# Masters Projects

## Research Study Group

## Stage 1 : Proposal

I was part of the Multiscale Materials group with Amal Alphonse, Simon Bignold, Abhishek Shukla and Matthew Thorpe and under the supervision of Dr Stefan Adams and Dr Christoph Ortner.

The proposal focused on different methods of calculating the free energy of a system. Another major aspect of the proposal was to extend the Cauchy–Born rule to finite temperature settings.

### Stage 2 : Implementation

For this stage I was in the Stochastic Finance group working again with Abhishek Shukla and Matthew Thorpe under the supervision of Dr Aleksandar Mijatović and later under Dr Vassili Kolokoltsov.

The goal of the project was the pricing of path-dependent options, in particular barrier options. The type of option considered are of the European type so they can only be exercised at maturity. An example of a barrier option is an up-and-in call option which can only be exercised if the price of the underlying have reached a certain level before expiry in which case it has the payoff of a European call option otherwise it has zero payoff. It can be shown the the price of an up-and-in call is given by

\(\mathbb{E}_x^*\left[e^{-rT}(S_T - K)^+ \mathbb{I}_{\{\max_{0\leq t\leq T} S_t \geq H\}}\right]\),

where the superscript * means that the expectation is with respect to the risk neutral measure and \(T\) is the expiry, \(K\) is the strike price, \(H\) is the barrier and \(r>0\) is the riskless interest rate. The price process \(S_t\) is assumed to follow a jump-diffusion model of the form

\( \frac{\mathrm{d} S_t}{S_{t-}} = \mu\mathrm{d}t + \sigma\mathrm{d}B_t + \mathrm{d}J_t \),

where \( J_t \) is a Lévy process.

The expectation above is in general difficult to compute analytically but it can be estimated by numerical means. The approach taken in this project was the method of Monte Carlo integration. The first step was to discretise the state space \( [0,\infty) \) of \( S_t \) by a finite grid \( \mathbb{G} \) and then approximating the price process \( S_t \) by a continuous time Markov chain \(X\) with state space \( \mathbb{G} \). This is achieved by constructing the transition rate matrix (Q-matrix) of the chain via the decomposition \( \Lambda = \Lambda_D + \Lambda_J \) where \( \Lambda_J \) is determined by the Lévy measure of \( J_t \) and \( \Lambda_d \) is chosen so that the first and second moments of \(X_t\) and \(S_t\) match.

Once the Q-matrix is constructed one can sample paths of the Markov chain and calculate the payoff of the option for that path. The price of the option is then estimated by the sample mean of the payoff. More details can be found here.