Explain the stochastic volatility in an option-pricing

Question

Explain the stochastic volatility in an option-pricing.

Answer 1

Two sources of randomness are in there stochastic volatility models, the stock return and the volatility also. One of the parameters in such models is the correlation among the two sources of randomness. Its correlation is classically negative so that a fall into the stock price is frequently accompanied by an increase in volatility. It results in a negative skew for implied volatility. Unluckily, this negative skew is not typically as pronounced like the real market skew. Such models can also describe the smile. Like a rule one pays for convexity. We notice this in the simple Black–Scholes world where we pay for gamma. Within the stochastic volatility world we can look at the second derivative of option value respecting volatility, and when it is positive we would expect to have to pay for such convexity – it is, option values will be relatively higher wherever such quantity is major. For a call or put in the world of constant volatility then we have

∂²V/∂σ² = S√(T - t) ((d₁d₂ e^-D(T-t) e^-d2₁^/2)/√(2Π) σ)

This function is plotted in given figure for S = 100, T − t = 1, σ = 0.2, r = 0.05 and D = 0. Observe how this is positive away by the money, and small at the money. (Certainly, this is a bit of a cheat since on one hand I am talking regarding random volatility and even using a formula which is only correct for constant volatility.)

Stochastic volatility models have superior potential for capturing dynamics, although the problem, as always, knows that stochastic volatility model to select and how to determine its parameters. When calibrated to market prices you will even usually determine that supposed constant parameters in your model stay changing.

Figure: ∂²V/∂σ² versus strike.

This is frequently the case with calibrated models and suggests as the model is still not correct, even if its complexity seems to be very promising.