Probability Theory

Stochastic Finance and the Black-Scholes Formula

Цели урока

  • Understand the geometric Brownian motion model for asset prices
  • Master the Girsanov theorem and the change of measure P to Q
  • Derive the Black-Scholes formula and interpret the option Greeks
  • Connect the BS model to delta-hedging and the partial differential equation

Предварительные знания

  • Brownian motion and Ito stochastic calculus
  • Martingales and the optional stopping theorem
  • Girsanov theorem on change of measure
  • Conditional expectation
  • Large Deviations Theory

Fischer Black, Myron Scholes, and Robert Merton solved in 1973 a problem the entire finance community had failed to crack: how to fairly price an option.

  • **CBOE:** daily options market turnover exceeds 1.2 trillion dollars - every trade relies on BS or its extensions
  • **Delta-hedging:** market-makers continuously rebalance portfolios by delta, creating synthetic options
  • **Risk management:** the Greeks (delta, gamma, vega) are used by banks for aggregated market risk control
  • **Volatility as an asset:** VIX 'fear index' is computed from market option prices by inverting BS

Geometric Brownian Motion and BS Derivation

In 1973 Fischer Black, Myron Scholes, and Robert Merton cracked a problem the entire finance community had failed to solve: how to fairly price an option. The formula earned Scholes and Merton the 1997 Nobel Prize. CBOE's daily options turnover exceeds 1.2 trillion dollars - every trade rests on BS or its extensions.

The BS PDE is the heat equation in reverse time with financial boundary conditions. The terminal condition V(S,T) = max(S-K, 0) for a call sets the initial data for the heat equation.

Why does stochastic calculus introduce the -sigma^2/2 correction in the solution of GBM?

Risk-Neutral Measure and Girsanov's Theorem

The central technical trick of BS is the change of measure. Under the physical measure P the stock grows at rate mu, under the risk-neutral measure Q at the risk-free rate r. Girsanov's theorem gives the explicit re-parametrization of Brownian motion. This makes the option price equal to the discounted expected payoff under Q - and in that expectation mu simply vanishes.

The equivalent martingale measure (EMM) is a central concept of modern mathematical finance. The Harrison-Pliska theorem (1979) formalizes: a market is free of arbitrage iff an EMM exists.

Why does the drift mu not appear in the Black-Scholes option price?

Option Greeks and Hedging

The Greeks are derivatives of the option price with respect to its parameters. Delta = partial_S C is the number of shares to hedge; gamma is the sensitivity of delta; vega is the response to volatility. In 1994 Scholes and Merton co-founded the hedge fund LTCM on these ideas, but in 1998 the fund collapsed with losses of 4.6 billion dollars: vega exposure and tail risks materialized under market stress that the BS model could not predict.

BS assumes constant volatility, no transaction costs, and continuous trading. In practice sigma is stochastic (Heston, SABR), jumps are modeled by jump-diffusion (Merton). The LTCM hedge collapsed in 1998 under stress scenarios the model never anticipated.

Black-Scholes bridges stochastic analysis and finance

The formula unites Brownian motion, the Girsanov theorem, partial differential equations, and martingale theory.

  • Brownian motion — Geometric BM is the basic model: stock price as the exponential of a Brownian motion with drift
  • Martingales — The discounted price under the risk-neutral measure Q is a martingale; the key no-arbitrage condition
  • Partial differential equations — The BS PDE reduces to the heat equation; the analytic solution yields the formula via Phi
  • Large deviations — Tail risks and the volatility smile are explained by deviations of real distributions from log-normal

Итоги

  • **GBM:** dS = mu*S dt + sigma*S dW; solution S_t = S_0*exp((mu - sigma^2/2)t + sigma*W_t)
  • **Girsanov:** change of measure P to Q removes mu, replacing it with r - hence mu's absence from the formula
  • **BS formula:** C = S_0*Phi(d_1) - K*e^{-rT}*Phi(d_2); parameters only r, sigma, T, K/S_0
  • **Delta-hedging:** Delta = Phi(d_1) - number of shares needed to replicate the option
  • **Greeks:** vega > 0, theta < 0; gamma is curvature; full P&L decomposition by risk factors
  • **Limitations:** constant sigma is an idealization; real markets exhibit smiles and stochastic volatility

What does positivity of vega (V > 0) mean for an option?

Stochastic Finance and the Black-Scholes Formula

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