Complete Black-Scholes Proofs: From First Principles

1 Part I: Mathematical Foundations and Definitions

This section establishes all the mathematical machinery needed for the Black-Scholes proofs. We assume knowledge of measure theory and basic stochastic calculus but will define all financial and probabilistic concepts explicitly.

1.0.1 Financial Market Framework

Definition 1.1 (Financial Market)

A financial market consists of:

1. A probability space \((\Omega, \mathcal{F}, \mathbb{P})\) where \(\Omega\) represents all possible market scenarios
2. A filtration \(\{\mathcal{F}_t\}_{t \geq 0}\) representing information available at time \(t\)
3. A finite collection of traded assets with price processes \(\{S^i_t\}_{i=0,1,\ldots,n}\)

Definition 1.2 (Brownian Motion)

A stochastic process \(\{W_t\}_{t \geq 0}\) on \((\Omega, \mathcal{F}, \mathbb{P})\) is a standard Brownian motion if:

1. \(W_0 = 0\) almost surely
2. \(W\) has independent increments: for \(0 \leq s < t\), \(W_t - W_s\) is independent of \(\mathcal{F}_s\)
3. \(W_t - W_s \sim N(0, t-s)\) for all \(0 \leq s < t\)
4. \(W\) has continuous sample paths almost surely

Definition 1.3 (Natural Filtration of Brownian Motion)

The natural filtration of Brownian motion is \[\mathcal{F}_t^W = \sigma(W_s : 0 \leq s \leq t)\] the \(\sigma\)-algebra generated by the Brownian motion up to time \(t\).

Definition 1.4 (Adapted Process)

A stochastic process \(\{X_t\}_{t \geq 0}\) is adapted to the filtration \(\{\mathcal{F}_t\}\) if \(X_t\) is \(\mathcal{F}_t\)-measurable for all \(t \geq 0\).

Definition 1.5 (Geometric Brownian Motion)

A process \(\{S_t\}_{t \geq 0}\) follows geometric Brownian motion with parameters \(\mu \in \mathbb{R}\) and \(\sigma > 0\) if it satisfies the stochastic differential equation: \[dS_t = \mu S_t \, dt + \sigma S_t \, dW_t\] with initial condition \(S_0 > 0\).

Remark 1.1

The explicit solution to geometric Brownian motion is: \[S_t = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)\] This can be verified using Itô’s lemma on \(f(t,x) = \ln x\).

1.0.2 Portfolio and Trading Strategy Concepts

Definition 1.6 (Trading Strategy)

A trading strategy is a pair of adapted processes \((\phi^0_t, \phi^1_t)\) where:

• \(\phi^0_t\) represents the number of units of the bond held at time \(t\)
• \(\phi^1_t\) represents the number of shares of stock held at time \(t\)

Both processes must be adapted to the filtration \(\{\mathcal{F}_t\}\).

Definition 1.7 (Portfolio Value)

The value of a portfolio with trading strategy \((\phi^0_t, \phi^1_t)\) at time \(t\) is: \[V_t = \phi^0_t B_t + \phi^1_t S_t\] where \(B_t\) is the bond price and \(S_t\) is the stock price.

Definition 1.8 (Self-Financing Strategy)

A trading strategy \((\phi^0_t, \phi^1_t)\) is self-financing if: \[dV_t = \phi^0_t dB_t + \phi^1_t dS_t\] This means no money is added or withdrawn from the portfolio; changes in value come only from price movements of held assets.

Definition 1.9 (Arbitrage Opportunity)

An arbitrage opportunity is a self-financing trading strategy with:

1. Initial value \(V_0 = 0\)
2. \(\mathbb{P}(V_T \geq 0) = 1\) for some time \(T > 0\)
3. \(\mathbb{P}(V_T > 0) > 0\)

Definition 1.10 (Arbitrage-Free Market)

A market is arbitrage-free if no arbitrage opportunities exist.

1.0.3 Martingale Theory for Finance

Definition 1.11 (Martingale)

An adapted process \(\{M_t\}_{t \geq 0}\) is a martingale with respect to filtration \(\{\mathcal{F}_t\}\) and probability measure \(\mathbb{P}\) if:

1. \(\mathbb{E}[|M_t|] < \infty\) for all \(t \geq 0\)
2. \(\mathbb{E}[M_t | \mathcal{F}_s] = M_s\) for all \(0 \leq s \leq t\)

Definition 1.12 (Equivalent Probability Measures)

Two probability measures \(\mathbb{P}\) and \(\mathbb{Q}\) on \((\Omega, \mathcal{F})\) are equivalent (written \(\mathbb{P} \sim \mathbb{Q}\)) if they have the same null sets: \[\mathbb{P}(A) = 0 \iff \mathbb{Q}(A) = 0 \text{ for all } A \in \mathcal{F}\]

Definition 1.13 (Radon-Nikodym Derivative)

If \(\mathbb{Q} \ll \mathbb{P}\) (Q is absolutely continuous with respect to P), then there exists a non-negative \(\mathcal{F}\)-measurable random variable \(Z\) such that: \[\mathbb{Q}(A) = \int_A Z \, d\mathbb{P} \text{ for all } A \in \mathcal{F}\] We write \(Z = \frac{d\mathbb{Q}}{d\mathbb{P}}\) and call \(Z\) the Radon-Nikodym derivative.

Theorem 1.1 (Girsanov’s Theorem - Statement)

Let \(\theta\) be an adapted process with \(\int_0^T \theta_s^2 \, ds < \infty\) almost surely. Define: \[Z_t = \exp\left(-\int_0^t \theta_s \, dW_s - \frac{1}{2}\int_0^t \theta_s^2 \, ds\right)\]

If \(\mathbb{E}[Z_T] = 1\), then:

1. The process \(Z_t\) is a martingale
2. The measure \(\mathbb{Q}\) defined by \(\frac{d\mathbb{Q}}{d\mathbb{P}} = Z_T\) is a probability measure equivalent to \(\mathbb{P}\)
3. Under \(\mathbb{Q}\), the process \(\tilde{W}_t = W_t + \int_0^t \theta_s \, ds\) is a Brownian motion

Definition 1.14 (Risk-Neutral Measure)

In a financial market with bond \(B_t = e^{rt}\) and stock following \[dS_t = \mu S_t \, dt + \sigma S_t \, dW_t\] a probability measure \(\mathbb{Q}\) equivalent to \(\mathbb{P}\) is called risk-neutral if the discounted stock price \(e^{-rt}S_t\) is a \(\mathbb{Q}\)-martingale.

1.0.4 Options and Derivatives

Definition 1.15 (European Option)

A European option is a financial contract that gives the holder the right (but not obligation) to:

• Call option: Buy an asset at strike price \(K\) at maturity time \(T\)
• Put option: Sell an asset at strike price \(K\) at maturity time \(T\)

Definition 1.16 (Option Payoff)

The payoff of a European option at maturity \(T\) is:

• Call: \(h(S_T) = (S_T - K)^+ = \max(S_T - K, 0)\)
• Put: \(h(S_T) = (K - S_T)^+ = \max(K - S_T, 0)\)

where \(S_T\) is the stock price at maturity.

Definition 1.17 (Option Price)

The price of an option at time \(t\) is denoted \(V(t, S_t)\) and depends on the current time \(t\) and current stock price \(S_t\).

Definition 1.18 (Replicating Portfolio)

A replicating portfolio for an option is a self-financing trading strategy \((\phi^0_t, \phi^1_t)\) such that the portfolio value at maturity equals the option payoff: \[V_T = \phi^0_T B_T + \phi^1_T S_T = h(S_T)\]

1.0.5 Fundamental Theorems of Asset Pricing

Theorem 1.2 (First Fundamental Theorem of Asset Pricing)

A market is arbitrage-free if and only if there exists a probability measure \(\mathbb{Q}\) equivalent to \(\mathbb{P}\) such that all discounted asset prices are \(\mathbb{Q}\)-martingales.

Theorem 1.3 (Second Fundamental Theorem of Asset Pricing)

An arbitrage-free market is complete if and only if the risk-neutral measure is unique.

