Discretization Of The Jump Diffusion Model Of Option Pricing

In this paper, we study how to price and hedge European and American option in the discrete time when there is jump in the diffusion process of the underlying securities. Our work is mainly based on a paper called "Jump Diffusion Option Valuation in Discrete Time" by Kaushik I. Amin published in 1993. In his paper, Amin added jump to a binomial model, distinguishing between local changes and jumps and establishing a discrete time model to price European and American options. He has built a simple discrete stock price space for the lognormal diffusion process with jumps described by Merton. We replicate his work and analyze how to hedge the option under his discrete framework. We give efficient computational procedure for the valuation of both European and American, call and put options, under two types of pre-specified jump distribution. Furthermore, we simulate the stock price and establish our hedging portfolios based on the trajectory of the stock price. We also compare the performance of our hedging portfolio with the spot option price. The result shows that our hedging is successful.The whole paper is divided into 7 parts and organized as follows:1. IntroductionIn this section, we state the brief history and main models for option pricing. We also explain why the Amin’s model is chosen, because the closed formula does not exist for continuous model if the jump distribution is neither lognormal nor discrete. What’s more, the closed formula is not available when early exercise is possible. Thus the discrete model is useful in these cases. Moreover, Amin’s discretization is simple and yet converges weakly to the continuous one. This ensures us that our result will also weakly converge to the true value of option under the continuous time space. The simplicity of the model also enables us to analyze the hedging strategy and compute them.2. Amin’s model3. Amin’s discretized jump diffusion processIn these two sections, we describe what Aim has done to establish the framework for our hedging analysis. We properly define the necessary parameters for our analysis in this part. Then we discuss two types of price variation:local changes only and jumps, suppose we invest in a portfolio composed of option, bonds and stocks. We derive the value of our portfolio under these two types of price change. Then we combine them to deduce the backward recursive formula for the option pricing.We also discrete the continuous time underlying process raised by Merton under a risk-neutral probability space. Then we give the transition probability from one date to the next.4. Hedging strategy under jump diffusion processIn this section, we analyze how to hedge an option under the discrete framework described above. Due to the occurrence of jumps, the market is incomplete, we cannot hedge perfectly. Thus, we employ a quantile hedging technique. We do not require our hedging portfolio will be successful at all possible state of stock price at next date; instead, we only hedge a certain percent of the possible outcomes. We limit ourselves to a bounded space of stock price at next date, ensuring that our hedging portfolio can cover the option price within this region. This can further be simplified as setting the value of our hedging portfolio at next date equals to the highest option value. By doing so, we can ensure that our portfolio value will be higher that all the other option values in the possible price state space. We also assume our hedging is self-financing. This means that our current portfolio based on current price should be equal to the next date portfolio based on current price. With these two equations, we can solve for the next date’s investment in bond and stock.5. Computation of option priceIn this section, we use Matlab code to compute the price for European call, European put, American call and American put options under lognormal jump and bivariate jump distributions separately. In the discrete backward recursive model, we establish a bounded state space of stock price. At the start of our computation, we assign the intrinsic value of the option at its time boundary and state boundaries. Then we go the one date previous to maturity. At each state, we truncate the possible price state into 4 standard deviation region, and calculate the discounted expectation of future option price based on stock price distribution. If the 4 standard deviation goes out of the bounded space, we assign the probability mass outside into the boundary state. And this process continues. Finally, we can get the option price at the initial date. If it is American option, we need to compare the discounted expectation of option on the next date with the immediate payoff at current date and use the bigger value. This explains why American option is always higher priced than the European one of the same type; it gives the option holder more flexibility.We also derive the closed formula for both European call and put options under the two types of jump distribution. The results are shown together with the one computed using the discrete model.We can see from the result that our discrete model works well.6. Simulation of hedging strategyIn this section, we use Matlab code to show how our hedging strategy works and how it performs.We first simulate the diffusion process of a random stock price. To do this, we first generate a random Poisson variable to simulate how many jumps have occurred during the whole time period. Then we generate the same number of uniformly distributed random variables over the whole period to simulate the exact jump time. Then we start from time 0. If the current time equals to a jump time, we generate a random jump, otherwise, we allow the stock price to go either uptick or downtick.After generating the stock price process, we go along this process to establish our quantile hedging portfolio. At each date, based on the current know price and possible state of the stock price in the next date, we choose our hedging portfolio for the next period. This process continues.Finally we compute the value of our hedging portfolio at each date and the option value, and compare them to see the hedging performance. The results proved that our strategy is successful. 7. The final part of this paper is conclusion. We summarize what we did and raise some problems for further discussion.