Stochastic Methods in Finance

Mathematical financial theory uses stochastic calculus for derivative pricing, risk management, and optimal investing. Girsanov's theorem and the Black-Scholes formula are central results.

Girsanov's Theorem and the Risk-Neutral Measure

The problem: In the real world, a stock grows at a rate μ > r (risk-free rate). For pricing, a "risk-neutral" measure Q is needed.

Girsanov's theorem: When switching measure from P to Q via $dQ/dP = e^{-\theta W_T - \theta^2 T/2}$: $\tilde{W}_t = W_t + \theta t$ is a Q-Brownian motion. For GBM with μ: $dS = \mu S,dt + \sigma S,dW \rightarrow dS = r S,dt + \sigma S,d\tilde{W}$ (at $\theta=(\mu - r)/\sigma$).

Pricing: $V_0(F) = e^{-rT} \mathbb{E}^Q[F_T]$ — price of a derivative with payoff $F_T$. This follows from the absence of arbitrage: the discounted price is a Q-martingale.

The Black-Scholes Formula

European call: $C_0 = \mathbb{E}^Q\left[e^{-rT} \max(S_T - K, 0)\right]$. For $S_T \sim \text{LN}$: $C_0 = S_0 N(d_1) - K e^{-rT} N(d_2)$. $d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$, $d_2 = d_1 - \sigma\sqrt{T}$.

Dynamic hedging: $\Delta = \partial C/\partial S = N(d_1)$. Continuous rebalancing of Δ shares in the stock plus risk-free asset → exact replication.

Stochastic Optimization (Merton)

The problem: Investor maximizes $\mathbb{E}[u(W_T)]$. Wealth dynamics: $dW = (r + \pi(\mu - r))W,dt + \pi W \sigma,dW$. $\pi$ — proportion in the stock.

Solution (CARA $u = -e^{-\gamma W}$): Optimal $\pi^* = \frac{\mu - r}{\gamma \sigma^2 W}$ — constant proportion of assets in the stock. The "Merton rule".

Hamilton-Jacobi-Bellman equation: $-V_t + r W V_W - \frac{(\mu - r)^2}{2\sigma^2} \frac{(V_W)^2}{V_{WW}} = 0$. Nonlinear PDE for the value function $V(W, t)$.

Exercise: (a) For BS formula with $S_0=100$, $K=100$, $T=1$, $r=5%$, $\sigma=20%$: compute $C_0$, $\Delta$, Vega ($\partial C/\partial \sigma$). (b) If $\sigma$ increases to $25%$: new price? Explain using Vega. (c) Merton portfolio: $\gamma=2$, $\mu=10%$, $r=5%$, $\sigma=20%$. Find $\pi^*$, $\mathbb{E}[W_T]/W_0$ for $T=5$.

Option Greeks: Full Table

All partial derivatives of the option price — "Greeks". Delta $\Delta = \partial C/\partial S = \Phi(d_1)$. Gamma $\Gamma = \partial^2 C/\partial S^2 = \varphi(d_1)/(S \sigma \sqrt{T})$: rate of change of delta. Theta $\Theta = \partial C/\partial t = -S \varphi(d_1)\sigma/(2\sqrt{T}) - r K e^{-rT}\Phi(d_2)$: time decay. Vega $\nu = \partial C/\partial \sigma = S \varphi(d_1)\sqrt{T}$: sensitivity to volatility. Rho $\rho = \partial C/\partial r = K T e^{-rT} \Phi(d_2)$.

Gamma-theta relationship: for a portfolio replicating a call: $\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma + rS\Delta = rC$. High gamma is compensated by high time decay theta.

Implied Volatility (IV) and Volatility Smile

IV — the value of $\sigma$ at which the BS formula matches the market price. IV $\neq$ realized volatility. Volatility smile: IV depends on strike $K$ — violation of BS assumptions. In reality: IV is higher for OTM options (tail risk). Volatility surface: IV$(K,T)$ — also depends on $T$ (term structure of volatility).

Post-BS models: stochastic volatility (Heston), local volatility (Dupire), jump model (Merton). Each calibrates to the market IV surface.

