| An option contract is an agreement between two parties to buy or sell an asset at acertain future time for a certain price. The long position who has bought the option has theright to exercise the option but does not have to do it. However, the writer who has sold theoption has potential obligation to fulfill the contract if the holder exercise theoption,without the right to ask the long position to exercise the option. Options have littlecredit risk when they are traded in the counter ; but that is different when they are tradedout of counter(OTC).Because there is no specific institution similar to clearinghouse tocompell the short position to fulfill obligation at maturity, the long position of option isexposed to market risk and credit risk. Johnson and Stulz (1987) firstly studied the optionpricing with credit risk, in which the option with credit risk is called vulnerable option. Soit is significant both in theory and reality to price vulnerable option accurately.Risk measurement is the basis of risk management. Option allows the holder to hedgerisk. But in fact, due to the changes of financial market and volatility of underlyingvariables, the value of option is unstable, especially when the underlying variables havefierce fluctuations or jumps in the wake of the arrival of new important information.Therefore, it is practically significant to measure the risk of option.In this dissertation, the pricing and risk measurement problems of European stockoption when the underlying assets following jump diffusion were mainly studied. By usingthe risk neutral valuation principle, the pricing formulae of standard European stock calland put options are obtained when the underlying stock price is depicted by jump diffusionprocess. Numerical analysis shows that the pricing formulae in the form of series areconvergent quickly and the value of standard European stock call option based on jumpdiffusion process are higher than that based on continuous diffusion process. The empiricalanalysis on pricing stock warrants in our country demonstrates that using the standardEuropean call option pricing formula deduced in this dissertation based on a jump diffusionprocess can estimate the value of warrants more accurately than the Black-Scholes formula.For the risk measurement, the VaR calculations of standard European stock option areobtained when the underlying stock price following jump diffusion and continuousdiffusion separately. Numerical analysis indicate that the VaR is an increasing function ofthe time to maturity, the stock price and its volatility, jump intensity; the VaR of standard European stock call option based on jump diffusion process is higher than that based oncontinuous diffusion process.In the pricing of vulnerable European stock option, the pricing problems when newinformation arrives were studied in detail. The pricing formulae for European option wereprovided under the combinations of the assumptions: stock price, corporation value andcorporation debt following continuous diffusion or jump diffusion process respectively. Allof these formulae in form of series are convergent, in addition, the judgement that the priceof vulnerable standard option is not higher than that of standard European option wasshown. Numerical experiments prove that the value of standard European option is largerthan that of the corresponding vulnerable European option.In the risk measurement of vulnerable European option, the VaR calculation formulaeare derived under the combinations of the assumptions: stock price following continuousdiffusion process or jump diffusion process, corporation value following continuousdiffusion process or jump diffusion process, corporation debt being constant and followingcontinuous diffusion process and jump diffusion process, separately. Some appropriatenumerical experiments are also carried out. |
