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# Option Pricing and Logarithmic Euler-Maruyama Convergence of Stochastic Delay Equations driven by Levy process

• Option Pricing and Logarithmic Euler-Maruyama Scheme of Stochastic Delay Equations with Jumps

• Author / Creator
Agrawal, Nishant
• In this thesis, we study the product formula for finitely many multiple Itˆo-Wiener integrals of Levy process, option pricing formula where the stock price is modelled by stochastic delay differential equation (SDDE) driven by Levy process and logarithmic Euler-Maruyama scheme for the SDDE. In the first part, we derive a product formula for finitely many multiple Itˆo-Wiener integrals of Levy process, expressed in terms of the associated Poisson random measure. The formula is compact and the proof is short and uses the exponential vectors and polarization techniques. In the second part of the thesis we discuss the option pricing when the underlying model follows SDDE. In this part, we obtain the existence, uniqueness, and positivity of the solution to SDDE with jumps. This equation is then applied to model the price movement of the risky asset in a financial market and the Black-Scholes formula for the price of European option is obtained together with the hedging portfolios. The option price is evaluated analytically at the last delayed period by using the Fourier transformation technique. However, in general, there is no analytical expression for the option price. To evaluate the price numerically, we then use the Monte-Carlo method. To this end, we need to simulate the delayed stochastic differential equations with jumps. We propose a logarithmic Euler-Maruyama scheme to approximate the equation and prove that all the approximations remain positive and the rate of convergence of the scheme is proved to be 0.5. Finally, in the last part of the thesis, we discuss the logarithmic Euler-Maruyama scheme and convergence of the logarithmic Euler-Maruyama scheme for a multi-dimensional SDDE’s.

• Subjects / Keywords
Fall 2021
• Type of Item
Thesis
• Degree
Doctor of Philosophy
• DOI
https://doi.org/10.7939/r3-jbmg-yp12