Financial modeling
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4. Let St be the price of a non-dividend paying stock. a) Let the price at time t[0,T] of the European put option with strike price K and expiry time T be given by the Black-Scholes equation pt=Ker(Tt)N(d2)StN(d1). Use this equation to derive the Theta of the put option. Show all your steps! [7] b) A trader has a short put option position that's delta-neutral but has a gamma of 800. In the market, there's a tradable call option with a delta of 0.8 and a gamma of 2; as well as a non-dividend paying underlying stock. To maintain the portfolio gamma-neutral and delta-neutral, what would be the trader's strategy? Formulae: Up-step risk-neutral probability: p=udertd, where u=et+qt;d=et+qt;t=titi1andq=dividendyield. Ito's Lemma: Suppose that the random process x is defined by the lto process dx(t)=a(x,t)dt+b(x,t)dz where z is a standard Brownian motion or Wiener process. Suppose that the process y(t) is defined by y(t)=G(x,t). Then y(t) satisfies the lto equation dy(t)(xGa+tG+21x22Gb2)dt+xGbdz. where z is a standard Brownian motion. Feynman-Kac stochastic representation formula: Assume that F is a solution of the following boundary value problem tF(t,x)+(t,x)xF(t,x)+212(t,x)x22F(t,x)F(T,x)=0,=(x). Assume furthermore that the process (s,X)xF(s,X) is in L2, where the process X satisfies the SDE dXt=(s,Xs)ds+(s,Xa)dWs,Xr=x. Then F has the representation F(t,x)Et,[(Xs)]. 4. Let St be the price of a non-dividend paying stock. a) Let the price at time t[0,T] of the European put option with strike price K and expiry time T be given by the Black-Scholes equation pt=Ker(Tt)N(d2)StN(d1). Use this equation to derive the Theta of the put option. Show all your steps! [7] b) A trader has a short put option position that's delta-neutral but has a gamma of 800. In the market, there's a tradable call option with a delta of 0.8 and a gamma of 2; as well as a non-dividend paying underlying stock. To maintain the portfolio gamma-neutral and delta-neutral, what would be the trader's strategy? Formulae: Up-step risk-neutral probability: p=udertd, where u=et+qt;d=et+qt;t=titi1andq=dividendyield. Ito's Lemma: Suppose that the random process x is defined by the lto process dx(t)=a(x,t)dt+b(x,t)dz where z is a standard Brownian motion or Wiener process. Suppose that the process y(t) is defined by y(t)=G(x,t). Then y(t) satisfies the lto equation dy(t)(xGa+tG+21x22Gb2)dt+xGbdz. where z is a standard Brownian motion. Feynman-Kac stochastic representation formula: Assume that F is a solution of the following boundary value problem tF(t,x)+(t,x)xF(t,x)+212(t,x)x22F(t,x)F(T,x)=0,=(x). Assume furthermore that the process (s,X)xF(s,X) is in L2, where the process X satisfies the SDE dXt=(s,Xs)ds+(s,Xa)dWs,Xr=x. Then F has the representation F(t,x)Et,[(Xs)]
