### Title page for ETD etd-09082007-155649

Type of Document Dissertation
Author Timsina, Tirtha Prasad
Author's Email Address ttimsina@gmail.com
URN etd-09082007-155649
Title Sensitivities in Option Pricing Models
Degree PhD
Department Mathematics
Sachs, Ekkehard W. Committee Chair
Borggaard, Jeffrey T. Committee Member
Cliff, Michael T. Committee Member
Day, Martin V. Committee Member
Keywords
• Sensitivity
• Optimization
• Jump diffusion
Date of Defense 2007-08-13
Availability unrestricted
Abstract
The inverse problem in finance consists of determining the unknown parameters of

the pricing equation from the values quoted from the market. We formulate the

inverse problem as a minimization problem for an appropriate cost function to minimize

the difference between the solution of the model and the market observations.

Efficient gradient based optimization requires accurate gradient estimation of the

cost function. In this thesis we highlight the adjoint method for computing gradients

of the cost function in the context of gradient based optimization and show its

importance. We derive the continuous adjoint equations with appropriate boundary

conditions for three main option pricing models: the Black-Scholes model, the Heston’s

model and the jump diffusion model, for European type options. These adjoint

equations can be used to compute the gradient of the cost function accurately for

parameter estimation problems.

The adjoint method allows efficient evaluation of the gradient of a cost function F(¾)

with respect to parameters ¾ where F depends on ¾ indirectly, via an intermediate

variable. Compared to the finite difference method and the sensitivity equation

method, the adjoint equation method is very efficient in computing the gradient

of the cost function. The sensitivity equations method requires solving a PDE

corresponding to each parameter in the model to estimate the gradient of the cost

function. The adjoint method requires solving a single adjoint equation once. Hence,

for a large number of parameters in the model, the adjoint equation method is very

efficient.

Due to its nature, the adjoint equation has to be solved backward in time. The

adjoint equation derived from the jump diffusion model is harder to solve due to its

non local integral term. But algorithms that can be used to solve the Partial Integro-

Differential Equation (PIDE) derived from jump diffusion model can be modified to

solve the adjoint equation derived from the PIDE.

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