1. Learning Objectives

By the end of this lesson, you will be able to:

  • Define and construct Brownian motion (Wiener process) and derive its key properties (continuity, quadratic variation, normality of increments).

  • Define and interpret stochastic differential equations (SDEs) and their integral representation using Ito integrals.

  • Derive and apply Ito’s Lemma to find the SDE of functions of stochastic processes.

  • Apply Ito’s Lemma to derive the Black-Scholes partial differential equation (PDE) from first principles.

  • Solve the Black-Scholes PDE to obtain the closed-form pricing formulas for European call and put options.

  • Implement the Black-Scholes formula in Python and compute the Greeks (Delta, Gamma, Vega, Theta, Rho).

  • Understand the limitations of the Black-Scholes model (constant volatility, log-normal returns) and the need for advanced models (stochastic volatility, jump-diffusion).

2. Brownian Motion (Wiener Process) – The Building Block of Continuous-Time Finance

A standard Brownian motion {W_t, t ≥ 0} is a continuous-time stochastic process with the following properties:

  1. W_0 = 0 (starts at zero).

  2. Independent increments: For 0 ≤ s < t, W_t - W_s is independent of F_s.

  3. Normal increments: W_t - W_s ∼ N(0, t-s).

  4. Continuous paths: t ↦ W_t is continuous almost surely.

2.1 Properties of Brownian Motion

  • Expectation: E[W_t] = 0 for all t.

  • Variance: Var(W_t) = t.

  • Covariance: Cov(W_s, W_t) = min(s, t).

  • Scalability: For any c > 0, {c W_{t/c²}} is also a Brownian motion.

  • Martingale Property: W_t is a martingale: E[W_t | F_s] = W_s for s < t.

2.2 Quadratic Variation of Brownian Motion
The quadratic variation of a process X_t over [0, t] is defined as:
[X, X]_t = lim_{||P|| → 0} Σ_{i=1}^{n} (X_{t_i} - X_{t_{i-1}})^2.
For Brownian motion: [W, W]_t = t.
This is a crucial property: Brownian motion has infinite first variation (paths are too rough) but finite quadratic variation. This is why the standard rules of calculus (where (dt)^2 = 0 and (dW)^2 ≈ dt) must be replaced by Ito calculus.

2.3 Financial Interpretation
Stock prices often follow a Geometric Brownian Motion (GBM):
dS_t = μ S_t dt + σ S_t dW_t.

  • μ: Drift (expected return).

  • σ: Volatility (standard deviation of returns).
    The solution (via Ito’s Lemma) is:
    S_t = S_0 exp( (μ - σ²/2)t + σ W_t ).
    This ensures that S_t > 0 and that ln(S_t/S_0) ∼ N( (μ - σ²/2)t, σ² t ).

3. Stochastic Differential Equations (SDEs) and Ito Integrals

An SDE is an equation of the form:
dX_t = a(t, X_t) dt + b(t, X_t) dW_t.

  • a(t, X_t): Drift term (deterministic component).

  • b(t, X_t): Diffusion term (stochastic component).
    The integral form is:
    X_t = X_0 + ∫_0^t a(s, X_s) ds + ∫_0^t b(s, X_s) dW_s.
    The first integral is a standard Riemann integral. The second is an Ito integral, which is defined as a limit of Riemann-Stieltjes sums over the partition {t_i}:
    ∫_0^t b(s, X_s) dW_s = lim_{||P|| → 0} Σ_{i=1}^{n} b(t_{i-1}, X_{t_{i-1}}) (W_{t_i} - W_{t_{i-1}}).
    Key Property of Ito Integral: It is a martingale. E[ ∫_0^t b_s dW_s ] = 0.

3.1 Ito Isometry
For any adapted process b_t with E[ ∫_0^t b_s² ds ] < ∞:
E[ (∫_0^t b_s dW_s)² ] = E[ ∫_0^t b_s² ds ].
This is used to compute the variance of stochastic integrals, which is essential for risk management.