Definition 1.19 (Complete Market)

A market is complete if every contingent claim (option payoff) can be replicated by a self-financing trading strategy.

Theorem 1.4 (Risk-Neutral Valuation Principle)

In an arbitrage-free complete market, the price of any attainable contingent claim \(h(S_T)\) at time \(t\) is: \[V(t, S_t) = e^{-r(T-t)} \mathbb{E}_\mathbb{Q}[h(S_T) | \mathcal{F}_t]\] where \(\mathbb{Q}\) is the unique risk-neutral measure.

1.1 Part II: The Black-Scholes Model Setup

1.1.1 Model Specification

We consider a financial market on a probability space \((\Omega, \mathcal{F}, \mathbb{P})\) with filtration \(\{\mathcal{F}_t\}_{t \geq 0}\) and two traded assets:

Definition 2.1 (Black-Scholes Market Model)

The Black-Scholes market consists of:

1. Risk-free bond: \(dB_t = rB_t \, dt\) with \(B_0 = 1\), giving \(B_t = e^{rt}\)
2. Risky stock: \(dS_t = \mu S_t \, dt + \sigma S_t \, dW_t\) with \(S_0 > 0\)

where:

• \(r > 0\) is the constant risk-free interest rate
• \(\mu \in \mathbb{R}\) is the stock’s expected return (drift)
• \(\sigma > 0\) is the stock’s volatility
• \(\{W_t\}_{t \geq 0}\) is a standard Brownian motion adapted to \(\{\mathcal{F}_t\}\)

Definition 2.2 (European Call Option in Black-Scholes Model)

We consider a European call option with:

• Strike price \(K > 0\)
• Maturity time \(T > 0\)
  
• Payoff at maturity: \(h(S_T) = (S_T - K)^+\)

Our goal is to find the option price \(V(t, S_t)\) for \(0 \leq t \leq T\).

1.1.2 Key Properties of the Model

Lemma 2.1 (Stock Price Solution)

The solution to the stock price SDE \(dS_t = \mu S_t \, dt + \sigma S_t \, dW_t\) is: \[S_t = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)\]

Proof of Lemma 2.1

Let \(Y_t = \ln S_t\). By Itô’s lemma with \(f(x) = \ln x\):

\[\begin{align} dY_t &= f'(S_t) dS_t + \frac{1}{2}f''(S_t)(dS_t)^2 \\ &= \frac{1}{S_t} dS_t + \frac{1}{2}\left(-\frac{1}{S_t^2}\right)(dS_t)^2 \\ &= \frac{1}{S_t}(\mu S_t \, dt + \sigma S_t \, dW_t) - \frac{1}{2S_t^2}(\sigma S_t)^2 dt \\ &= \mu \, dt + \sigma \, dW_t - \frac{\sigma^2}{2} dt \\ &= \left(\mu - \frac{\sigma^2}{2}\right) dt + \sigma \, dW_t \end{align}\]

Integrating from \(0\) to \(t\): \[Y_t = Y_0 + \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\]

Since \(Y_t = \ln S_t\) and \(Y_0 = \ln S_0\): \[\ln S_t = \ln S_0 + \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\]

Therefore: \[S_t = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)\]

Corollary 2.1 (Log-Normal Distribution of Stock Price)

Under \(\mathbb{P}\), we have: \[\ln(S_t/S_0) \sim N\left(\left(\mu - \frac{\sigma^2}{2}\right)t, \sigma^2 t\right)\]

1.2 Part III: Proof via Martingale Approach

1.2.1 Step 1: Construction of Risk-Neutral Measure

Theorem 3.1 (Existence of Risk-Neutral Measure in Black-Scholes)

There exists a unique probability measure \(\mathbb{Q}\) equivalent to \(\mathbb{P}\) such that the discounted stock price \(e^{-rt}S_t\) is a \(\mathbb{Q}\)-martingale.

Proof of Theorem 3.1

Step 1: Define the market price of risk: \[\theta = \frac{\mu - r}{\sigma}\]

This is the constant that will appear in Girsanov’s theorem.

Step 2: Define the Radon-Nikodym density process: \[\begin{align} Z_t &= \exp\left(-\theta W_t - \frac{1}{2}\theta^2 t\right) \\ &= \exp\left(-\frac{\mu - r}{\sigma} W_t - \frac{1}{2}\left(\frac{\mu - r}{\sigma}\right)^2 t\right) \end{align}\]

Step 3: Verify that \(Z_t\) is a martingale and \(\mathbb{E}[Z_T] = 1\).

Since \(\theta\) is constant, we can compute: \[\mathbb{E}[Z_t] = \mathbb{E}\left[\exp\left(-\theta W_t - \frac{1}{2}\theta^2 t\right)\right]\]

Since \(W_t \sim N(0,t)\) under \(\mathbb{P}\): \[\begin{align} \mathbb{E}[Z_t] &= \int_{-\infty}^{\infty} \exp\left(-\theta w - \frac{1}{2}\theta^2 t\right) \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{w^2}{2t}\right) dw \\ &= \exp\left(-\frac{1}{2}\theta^2 t\right) \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{w^2 + 2\theta tw}{2t}\right) dw \\ &= \exp\left(-\frac{1}{2}\theta^2 t\right) \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{(w + \theta t)^2 - \theta^2 t^2}{2t}\right) dw \\ &= \exp\left(-\frac{1}{2}\theta^2 t\right) \exp\left(\frac{1}{2}\theta^2 t\right) \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{(w + \theta t)^2}{2t}\right) dw \\ &= 1 \end{align}\]

The last integral equals 1 since it’s the integral of a normal density.

Step 4: Define \(\mathbb{Q}\) by \(\frac{d\mathbb{Q}}{d\mathbb{P}} = Z_T\).

Step 5: Apply Girsanov’s theorem. Under \(\mathbb{Q}\), the process: \[\tilde{W}_t = W_t + \int_0^t \theta \, ds = W_t + \theta t\] is a \(\mathbb{Q}\)-Brownian motion.

Step 6: Transform the stock price dynamics. Under \(\mathbb{Q}\): \[\begin{align} dS_t &= \mu S_t \, dt + \sigma S_t \, dW_t \\ &= \mu S_t \, dt + \sigma S_t (d\tilde{W}_t - \theta \, dt) \\ &= \mu S_t \, dt + \sigma S_t \, d\tilde{W}_t - \sigma S_t \theta \, dt \\ &= (\mu - \sigma\theta) S_t \, dt + \sigma S_t \, d\tilde{W}_t \\ &= \left(\mu - \sigma \cdot \frac{\mu - r}{\sigma}\right) S_t \, dt + \sigma S_t \, d\tilde{W}_t \\ &= r S_t \, dt + \sigma S_t \, d\tilde{W}_t \end{align}\]

Step 7: Verify the martingale property. The discounted stock price is: \[e^{-rt}S_t\]

Its differential is: \[\begin{align} d(e^{-rt}S_t) &= -re^{-rt}S_t \, dt + e^{-rt} dS_t \\ &= -re^{-rt}S_t \, dt + e^{-rt}(rS_t \, dt + \sigma S_t \, d\tilde{W}_t) \\ &= e^{-rt}\sigma S_t \, d\tilde{W}_t \end{align}\]

Since this has no \(dt\) term, \(e^{-rt}S_t\) is a \(\mathbb{Q}\)-martingale.

Why Do We Need the Risk-Neutral Measure \(\mathbb{Q}\) at All?

This fundamental question gets to the heart of mathematical finance. The answer reveals why the risk-neutral measure is not just a mathematical convenience, but an essential concept for correct option pricing.

1.2.1.0.1 The Fundamental Problem with the Physical Measure \(\mathbb{P}\)

Under the physical measure \(\mathbb{P}\), the stock follows: \[dS_t = \mu S_t dt + \sigma S_t dW_t\]

If we naively tried to price the option directly under \(\mathbb{P}\): \[V(0, S_0) = e^{-rT} \mathbb{E}_\mathbb{P}[(S_T - K)^+]\]

This formula is fundamentally incorrect! Here’s why:

1.2.1.0.1.1 Issue 1: The Discount Rate Problem

The expectation under \(\mathbb{P}\) gives us the expected payoff in the “real world” where the stock has expected return \(\mu\). But what discount rate should we use?

