FInancial modeling

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formulae

5. Let {Wt=W(t),t0} be the standard Brownian motion. a) Outline 2 properties of the standard Brownian motion. [2] b) Show that the stochastic process Xt=Wt2t is a martingale with respect to the filtration Ft associated with Wt. 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)]. 5. Let {Wt=W(t),t0} be the standard Brownian motion. a) Outline 2 properties of the standard Brownian motion. [2] b) Show that the stochastic process Xt=Wt2t is a martingale with respect to the filtration Ft associated with Wt. 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)]