4. Ito’s Lemma – The Stochastic Chain Rule

Ito’s Lemma is the fundamental theorem of stochastic calculus. It tells us how to differentiate functions of stochastic processes.

4.1 Statement of Ito’s Lemma
Let X_t be an Ito process: dX_t = a_t dt + b_t dW_t.
Let f(t, x) be a twice continuously differentiable function in x and once in t.
Then Y_t = f(t, X_t) is also an Ito process, and its differential is:
df(t, X_t) = ( ∂f/∂t + a_t ∂f/∂x + (1/2) b_t² ∂²f/∂x² ) dt + b_t ∂f/∂x dW_t.

4.2 Derivation of Ito’s Lemma (Heuristic)
Using a Taylor expansion up to second order:
df = ∂f/∂t dt + ∂f/∂x dX + (1/2) ∂²f/∂x² (dX)² + (1/2) ∂²f/∂t² (dt)² + ...
Substitute dX = a dt + b dW:
(dX)² = a² (dt)² + 2ab dt dW + b² (dW)².
Using the rules (dt)² = 0, dt dW = 0, and (dW)² = dt, we get (dX)² = b² dt.
Substituting yields Ito’s Lemma.

4.3 Financial Application – Geometric Brownian Motion
Let X_t = ln(S_t), where S_t follows dS_t = μ S_t dt + σ S_t dW_t.
We have a_t = μ S_t, b_t = σ S_t, f(t, x) = ln(x).
Compute:
∂f/∂t = 0, ∂f/∂x = 1/x, ∂²f/∂x² = -1/x².
Apply Ito’s Lemma:
d(ln S_t) = ( 0 + μ S_t * (1/S_t) + (1/2) σ² S_t² * (-1/S_t²) ) dt + σ S_t * (1/S_t) dW_t
= ( μ - σ²/2 ) dt + σ dW_t.
Integrating from 0 to t:
ln(S_t/S_0) = ( μ - σ²/2 ) t + σ W_t.
Taking exponentials:
S_t = S_0 exp( (μ - σ²/2)t + σ W_t ).
This is the solution to the GBM SDE.

5. Derivation of the Black-Scholes PDE – From First Principles

We derive the Black-Scholes PDE using a delta-hedging argument.

5.1 Assumptions

  1. The stock price follows GBM: dS_t = μ S_t dt + σ S_t dW_t.

  2. No transaction costs, no dividends.

  3. Risk-free interest rate r is constant.

  4. Continuous trading is possible.

  5. No arbitrage opportunities.

5.2 The Option Price Process
Let V(t, S_t) be the price of a European option (call or put) at time t with stock price S_t. By Ito’s Lemma:
dV = ( ∂V/∂t + μ S ∂V/∂S + (1/2) σ² S² ∂²V/∂S² ) dt + σ S ∂V/∂S dW_t.

5.3 Constructing the Riskless Hedge Portfolio
Construct a portfolio Π consisting of:

  • Long 1 option contract: +V.

  • Short Δ shares of the stock: -Δ S.
    The portfolio value is: Π = V - Δ S.
    The change in the portfolio is:
    dΠ = dV - Δ dS.
    Substitute dV and dS:
    dΠ = ( ∂V/∂t + μ S ∂V/∂S + (1/2) σ² S² ∂²V/∂S² ) dt + σ S ∂V/∂S dW_t - Δ( μ S dt + σ S dW_t ).
    dΠ = ( ∂V/∂t + μ S (∂V/∂S - Δ) + (1/2) σ² S² ∂²V/∂S² ) dt + σ S (∂V/∂S - Δ) dW_t.

5.4 Eliminating the Stochastic Term (Delta Hedging)
Choose Δ = ∂V/∂S (the delta of the option). This eliminates the dW_t term, making the portfolio riskless:
dΠ = ( ∂V/∂t + (1/2) σ² S² ∂²V/∂S² ) dt.

