# Problems with local volatility models (vs stochastic volatility models)

Why is pricing with local volatility models are problem with exotics, mainly due to "the volatility surface is the market's current view of volatility and this will change in the future meaning the exotic options will no longer be consistent with market prices" (from Quant Job Interview Questions and Answers)

What does it mean by the vol surface is the current view of vol (I didnt think vol models were predictive of the future anyway) and why is this better if you use stochastic volatility models instead?

• Summaries of the conclusions are best, to get a headline idea. People can dig deeper into individual issues if they want to – Trajan Dec 12 '18 at 17:00

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.

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).
• The leads to a flattening of the forward smile is not obvious why this is true at all – Trajan May 7 '18 at 18:20
• Also it is not clear to me what desirable smile dynamics is – Trajan May 7 '18 at 18:22
• The forward vol/smile are given by the local vol function $\sigma(S,t)$. This function is deterministic and is calibrated on vanilla options prices. Vanilla options implied vol is flat for long maturities (see here for an explanation of this empirical fact: quant.stackexchange.com/questions/16670/…), which gives a flat local vol function for times in the future. – byouness May 7 '18 at 19:07
• Desirable smile dynamics are characteristics that the implied volatility actually exhibits. e.g. at a given time, the vol is decreasing as a function of the option maturity, however for a given option maturity, although the level and orientation of the smile curve might change over time, its rough shape stays very much the same, etc. Everything of course depends on what you want to use the model for. It has to be accurate for the characteristics that your product is sensitive to. – byouness May 7 '18 at 19:21
• It doesn't. The difference is in SV models the vol is random driven by its own Brownian while in LV models it's not, the randomness of vol comes only from the stock, via a deterministic function. SV models give realistic scenarios for the evolution of the vol surface as a result, while LV models don't. – byouness May 11 '18 at 21:13

The following paper is helpful for understanding the point you raise:

Hagan et al.: Managing Smile Risk, January 2002, Wilmott 1:84-108

The main point is given in the paper:

[...] the dynamics of the market smile predicted by local vol models is opposite of observed market behavior: when the price of the underlying decreases, local vol models predict that the smile shifts to higher prices; when the price increases, these models predict that the smile shifts to lower prices. Due to this contradiction between model and market, delta and vega hedges derived from the model can be unstable and may perform worse than naive Black-Scholes’ hedges.

You can find the details on page 5ff.