• If \(\mu > r\): The stock is expected to grow faster than the risk-free rate
• We might think we should discount at rate \(\mu\), not \(r\)
• But the option’s risk profile is completely different from the stock’s risk profile
• The appropriate discount rate for the option depends on its specific risk characteristics

The key insight: We don’t know what discount rate to use for the option under \(\mathbb{P}\) because the option’s risk premium is unclear and depends on complex risk preferences.

1.2.1.0.1.2 Issue 2: Risk Premium Confusion

Under the physical measure \(\mathbb{P}\): - The stock earns expected return \(\mu\) - The bond earns certain return \(r\) - The difference \(\mu - r\) is the equity risk premium

But the option has a completely different risk profile than the stock! Its appropriate risk premium is not obvious and would require knowledge of: - Investor risk preferences - The option’s correlation with market factors - The option’s beta relative to various risk factors

1.2.1.0.2 Why the Risk-Neutral Measure \(\mathbb{Q}\) Solves This Elegantly

The brilliant insight is to change to a probability measure where all assets have the same expected return equal to the risk-free rate.

1.2.1.0.2.1 Under \(\mathbb{Q}\):

• Stock expected return: \(r\)
• Bond expected return: \(r\)
• Any derivative’s expected return: \(r\)

This means we can discount everything at the risk-free rate \(r\)!

\[V(0, S_0) = e^{-rT} \mathbb{E}_\mathbb{Q}[(S_T - K)^+]\]

Now the formula is correct because: 1. The expected return of the option under \(\mathbb{Q}\) is exactly \(r\) 2. So we discount at rate \(r\) 3. The mathematics works out perfectly 4. No risk premiums need to be determined

1.2.1.0.3 The Economic Intuition

Think of the two measures this way:

Physical measure \(\mathbb{P}\): “What will actually happen in the real world?” - Stocks have risk premiums reflecting investor risk aversion - Different assets have different expected returns - Discount rates are asset-specific and difficult to determine - Requires knowledge of risk preferences and market prices of risk

Risk-neutral measure \(\mathbb{Q}\): “What would happen in a hypothetical world where all investors are risk-neutral?” - All assets earn the risk-free rate in expectation - No risk premiums exist - Everything can be discounted at the same rate \(r\) - Completely bypasses the need to know risk preferences

1.2.1.0.4 The Arbitrage Connection

The risk-neutral measure exists precisely because there are no arbitrage opportunities. Here’s the fundamental logic:

1. No arbitrage ⟺ Risk-neutral measure exists (First Fundamental Theorem of Asset Pricing)
2. If we can replicate the option with a portfolio of stock and bonds, then the option price must equal the portfolio value (no arbitrage principle)
3. The replicating portfolio approach automatically gives us the risk-neutral valuation
4. The \(\mathbb{Q}\) measure is the unique measure that makes this work

1.2.1.0.5 A Concrete Example

Consider a simple one-period binomial model:

Setup: - Stock price: \(S_0 = 100\) - Up move: \(S_1 = 120\) with probability \(p = 0.6\) under \(\mathbb{P}\) - Down move: \(S_1 = 80\) with probability \(1-p = 0.4\) under \(\mathbb{P}\) - Risk-free rate: \(r = 5\%\)

Under \(\mathbb{P}\): Expected stock return is \(0.6 \times 20\% + 0.4 \times (-20\%) = 4\%\)

For a call option with \(K = 100\): - Payoff if stock goes up: \(\max(120-100, 0) = 20\) - Payoff if stock goes down: \(\max(80-100, 0) = 0\)

1.2.1.0.5.1 Wrong Approach (using \(\mathbb{P}\)):

\[V_0 = \frac{0.6 \times 20 + 0.4 \times 0}{1.05} = \frac{12}{1.05} = 11.43\]

This is incorrect because we’re using the wrong probabilities for discounting at the risk-free rate.

1.2.1.0.5.2 Correct Approach (using \(\mathbb{Q}\)):

First, find the risk-neutral probabilities. Under \(\mathbb{Q}\), the stock must have expected return equal to the risk-free rate \(r = 5\%\):

\[q \times 120 + (1-q) \times 80 = 100 \times 1.05 = 105\] \[40q + 80 = 105\] \[q = 0.625\]

Now we can price correctly: \[V_0 = \frac{0.625 \times 20 + 0.375 \times 0}{1.05} = \frac{12.5}{1.05} = 11.90\]

The risk-neutral approach gives the unique arbitrage-free price!

1.2.1.0.5.3 Verification by Replication:

We can verify this is correct by constructing a replicating portfolio: - Buy \(\Delta\) shares of stock - Invest \(B\) in bonds

Portfolio value in up state: \(120\Delta + 1.05B = 20\) Portfolio value in down state: \(80\Delta + 1.05B = 0\)

Solving: \(\Delta = 0.5\), \(B = -38.10\)

Initial portfolio cost: \(100 \times 0.5 - 38.10 = 11.90\) ✓

1.2.1.0.6 Why This Matters for Black-Scholes

In the Black-Scholes framework:

1. Physical measure: Stock has drift \(\mu\), but option pricing would require determining the option’s risk premium
2. Risk-neutral measure: Stock has drift \(r\), and option pricing becomes a pure expectation calculation
3. Girsanov’s theorem: Provides the mathematical machinery to change from \(\mathbb{P}\) to \(\mathbb{Q}\)
4. Hedging connection: The \(\mathbb{Q}\) measure emerges naturally from the delta-hedging argument

1.2.1.0.7 Summary: Why We Need \(\mathbb{Q}\)

The risk-neutral measure is essential because it:

1. Eliminates risk premium confusion - all assets earn rate \(r\) in expectation
2. Provides the correct discount rate - always the risk-free rate \(r\)
3. Gives the unique arbitrage-free approach - guaranteed by fundamental theorems
4. Connects to replication strategies - matches the hedging-based derivation perfectly
5. Makes pricing computationally tractable - turns complex pricing into expectation calculations
6. Bypasses investor preferences - no need to know risk aversion parameters
7. Ensures market completeness - works for any derivative in a complete market

The key insight: The risk-neutral measure is not about what will actually happen in reality - it’s about finding the unique arbitrage-free price in a mathematically elegant and practically implementable way. It transforms the complex problem of determining risk-adjusted discount rates into the simpler problem of computing expectations under an artificial but mathematically convenient probability measure.

1.2.2 Step 2: Stock Price Under Risk-Neutral Measure

Lemma 3.1 (Stock Price Solution Under \(\mathbb{Q}\))

Under the risk-neutral measure \(\mathbb{Q}\), the stock price has the explicit form: \[S_t = S_0 \exp\left(\left(r - \frac{\sigma^2}{2}\right)t + \sigma \tilde{W}_t\right)\] where \(\tilde{W}_t\) is a \(\mathbb{Q}\)-Brownian motion.

Proof of Lemma 3.1

Under \(\mathbb{Q}\), the stock follows: \[dS_t = rS_t \, dt + \sigma S_t \, d\tilde{W}_t\]

Using the same technique as in the original measure, let \(Y_t = \ln S_t\): \[\begin{align} dY_t &= \frac{1}{S_t} dS_t - \frac{1}{2S_t^2}(dS_t)^2 \\ &= \frac{1}{S_t}(rS_t \, dt + \sigma S_t \, d\tilde{W}_t) - \frac{1}{2S_t^2}(\sigma S_t)^2 dt \\ &= r \, dt + \sigma \, d\tilde{W}_t - \frac{\sigma^2}{2} dt \\ &= \left(r - \frac{\sigma^2}{2}\right) dt + \sigma \, d\tilde{W}_t \end{align}\]

Integrating: \[Y_t = Y_0 + \left(r - \frac{\sigma^2}{2}\right)t + \sigma \tilde{W}_t\]

Therefore: \[S_t = S_0 \exp\left(\left(r - \frac{\sigma^2}{2}\right)t + \sigma \tilde{W}_t\right)\]

1.2.3 Step 3: Risk-Neutral Valuation

Theorem 3.2 (Risk-Neutral Option Pricing)

The price of the European call option at time \(t\) is: \[V(t, S_t) = e^{-r(T-t)} \mathbb{E}_\mathbb{Q}[(S_T - K)^+ | \mathcal{F}_t]\]

Proof of Theorem 3.2

This follows directly from the risk-neutral valuation principle. In an arbitrage-free complete market, the option price equals the discounted expected payoff under the risk-neutral measure.

