##### 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 Hestonâ 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(Â¾)
with respect to parameters Â¾ where F depends on Â¾ 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.