Abstract
Financial markets are known to be far from deterministic but stochastic and hence time dependent correlation tends to suit the markets. We price for European Options by using three dimensional assets under stochastic correlation. The pricing equations under constant correlation and stochastic correlation are derived numerically by using finite difference method called the Crank Nicolson method. We compare the pricing equations when the correlation is stochastic and constant by using real data from emerging financial markets, that is, exchange rates data for Kenya as the domestic currency and South Africa as the foreign currency. Pricing equation for the European option with stochastic correlation performed better than that with constant correlation.
Highlights
The long history of option pricing began in 1900 when the French mathematician Louis Bachelier deduced an option pricing formula based on the assumption that stock prices followed a Brownian motion with zero drift [1]
Most models used in the pricing of multidimensional derivatives consider constant correlation among their components but empirical facts suggest that correlation varies over time
Data from the daily closing exchange rates of Kenya and South Africa was used which was got from OANDA starting from 1 January 2010 to 31 December 2015 and in total 1837 observations
Summary
The long history of option pricing began in 1900 when the French mathematician Louis Bachelier deduced an option pricing formula based on the assumption that stock prices followed a Brownian motion with zero drift [1]. In [7], closed-form approximation as well as a measure of the error for the price of two dimensional derivatives under the assumptions of stochastic correlation and constant volatility was provided They provided a simulations-free approximation to the price of Spread Options and Quantos Options under non-constant correlation. [6] dealt with the stochastic modelling of correlation in finance where they illustrated the evidence that the correlation was hardly a deterministic quantity with the analysis of correlation between daily returns time series of S and P Index and Euro/USD exchange rates They determined a transition density function of the stochastic correlation processes in closed form and computed the price of a quantity adjusting option (Quanto). The study is divided into four sections, that is, pricing with constant correlation, pricing when the correlation is stochastic, numerical results and conclusion
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