1. What does it mean by the vol surface is the current view of vol?
The local volatility model is calibrated to vanillas prices (and equivalently their implied volatilities), which reflect the market's view of the volatility, in order to use it to use it to price other options that one will hedge with the vanillas.
Where a Black-Scholes model (no smile) will not be able to match the options implied volatilities at all strikes (smile). Local volatility models will.
Given a continuous surface of call options prices, that is twice-differentiable in strike and once in time, Dupire's formula gives the unique risk-neutral diffusion (no jumps) process that is compatible with european option prices:
- $dS_t = (r − q)S_t dt + σ(t, S_t)S_tdW_t$
- with: $\sigma(t, S)^2 = 2 \frac{\frac{\partial C}{\partial T} + qC +(r-q)K\frac{\partial C}{\partial K}}{K^2\frac{\partial^2C}{\partial K^2}} |_{K = S, T = t}$
- where $r$ is the interest rate, $q$ the div yield, $C$ the function giving the call price, $K$ the strike and $T$ the expiry.
For more info, see Dupire's and Derman and Kani's seminal papers:
2. Why is this better if you use stochastic volatility models instead?
The local volatility models will be able to match the value of the smile as of today, but because the smile flattens for long maturities, the model gives an almost constant smile for these maturities, leading to a flattening of the forward smile (i.e. smile in the future), which is unrealistic.
This is not desirable when the exotic option you are concerned with depends on the forward smile (e.g. ratchet option). In this case, one needs a model which will give realistic smile dynamics.
Stochastic volatility models give more realistic dynamics of the volatility smile. However, they come with their issues/challenges.
For example, they may be harder to calibrate than local vol models. Furthermore, they may sometimes not exhibit enough smile for options with short maturities. To overcome this second issue, stochastic volatility models are either:
- combined with jumps in the underlying.
- combined with local volatility (local-stochastic vol models).