巨灾权益连结卖权定价模型研究:基于随机波动率与随机利率条件

Pricing model of catastrophe equity put option:Based on the conditions of stochastic volatility and stochastic interest rate

巨灾权益连结卖权作为一种新型复合期权,它能够保证保险公司不会同时承受巨灾导致的赔付风险与公司股价下跌造成的流动性风险,其引发了广大学者的研究。但现有卖权定价研究仍存在一些不足,如未考虑随机波动率的情况、没有综合考虑随机利率和随机波动率的影响、巨灾对股价影响函数的设定不合理、未确定巨灾风险的等价鞅测度等。为了克服这些不足,本文在研究巨灾权益连结卖权定价问题时,考虑了随机利率和随机波动率影响因素,利用Vasicek模型刻画利率期限结构,同时假设单位时间内巨灾损失对股价收益率影响是恒定的,提出了一种更加合理的巨灾损失影响函数形式。在此基础上,本文放宽了巨灾风险分布不受测度变换的假设,确定了受巨灾影响的股价过程的等价鞅测度,从而得到了巨灾权益连结卖权的定价公式。最后,文章运用蒙特卡罗模拟的方法对巨灾权益连结卖权价值的影响因素进行了敏感性分析,以验证本文模型的合理性。

The catastrophe equity put option is a kind of compound option, which enables the buyer (usually an insurance company) to sell a certain share of the convertible stock to the issuer at the predetermined price. However, this right can be executed if the accumulated loss exceeds a certain amount. The catastrophe equity put option is designed to ensure that an insurer would not suffer from enormous compensation or great fall in share value when a catastrophe occurs. From the mechanism of this put option, it is not difficult to see that the benefits not only depend on the stock prices but also depend on the cumulative loss. This new-styled option attracts great attention from the researchers. There are still some aspects to be improved in the current research. Firstly, the volatility is considered as constant and invariant, which is not in accord with reality. Secondly, the impact of stochastic interest rate and stochastic volatility are not considered at the same time. Thirdly, the influence function of catastrophe is set unreasonable and the formulation to reflect the relationship between share value and losses is not reasonable. Besides, in current research distribution of catastrophic losses is assumed to be invariant in different measures, which also deviates from the actual situation. Finally, the equivalent martingale measure of the stock price affected by catastrophe risk is not yet determined. In our research, because the price of catastrophe equity put option is more sensitive to interest rate, and volatility rate, the combination of both rates can depict the randomness effectively. We introduce the Vasicek model to solve the problem of interest rate term structure and study the problem of pricing the double trigger catastrophe equity put option with stochastic volatility and stochastic interest rate in order to improve the precision of option pricing model. In addition, there are also many defects in setting catastrophic influence function and parameters. Existing researches cannot propose a reasonable catastrophic influence function expression which can reflect the catastrophic impact on stock prices. Meanwhile, the hypothesis is not reasonable when determining the specific parameters of a catastrophic influence function expression. Therefore, we do not only consider a random stochastic volatility and stochastic interest model but also propose a new influence function to model the relation between losses and the share value, assuming that the effects of losses occurring in each unit time on share value are the same rather than the impacts of unit catastrophe losses on stock prices are the same, which can reflect the relationship between the impact of catastrophe and time-varying. In addition, we relax the assumption that catastrophe risk is not affected by the measure changed. The stock price that is affected by the catastrophe risk is not a martingale process under the real measure, but it can be transformed into a martingale process by the equivalent martingale measure transformation. In this research, we use the Esscher transform method, based on which we determine the equivalent martingale measure, and obtain the pricing formula for catastrophe equity put option. To verify the reasonability of our model, we employ the Monte Carlo simulation to perform sensitivity analysis on the effect of some extraneous variables that affect the price of catastrophe equity put option. The chapters of this paper are arranged as follows: In the second section, we give the model of double trigger compound option pricing, including stochastic stock price, stochastic interest rate, stochastic volatility model, the distribution of catastrophe losses, and the impact function of catastrophe on stock price and so on. In the third section, based on the model, an analytical method is presented by using the Girsanov theorem. In the fourth section, this paper uses the method of Monte Carlo simulation to analyze the influencing factors of the value of the double trigger catastrophe equity put option, and verifies the rationality of this option pricing model. In the last section, the conclusions are reviewed, and future research is proposed. This paper combines the special characteristics of the double trigger catastrophe equity put option, the pricing method of the option is studied in detail. The three main innovations are as follows:(i) The condition of stochastic volatility is added to the existing double trigger catastrophe equity put option pricing model. We established a three-factor stochastic model considered the stochastic stock price, stochastic interest rate and stochastic volatility, which can provide more practical results.(ii) The double trigger catastrophe equity put option pricing model is fully combined the research results of the equivalent martingale measure theory, and reflects the catastrophe characteristics of the double trigger catastrophe equity put option.(iii) This paper discusses the characteristics of the impact of the catastrophe on stock prices and puts forward a new catastrophic influence function that has improved the previous researches.The conclusions show that:(i) When the influence coefficient of the loss is small, the impact of the loss is not obvious, and the trading of catastrophe equity put options is not too important. At the same time, the stability of the transaction is also poor. When the influence coefficient of the loss is large, the trading of catastrophe equity put options has a very important significance. In addition, the stability of the transaction is relatively good.(ii) When the fluctuation coefficient is low, the impact of the growth on the volatility is positive, that is to say, the increase in the volatility coefficient leads to the increase in the volatility, which leads to the higher value of the catastrophe equity put option. However, when the fluctuation coefficient reaches a certain critical point, with the increase of the coefficient of fluctuation, the effect of the growth on the volatility is negative. With the decrease in volatility, the value of the catastrophe equity put option declines further.(iii) The higher initial volatility leads to a higher level of volatility at the beginning. Therefore, the value of the catastrophe equity put option is high.(iv) The rising of fluctuation in interest rate leads to the increase in the probability of profit of the catastrophe equity put option contract, which leads to the rise of the value of the catastrophe equity put option contract. In this paper, random stochastic volatility and stochastic interest are incorporated into the model simultaneously. The accuracy of pricing on catastrophe equity put option will be improved significantly in theory. However, due to the lack of relevant historical data, it is difficult to specify the increased degree of price accuracy, which needs to wait for the insurance company to issue some catastrophe equity put options to verify in the future.