The vanna - volga method for derivatives pricing.

This Master thesis highlights some basic features and applications of the vanna-volga method and its accuracy when pricing plain vanillas and simple barrier options. In the paper we derive formulas for premiums of vanilla FX options using two versions of the vanna-volga method – the exact vanna-volga method and the simplified vanna-volga method. We review a very common vanna-volga variation used to price the first-generation exotics and the application of the vanna-volga method to construct the implied volatility surface. Furthermore, we briefly discuss a popular stochastic volatility model that aims to take the smile effect into account – the Heston model. Its accuracy and efficiency is further compared with that of the vanna-volga method. In the part of the thesis, which is devoted to calibration results, we compare the results obtained by the exact vanna-volga method, the simplified vanna-volga method and the Heston model. We also investigate the accuracy of the vanna-volga method applied to barrier options. All the plots and graphs in this thesis were produced by programs implemented by the author in MATLAB. These programs are available on request.vanna- volga method; implied volatility; volatility smile; Heston model

FX Smile in the Heston Model

The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is non-negative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options. In this article we adapt the original work of Heston (1993) to a foreign exchange (FX) setting. We discuss the computational aspects of using the semi-analytical formulas, performing Monte Carlo simulations, checking the Feller condition, and option pricing with FFT. In an empirical study we show that the smile of vanilla options can be reproduced by suitably calibrating three out of five model parameters.Comment: Chapter prepared for the 2nd edition of Statistical Tools for Finance and Insurance, P.Cizek, W.Haerdle, R.Weron (eds.), Springer-Verlag, forthcoming in 201

FX Smile in the Heston Model

The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is nonnegative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options. In this article we adapt the original work of Heston (1993) to a foreign exchange (FX) setting. We discuss the computational aspects of using the semi-analytical formulas, performing Monte Carlo simulations, checking the Feller condition, and option pricing with FFT. In an empirical study we show that the smile of vanilla options can be reproduced by suitably calibrating three out of five model parameters.Heston model; vanilla option; stochastic volatility; Monte Carlo simulation; Feller condition; option pricing with FFT;

The vanna - volga method for derivatives pricing.

This Master thesis highlights some basic features and applications of the vanna-volga method and its accuracy when pricing plain vanillas and simple barrier options. In the paper we derive formulas for premiums of vanilla FX options using two versions of the vanna-volga method – the exact vanna-volga method and the simplified vanna-volga method. We review a very common vanna-volga variation used to price the first-generation exotics and the application of the vanna-volga method to construct the implied volatility surface. Furthermore, we briefly discuss a popular stochastic volatility model that aims to take the smile effect into account – the Heston model. Its accuracy and efficiency is further compared with that of the vanna-volga method. In the part of the thesis, which is devoted to calibration results, we compare the results obtained by the exact vanna-volga method, the simplified vanna-volga method and the Heston model. We also investigate the accuracy of the vanna-volga method applied to barrier options. All the plots and graphs in this thesis were produced by programs implemented by the author in MATLAB. These programs are available on request

The vanna - volga method for derivatives pricing.

This Master thesis highlights some basic features and applications of the vanna-volga method and its accuracy when pricing plain vanillas and simple barrier options. In the paper we derive formulas for premiums of vanilla FX options using two versions of the vanna-volga method – the exact vanna-volga method and the simplified vanna-volga method. We review a very common vanna-volga variation used to price the first-generation exotics and the application of the vanna-volga method to construct the implied volatility surface. Furthermore, we briefly discuss a popular stochastic volatility model that aims to take the smile effect into account – the Heston model. Its accuracy and efficiency is further compared with that of the vanna-volga method. In the part of the thesis, which is devoted to calibration results, we compare the results obtained by the exact vanna-volga method, the simplified vanna-volga method and the Heston model. We also investigate the accuracy of the vanna-volga method applied to barrier options. All the plots and graphs in this thesis were produced by programs implemented by the author in MATLAB. These programs are available on request

HESTONFFTVANILLA: MATLAB function to evaluate European FX option prices in the Heston (1993) model using the FFT approach of Carr and Madan (1999).

