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.

            Definition: A stochastic process \( X_t \) is a martingale if:
            \[
            \mathbb{E}[X_{t+1} \mid \mathcal{F}_t] = X_t
            \]
            where \( \mathcal{F}_t \) represents the information up to time \( t \).
        

            Additionally, \( \mathcal{F}_T \) can be represented as the collection \( X_0, X_1, \dots, X_T \), encapsulating all past states.
        

Conditions for Martingales

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

$ \mathbb{E}[|X_t|] < \infty $ for all $ t \geq 0 $ (finite expectation).
$ \mathbb{E}[X_{t+1} \mid \mathcal{F}_t] = X_t $ (martingale property).
$ 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] $.

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 $: \[ M_{t+1} = (S_t + X_{t+1})^2 - \frac{\sigma^2 (t+1)}{4}. \]
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 $: \[ \mathbb{E}[M_{t+1} \mid \mathcal{F}_t] = \mathbb{E}[S_t^2 + 2S_t X_{t+1} + X_{t+1}^2 \mid \mathcal{F}_t] - \frac{\sigma^2 (t+1)}{4}. \]
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 $: \[ \mathbb{E}[M_{t+1} \mid \mathcal{F}_t] = S_t^2 + \sigma^2 - \frac{\sigma^2 (t+1)}{4}. \]
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: \[ C - P = 10 - 10 = 0 \implies C = P \]

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:

\[ C - P = 3 - 1 = 2 \neq 0 \]

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 = $-\$10$
Outflow for put = $-\$1$
Inflow from selling call = $+\$3$

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}} > \$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:

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

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

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

Why This Is Arbitrage

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

            Key Takeaway: When the market prices of calls and puts deviate from the relationship:
            \[
            C - P = S - K e^{-rT}
            \]
            savvy traders can construct risk-free trades to lock in guaranteed profit. This forces the market to correct the prices, restoring put-call parity and eliminating 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 $:

\[ \mathbb{E}_Q[S_T \cdot 1\{S_T > K\}] = \int_K^\infty S_T f_Q(S_T) dS_T \]

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:

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

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

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

Substitute Into the Put-Call Parity Formula

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

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

Click to see work expanded

Distribute the Negative Sign in the Put Term:

Expand the negative sign over $ P_0 $:

\[ C_0 - P_0 = e^{-rT} \mathbb{E}_Q[S_T \cdot 1\{S_T > K\}] - e^{-rT} K Q(S_T > K) - e^{-rT} \mathbb{E}_Q[K \cdot 1\{S_T \leq K\}] + e^{-rT} \mathbb{E}_Q[S_T \cdot 1\{S_T \leq K\}] \]

Simplify to:

\[ C_0 - P_0 = e^{-rT} \mathbb{E}_Q[S_T \cdot 1\{S_T > K\}] + e^{-rT} \mathbb{E}_Q[S_T \cdot 1\{S_T \leq K\}] - e^{-rT} K Q(S_T > K) - e^{-rT} \mathbb{E}_Q[K \cdot 1\{S_T \leq K\}] \]

$ S_T $:

Notice that the first two terms involve $ S_T $ over disjoint regions $ S_T > K $ and $ S_T \leq K $:

\[ e^{-rT} \mathbb{E}_Q[S_T \cdot 1\{S_T > K\}] + e^{-rT} \mathbb{E}_Q[S_T \cdot 1\{S_T \leq K\}] = e^{-rT} \mathbb{E}_Q[S_T] \]

So now the expression becomes:

\[ C_0 - P_0 = e^{-rT} \mathbb{E}_Q[S_T] - e^{-rT} K Q(S_T > K) - e^{-rT} \mathbb{E}_Q[K \cdot 1\{S_T \leq K\}] \]

Combine Terms for $ K $:

Now focus on the $ K $-related terms:

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

Factor out $ -e^{-rT} K $:

\[ -e^{-rT} K (Q(S_T > K) + Q(S_T \leq K)) \]

Since $ Q(S_T > K) + Q(S_T \leq K) = 1 $, this simplifies to:

\[ -e^{-rT} K \]

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 \leq 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.

Definition (Martingale/Submartingale/Supermartingale):

Let $ (X_t)_{t \ge 0} $ be a stochastic process adapted to a filtration $ (\mathcal{F}_t)_{t \ge 0} $ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. We say $ X_t $ is:

A martingale if \[ \mathbb{E}[X_{t+1} \mid \mathcal{F}_t] = X_t \quad \text{for all } t \ge 0, \] and $ \mathbb{E}[|X_t|] < \infty $ for all $ t $.
A submartingale if \[ \mathbb{E}[X_{t+1} \mid \mathcal{F}_t] \; \ge \; X_t \quad \text{for all } t \ge 0. \]
A supermartingale if \[ \mathbb{E}[X_{t+1} \mid \mathcal{F}_t] \; \le \; X_t \quad \text{for all } t \ge 0. \]

The intuition is that a submartingale “tends to move upwards” in expectation (even after conditioning on the present), while a supermartingale “tends to move downwards.” A martingale is precisely balanced between these two.

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!

