Local and stochastic volatility

Volatility is not a constant

Black-Scholes uses one constant $\sigma$. If it were right, every option on a stock would have the same implied volatility. In real markets it does not: implied volatility changes with the strike, the smile or skew, and with the maturity. Two ideas fix this. In local volatility the number $\sigma(S,t)$ is a function of price and time, and Dupire's formula picks that function so the model matches today's whole surface of option prices. In stochastic volatility the variance $v$ is itself a random process. The best-known example is the Heston model, where the variance is pulled back to a long-run level.

In Heston the variance is a CIR process, so everything from the last topic applies: it is pulled to $\theta$ at speed $\kappa$ and stays positive if $2\kappa\theta\ge\xi^{2}$. The correlation $\rho$ links price shocks and variance shocks. For stocks it is negative: when the price falls, volatility rises. This creates the skew, with higher implied volatility for low strikes. The two approaches differ in what they say about the future. In local volatility the volatility is a function of the price, so it moves in lockstep with it. In stochastic volatility it has its own shocks, which is closer to what markets show.

• Smile and skew: implied volatility differs by strike and maturity, so one constant $\sigma$ cannot fit.
• Local volatility: $\sigma(S,t)$ is a function of price and time, fitted to today's option surface.
• Stochastic volatility: the variance has its own shocks, and $\rho<0$ gives the skew of stocks.

Lab · what's graded

• The long-run volatility15 pts
• The expected variance15 pts
• The Feller margin15 pts
• The sign of the correlation15 pts
• Match the vocabulary15 pts
• Building a local volatility model10 pts
• Why one volatility is not enough15 pts