Local and Stochastic Volatility Models for Exotic Option Pricing

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


# 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?

## Answer by byouness (score 19, accepted)

https://quant.stackexchange.com/a/39665

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 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:



- The Volatility Smile and Its Implied Tree: http://www.cmat.edu.uy/~mordecki/hk/derman-kani.pdf

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).

## Answer by vonjd (score 9)

https://quant.stackexchange.com/a/39657

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.

The following questions (and answers therein) may also be helpful:

- Local Volatility vs. Stochastic Volatility

- For pricing, what types of Exotic Options are suitable using Local Volatility Model or a Stochastic Volatility Model?

## Answer by Alistar (score 5)

https://quant.stackexchange.com/a/42976

The following source contains detailed answers to your questions in a research paper from ETH Zürich. van der Weijst, Roel (2017). "Numerical Solutions for the Stochastic Local Volatility Model" http://resolver.tudelft.nl/uuid:029cbbc3-d4d4-4582-8be2-e0979e9f6bc3