1.2.4 Step 4: Computing the Expectation

Theorem 3.3 (Black-Scholes Formula via Martingale Method)

The European call option price is: \[V(t, S_t) = S_t \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2)\] where: \[\begin{align} d_1 &= \frac{\ln(S_t/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}} \\ d_2 &= d_1 - \sigma\sqrt{T-t} = \frac{\ln(S_t/K) + (r - \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}} \end{align}\] and \(\Phi\) is the cumulative distribution function of the standard normal distribution.

Proof of Theorem 3.3

Step 1: Express the stock price at maturity. Under \(\mathbb{Q}\), using the strong Markov property: \[S_T = S_t \exp\left(\left(r - \frac{\sigma^2}{2}\right)(T-t) + \sigma (\tilde{W}_T - \tilde{W}_t)\right)\]

Since \(\tilde{W}_T - \tilde{W}_t \sim N(0, T-t)\) under \(\mathbb{Q}\), let \(Z \sim N(0,1)\). Then: \[S_T = S_t \exp\left(\left(r - \frac{\sigma^2}{2}\right)(T-t) + \sigma\sqrt{T-t} \cdot Z\right)\]

Step 2: Set up the expectation. \[\begin{align} V(t, S_t) &= e^{-r(T-t)} \mathbb{E}_\mathbb{Q}[(S_T - K)^+ | \mathcal{F}_t] \\ &= e^{-r(T-t)} \mathbb{E}_\mathbb{Q}\left[\left(S_t e^{(r-\sigma^2/2)(T-t) + \sigma\sqrt{T-t} Z} - K\right)^+\right] \end{align}\]

Step 3: Find the exercise region. The option is exercised when \(S_T > K\), i.e., when: \[S_t e^{(r-\sigma^2/2)(T-t) + \sigma\sqrt{T-t} Z} > K\]

Taking logarithms: \[\begin{align} &(r-\sigma^2/2)(T-t) + \sigma\sqrt{T-t} Z > \ln(K/S_t) \\ &Z > \frac{\ln(K/S_t) - (r-\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}} = -d_2 \end{align}\]

Step 4: Evaluate the integral. \[\begin{align} V(t, S_t) &= e^{-r(T-t)} \int_{-d_2}^{\infty} \left(S_t e^{(r-\sigma^2/2)(T-t) + \sigma\sqrt{T-t} z} - K\right) \\ &\quad \times \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz \\ &= e^{-r(T-t)} S_t e^{(r-\sigma^2/2)(T-t)} \int_{-d_2}^{\infty} e^{\sigma\sqrt{T-t} z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz \\ &\quad - e^{-r(T-t)} K \int_{-d_2}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz \end{align}\]

Step 5: Evaluate the first integral. For the first integral, complete the square in the exponent: \[\sigma\sqrt{T-t} z - \frac{z^2}{2} = -\frac{1}{2}(z - \sigma\sqrt{T-t})^2 + \frac{\sigma^2(T-t)}{2}\]

Therefore: \[\begin{align} &\int_{-d_2}^{\infty} e^{\sigma\sqrt{T-t} z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz \\ &= e^{\sigma^2(T-t)/2} \int_{-d_2}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-(z-\sigma\sqrt{T-t})^2/2} dz \\ &= e^{\sigma^2(T-t)/2} \int_{-d_2-\sigma\sqrt{T-t}}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-u^2/2} du \\ &= e^{\sigma^2(T-t)/2} \Phi(d_2 + \sigma\sqrt{T-t}) \\ &= e^{\sigma^2(T-t)/2} \Phi(d_1) \end{align}\]

where we used the substitution \(u = z - \sigma\sqrt{T-t}\) and the fact that: \[d_2 + \sigma\sqrt{T-t} = d_1\]

Step 6: Evaluate the second integral. \[\int_{-d_2}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz = \Phi(d_2)\]

Step 7: Combine the results. \[\begin{align} V(t, S_t) &= e^{-r(T-t)} S_t e^{(r-\sigma^2/2)(T-t)} e^{\sigma^2(T-t)/2} \Phi(d_1) - e^{-r(T-t)} K \Phi(d_2) \\ &= S_t e^{-r(T-t)} e^{(r-\sigma^2/2)(T-t)} e^{\sigma^2(T-t)/2} \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2) \\ &= S_t e^{-r(T-t) + r(T-t) - \sigma^2(T-t)/2 + \sigma^2(T-t)/2} \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2) \\ &= S_t \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2) \end{align}\]

1.3 Part IV: Proof via PDE Approach

1.3.1 Step 1: Delta Hedging Strategy

Definition 4.1 (Delta Hedging Portfolio)

Consider a portfolio \(\Pi_t\) consisting of:

• Long position: 1 unit of the option with value \(V(t, S_t)\)
• Short position: \(\Delta_t\) shares of the stock
• Cash position: The remaining value invested in bonds

The portfolio value is: \(\Pi_t = V(t, S_t) - \Delta_t S_t\)

Definition 4.2 (Delta)

The delta \(\Delta_t\) represents the hedge ratio - the number of shares of stock to short for each option held. We will determine \(\Delta_t\) to make the portfolio risk-free.

1.3.2 Step 2: Application of Itô’s Lemma to Option Value

Lemma 4.1 (Stochastic Differential of Option Value)

Assume the option price has the form \(V(t, S_t)\) where \(V \in C^{1,2}([0,T] \times (0,\infty))\). Then by Itô’s lemma: \[\begin{align} dV &= \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS_t + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} (dS_t)^2 \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt + \sigma S_t \frac{\partial V}{\partial S} dW_t \end{align}\]

Proof of Lemma 4.1

Since \(dS_t = \mu S_t \, dt + \sigma S_t \, dW_t\), we have: \((dS_t)^2 = (\mu S_t \, dt + \sigma S_t \, dW_t)^2 = \sigma^2 S_t^2 (dW_t)^2 = \sigma^2 S_t^2 dt\)

where we used the fact that \((dW_t)^2 = dt\), \(dt \cdot dW_t = 0\), and \((dt)^2 = 0\).

Applying Itô’s lemma: \[\begin{align} dV &= \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS_t + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} (dS_t)^2 \\ &= \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} (\mu S_t \, dt + \sigma S_t \, dW_t) + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} \sigma^2 S_t^2 dt \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt + \sigma S_t \frac{\partial V}{\partial S} dW_t \end{align}\]

1.3.3 Step 3: Portfolio Dynamics

Lemma 4.2 (Portfolio Value Dynamics)

The change in portfolio value is: \[\begin{align} d\Pi_t &= dV - \Delta_t dS_t \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \left(\frac{\partial V}{\partial S} - \Delta_t\right) + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt \\ &\quad + \sigma S_t \left(\frac{\partial V}{\partial S} - \Delta_t\right) dW_t \end{align}\]

Proof of Lemma 4.2

We have: \[\begin{align} d\Pi_t &= dV - \Delta_t dS_t \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt + \sigma S_t \frac{\partial V}{\partial S} dW_t \\ &\quad - \Delta_t (\mu S_t dt + \sigma S_t dW_t) \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} - \Delta_t \mu S_t\right] dt \\ &\quad + \sigma S_t \left(\frac{\partial V}{\partial S} - \Delta_t\right) dW_t \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \left(\frac{\partial V}{\partial S} - \Delta_t\right) + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt \\ &\quad + \sigma S_t \left(\frac{\partial V}{\partial S} - \Delta_t\right) dW_t \end{align}\]

1.3.4 Step 4: Eliminating Risk

Definition 4.3 (Delta-Neutral Portfolio)

To eliminate the stochastic component, we choose: \(\Delta_t = \frac{\partial V}{\partial S}\)

This choice makes the portfolio instantaneously risk-free.

