利用控制变量求解障碍期权？

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利用利率为时间的函数作为控制变量，来求解随机利率的障碍期权定价。

但是不知道利率为时间的函数时，障碍期权的期望怎么求解？
已经解出了解析解V(S,t)，但是是标的资产价格和时间的函数，不知道这两项应该带入什么？
标的资产价格是带入离散的路径价格均值吗？
前面离散化的（S-K）+是终值的,那么时间是带入0还是T？
很急，谢谢各位

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关键词：控制变量 障碍期权 资产价格 期权定价 不知道 障碍

not sure understand your question.

The barrier option's price is determined by current time to maturity and current stock price. Nothing to do with the past path.

Chemist_MZ 发表于 2013-4-15 12:00
not sure understand your question.

The barrier option's price is determined by current time to ma ...

i want to use the expectation of the barrier option in the control variate method,Y(b)=Y-b(X-E(X)).but i don't know how to calculate E(X).I have the formula of the barrier option,but it is the function of the underlying price and time,I don't know waht two values should be if i want to calculate the expectation.

2009119817 发表于 2013-4-15 13:53
i want to use the expectation of the barrier option in the control variate method,Y(b)=Y-b(X-E(X) ...

For control variant method, E(X) should be some thing we know explicitly. It has many choices

Your task now is to simulate Y, which is the option's price under stochastic interest rate framework.

What E(X) you can choose is the Barrier option price under constant interest rate model.

Of course, in theory, you can also choose the standard European Call for X, but that may not make much improvement on the simulation accuracy

Barrier option price under constant interest rate model, everything is the same as standard BS model, current time to maturity and current stock price.
I you feel confused of the terminal or starting point. You can just simulate the discounted payoff. The average discounted payoff is the option's price. E(X) is the expectation of discounted payoff, Y is the discounted payoff in the stochastic interest rate framwork and so does X

Chemist_MZ 发表于 2013-4-15 14:05
For control variant method, E(X) should be some thing we know explicitly. It has many choices

Y ...

i still don't know what is the meaning of  ''E(X) is in the close form''. the formula of the barrier option price with deterministic interest rate is the function of S and t. if S is the average value of underlying price at  time T I simulated, what will t be if I want to calculate E(X)?can you help me explain it more explicitly?
thanks very much!

2009119817 发表于 2013-4-15 14:24
i still don't know what is the meaning of  ''E(X) is in the close form''. the formula of the barri ...

Ok, I give a simple example

suppose you want to get the price of an exotic option such as barrier or look back.

The naive MC is just simulate, say 10000 stock paths and calculate the corresponding payoff, then average all the paths and get the price.

The control variant method is you choose a control variant X. The  control variate's expectation should be known. In this example, I choose european call option which has a well-known BS formula to calcuate E(X) which is the close form solution of the european call.

What you now need to do is that simulate the payoff of the barrier option Y and the european call option X simutanously. Then you construct a new variable Y(b)=Y+b(X-E(X)), E(X) is known, you have the current stock price, the time to maturity and all the other necessary parameters. b is also available via some algorithm. So now you average Y(b) and get the barrier option price

Y(b) has a less variance than Y if you choose the control variant properly
In your case, you want to get the barrier option' price under stochastic interest rate, so what I recommand, is you choose the barrier option under constant interest rate framework as the control variant. It may or may not work. Anyway, you can make a try. You can also use interest rate as an control variant as you proposed, but it may not work.

It's too late, I should go to bed. Hope its clear now. If it is still not clear, you can check some related books or just google or baidu it out.
Best,

Chemist_MZ 发表于 2013-4-15 14:42
ok, I give a simple example

suppose you want to get the price of an exotic option such as barri ...

thanks
but what i don't understand is how to use black scholes formula to calculate the E(X) .The black scholes formula is the function of S and t.what is the vaule of S and t?

2009119817 发表于 2013-4-15 14:50
thanks
but what i don't understand is how to use black scholes formula to calculate the E(X) .Th ...

oh...

What really confused you is this?

OK, as I have answered you in the other message

BS formula

C=SN(d1)-Kexp(-rT)N(d2)

d1=(log(S/K)+(r+0.5*sigma^2)T)/(sigma*sqrt(T)); d2=d1-sigma*sqrt(T)

option pricing is always price the price now, not in the future. What you should use are all the current information

S is the current stock price as you can observe. In MC simulation, this is the initial point where the stock path evolves. T is time to maturity, say you want to price a option expires three month later, then T is 0.25, all other parameters are known.

done~