Abstract
The problem of calibration of local volatility model of Dupire has been formalized. It uses genetic algorithm as alternative to regularization approach with further application of gradient descent algorithm. Components that solve Dupire’s partial differential equation that represents dynamics of underlying asset’s price within Dupire model have been built. This price depends in particular on values of volatility parameters. Local volatility is parametrized in two dimensions (by Dupire model): time to maturity of the option and strike price (execution price). On maturity axis linear interpolation is used while on strike axis we use B-Splines. Genetic operators of mutation and selection are then applied to parameters of B-Splines. Resulting parameters allow us to obtain the values of local volatility both in knot points and intermediate points using interpolation techniques. Then we solve Dupire equation and calculate model values of option prices. To calculate cost function we simulate market values of option prices using classic Black-Scholes model. An experimental research to compare simulated market volatility and volatility obtained by means of calibration of Dupire model has been conducted. The goal is to estimate the precision of the approach and its usability in practice. To estimate the precision of obtained results we use a measure based on average deviation of modeled local volatility from values used to simulate market prices of the options. The research has shown that the approach to calibration using genetic algorithm of optimization requires some additional manipulations to achieve convergence. In particular it requires non-uniform discretization of the space of model parameters as well as usage of de Boor interpolation. Value 0.07 turns out to be the most efficient mutation parameter. Using this parameter leads to quicker convergence. It has been proved that the algorithm allows precise calibration of local volatility surface from option prices.
Highlights
Since the famous Black-Scholes model was introduced in 1973, option markets have evolved to become autonomous, organized markets with a fairly high degree of liquidity, especially for index options and foreign exchange options
In the last section we provide numerical results obtained by running the genetic algorithm to calibrate local volatility surface
The result of the calibration is a local volatility surface introduced on Figure 6 and Figure 7
Summary
Since the famous Black-Scholes model was introduced in 1973, option markets have evolved to become autonomous, organized markets with a fairly high degree of liquidity, especially for index options and foreign exchange options. In such markets, the market prices of a series of liquid options, which are often call or put options, are readily observed. In reality the volatility varied among different strikes and maturity dates for benchmark options. It has led to development of generalized models, including local and stochastic volatility models
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