Applications: Black-Scholes & Filtering
Applications: Black-Scholes & Filtering
The Black-Scholes-Merton model uses GBM for the stock price and Ito's lemma to derive a PDE whose solution gives a no-arbitrage price for European options. The key insight is delta hedging: a self-financing portfolio in the stock and bond replicates the option payoff, eliminating risk. The Girsanov theorem enables a change of measure to the risk-neutral world where discounted prices are martingales.
The Kalman-Bucy filter addresses a complementary problem: estimating an unobserved state from noisy observations. It provides the optimal (minimum mean-square error) linear filter as a stochastic differential equation, combining prediction and update steps continuously in time.
Black-Scholes PDE
Under $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$ and risk-free rate $r$, an option price $V(t,S)$ satisfies $$\partial_t V+rS\partial_S V+\tfrac{1}{2}\sigma^2 S^2\partial_{SS}V-rV=0.$$ For a European call with payoff $(S_T-K)^+$, the solution is $$V=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2),$$ where $d_{1,2}=\frac{\ln(S_0/K)+(r\pm\sigma^2/2)T}{\sigma\sqrt{T}}$.
Girsanov & Risk-Neutral Pricing
Define $\tilde{W}_t=W_t+\frac{\mu-r}{\sigma}t$ (market price of risk $\lambda=(\mu-r)/\sigma$). By Girsanov, under the measure $\tilde{P}$ with Radon-Nikodym derivative $e^{-\lambda W_T-\lambda^2 T/2}$, $\tilde{W}_t$ is BM and $dS_t=rS_t\,dt+\sigma S_t\,d\tilde{W}_t$. The price of any derivative with payoff $X$ is $V_0=e^{-rT}\tilde{E}[X]$.
Example 1 — Black-Scholes Greeks
For a European call: $\Delta=\partial_S V=\Phi(d_1)$ (delta), $\Gamma=\partial_{SS}V=\phi(d_1)/(S\sigma\sqrt{T})$ (gamma), $\Theta=\partial_t V=-S\phi(d_1)\sigma/(2\sqrt{T})-rKe^{-rT}\Phi(d_2)$ (theta). Delta-hedging: hold $\Delta$ shares and $V-\Delta S$ in bonds to form a riskless portfolio.
Example 2 — Kalman-Bucy Filter
State: $dX_t=A X_t\,dt+\sigma_X\,dW_t^X$. Observation: $dY_t=C X_t\,dt+\sigma_Y\,dW_t^Y$. The optimal filter $\hat{X}_t=E[X_t|\mathcal{F}_t^Y]$ satisfies $d\hat{X}_t=(A-P_t C^2/\sigma_Y^2)\hat{X}_t\,dt+(P_t C/\sigma_Y^2)\,d\nu_t$ where $\nu_t=Y_t-\int_0^t C\hat{X}_s\,ds$ is the innovations process and $P_t$ solves the Riccati ODE $\dot{P}=2AP-P^2C^2/\sigma_Y^2+\sigma_X^2$.
Practice
- Verify that $V=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2)$ satisfies the Black-Scholes PDE at expiry.
- Explain put-call parity $C-P=S_0-Ke^{-rT}$ using risk-neutral pricing.
- State what the innovations process $\nu_t$ represents and why it is a martingale.
Show Answer Key
1. At expiry $T$: $V(S,T)=\max(S-K,0)$. For $S>K$, $\Phi(d_1)\to1$, $\Phi(d_2)\to1$, so $V\to S-K$. For $S 2. By risk-neutral pricing: $C=e^{-rT}E^Q[\max(S_T-K,0)]$ and $P=e^{-rT}E^Q[\max(K-S_T,0)]$. Then $C-P=e^{-rT}E^Q[S_T-K]=e^{-rT}(S_0 e^{rT}-K)=S_0-Ke^{-rT}$. 3. The innovations process $\nu_t=dY_t-\hat{h}_t\,dt$ is the difference between the observed signal increment and its conditional expectation. It is a martingale because it captures only the 'new information' not already in the filtration.