Lemma 4.3 (Risk-Free Portfolio Dynamics)

With \(\Delta_t = \frac{\partial V}{\partial S}\), the portfolio dynamics become: \(d\Pi_t = \left[\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt\)

1.3.5 Step 5: No-Arbitrage Condition

Theorem 4.1 (Risk-Free Rate Condition)

Since the portfolio is risk-free, it must earn the risk-free rate to prevent arbitrage: \(d\Pi_t = r\Pi_t dt\)

Substituting \(\Pi_t = V - \Delta_t S_t = V - \frac{\partial V}{\partial S} S_t\): \(d\Pi_t = r\left(V - \frac{\partial V}{\partial S} S_t\right) dt\)

Proof of Theorem 4.1

If a risk-free portfolio earned more than the risk-free rate, we could borrow at rate \(r\), invest in the portfolio, and earn a risk-free profit (arbitrage). If it earned less, we could short the portfolio, lend at rate \(r\), and again earn risk-free profit.

1.3.6 Step 6: Deriving the Black-Scholes PDE

Theorem 4.2 (Black-Scholes Partial Differential Equation)

The option price \(V(t,S)\) satisfies the Black-Scholes PDE: \(\frac{\partial V}{\partial t} + rS \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0\)

with terminal condition: \(V(T,S) = (S-K)^+\)

Proof of Theorem 4.2

Equating the two expressions for \(d\Pi_t\): \(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} = r\left(V - \frac{\partial V}{\partial S} S_t\right)\)

Rearranging: \(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} = rV - rS_t \frac{\partial V}{\partial S}\)

Therefore: \(\frac{\partial V}{\partial t} + rS_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} - rV = 0\)

The terminal condition comes from the option payoff at maturity.

1.3.7 Step 7: Solving the PDE

To solve this PDE, we use a change of variables to transform it into the heat equation.

Definition 4.4 (Variable Transformation)

Let: \[\begin{align} \tau &= T - t \quad \text{(time to maturity)} \\ x &= \ln(S/K) \quad \text{(log-moneyness)} \\ u(\tau, x) &= e^{r\tau} \frac{V(T-\tau, Ke^x)}{K} \end{align}\]

Lemma 4.4 (Transformed PDE)

The function \(u(\tau, x)\) satisfies the diffusion equation: \(\frac{\partial u}{\partial \tau} = \frac{1}{2}\sigma^2 \frac{\partial^2 u}{\partial x^2} + \left(r - \frac{\sigma^2}{2}\right) \frac{\partial u}{\partial x}\)

with initial condition: \(u(0, x) = (e^x - 1)^+\)

Proof of Lemma 4.4

We need to compute the partial derivatives of \(V\) in terms of \(u\). We have: \(V(t,S) = K e^{-r(T-t)} u(T-t, \ln(S/K))\)

Let \(\tau = T-t\) and \(x = \ln(S/K)\). Then:

\[\begin{align} \frac{\partial V}{\partial t} &= K e^{-r\tau} \left[r u - \frac{\partial u}{\partial \tau}\right] \\ \frac{\partial V}{\partial S} &= K e^{-r\tau} \frac{1}{S} \frac{\partial u}{\partial x} \\ \frac{\partial^2 V}{\partial S^2} &= K e^{-r\tau} \frac{1}{S^2} \left[\frac{\partial^2 u}{\partial x^2} - \frac{\partial u}{\partial x}\right] \end{align}\]

Substituting into the Black-Scholes PDE and simplifying (after dividing by \(K e^{-r\tau}\)): \(-\frac{\partial u}{\partial \tau} + \left(r - \frac{\sigma^2}{2}\right) \frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2 \frac{\partial^2 u}{\partial x^2} = 0\)

Therefore: \(\frac{\partial u}{\partial \tau} = \frac{1}{2}\sigma^2 \frac{\partial^2 u}{\partial x^2} + \left(r - \frac{\sigma^2}{2}\right) \frac{\partial u}{\partial x}\)

The initial condition at \(\tau = 0\) (i.e., \(t = T\)) is: \(u(0, x) = \frac{(Ke^x - K)^+}{K} = (e^x - 1)^+\)

Theorem 4.3 (Solution via Fundamental Solution)

The solution to the transformed PDE is: \(u(\tau, x) = \int_{-\infty}^{\infty} (e^y - 1)^+ G(\tau; x, y) dy\)

where \(G(\tau; x, y)\) is the fundamental solution: \(G(\tau; x, y) = \frac{1}{\sqrt{2\pi\sigma^2\tau}} \exp\left(-\frac{(y - x - (r-\sigma^2/2)\tau)^2}{2\sigma^2\tau}\right)\)

1.3.8 Step 8: Computing the Solution

Theorem 4.4 (Explicit Solution of the PDE)

After computing the integral (which follows the same steps as in the martingale approach), we obtain: \(u(\tau, x) = e^x \Phi(d_1) - \Phi(d_2)\)

where: \[\begin{align} d_1 &= \frac{x + (r + \sigma^2/2)\tau}{\sigma\sqrt{\tau}} \\ d_2 &= \frac{x + (r - \sigma^2/2)\tau}{\sigma\sqrt{\tau}} = d_1 - \sigma\sqrt{\tau} \end{align}\]

Theorem 4.5 (Black-Scholes Formula via PDE Method)

Transforming back to the original variables, the European call option price is: \(V(t,S) = S \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2)\)

where: \[\begin{align} d_1 &= \frac{\ln(S/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}} \\ d_2 &= d_1 - \sigma\sqrt{T-t} \end{align}\]

Proof of Theorem 4.5

We have: \(V(t,S) = K e^{-r(T-t)} u(T-t, \ln(S/K))\)

The transformation from the \(u\) solution to the \(V\) solution follows the same integration techniques as in the martingale approach. The key insight is that both methods yield identical results, demonstrating the equivalence of risk-neutral valuation and PDE approaches.

1.4 Verification and Conclusion

Theorem 5.1 (Equivalence of Both Methods)

Both the martingale approach and the PDE approach yield the identical Black-Scholes formula: \(V(t,S) = S \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2)\)

This demonstrates the deep connection between:

• Risk-neutral valuation (martingale theory)
• Dynamic hedging and partial differential equations
  
• Stochastic calculus and deterministic analysis

1.4.1 Economic Interpretation

Remark 5.1 (Economic Interpretation)

The Black-Scholes formula has a clear economic interpretation:

• \(S \Phi(d_1)\): Expected value of the stock if exercised, weighted by probability of exercise
• \(K e^{-r(T-t)} \Phi(d_2)\): Present value of strike price, weighted by probability of exercise
  
• \(\Phi(d_2)\): Risk-neutral probability that option finishes in-the-money
• \(\Delta = \frac{\partial V}{\partial S} = \Phi(d_1)\): Hedge ratio (number of shares to hold per option)

1.4.2 Key Insights

Remark 5.2 (Key Insights)

The derivation reveals several fundamental insights:

1. The option price depends only on the risk-free rate \(r\), not the actual expected return \(\mu\) of the stock
2. Perfect hedging is possible in continuous time with continuous trading
3. The same formula emerges from completely different mathematical approaches
4. The risk-neutral measure transforms the pricing problem into an expectation calculation

1.4.3 Greeks and Risk Management

Definition 5.1 (The Greeks)

The Greeks measure sensitivities of the option price to various parameters:

Delta: \(\Delta = \frac{\partial V}{\partial S} = \Phi(d_1)\) - Measures sensitivity to stock price changes - Represents the hedge ratio

Gamma: \(\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\phi(d_1)}{S\sigma\sqrt{T-t}}\) - Measures rate of change of delta - Important for portfolio convexity

Theta: \(\Theta = \frac{\partial V}{\partial t} = -\frac{S\phi(d_1)\sigma}{2\sqrt{T-t}} - rKe^{-r(T-t)}\Phi(d_2)\) - Measures time decay - Always negative for long call options

Vega: \(\nu = \frac{\partial V}{\partial \sigma} = S\phi(d_1)\sqrt{T-t}\) - Measures sensitivity to volatility changes - Always positive for long options

Rho: \(\rho = \frac{\partial V}{\partial r} = K(T-t)e^{-r(T-t)}\Phi(d_2)\) - Measures sensitivity to interest rate changes

where \(\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}\) is the standard normal density.