HESTONFFTVANILLA returns the price of a European Call or Put option given spot price S, strike K, time to maturity (in years) T, domestic R and foreign RF interest rates, rate of mean reversion KAPPA, average level of volatility THETA, volatility of volatility SIGMA, correlation between the Wiener increments driving the spot and vol processes RHO and initial volatility VO.Option premium, FX option, Stochastic volatility, Heston (1993) model, Carr and Madan (1999) FFT approach.

STF2HES: MATLAB functions for "FX smile in the Heston model"

These functions are Matlab implementations of the concepts brought forward in Chapter 4 "FX smile in the Heston model" of "Statistical Tools for Finance and Insurance (2nd ed.)" edited by P.Cizek, W.Haerdle and R.Weron, published by Springer, 2011. The zip file includes 9 functions: GarmanKohlhagen.m, HestonFFTVanilla.m, HestonVanilla.m, HestonVanillaFitSmile.m, HestonVanillaLipton.m, HestonVanillaSmile.m, pdfHeston.m, simGBM.m, simHeston.m. For sample applications see the STF2HES_EX.zip example scripts at http://ideas.repec.org/c/wuu/hscode/zip10001.html.Option premium, FX option, Volatility smile, Stochastic volatility, Heston (1993) model, Carr and Madan (1999) FFT approach, Lipton (2002) approach, Garman and Kohlhagen (1983) model, Calibration

GARMANKOHLHAGEN: MATLAB function to evaluate European FX option prices in the Garman and Kohlhagen (1983) model.

GARMANKOHLHAGEN returns FX option price, (spot) delta or strike depending on the value of the TASK (= 0,1,2) parameter in the Garman and Kohlhagen (1983) option pricing model. The remaining input parameters are: FX spot S, strike/spot delta K, volatility VOL, domestic and foreign riskless interest rates RD and RF (annualized), time to expiry (in years) TAU and option type (Call/Put).Option premium, FX option, Garman and Kohlhagen (1983) model.

HESTONVANILLASMILE: MATLAB function to compute the volatility smile implied by the Heston (1993) option pricing model.

HESTONVANILLASMILE returns a vector of volatilities given a vector of strikes STRIKES, spot price SPOT, initial volatility V0, vol of vol VV, domestic and foreign interest rates RD and RF, time to maturity (in years) TAU, mean reversion KAPPA, long-run mean THETA, market price of risk LAMBDA, correlation RHO and option type (Call/Put).Option premium, FX option, Volatility smile, Stochastic volatility, Heston (1993) model, Garman and Kohlhagen (1983) model.

SIMHESTON: MATLAB function to simulate trajectories of the spot price and volatility processes in the Heston (1993) model.

SIMHESTON returns a 2-column array, containing the simulated trajectories of the spot price S(t) and volatility v(t) for t=0:DELTA:N, in the model: dS(t) = mu*S(t)*dt + v^0.5*S(t)*dW1(t), dv(t) = kappa*(theta - v(t))*dt + sigma*(v(t)^0.5)*dW2(t), Cov[dW1(t),dW2(t)] = rho*dt, given starting value of the spot price process S0, starting value of the volatility process V0, drift MU, speed of mean reversion of the volatility process KAPPA,long-term mean of the volatility process THETA, volatility SIGMA, correlation between the spot price and volatility processes RHO, time endpoint N, a 2-column vector of normally distributed pseudorandom numbers NO and a flag denoting used simulation scheme (Quadratic-Exponential scheme, Euler scheme with absorption or reflection for the volatility process).FX option, Stochastic volatility, Heston (1993) model, Sample trajectory, Quadratic-Exponential scheme, Euler scheme, absorption, reflection.