Let CE(St, t) be the price at time t of the European call option with strike...

 

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Let CE(St, t) be the price at time t of the European call option with strike price K and expiration date T. We have shown that the price f(St, t) any derivative with underlying security price St must satisfy the BSM p.d.e. af t 1 508202 f s 5 +(r-q)sof -rf = 0 s This equation has infinitely many solutions. In order to get the unique solution giving CE (St, t), we need to impose final and boundary conditions. In the case when q=0 these are : C(ST, T) = max{ST - K, 0} [Final condition: the payoff.] C(0, t) = 0. [Boundary condition at 0: no one will buy the call on a security whose price is zero] C(St, t) St, as Sto. [Boundary condition at infinity: if at any time the market believes the security will increase without bound, then everyone will want to buy the call, which increases until it reaches S(t).] Question 1. Using the change of variables, s = Ke x = ln t = T = /2 = T = /2 (T-t) C(s, t) = Kv(x, T) v(x, t) = a = 0/21 show that the BSM p.d.e (q = 0) can be changed into the following constant coefficient equation: T C(s, t) K 8v x + (a 1); Ju T v = Question 2. Show that by letting v(x,7) = e(a1)x}(a+)u(x,7) the previous equation becomes the heat equation : Ju x x . Let CE(St, t) be the price at time t of the European call option with strike price K and expiration date T. We have shown that the price f(St, t) any derivative with underlying security price St must satisfy the BSM p.d.e. af t 1 508202 f s 5 +(r-q)sof -rf = 0 s This equation has infinitely many solutions. In order to get the unique solution giving CE (St, t), we need to impose final and boundary conditions. In the case when q=0 these are : C(ST, T) = max{ST - K, 0} [Final condition: the payoff.] C(0, t) = 0. [Boundary condition at 0: no one will buy the call on a security whose price is zero] C(St, t) St, as Sto. [Boundary condition at infinity: if at any time the market believes the security will increase without bound, then everyone will want to buy the call, which increases until it reaches S(t).] Question 1. Using the change of variables, s = Ke x = ln t = T = /2 = T = /2 (T-t) C(s, t) = Kv(x, T) v(x, t) = a = 0/21 show that the BSM p.d.e (q = 0) can be changed into the following constant coefficient equation: T C(s, t) K 8v x + (a 1); Ju T v = Question 2. Show that by letting v(x,7) = e(a1)x}(a+)u(x,7) the previous equation becomes the heat equation : Ju x x .

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1 Using the change of variables s Kex t T Cst Kvx ... View the full answer