1.4.4 Extensions and Limitations

Remark 5.3 (Model Assumptions and Limitations)

The Black-Scholes model relies on several strong assumptions:

Assumptions: - Constant risk-free rate and volatility - No dividends - European exercise only - No transaction costs - Continuous trading possible - Log-normal stock price distribution

Limitations: - Volatility smile/skew not captured - Jump risk ignored - Early exercise features not handled - Transaction costs can be significant - Liquidity constraints in practice

Extensions: - American options (early exercise) - Stochastic volatility models (Heston, etc.) - Jump-diffusion models (Merton) - Dividend-paying stocks - Time-dependent parameters

1.4.5 Computational Implementation

Example 5.1 (Python Implementation)

import numpy as np
from scipy.stats import norm

def black_scholes_call(S, K, T, r, sigma):
    """
    Calculate Black-Scholes call option price
    
    Parameters:
    S: Current stock price
    K: Strike price  
    T: Time to expiration
    r: Risk-free rate
    sigma: Volatility
    """
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    
    call_price = S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)
    
    return call_price

def calculate_greeks(S, K, T, r, sigma):
    """Calculate the Greeks"""
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    
    delta = norm.cdf(d1)
    gamma = norm.pdf(d1) / (S*sigma*np.sqrt(T))
    theta = (-S*norm.pdf(d1)*sigma/(2*np.sqrt(T)) 
             - r*K*np.exp(-r*T)*norm.cdf(d2))
    vega = S*norm.pdf(d1)*np.sqrt(T)
    rho = K*T*np.exp(-r*T)*norm.cdf(d2)
    
    return {'delta': delta, 'gamma': gamma, 'theta': theta, 'vega': vega, 'rho': rho}

## Example usage
S0 = 100    ## Current stock price
K = 105     ## Strike price
T = 0.25    ## 3 months to expiration
r = 0.05    ## 5% risk-free rate  
sigma = 0.2 ## 20% volatility

price = black_scholes_call(S0, K, T, r, sigma)
greeks = calculate_greeks(S0, K, T, r, sigma)

print(f"Call option price: ${price:.2f}")
print(f"Delta: {greeks['delta']:.4f}")
print(f"Gamma: {greeks['gamma']:.4f}")

This complete treatment of the Black-Scholes model provides both theoretical rigor and practical applicability, showing how mathematical finance bridges pure mathematics and real-world financial applications.

1.5 Introduction

The Feynman-Kac theorem provides a fundamental connection between partial differential equations and stochastic processes. While our previous Black-Scholes derivations used either pure martingale methods or pure PDE methods, the Feynman-Kac theorem elegantly unifies both approaches and provides the most direct path from the Black-Scholes PDE to the risk-neutral valuation formula.

Why Feynman-Kac Matters

The Feynman-Kac theorem shows that the solution to certain PDEs can be represented as expectations of stochastic processes. This is exactly what we need in finance: we derive a PDE from no-arbitrage arguments, then use Feynman-Kac to get the risk-neutral pricing formula.

1.6 Part I: The Feynman-Kac Theorem

1.6.1 Statement of the Theorem

Theorem 1.1 (Feynman-Kac Theorem)

Consider the parabolic PDE: \[\frac{\partial u}{\partial t} + \frac{1}{2}\sigma^2(t,x) \frac{\partial^2 u}{\partial x^2} + b(t,x) \frac{\partial u}{\partial x} - r(t,x) u + f(t,x) = 0\]

for \((t,x) \in [0,T) \times \mathbb{R}\), with terminal condition: \[u(T,x) = g(x)\]

Assumptions: 1. \(\sigma(t,x) > 0\) (non-degeneracy) 2. \(b(t,x)\), \(\sigma(t,x)\), \(r(t,x)\), \(f(t,x)\) satisfy appropriate regularity conditions 3. The associated SDE has a unique strong solution 4. Polynomial growth conditions on coefficients

Then the solution is given by: \[u(t,x) = \mathbb{E}\left[ e^{-\int_t^T r(s,X_s) ds} g(X_T) + \int_t^T e^{-\int_t^s r(u,X_u) du} f(s,X_s) ds \,\Big|\, X_t = x \right]\]

where \(X_s\) is the solution to the SDE: \[dX_s = b(s,X_s) ds + \sigma(s,X_s) dW_s, \quad X_t = x\]

Complete Proof of Feynman-Kac Theorem

The proof uses Itô’s lemma and the martingale representation theorem.

Step 1: Define the discounted process Let \(Y_s = e^{-\int_t^s r(u,X_u) du} u(s,X_s)\) for \(s \in [t,T]\).

Step 2: Apply Itô’s lemma to \(Y_s\) We need to compute \(dY_s\). Using the product rule and Itô’s lemma:

\[\begin{align} dY_s &= d\left(e^{-\int_t^s r(u,X_u) du}\right) u(s,X_s) + e^{-\int_t^s r(u,X_u) du} du(s,X_s) \\ &\quad + d\left(e^{-\int_t^s r(u,X_u) du}\right) du(s,X_s) \end{align}\]

Step 3: Compute each term

For the discount factor: \[d\left(e^{-\int_t^s r(u,X_u) du}\right) = -r(s,X_s) e^{-\int_t^s r(u,X_u) du} ds\]

For \(u(s,X_s)\), by Itô’s lemma: \[\begin{align} du(s,X_s) &= \frac{\partial u}{\partial s} ds + \frac{\partial u}{\partial x} dX_s + \frac{1}{2}\frac{\partial^2 u}{\partial x^2} (dX_s)^2 \\ &= \left[\frac{\partial u}{\partial s} + b(s,X_s)\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2(s,X_s)\frac{\partial^2 u}{\partial x^2}\right] ds \\ &\quad + \sigma(s,X_s)\frac{\partial u}{\partial x} dW_s \end{align}\]

Step 4: Substitute the PDE From our PDE, we know: \[\frac{\partial u}{\partial s} + b(s,X_s)\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2(s,X_s)\frac{\partial^2 u}{\partial x^2} = r(s,X_s)u - f(s,X_s)\]

Step 5: Combine terms \[\begin{align} dY_s &= e^{-\int_t^s r(u,X_u) du} \left[-r(s,X_s)u(s,X_s) + r(s,X_s)u(s,X_s) - f(s,X_s)\right] ds \\ &\quad + e^{-\int_t^s r(u,X_u) du} \sigma(s,X_s)\frac{\partial u}{\partial x} dW_s \\ &= -e^{-\int_t^s r(u,X_u) du} f(s,X_s) ds + e^{-\int_t^s r(u,X_u) du} \sigma(s,X_s)\frac{\partial u}{\partial x} dW_s \end{align}\]

Step 6: Integrate from \(t\) to \(T\) \[Y_T - Y_t = -\int_t^T e^{-\int_t^s r(u,X_u) du} f(s,X_s) ds + \int_t^T e^{-\int_t^s r(u,X_u) du} \sigma(s,X_s)\frac{\partial u}{\partial x} dW_s\]

Step 7: Take expectation The stochastic integral has zero expectation (under appropriate integrability conditions): \[\mathbb{E}[Y_T | X_t = x] = Y_t - \mathbb{E}\left[\int_t^T e^{-\int_t^s r(u,X_u) du} f(s,X_s) ds \,\Big|\, X_t = x\right]\]

Step 8: Substitute definitions \[\begin{align} \mathbb{E}\left[e^{-\int_t^T r(u,X_u) du} u(T,X_T) \,\Big|\, X_t = x\right] &= u(t,x) \\ &\quad - \mathbb{E}\left[\int_t^T e^{-\int_t^s r(u,X_u) du} f(s,X_s) ds \,\Big|\, X_t = x\right] \end{align}\]

Step 9: Use terminal condition \(u(T,x) = g(x)\) \[u(t,x) = \mathbb{E}\left[e^{-\int_t^T r(u,X_u) du} g(X_T) + \int_t^T e^{-\int_t^s r(u,X_u) du} f(s,X_s) ds \,\Big|\, X_t = x\right]\]

This completes the proof.

