Abstract
Quantitative finance is concerned about applying mathematics to financial markets.This thesis is a collection of essays that study different problems in this field: How efficient are option price approximations to calibrate a stochastic volatilitymodel? (Chapter 2)
How different is the discretely sampled realized variance from the continuouslysampled realized variance? (Chapter 3)
How can we do static hedging for a payoff with two assets? (Chapter 4)
Can we apply fast Fourier Transform methods to efficiently use interest rateMarkov-functional models? Can we extend them to accommodate othertypes of dynamics? (Chapter 5)
How can we formulate a simple free-arbitrage model to price correlationswaps? (Chapter 6)
A summary of the work presented in this thesis:Approximation Behooves CalibrationIn this paper we show that calibration based on an expansion approximation for option prices in the Heston stochastic volatility model gives stable, accurate, and fast results for S&P500-index option data over the period 2005 to 2009.Discretely Sampled Variance Options: A Stochastic Approximation ApproachIn this paper, we expand Drimus and Farkas (2012) framework to price variance options on discretely sampled variance. We investigated the impact of their assumptions and we present an adjustment for their formula. Our adjustment provides a better approximation to price discretely sampled realized variance options under different market scenariosStatic Hedging for Two-Asset OptionsIn this paper we derive a static spanning relation between an option written on two different assets and a continuum of two-asset binary options written on the same assets. Our Monte Carlo simulations show that under a continuous price dynamics discretized static hedges are possible with small hedging errors.A Jump Markov-functional Interest Rate Model with Fast Fourier TransformThis paper shows how the fast Fourier Transform (FFT) may be used to calibrate an interest rate Markov-functional (MF) model when the characteristic function of the Markov process is analytically known. We added a compound Poisson process to the standard Markov process to create a new MF model - the Jump MF model. We show how can we apply the FFT methodology to calibrate the Jump MF model. Finally, we apply the model to price barrier caplets and floorlets.Pricing and Hedging Correlation Swaps with a Two-Factor ModelIn this paper, we introduce a new model to price and hedge correlation swaps. The model is based on an approximation for the realized correlation payoff, written as the ratio between two quantities, which can be replicated from traded assets in the variance swap markets. The model allow us to derive simple analyticalformulas for the price of a correlation swap. Finally, numerical evaluations of correlation swap prices are presented.
BibliographyDrimus, Gabriel and Farkas, Walter. Valuation of options on discretely sampledvariance: A general analytic approximation. Working paper available atssrn.com, 2012.
How different is the discretely sampled realized variance from the continuouslysampled realized variance? (Chapter 3)
How can we do static hedging for a payoff with two assets? (Chapter 4)
Can we apply fast Fourier Transform methods to efficiently use interest rateMarkov-functional models? Can we extend them to accommodate othertypes of dynamics? (Chapter 5)
How can we formulate a simple free-arbitrage model to price correlationswaps? (Chapter 6)
A summary of the work presented in this thesis:Approximation Behooves CalibrationIn this paper we show that calibration based on an expansion approximation for option prices in the Heston stochastic volatility model gives stable, accurate, and fast results for S&P500-index option data over the period 2005 to 2009.Discretely Sampled Variance Options: A Stochastic Approximation ApproachIn this paper, we expand Drimus and Farkas (2012) framework to price variance options on discretely sampled variance. We investigated the impact of their assumptions and we present an adjustment for their formula. Our adjustment provides a better approximation to price discretely sampled realized variance options under different market scenariosStatic Hedging for Two-Asset OptionsIn this paper we derive a static spanning relation between an option written on two different assets and a continuum of two-asset binary options written on the same assets. Our Monte Carlo simulations show that under a continuous price dynamics discretized static hedges are possible with small hedging errors.A Jump Markov-functional Interest Rate Model with Fast Fourier TransformThis paper shows how the fast Fourier Transform (FFT) may be used to calibrate an interest rate Markov-functional (MF) model when the characteristic function of the Markov process is analytically known. We added a compound Poisson process to the standard Markov process to create a new MF model - the Jump MF model. We show how can we apply the FFT methodology to calibrate the Jump MF model. Finally, we apply the model to price barrier caplets and floorlets.Pricing and Hedging Correlation Swaps with a Two-Factor ModelIn this paper, we introduce a new model to price and hedge correlation swaps. The model is based on an approximation for the realized correlation payoff, written as the ratio between two quantities, which can be replicated from traded assets in the variance swap markets. The model allow us to derive simple analyticalformulas for the price of a correlation swap. Finally, numerical evaluations of correlation swap prices are presented.
BibliographyDrimus, Gabriel and Farkas, Walter. Valuation of options on discretely sampledvariance: A general analytic approximation. Working paper available atssrn.com, 2012.
| Original language | English |
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| Publisher | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
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| Number of pages | 152 |
| Publication status | Published - 2015 |