From: "Noel Vaillant" Subject: Re: Stochastic calculus/Brownian motion question (correction) Date: Tue, 7 Dec 1999 22:33:00 -0000 Newsgroups: sci.math Keywords: Ito's formula, derivatives in stochastic processes A question of financial mathematics on sci.math ! Great. :-) ... Gt=f(St), where f(x)=sqrt(x)=x^1/2 Ito's formula can be written as follows: dGt=f'(St)dSt + (1/2)f''(St)dt where t is the quadratic variation of St. f'(x)=1/2x^(-1/2) f''(x)=-1/4x^(-3/2) to work out dt, calculate (dSt)^2 and forget about all terms, except the one in dZt^2. Replace dZt^2 by dt.... (dSt)^2 = b^2S^2dt Finally, dGt=1/2(St)^(-1/2)aStdt +1/2(St)^(-1/2)bStdZt -1/8(St)^(-3/2)b^2(St)^2dt =1/2a(St)^(1/2)dt -1/8b^2(St)^(1/2)dt 1/2b(St)^(1/2)dZt So dGt=Gt[(1/2a-1/8b^2)dt + 1/2bdZt] Gt is also lognormal with drift (1/2a-1/8b^2) and vol 1/2b I hope this is correct. Regards. Noel. ------------------------------------------- Dr Noel Vaillant http://www.probability.net vaillant@probability.net > > dS=aSdt+bSdz > > where S is a stock price, a & b are constants and z is a standardized > weiner process. If > G(t)=S^1/2(t), what is dG? I also need the interim steps required to > get to the answer... Any help would be greatly appreciated! ============================================================================== From: Hankel O'Fung Subject: Re: Stochastic calculus/Brownian motion question (correction) Date: Wed, 08 Dec 1999 11:35:40 +0800 Newsgroups: sci.math To: disced@alcor.concordia.ca disced@alcor.concordia.ca wrote: > Here's my question (application of Ito's lemma): > > dS=aSdt+bSdz > > where S is a stock price, a & b are constants and z is a standardized > weiner process. If > G(t)=S^1/2(t), what is dG? Ito's lemma says that, if dS = A(S,t)dt + B(S,t)dz, and G = G(S), then dG = B(dG/dS) dz + [A (dG/dS) + (1/2)A^2 (d^2G/dS^2)] dt. Now put G=G(S)=sqrt(S), A=A(S,t)=aS and B=B(S,t)=bS into the above equation. It is easy to get dG = (b/2)G dz + [(a/2) - (b^2 /8)]G dt. > I also need the interim steps required to > get to the answer... Uhhh, you should do this by yourself. -- Cheers, Hankel http://www.acad.polyu.edu.hk/~master