Local volatility

Local volatility is a term used in quantitative finance to denote the set of diffusion coefficients, $sigma(S\_T,T)$, that are consistent with the set of market prices for all option prices on a given underlier. This model is used to calculate values of exotic options which are consistent with observed prices of vanilla options.

The concept of a local volatility was originated by Emanuel Derman and Iraj Kani as part of the implied volatility tree model.cite paper | author=Derman, E., Iraj Kani | title=The Volatility Smile and its Implied Tree | publisher=Goldman-Sachs | format=PDF | date=January, 2004 | url=http://www.ederman.com/new/docs/gs-volatility_smile.pdf | accessdate=2007-06-01] As described and implemented by Derman and Kani, the local volatility function models the instantaneous volatility to use at each node in a binomial options pricing model such that the tree will produce a set of option valuations that are consistent with the option prices observed in the market for all strikes and expirations.

The Derman-Kani model was formulated in a binomial model with discrete time and stock-price steps. The key continuous-time equations used in local volatility models were developed by Bruno Dupire in 1994. Dupire's equation states

:$frac\{partial\; C\}\{partial\; T\}\; =\; frac\{1\}\{2\}\; sigma^2(K,T;\; S\_0)K^2\; frac\{partial^2C\}\{partial\; K^2\}$

Local volatility models are useful in any options market in which the underlier's volatility is predominantly a function of the level of the underlier, interest-rate derivatives for example. Time-invariant local volatilities are supposedly inconsistent with the dynamics of the equity index implied volatility surface,cite journal | author=Dumas, B., J. Fleming, R. E. Whaley | title=Implied volatility functions: Empirical tests | journal=The Journal of Finance | volume=53 | date=1998] but see cite journal | author=Crepey, S | title=Delta-hedging Vega Risk | journal=Quantitative Finance | volume=4 | date=2004, who claims that such models provide the best average hedge for equity index options. Local volatility models are nonetheless useful in the formulation of stochastic volatility models.cite book | author=Gatheral, J. | title=The Volatility Surface: A Practioners's Guide | publisher = Wiley Finance | date=2006 | isbn=13 978-0-471-79251-2 ]

Local volatility models have a number of attractive features. Because the only source of randomness is the stock price, local volatility models are easy to calibrate.

In local volatility models the volatility is deterministic function of the random stock price. Local volatility models are not very well used to price cliquet options or forward start options, whose values depend specifically on the random nature of volatility itself.

References

#if: {colwidth|}| style="-moz-column-width:{colwidth}; column-width:{colwidth};" | #if: {1|}| style="-moz-column-count:{1}; column-count:{1} ;" |>