5.5 No Arbitrage Principle
A riskless portfolio must earn the risk-free rate r:
dΠ = r Π dt = r (V - Δ S) dt = r (V - S ∂V/∂S) dt.

5.6 The Black-Scholes PDE
Equate the two expressions for dΠ:
∂V/∂t + (1/2) σ² S² ∂²V/∂S² = r V - r S ∂V/∂S.
Rearrange:
∂V/∂t + r S ∂V/∂S + (1/2) σ² S² ∂²V/∂S² - r V = 0.
This is the Black-Scholes Partial Differential Equation (PDE).

6. Solving the Black-Scholes PDE – The Closed-Form Solution

The Black-Scholes PDE is a parabolic PDE. We solve it with boundary conditions specific to European calls and puts.

6.1 Boundary Conditions – European Call Option

  • At maturity t = T: V(T, S_T) = max(S_T - K, 0) (call payoff).

  • As S → 0: V(t, 0) = 0 (option worthless if stock is zero).

  • As S → ∞: V(t, S) ≈ S - K e^{-r(T-t)} (deep in-the-money).

6.2 Transformation to the Heat Equation
The PDE is solved by transforming variables:
τ = T - t (time to maturity).
x = ln(S/K) + (r - σ²/2) τ.
V(t, S) = K e^{-r τ} u(x, τ).
Substituting into the Black-Scholes PDE yields the standard heat equation:
∂u/∂τ = (σ²/2) ∂²u/∂x².

6.3 The Black-Scholes Formula for a European Call
Solving the heat equation with the payoff function u(x, 0) = max(e^x - 1, 0) yields:
C(t, S) = S N(d_1) - K e^{-r(T-t)} N(d_2).
where:
d_1 = ( ln(S/K) + (r + σ²/2)(T-t) ) / ( σ sqrt(T-t) ).
d_2 = d_1 - σ sqrt(T-t) = ( ln(S/K) + (r - σ²/2)(T-t) ) / ( σ sqrt(T-t) ).
N(z) is the standard normal cumulative distribution function: N(z) = (1/sqrt(2π)) ∫_{-∞}^{z} e^{-x²/2} dx.

6.4 The Black-Scholes Formula for a European Put
Using put-call parity: C - P = S - K e^{-r(T-t)}, we get:
P(t, S) = K e^{-r(T-t)} N(-d_2) - S N(-d_1).

6.5 Interpretation of d_1 and d_2

  • d_1 is the probability (under the risk-neutral measure) that the option will be exercised, weighted by the stock numeraire.

  • N(d_2) is the probability (under the risk-neutral measure) that the option will be exercised, weighted by the money market numeraire.

7. The Greeks – Risk Sensitivities

The Greeks measure the sensitivity of the option price to various parameters. They are essential for hedging.

7.1 Delta (Δ) – Sensitivity to Stock Price
Δ = ∂V/∂S = N(d_1) for a call, and Δ = N(d_1) - 1 for a put.
Interpretation: Delta is the hedge ratio. A delta of 0.6 means the option moves like 0.6 shares of stock.

7.2 Gamma (Γ) – Sensitivity of Delta to Stock Price
Γ = ∂²V/∂S² = N'(d_1) / (S σ sqrt(T-t)).
Interpretation: Gamma measures the convexity of the option. High gamma means delta changes rapidly; the option needs frequent rebalancing.

7.3 Vega (ν) – Sensitivity to Volatility
ν = ∂V/∂σ = S sqrt(T-t) N'(d_1) (same for calls and puts).
Interpretation: Vega measures exposure to volatility risk. Options are long vega (profit from increasing volatility).

7.4 Theta (Θ) – Sensitivity to Time Decay
Θ = ∂V/∂t = - (S σ N'(d_1)) / (2 sqrt(T-t)) - r K e^{-r(T-t)} N(d_2) (call) or N(-d_2) for put.
Interpretation: Theta is negative (time decay). Options lose value as time passes.

