Financial Mathematics

The daily trading volume of financial derivatives exceeds $6 trillion. The entire industry is built on stochastic process mathematics: the Black-Scholes formula, martingale measures, the Ito integral. Understanding this mathematics opens doors to quantitative finance-one of the highest-paid applications of mathematics.

• **Quantitative Finance**-option pricing, Value at Risk (VaR), credit risk models-all built on stochastic processes
• **Crypto exchanges**-BTC/ETH options use Black-Scholes-style models adjusted for heavy tails
• **Insurance**-valuing long-term liabilities (pension funds, life insurance) via interest rate models (Vasicek, CIR)

Предварительные знания

• Ito Stochastic Integrals

The Black-Scholes Model

In 1973 Fischer Black, Myron Scholes, and Robert Merton solved the problem: how should an option be priced-the right (but not the obligation) to buy a stock in T days at price K? Their model assumes the stock price S(t) follows geometric Brownian motion: dS = μS dt + σS dB. The key insight was the **hedge argument**: by holding a portfolio of stock and option, all risk is eliminated and earn the risk-free rate.

**Black-Scholes model:** dS = μS dt + σS dB (stock price) **Black-Scholes PDE:** ∂V/∂t + (1/2)σ²S²∂²V/∂S² + rS∂V/∂S - rV = 0 **European Call formula:** C = S·N(d₁) - K·e^{-rT}·N(d₂) d₁ = (ln(S/K) + (r + σ²/2)T) / (σ√T) d₂ = d₁ - σ√T where r is the risk-free rate and N is the standard normal CDF.

The Black-Scholes PDE is a heat equation in financial variables. The substitution X = ln(S/K) converts it to classical diffusion. The boundary condition is the option payoff at expiry: V(S,T) = max(S-K, 0) for a call. This yields a closed-form formula-rare in financial mathematics.

The Risk-Neutral Measure

The central idea of financial mathematics: the price of any derivative = discounted expectation of its payoff under the **risk-neutral measure** Q. This is not the real-world measure P (where the stock grows at rate μ) but an auxiliary measure under which all assets grow at the risk-free rate r. The change of measure P → Q is carried out via Girsanov's lemma.

**Risk-neutral measure Q:** There exists a measure Q equivalent to P such that: - Under Q: dS = rS dt + σS dB^Q (μ replaced by r) - Discounted price e^{-rt}S(t) is a martingale under Q - Price of derivative: V(0) = e^{-rT} E^Q[Payoff(S(T))] **Girsanov's lemma:** B^Q(t) = B(t) + ((μ-r)/σ)t is a BM under Q. dQ/dP = exp(-θ²T/2 - θB(T)), θ = (μ-r)/σ (market price of risk).

**Fundamental theorem of asset pricing:** A market is arbitrage-free if and only if a risk-neutral measure Q exists. A market is complete (every derivative is hedgeable) if and only if Q is unique. Black-Scholes is a complete market with a unique Q. Real markets (with jumps, stochastic volatility) are incomplete; Q is not unique.

Option Pricing

Beyond European calls and puts there are hundreds of option types. **Binary** options (pay $1 if S(T) > K), **barrier** options (knocked out when the barrier is crossed), **Asian** options (payoff based on average S)-for each type the risk-neutral approach gives a formula or pricing algorithm.

**Option types and pricing methods:** **European:** closed-form formula (Black-Scholes) **American:** no closed form; binomial trees or numerical PDE **Barrier:** conditional expectations; some have closed forms **Asian:** Monte Carlo or approximations (average is not lognormal) **Lookback:** depends on min/max of path; Monte Carlo General principle: V = e^{-rT} E^Q[Payoff(path)]

**Implied volatility:** Markets trade options at prices that correspond to different σ for different K and T-the "volatility smile/skew". This violates Black-Scholes assumptions (constant σ) and signals that real dynamics are more complex: there are jumps, stochastic volatility, and heavy tails.

Delta Hedging

An option seller has taken on the risk of paying max(S(T)-K, 0). How do they protect themselves? **Delta hedging**: hold Δ = ∂C/∂S shares for each option sold. When the stock moves, the share position offsets the change in option value. This continuous rebalancing is the source of the option seller's income-they collect the option's "time value".

**Delta hedging:** Δ = ∂C/∂S = N(d₁) (for a European Call) Portfolio: short 1 option + long Δ shares P&L = C(S+dS) - C(S) - Δ·dS ≈ 0 (first order) **Gamma** (Γ = ∂²C/∂S²): second derivative-measures how fast the hedge changes. **Hedging P&L** ≈ (Γ/2)(dS)² - Θ dt where Θ = -∂C/∂t (time decay).

**Delta hedging is a continuous process.** With discrete rebalancing, "hedging errors" accumulate-the quadratic variation of the error ≈ (Γ/2)·σ²S²·dt. This explains why option sellers profit through "gamma trading": they sell time value (theta) and hedge the delta, earning on gamma.

Key Ideas

• **Black-Scholes model**-GBM for stock price; formula C = S·N(d₁) - Ke^{-rT}·N(d₂); price is independent of μ
• **Risk-neutral measure Q**-all assets grow at rate r; V = e^{-rT}E^Q[Payoff]; change of measure via Girsanov
• **Option pricing**-European: closed form; American/Asian/Barrier: Monte Carlo or PDE
• **Delta hedging**-hold Δ=N(d₁) shares; eliminates linear risk; gamma risk remains

Related Topics

Financial mathematics is the core application of stochastic processes:

• Brownian Motion and Ito Integral — GBM and Ito's formula are the foundation of the Black-Scholes derivation
• Martingales — The discounted price is a martingale under Q; optional stopping theorem = no-arbitrage
• MCMC and Sampling — Monte Carlo option pricing is a direct application of sampling methods

Вопросы для размышления

• Black-Scholes assumes constant volatility. The market shows a "smile": σ(K) is higher for out-of-the-money options. What does this reveal about the real distribution of S(T)?
• In 2008, asset correlations spiked dramatically. Why does this break delta-hedging assumptions?
• Delta of an at-the-money option is ≈ 0.5 when S = K. Why is 0.5 intuitively the right answer?

Связанные уроки

• calc-25-green-theorem