Research On The Valuation Of Stock Loans In A Jump-Diffusion Model

A stock loan is a tool of investors financing from banks. It makes the investors hold their stocks, and increases the capital efficiency of the bank. Stock loan is an important financing tool in a mature financial market. Stock loan is issued in 2000 in China, which is significant to the stock market and the financial market.There are many results and discussion about laws and supervision about the business in China. From the mathematical point of view, both of the borrower and the bank pursue their own maximum profit. Higher loan-to-value ratio will decrease the will of the borrowers, but lower loan-to-value ratio will increase the default risk of the borrowers. Also, the loan rates and fees are important because they are the main income of the bank in a stock loan business . Then what values of the loan-to-value ratio, loan rate and fee are fair to both the borrower and the bank? And when is the best time for the borrower to redeem the stocks? Domestic scholars mainly studied the value of the loan-to-value ratio from the risk management. Xia and Zhou [2]gave the first results on the valuation of stock loans by the theory of options pricing, in which they assumed the stock price follows by a classical geometric Brownian model and proved the stock loan equals to an American call option with a possibly negative interest rate. They used a pure probabilistic approach, and gave the fair relationships between loan-to-value ratio, loan rate and fees. Zhang and Zhou extended the above results to stock loans with Regime Switching in [10] by the variational inequality approach. By looking for the necessary condition of the solution, they excluded the solutions does not meet the condi-tions and finally get the unique solution of the valuation of stock loans. Xing Tie extended Xia and Zhou's results to stock loans with a positive exponential jump-diffusion model in [7] and got the valuation by theories of optimal stopping time and the bankruptcy theory. In this paper we extend Zhang and Zhou's results to a negative jump-diffusion model, and follow from looking for the necessary condition of the unique solution. Furthermore, we use the same method for a positive exponential jump-diffusion model studied by Xing Tie in [7], and obtain the valuation of the unresponding stock loan. Finally, from our results in this thesis, we show the fair relationship of the loan-to-value ratio, loan rate and fees in two cases.Generally, the stock price is not continuous in real world. It may suddenly jump down because of the bankruptcy or bad performance of the issued company. Here, we assume the stock price follows from a negative jump-diffusion model. Assume that in probability space (Ω, 0. Let St=e-γtSt0, then S0= S00, and where r= r-γ< 0.In fact, the model established by (1.1)and(1.2)is a special case of double exponential jump-diffusion model posed by S. G. Kou in[18], where p= 0. Then the value of the stock loan isν(S0)Using Zhang and Zhou's method for reference, we obtain the valuation of the stock loan with negative exponential jump-diffusion model.Theorem 1.1. In the negative exponential jump-diffusion model, the valuation of the stock loan is where k> 1 is one of the solution to the equationAs to the positive exponential jump-diffusion model, in which the stock price satisfied the following SDE Let Uk=Yk-1, and the density function of Yk is we also use this method and obtain the valuation of the stock loan.Theorem 1.2. In the positive exponential jump-diffusion model, the valuation of the stock loan is where k1,k2are the solutions to the function andThe above results are the same as Xing Tie's in some condition.An Analysis of the results obtained is that if the stock price follows from a negative jump-diffusion model, the loan-to-value ratio, loan rate and fees should have the relationship that and q>k-1/k S0, and that if the stock price follows from a positive jump-diffusion model, the loan-to-value ratio, loan rate and fees should have the relationship thatγ> r+1/2σ2+λ/(η-1)2 and Then the stock loan business is fair to both the borrower and the bank. Besides, we give the sensitivity analysis of (1.5) and (1.8) on the parametersλ,ηandσ.