7.5 Rho (ρ) – Sensitivity to Interest Rate
ρ = ∂V/∂r = K (T-t) e^{-r(T-t)} N(d_2) (call), ρ = -K (T-t) e^{-r(T-t)} N(-d_2) (put).

7.6 AI Application of Greeks

  • Delta Hedging: An AI model can be trained to predict the optimal rebalancing frequency to minimise transaction costs while maintaining a delta-neutral portfolio.

  • Volatility Forecasting: Vega can be used to hedge volatility risk. An AI model that forecasts future volatility (σ_forecast) can dynamically adjust the option portfolio’s vega exposure.

  • Gamma Scalping: A strategy that profits from large price movements. An AI model can trigger Gamma scalping trades when Gamma exceeds a threshold.

8. Limitations of Black-Scholes and Advanced Models

The Black-Scholes model has significant limitations that AI can address.

8.1 Constant Volatility (Volatility Smile/Smirk)
Empirically, implied volatility varies with strike price and maturity (volatility smile). This violates the constant volatility assumption.
AI Solution: Train a neural network to predict the implied volatility surface from market data (strike, maturity, moneyness). Feed the predicted volatility into Black-Scholes to price options more accurately.

8.2 Log-Normal Returns
Empirical returns have fat tails (excess kurtosis) and negative skewness.
AI Solution: Use Stochastic Volatility Models (Heston model) where volatility itself is a stochastic process:
dS_t = μ S_t dt + sqrt(V_t) S_t dW_t^1.
dV_t = κ(θ - V_t) dt + ξ sqrt(V_t) dW_t^2.
Correlation: dW_t^1 dW_t^2 = ρ dt.
AI can calibrate Heston parameters from market data using MLE or deep learning.

8.3 Jump-Diffusion Models (Merton)
Add jump processes to capture sudden market crashes:
dS_t = (μ - λ κ) S_t dt + σ S_t dW_t + J_t S_t dN_t.

  • N_t: Poisson process with intensity λ.

  • J_t: Jump size (log-normal).
    AI models can learn the jump intensity λ and jump size distribution from historical data.

8.4 Machine Learning for Option Pricing
Instead of assuming a parametric model, train a neural network directly on market option prices:
V(t, S, K, T, σ) ≈ NN(t, S, K, T, σ).
This is a data-driven approach that learns the true pricing function without distributional assumptions. This is the frontier of AI in derivatives pricing.

9. Summary for the AI Practitioner

  1. Brownian motion is the fundamental building block. It has independent, normally distributed increments and quadratic variation [W, W]_t = t.

  2. Ito’s Lemma is the stochastic chain rule. It introduces the (1/2) σ² S² V_{SS} term (the convexity adjustment). This is why options have time value.

  3. The Black-Scholes PDE is derived from a delta-hedged, riskless portfolio. It is a deterministic PDE that must hold for all derivative prices under no-arbitrage.

  4. The Black-Scholes formula gives closed-form prices for European options. C = S N(d_1) - K e^{-rT} N(d_2).

  5. The Greeks are derivatives of the option price. Delta is for hedging; Gamma measures convexity; Vega measures volatility risk.

  6. Black-Scholes has limitations: constant volatility, log-normal returns, no jumps. AI addresses these via implied volatility surfaces, stochastic volatility, jump-diffusion, and neural network surrogates.

  7. AI in option pricing: Use LSTMs/Transformers to predict implied volatility surfaces. Use reinforcement learning to optimise delta-hedging strategies with transaction costs.

In Lesson 2.7, we will cover Matrix Factorisations and Numerical Linear Algebra for Finance, including Cholesky decomposition, QR decomposition, and efficient computation of covariance inverses for large portfolios. In Lesson 2.8, we will cover Information Theory, Entropy, and KL Divergence, which are essential for understanding loss functions, model selection, and adversarial training in financial AI.

 
 
 
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