波动率的三类模型
波动率模型主要有以下三类
- Local volatility, where the forward/spot price of the underlying solves the SDE dF = sigma(F)F dW for a (deter-
ministic) function sigma. A special model is the CEV model with sigma(F) = sigma_beta * F ^(beta-1). Problems are that either one use a fully calibrated model (Dupire's formula) but one no longer has closed formula and the resulting
volatility is unstable; or if one use a given model (such as the CEV model), it is hard to calibrate it to the market smile. In addition, the sticky Greek phenomenon is ampli cated. And the dynamics of the smile is wrong.
- Jump models, where the underlying solves a jump-di usion equation. Jump models fit the smile very well
in the short-term (because the impact of the jump is essentially in the short run). So they are relevant
for short term options. But, they are not satisfactory for long-term options because there is no volatility
risk (the volatility remains deterministic).
- Stochastic volatility models assume that the volatility is a di ffusion process correlated to the spot. It is
easily calibrated and allows for a satisfactory handling of the volatility risk. It is less satisfactory for
short-term options.
