Monte Carlo Simulation Of Heston Model In Matlab(1)

Monte Carlo Simulation of Heston Model in MATLAB GUI and its Application to Options BACHELOR THESIS IN MATHEMATICS /APPLIED MATHEMATICS DEPARTMENT OF MATHEMATICS AND PHYSICS MÄLARDALEN UNIVERSITY Author Amir Kheirollah Supervisor Robin Lundgren Examiner Dmitrii Silvestrov

Abstract ,[object Object],[object Object],[object Object],[object Object],[object Object]

Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Theoretical Background ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Monte Carlo Simulation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],The method samples randomly from a universe of possible outcomes and takes the fraction of random draws that fall in a given set as an estimate of the set’s measure. The Law of Large Numbers guarantees the convergence of the estimation to the correct value as the number of draws increases.

An Overview on Black-Scholes Model ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Heston Model – Genral Form of SDEs ,[object Object],[object Object],[object Object],[object Object],(1) (2)

The Heston Model ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],(3) (4)

The Heston Model ,[object Object],[object Object],[object Object],W 1 and W 2 In equations (3) and (4) Replaced by Where epsilons are correlated (5) (6)

The Heston Model – Discretization Versus Randomization ,[object Object],[object Object],[object Object],ε = Ax + µ (7)

The Heston Model – Mean Reversion ,[object Object],[object Object],[object Object]

The Heston Model – Application of Monte Carlo Simulation and Cholesky Decomposition ,[object Object],[object Object]

The Heston Model – Application of Monte Carlo Simulation and Cholesky Decomposition ,[object Object],[object Object],[object Object],[object Object],[object Object]

Options under Study ,[object Object],[object Object],[object Object],[object Object],[object Object],1) 2) 3) where

Empirical Investigation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Empirical Investigation – The Table Table 1: Data for Empirical Investigation Category Number of Iterations Black-Scholes (BLS) Price Heston Model Price Heston Model STD Heston Model MIN Heston Model MAX Heston Model COV Heston Model’s Relative Error from BLS 1 10 11,0666 11,3257 1,2997 7,8744 14,0729 1,6893 0,023413 2 100 11,0666 11,0667 0,4381 10,0278 12,2727 0,1919 9,04E-06 3 1000 11,0666 11,0632 0,1146 10,6927 11,2826 0,0131 0,000307 4 5000 11,0666 11,067 0,0567 10,9516 11,2267 0,0032 3,61E-05 5 10000 11,0666 11,0568 0,0402 10,9635 11,1486 0,0016 0,000886 6 25000 11,0666 11,0599 0,023 11,0006 11,1159 5,30E-04 0,000605 7 50000 11,0666 11,0601 0,0156 11,0249 11,0929 2,44E-04 0,000587

Empirical Investigation ,[object Object],[object Object],[object Object],[object Object],European call option priced by both methods. As number of iterations increases the price of Heston Model converges To price obtaind by Black-Scholes.

Empirical Investigation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Standard Deviation

Empirical Investigation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Relative Error of Heston Model from Black-Scholes Model Covariance Between Prices Respective Number of Iterations.

Empirical Investigation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Empirical Investigation Data for Empirical Investigation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],10000 Iterations Price STD MIN MAX Asian Option 11,03488 0,025823128 10,972 11,095 Look Back Option 14,06824 0,028532498 14,001 14,139 European Call Option 11,05677 0,04028882 10,963 11,149

MATLAB GUI Manual – Graphical Representation Opening window of the program HestonGUI.

MATLAB GUI Manual – User Manual The pattern considered in example.

MATLAB GUI Manual – User Manual ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

MATLAB GUI Manual – Error-dialogue Windows Error-dialogue declaring the input can not be negative or non-numeric. Error-dialogue declaring the input can not be taken other than the interval. Error-dialogue declaring the input can not take zero, non-numeric or negative values. Error-dialogue windows of the program.

Conclusion ,[object Object],[object Object],[object Object],[object Object]

Monte Carlo Simulation Of Heston Model In Matlab(1)

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