt}{S}_t$ is a true martingale under the risk-neutral measure $Q$. If this condition fails—for instance, if calls or puts are mispriced relative to put-call parity—then the discounted price process (or the portfolio you create using those instruments) can exhibit supermartingale or submartingale behavior instead.

Why does that matter? If a discounted price process is a supermartingale, it tends to drift downward in expectation (after discounting), which signals a profitable short-selling opportunity for traders. Conversely, if it is a submartingale, it tends to drift upward, making it advantageous to hold a long position. Either scenario violates the “fair game” intuition behind martingales and thus uncovers arbitrage opportunities.

Submartingales, Supermartingales, and Their Real Analysis Foundations

In our discussion of no-arbitrage and put-call parity, we relied on the condition $$S_0=e^{-rT}{\mathbb{E}}_Q[S_T],$$ which ensures that the discounted stock price process $(e^{-rt}{S}_t)$ is a martingale under the risk-neutral measure $Q$. But what happens if this condition is violated? It means $(e^{-rt}{S}_t)$ can become a submartingale or supermartingale, thereby hinting at potential arbitrage opportunities. Below, we make these notions precise.

Doob’s Decomposition: Linking (Sub)Supermartingales to Martingales

A powerful result from real analysis and probability theory—Doob’s Decomposition Theorem—states that any submartingale $Y_t$ can be written uniquely as $$Y_t=M_t+A_t,$$ where $M_t$ is a martingale, and $A_t$ is an almost surely nondecreasing, predictable process with $A_0=0$. (Intuitively, “predictable” means $A_t$ is determined by the past and present, not the future.) Similarly, for a supermartingale $Y_t$, one can write $$Y_t=M_t-B_t,$$ where $B_t$ is a nondecreasing predictable process.

This decomposition highlights that a submartingale differs from a martingale by an “increasing drift” component $A_t$, while a supermartingale has a “decreasing drift” component $B_t$. In finance terms:

• If $e^{-rt}{S}_t$ (the discounted stock price) is a submartingale, you effectively expect it to drift upwards (net of discounting). This suggests the market undervalues the asset today, creating profitable long positions.
• If $e^{-rt}{S}_t$ is a supermartingale, it is expected to drift downwards. This implies the market has overpriced the asset, making a short strategy lucrative.

Real Analysis Ingredients: Fatou’s Lemma, Dominated Convergence, etc.

The construction and proof of these (sub)supermartingale decompositions—and the fundamental theorems linking no-arbitrage to martingale pricing—rely heavily on measure-theoretic tools such as:

• Fatou’s Lemma: Ensures the lower bound on limits of expectations, used to prove convergence theorems for submartingales.
• Dominated Convergence Theorem: Allows exchanging limits and integrals (or expectations) when variables are uniformly dominated by an integrable random variable. This is crucial in option pricing arguments and in the rigorous justification of the Optional Stopping Theorem.
• Monotone Convergence Theorem: Another key lemma when dealing with nondecreasing sequences of random variables in submartingale or supermartingale contexts.
• Optional Stopping Theorem: A cornerstone in martingale theory, often invoked to demonstrate that stopping a martingale at a bounded stopping time preserves its expectation. However, in the case of unbounded stopping times (like hitting times in random walks), caution is needed!

Arbitrage Implications

Translating these results back to finance, no-arbitrage precisely requires that the discounted price process $(e^{-rt}{S}_t)$ is a true martingale under some equivalent measure $Q$. If, instead, that discounted process were a sub- or supermartingale, it would imply $${\mathbb{E}}_Q[e^{-r(t+1)}{S}_{t+1}\mid {\mathcal{F}}_t]>e^{-rt}{S}_t\text{(submartingale)}$$ or $${\mathbb{E}}_Q[e^{-r(t+1)}{S}_{t+1}\mid {\mathcal{F}}_t]<e^{-rt}{S}_t\text{(supermartingale)}.$$ Either case reveals a profit opportunity (by either going long or short accordingly), thus signaling arbitrage.

In essence, put-call parity violations are a simpler, concrete manifestation of such sub/supermartingale behavior. When put and call prices deviate from $$C-P=S_0-K{e}^{-rT},$$ one can construct a portfolio that yields a guaranteed profit in all scenarios—exactly the hallmark of arbitrage. Hence, the market quickly corrects the mispricing, restoring the martingale property of discounted asset prices.