1.6.2 Key Insights of Feynman-Kac

Remark 1.1 (PDE-SDE Duality)

The Feynman-Kac theorem reveals a fundamental duality:

PDE Side: Deterministic partial differential equation with boundary conditions SDE Side: Stochastic differential equation with expectation of terminal payoff

This duality is the mathematical foundation of: - Risk-neutral pricing in finance - Monte Carlo methods for PDE solving - Connection between heat equations and Brownian motion - Quantum mechanics (Feynman path integrals)

1.7 Part II: Application to Black-Scholes

1.7.1 Black-Scholes PDE Setup

Definition 2.1 (Black-Scholes PDE)

From our hedging argument, we derived that the option price \(V(t,S)\) satisfies: \[\frac{\partial V}{\partial t} + rS \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0\]

with terminal condition: \[V(T,S) = h(S) = (S-K)^+ \text{ for call options}\]

Key Assumptions: 1. Constant risk-free rate \(r\) 2. Constant volatility \(\sigma\)
3. No dividends 4. Continuous trading possible 5. No transaction costs 6. European exercise only

1.7.2 Transforming to Standard Feynman-Kac Form

Definition 2.2 (Variable Transformation)

To apply Feynman-Kac, we make the substitution \(x = \ln S\) to transform the PDE.

Let \(u(t,x) = V(t,e^x)\). Then: - \(S = e^x\) - \(\frac{\partial V}{\partial S} = \frac{1}{S}\frac{\partial u}{\partial x} = e^{-x}\frac{\partial u}{\partial x}\) - \(\frac{\partial^2 V}{\partial S^2} = e^{-2x}\left(\frac{\partial^2 u}{\partial x^2} - \frac{\partial u}{\partial x}\right)\) - \(\frac{\partial V}{\partial t} = \frac{\partial u}{\partial t}\)

Detailed Variable Transformation

Step 1: Express derivatives in terms of \(u(t,x)\).

Since \(V(t,S) = u(t,\ln S)\), by the chain rule: \[\frac{\partial V}{\partial S} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial S} = \frac{\partial u}{\partial x} \frac{1}{S}\]

For the second derivative: \[\begin{align} \frac{\partial^2 V}{\partial S^2} &= \frac{\partial}{\partial S}\left(\frac{1}{S}\frac{\partial u}{\partial x}\right) \\ &= -\frac{1}{S^2}\frac{\partial u}{\partial x} + \frac{1}{S}\frac{\partial}{\partial S}\left(\frac{\partial u}{\partial x}\right) \\ &= -\frac{1}{S^2}\frac{\partial u}{\partial x} + \frac{1}{S}\frac{\partial^2 u}{\partial x^2}\frac{\partial x}{\partial S} \\ &= -\frac{1}{S^2}\frac{\partial u}{\partial x} + \frac{1}{S^2}\frac{\partial^2 u}{\partial x^2} \\ &= \frac{1}{S^2}\left(\frac{\partial^2 u}{\partial x^2} - \frac{\partial u}{\partial x}\right) \end{align}\]

Step 2: Substitute into the Black-Scholes PDE. \[\begin{align} &\frac{\partial u}{\partial t} + rS \cdot \frac{1}{S}\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2 S^2 \cdot \frac{1}{S^2}\left(\frac{\partial^2 u}{\partial x^2} - \frac{\partial u}{\partial x}\right) - ru = 0 \\ &\frac{\partial u}{\partial t} + r\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2\left(\frac{\partial^2 u}{\partial x^2} - \frac{\partial u}{\partial x}\right) - ru = 0 \\ &\frac{\partial u}{\partial t} + \left(r - \frac{\sigma^2}{2}\right)\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 u}{\partial x^2} - ru = 0 \end{align}\]

Step 3: Rearrange to standard form. \[\frac{\partial u}{\partial t} + \frac{1}{2}\sigma^2\frac{\partial^2 u}{\partial x^2} + \left(r - \frac{\sigma^2}{2}\right)\frac{\partial u}{\partial x} - ru = 0\]

This is now in the standard Feynman-Kac form with: - \(b(t,x) = r - \frac{\sigma^2}{2}\) (drift coefficient) - \(\sigma(t,x) = \sigma\) (diffusion coefficient)
- \(r(t,x) = r\) (discount rate) - \(f(t,x) = 0\) (source term)

1.7.3 Applying Feynman-Kac to Black-Scholes

Theorem 2.1 (Black-Scholes via Feynman-Kac)

The transformed Black-Scholes PDE: \[\frac{\partial u}{\partial t} + \frac{1}{2}\sigma^2\frac{\partial^2 u}{\partial x^2} + \left(r - \frac{\sigma^2}{2}\right)\frac{\partial u}{\partial x} - ru = 0\]

with terminal condition \(u(T,x) = h(e^x) = (e^x - K)^+\), has the solution: \[u(t,x) = e^{-r(T-t)} \mathbb{E}[h(e^{X_T}) | X_t = x]\]

where \(X_s\) satisfies the SDE: \[dX_s = \left(r - \frac{\sigma^2}{2}\right) ds + \sigma dW_s, \quad X_t = x\]

Therefore, the Black-Scholes option price is: \[V(t,S) = u(t,\ln S) = e^{-r(T-t)} \mathbb{E}[h(S_T) | S_t = S]\]

where \(S_T = S_t \exp\left(\left(r - \frac{\sigma^2}{2}\right)(T-t) + \sigma(W_T - W_t)\right)\).

Detailed Application of Feynman-Kac

Step 1: Identify the parameters for Feynman-Kac. From our transformed PDE: - \(b(t,x) = r - \frac{\sigma^2}{2}\) - \(\sigma(t,x) = \sigma\)
- \(r(t,x) = r\) - \(f(t,x) = 0\) - \(g(x) = (e^x - K)^+\)

Step 2: Write the associated SDE. \[dX_s = \left(r - \frac{\sigma^2}{2}\right) ds + \sigma dW_s\]

Step 3: Apply the Feynman-Kac formula. Since \(f(t,x) = 0\), the formula simplifies to: \[u(t,x) = \mathbb{E}\left[e^{-r(T-t)} g(X_T) \,\Big|\, X_t = x\right]\]

Step 4: Solve the SDE. The solution to the linear SDE is: \[X_T = X_t + \left(r - \frac{\sigma^2}{2}\right)(T-t) + \sigma(W_T - W_t)\]

Step 5: Transform back to stock price. Since \(X = \ln S\), we have: \[\begin{align} \ln S_T &= \ln S_t + \left(r - \frac{\sigma^2}{2}\right)(T-t) + \sigma(W_T - W_t) \\ S_T &= S_t \exp\left(\left(r - \frac{\sigma^2}{2}\right)(T-t) + \sigma(W_T - W_t)\right) \end{align}\]

Step 6: Final formula. \[V(t,S_t) = e^{-r(T-t)} \mathbb{E}[(S_T - K)^+ | S_t]\]

This is exactly the risk-neutral valuation formula!

1.7.4 The Key Insight: Automatic Risk-Neutral Measure

Remark 2.1 (Feynman-Kac Reveals the Risk-Neutral Measure)

The beauty of the Feynman-Kac approach is that it automatically produces the risk-neutral measure without explicitly constructing it via Girsanov’s theorem.

