Pricing Leveraged ETFS Options under Heston Dynamics
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The aim of this thesis is to derive a pricing formula for options on leverage exchange-traded funds (LETFs) with the assumption that the underlying index follows the Heston model dynamics. In order to price options for LETFs, we first establish a relationship between the price of an LETF and the value of its underlying index. This relationship is dependent on the leverage ratio of the LETF and the volatility of the underlying index. Through empirical analysis, we are able to justify the accuracy of this link between an LETF and its underlying index. Furthermore, this link provides useful information on the behaviour of LETFs which is studied in depth. We also use an optimization technique to provide empirically estimated leverage ratios for various LETFs of VIX and several equities to understand their behaviour under different market conditions. The option pricing formula is derived by defining the joint moment-generating function of the underlying index and its volatility and linking this function to the characteristic function of an LETF. The Carr-Madan Fourier transform method is utilized to obtain a closed-form solution of option prices in the form of an integral. We then numerically calculate the call option prices for specific parameters. We perform extensive analysis on our model. The call prices calculated from our option pricing formula are compared with those obtained by Monte-Carlo simulations and the results are consistent, justifying the use of our model. Finally, we perform sensitivity analysis to analyze the effect of various parametric changes on our model.