Click to see how these Real Analysis concepts are applied

1) Fatou’s Lemma in Pricing and Risk Measures

Fatou’s Lemma states that for any sequence of nonnegative measurable functions $ \{f_n\} $: \[ \mathbb{E}\left[\liminf_{n \to \infty} f_n \right] \; \le \; \liminf_{n \to \infty} \mathbb{E}[f_n]. \] In finance, we often deal with nonnegative payoff processes (e.g., call and put payoffs). When analyzing limit behaviors—such as the limiting price of an option sequence or the payoff under extreme conditions—Fatou’s Lemma provides a lower bound on expected values. This is essential when proving, for instance, that certain limit operations preserve no-arbitrage or when showing the existence of a fair pricing measure.

2) Dominated Convergence Theorem (DCT) in Option Pricing

The Dominated Convergence Theorem allows us to interchange the limit and the expectation operator, provided there is a dominating integrable function. In finance:

Option Pricing: We often approximate continuous processes (like Brownian motion) by discrete models. DCT justifies taking a limit of discrete-time pricing models without losing rigor. If you have a dominating function—for example, an exponential bound on the payoff—DCT ensures \[ \lim_{n \to \infty} \mathbb{E}[f_n] \;=\; \mathbb{E}[\lim_{n \to \infty} f_n]. \]
Risk Measures: When evaluating Value at Risk (VaR) or Expected Shortfall (ES), we deal with probabilities of large losses. DCT is crucial for exchanging limits and expectations in tail-related integrals, thus letting us handle continuous approximations of risk measures exactly.

3) Optional Stopping and Martingale Techniques

The Optional Stopping Theorem (OST) says that for a suitable stopping time $\tau$ (with integrability/boundedness conditions) and a martingale $\{M_t\}$, \[ \mathbb{E}[M_\tau] = \mathbb{E}[M_0]. \] While OST itself is often discussed in probability textbooks, applying it in a rigorous measure-theoretic setting also depends on real-analysis results like dominated or monotone convergence to ensure conditions on integrability are met. For instance:

Barrier Options: OST can be used to compute the fair price of barrier options by cleverly choosing $\tau$ to be the hitting time of a barrier. Real-analysis tools guarantee that you can pass the limit inside the expectation when the barrier gets small or large.
Martingale Representation Theorem: Deriving a hedge or replication strategy for derivatives often uses the representation of any square-integrable martingale in terms of an integral with respect to Brownian motion. Monotone/Dominated Convergence are key steps in justifying these integral representations.

4) Monotone Convergence in Sub-/Supermartingale Analysis

The Monotone Convergence Theorem helps when dealing with a nondecreasing sequence of random variables. In the analysis of submartingales, for example, we often take increasing partial sums or truncated processes $ Y_t^n = \min(Y_t, n) $. As $n \to \infty$, these truncated processes converge monotonically to $Y_t$. MCT then enables us to pass the limit inside the expectation: \[ \mathbb{E}\left[ \lim_{n \to \infty} Y_t^n \right] = \lim_{n \to \infty} \mathbb{E}[Y_t^n]. \] In finance, submartingales can correspond to processes that exhibit an “upward drift” after discounting—potentially pointing to mispricing that invites arbitrage. MCT ensures we handle these partial sums or truncated payoffs correctly when we prove or disprove no-arbitrage in limiting cases.

Putting It All Together in Finance

In short, these measure-theoretic pillars—Fatou’s Lemma, Dominated Convergence, Monotone Convergence, and Optional Stopping—each play a vital role in guaranteeing that our mathematical manipulations, limits, and expectation interchanges are justified. They ensure that:

No “illicit” steps are taken when deriving option prices or risk measures in the limit.
We have a well-defined and sound foundation for the martingale (or sub-/supermartingale) arguments that underpin no-arbitrage proofs and pricing formulas.
We can handle boundary behaviors (like extreme strike prices or infinite time horizons) without introducing spurious arbitrage or mathematically unjustified results.

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 \[ \mathbb{E}_Q[e^{-r(t+1)} S_{t+1} \mid \mathcal{F}_t] \;>\; e^{-rt} S_t \quad\text{(submartingale)} \] or \[ \mathbb{E}_Q[e^{-r(t+1)} S_{t+1} \mid \mathcal{F}_t] \;<\; e^{-rt} S_t \quad\text{(supermartingale)}. \] 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.

Conclusion: Maintaining the martingale condition (under the appropriate measure) is equivalent to ruling out arbitrage. The submartingale or supermartingale “drift” components capture systematic opportunities for profit (or for hedging) that real-world markets cannot sustain for long. Finance can have some pretty cool mathemtical derivations inspired from Statisitcs and Physics. Concepts like Brownian motion and stochastic processes borrow heavily from physics, particularly in modeling random phenomena such as particle diffusion.

In practice, as soon as an arbitrage strategy is identified (for example, by finding a mismatch in put-call parity), traders will exploit it, driving prices back to levels consistent with the martingale (no-arbitrage) condition. Hence, the absence of arbitrage in well-functioning markets is precisely what enforces the equality \[ S_0 = e^{-rT}\mathbb{E}_Q[S_T] \] and preserves the martingale property of discounted asset prices under the risk-neutral measure.