What happens: 1. We start with the Black-Scholes PDE (derived from no-arbitrage hedging) 2. Feynman-Kac gives us the stochastic representation 3. The resulting SDE for \(X_t = \ln S_t\) has drift \(r - \sigma^2/2\) 4. This corresponds to the stock having drift \(r\) under the measure induced by the Feynman-Kac expectation 5. This measure is exactly the risk-neutral measure \(\mathbb{Q}\)!

The connection: - PDE approach: Hedging → Black-Scholes PDE → Feynman-Kac → Risk-neutral expectation - Martingale approach: No-arbitrage → Risk-neutral measure → Expectation formula

Both paths lead to the same destination, but Feynman-Kac provides the bridge.

1.8 Part III: Complete Assumptions and Regularity Conditions

1.8.1 Mathematical Assumptions for Feynman-Kac

Assumption 3.1 (Regularity Conditions for Feynman-Kac)

For the Feynman-Kac theorem to apply to the Black-Scholes PDE, we need:

Coefficient Regularity: 1. \(b(t,x) = r - \sigma^2/2\) is constant (trivially satisfies Lipschitz and linear growth) 2. \(\sigma(t,x) = \sigma > 0\) is constant and positive (non-degeneracy) 3. \(r(t,x) = r \geq 0\) is constant (ensures well-defined discounting)

Domain and Boundary Conditions: 4. Domain: \((t,x) \in [0,T) \times \mathbb{R}\) (unbounded spatial domain) 5. Terminal condition: \(u(T,x) = (e^x - K)^+\) has polynomial growth 6. No spatial boundary conditions needed (due to infinite domain)

Solution Existence and Uniqueness: 7. The associated SDE \(dX_s = (r-\sigma^2/2)ds + \sigma dW_s\) has unique strong solution 8. The solution has finite moments of all orders 9. The PDE has a unique classical solution in the class of functions with polynomial growth

Verification of Black-Scholes Assumptions

Assumption 1-3 (Coefficient Properties): ✓ All coefficients are constants, so they trivially satisfy: - Lipschitz continuity: \(|b(t,x) - b(t,y)| \leq L|x-y|\) (with \(L=0\)) - Linear growth: \(|b(t,x)| \leq C(1+|x|)\) (with \(C = |r-\sigma^2/2|\)) - Non-degeneracy: \(\sigma > 0\)

Assumption 4 (Domain): ✓ The log-price \(x = \ln S\) can take any real value, making \(\mathbb{R}\) the natural domain.

Assumption 5 (Terminal Condition): ✓ \((e^x - K)^+ \leq e^{|x|} + K\) has exponential growth, which satisfies polynomial growth conditions for the theorem.

Assumption 7 (SDE Solution): ✓ The SDE \(dX_s = \mu ds + \sigma dW_s\) with constant coefficients has the explicit solution: \[X_t = X_0 + \mu t + \sigma W_t\] This clearly has a unique strong solution.

Assumption 8 (Finite Moments): ✓ Since \(X_t\) is Gaussian, it has finite moments of all orders.

Assumption 9 (PDE Solution): ✓ The Black-Scholes PDE is a well-studied parabolic PDE with known classical solutions.

1.8.2 Financial Market Assumptions

Assumption 3.2 (Black-Scholes Market Assumptions)

The financial market model requires:

Asset Price Dynamics: 1. Stock follows geometric Brownian motion: \(dS_t = \mu S_t dt + \sigma S_t dW_t\) 2. Constant risk-free rate \(r > 0\) 3. Constant volatility \(\sigma > 0\) 4. No dividends paid during option life 5. Stock price always positive: \(S_t > 0\)

Market Structure: 6. Frictionless market (no transaction costs, bid-ask spreads) 7. Continuous trading possible at all times 8. Unlimited borrowing and lending at rate \(r\) 9. Short-selling allowed with full proceeds available 10. No restrictions on position sizes

Information and Arbitrage: 11. All market participants have same information (complete information) 12. No arbitrage opportunities exist 13. Market is complete (all contingent claims can be replicated) 14. Risk-free asset and risky stock span the entire market

Option Characteristics: 15. European exercise only (no early exercise) 16. Strike price \(K > 0\) fixed 17. Maturity \(T > 0\) fixed 18. Option payoff \((S_T - K)^+\) depends only on terminal stock price

1.8.3 Comparison with Alternative Approaches

Remark 3.1 (Strengths and Weaknesses of Different Approaches)

Martingale Approach: - Strengths: Direct probabilistic interpretation, connects to fundamental theorems - Weaknesses: Requires explicit construction of risk-neutral measure via Girsanov

PDE Approach:
- Strengths: Uses familiar hedging intuition, connects to numerical methods - Weaknesses: Requires solving PDE, less direct connection to probability

Feynman-Kac Approach: - Strengths: Unifies PDE and probability, automatic risk-neutral measure, generalizes easily - Weaknesses: Requires understanding of both PDE and SDE theory, more abstract

When to Use Each: - Martingale: When risk-neutral measure is natural starting point - PDE: When hedging interpretation is most important
- Feynman-Kac: When connecting different mathematical frameworks or generalizing to complex payoffs

1.9 Part IV: Extensions and Applications

1.9.1 Generalizations of Feynman-Kac in Finance

Example 4.1 (American Options via Optimal Stopping)

For American options, Feynman-Kac extends to optimal stopping problems:

\[V(t,S) = \sup_{\tau \geq t} \mathbb{E}\left[e^{-r(\tau-t)} h(S_\tau) \,\Big|\, S_t = S\right]\]

where the supremum is over all stopping times \(\tau \in [t,T]\).

This leads to free boundary problems and variational inequalities.

Example 4.2 (Path-Dependent Options)

For Asian options with payoff depending on \(\int_0^T S_u du\), we extend the state space:

Let \(Y_t = \int_0^t S_u du\). Then \((S_t, Y_t)\) follows a 2D system: \[\begin{align} dS_t &= rS_t dt + \sigma S_t dW_t \\ dY_t &= S_t dt \end{align}\]

Feynman-Kac applies to the corresponding 2D PDE.

Example 4.3 (Jump-Diffusion Models)

For jump-diffusion processes: \[dS_t = \mu S_t dt + \sigma S_t dW_t + S_{t-} \int h(z) \tilde{N}(dt,dz)\]

The Feynman-Kac theorem extends to integro-differential equations (PIDEs).

1.9.2 Computational Advantages

Remark 4.1 (Monte Carlo and PDE Methods)

The Feynman-Kac representation enables two complementary numerical approaches:

Monte Carlo Simulation: - Simulate the SDE paths - Compute sample average of discounted payoffs - Converges by Law of Large Numbers

PDE Finite Differences: - Discretize the PDE on a grid - Solve numerically using implicit/explicit schemes - Converges to PDE solution

Both methods solve the same problem via Feynman-Kac duality!

1.10 Conclusion

Theorem 4.1 (Unified Black-Scholes Result)

The Black-Scholes option pricing formula: \[V(t,S) = S\Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2)\]

can be derived through three equivalent approaches:

1. Martingale: Risk-neutral measure → Expectation calculation
2. PDE: Hedging portfolio → Black-Scholes PDE → Analytical solution
   
3. Feynman-Kac: PDE → Stochastic representation → Risk-neutral expectation

All three methods yield identical results, demonstrating the deep mathematical unity underlying modern quantitative finance.

Final Insights

The Feynman-Kac approach reveals that: - PDEs and stochastic processes are dual representations of the same underlying mathematics - The risk-neutral measure emerges naturally from the PDE structure - Option pricing connects heat equations, Brownian motion, and financial hedging - Monte Carlo and finite difference methods are fundamentally solving the same problem

This unity suggests that: - Different mathematical tools often lead to the same financial insights - The choice of method depends on computational needs and theoretical focus - Understanding multiple approaches deepens comprehension of the underlying economics

The Feynman-Kac theorem thus provides not just another derivation method, but a unifying perspective that bridges deterministic and stochastic approaches to mathematical finance.

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This derivation completes the trilogy of Black-Scholes approaches, showing how partial differential equations, stochastic processes, and martingale theory all contribute to our understanding of option pricing.