Merton Portfolio and Utility Maximization Problem

In continuous time: the investor maximizes $\mathbb{E}[\int_0^T e^{-\rho t}U(c_t),dt + B(W_T)]$. Hamilton-Jacobi-Bellman (HJB) equation: $-\partial V/\partial t = \max_{\pi,c} {U(c) + \partial V/\partial W (\pi\sigma(\mu - r) + r)W - c + \frac{1}{2}(\pi \sigma W)^2 \partial^2 V/\partial W^2}$. For power utility $U(W) = W^{1-\gamma}/(1-\gamma)$: optimal share $\pi^* = (\mu-r)/(\gamma \sigma^2)$ — constant (Merrill effect). Consumption $c^* = \rho/\gamma \cdot W$ — proportional to wealth.

Stochastic Control and the Principle of Dynamic Programming

Bellman's principle: $V(x, t) = \max_u \mathbb{E}\left[\int_t^T r(X_s, u_s),ds + g(X_T) | X_t = x\right]$. HJB equation: $0 = \partial V/\partial t + \max_u { r(x, u) + L^u V}$, where $L^u$ is the generator. For linear-quadratic control (LQR): $V = x^\top P(t)x + q(t)$, $P$ satisfies the matrix Riccati equation.

The Markowitz Portfolio Problem in Continuous Time

Merton problem (1969): Maximize $\mathbb{E}[U(W_T)]$ under dynamic management. For logarithmic utility $U(W) = \ln(W)$: $\pi^* = (\mu - r)/\sigma^2$ (risk-free asset $r$, risky with drift $\mu$, volatility $\sigma$). This is the "Kelly criterion" in continuous time. The "equal weight" strategy is close to Kelly when parameters are unknown.

Extreme Value Theory in Finance

Generalized extreme value (GEV) distribution: the maximum of $n$ i.i.d. variables after normalization → Gumbel, Weibull, Fréchet. For heavy tails (Pareto, t-distribution): limit is Fréchet. Peaks-over-threshold (POT) models: exceedances over high threshold $u \sim$ GPD (generalized Pareto distribution). Shape parameter $\xi > 0$ — heavy tail. Applied to estimation of extreme risks: 100-year losses, flooding, credit losses.

Credit Risk and Structural Models

Merton model (1974): company assets $A_t \sim$ GBM. Default occurs at $A_T < D$ (debt threshold). Equity: $E = \text{Call}(A_T, D)$. Default probability: $P(A_T < D) = N(-d_2)$, where $d_2$ is as in B-S. Distance-to-Default $= \frac{\ln(A/D) + (\mu - \sigma^2/2)T}{\sigma\sqrt{T}}$. KMV model (Moody's): extension of Merton with market value of assets estimated via iterative algorithm. Limitation: assets are unobservable, $\sigma_A$ is iteratively inferred from $\sigma_E$.

Numerical Example: Option Pricing by Black-Scholes

Task: European call option: $S_0=100$, $K=100$, $r=0.05$, $\sigma=0.20$, $T=1$ year.

Step 1: $d_1 = [\ln(S_0/K) + (r + \sigma^2/2)T]/(\sigma\sqrt{T}) = [\ln(1) + (0.05 + 0.02)\cdot 1]/0.20 = 0.07/0.20 = 0.350$.

Step 2: $d_2 = d_1 - \sigma\sqrt{T} = 0.350 - 0.20 = 0.150$.

Step 3: $N(0.350) \approx 0.6368$, $N(0.150) \approx 0.5596$. Call price: $C = 100 \cdot 0.6368 - 100 \cdot e^{-0.05} \cdot 0.5596 = 63.68 - 100 \cdot 0.9512 \cdot 0.5596 = 63.68 - 53.22 \approx 10.46$ rubles.

Step 4: Greeks: $\Delta = N(d_1) = 0.637$ (change in $C$ for $S$ up by 1 ruble); Vega $= S_0 \sqrt{T} \cdot \varphi(d_1) = 100 \cdot 1 \cdot 0.378 \approx 37.8$ (change in $C$ for $\sigma$ up by 1). Merton model: Credit Spread $= -(1/T) \cdot \ln(N(d_2) + D/A \cdot N(-d_1)) \approx 0.14%$ at $A/D = 2$.