Pricing Timer Option With Stochastic Volatility

In financial markets,the volatility derivatives are the financial productions on account of volatility index directly,such as the volatility options,variance swap and volatility swap and so on.In 2007,as a kind of innovative volatility derivatives,timer options had been launched by investment department of Societe Generale and then traded in financial markets.One of the characteristics of timer option is that investors could fix the threshold of accumulative volatility by predicting the trend of volatility of the stock price or the index and then conduct a financial trading activities like hedging risks by taking advantage of derivatives.Another characteristic is that one need ensure the joint distribution function of stopping time and volatility to solve the pricing problem of timer option.The holders will exercise options when realized variance accumulation reach the pre-determined level.So the timer options have a random execution time.Compared with the traditional American or European options and underlying derivatives,timer option is a tool which could be more effective and pure control the revealing risk of the underlying assets.Therefore,not only timer option is an important means to obtain the returns,but also one of the instruments can effective control the volatility risk of the underlying asset.In this thesis,we will consider pricing problem of timer option under the Hull-White model and obtain analytical solution.The key is to find the high-dimensional probability density.In the present,one can find pricing kernel or transition probability about some special exotic options,such as barrier options and Asian options.However,it is difficult to get the solution of timer option because of the model complexity.In the pricing option problem,the classical Black-Scholes formula shows the price of the standard European call option as follows:where S is the price of the underlying asset and follows geometric Brownian motion process,N is the standard normal cumulative distribution function,In this thesis,we will depict the price of underlying asset and stochastic volatility process by the Hull-White model.It means that under the risk neutral probability measure Q,the price of underlying asset St and the variance vt satisfy stochastic differential equations as follows:where r is the risk-free interest rate,k is variance return rate parameters,? is the volatility of the variance,WtS and Wtv are two standard Brownian motion with the coefficient ?.In chapter 3,we will apply two-dimensional Feynman-Kac theorem and base on the classical Black-Scholes-Merton pricing formulae on the above model.Then we will transform the original model to the standard Bessel process with time-change tech-nique.According to the related properties of Bessel process,we will adopt probabilistic method to give perpetual timer option closed-form analytical pricing formula by using modified Bessel function.Firstly,there are some variables on pricing timer option,passage time on accumulative realized variance ?B and realized variance accumulation It,Proposition 1 Let x = lnS.Then the pricing function of perpetual timer option satisfy the partial differential equation and terminal conditionsTheorem 1 Considering two-dimensional stochastic system,assuming that U(I,x,v)satisfy(0.0.18)with the terminal condition U(B,x,v)=h(x),then the pricing function of perpetual timer option U(I,x,v)?C1,2,2 has According to the theorem 1,firstly we transform the model with time-change technique from x and v to X and z and obtain the following results.Theorem 2 Assume thatThen we have the conditional Black-Scholes-Merton style pricing formula on perpetual call timer option where N(·)is the standard normal cumulative distribution,and Furthermore assuming pzB,(?)B,?B(z,h,g)is joint Probability density function on(zB,(?)B,?B),then where whereCompared with the methods earlier,we have the Laplace transformed joint density function on three-dimension variables,then we obtain the closed-form analytical formu-lae with stochastic volatility under the inverse Laplace transformation.The closed-form analytical solution has some advantages,especially on highly efficient computation.In this chapter,we also give a pricing formula for a type of special options.We consider that the payoff function for this type of derivatives at exercise time T,where I is the indicator function.Then we get the option price function U as followswhereThe joint density function are obtained by the properties of Bessel process,and whereCorollary 1 Assume that the logarithmic the underlying asset price x and s-tochastic volatility v satisfy Hull-White stochastic volatility model.The payoff function is U(T)=(ST-K)+??B>T,then a kind of special option price function U satisfy the conditional expectation(0.0.20),and the joint density function(0.0.21).From the above solving process and pricing formulae,we see that the pricing formulae could effectively deal with the pricing option problem at continuous time with stochastic volatility using Bessel process.In numerical experiment part of chapter 3,we analyze the joint density function of analytical solution of timer options under Hull-White model.It is applicable to the small volatility of volatility in the market.In order to further study the pricing formulae,we consider the exercise time as ?B?T.In the chapter 3,the property of Bessel process is the key for us to get closed-form analytical solution for perpetual timer option.But this difficulty to solve this problem is increased when the exercise time is uncertain.In chapter 4,we consider the problem from quantum mechanics under the path integral framework.We apply the Schrodinger physics equation into pricing timer option with general exercise time.Finally,we give the perpetual timer option formula under the path integral.The Langrangian z derived from Schrodinger equation is as followsBy Lemma 1,we have the transition probability function whereg(u;B)is the inverse Laplace transformation of g((?)?du)and Mm,n(x)is Kummer function.Furthermore whereSimilarly,we can also obtain U2 with the exercise time T.Firstly,making use of change of xT,vT to ?T,?T Then by the Feynman path integral method,the Lan-grangian ? isLemma 2 Considering exponential integral L[?,?]on[0,T],then we have By the Lemma 2,the transition probability where where and D?+1 is a cylindrical of parabolic function.Hence,we have thatTheorem 3 Assume that the exercise time is ?B?T,the payoff is U(?B?T,x?B?T)=max(ex?B?T-K,0).Then we have the price option at t=0 where U1 is expressed by(0.0.25)whose transition probability is(0.0.23)and U2 is expressed by(0.0.29)whose transition probability is(0.0.28).P(?B<T)and P(?B>T)can be known by(0.0.26).Theorem 4 Under the Hull-White model,assume that the strike price is K,variance budget is B.Then the pricing function of perpetual call timer option is where and the transform probability function P(ZB,?B|z0,0)is expressed in Lemma 1 and g(u;B)is obtained by(0.0.24)expression.