reference book: Brownian motion calculus by Ubbo

[8.8.1] For the exchange option in Section 8.3, verify c(0) by evaluating E as in Section 6.5. [8.8.2] For the option on a bond in Section 8.3, verify c(0) by evaluating E as in Section 6.5. [8.8.3] For the European call under a stochastic interest rate in Section 8.3, verify c(0) by evaluating E as in Section 6.5. 8.3.1 Exchange Option S2(0)c(0)=EP{max[F(T)1,0]} Here S(T) becomes F(T)=S1(T)/S2(T),K becomes 1,2T becomes F2T where F2=12+22212. The expected value expression becomes S2(0)S1(0)N(d~1)N(d~2)whered~1=defFTln[S2(0)S1(0)]+21F2T 188 Brownian Motion Calculus and d~2=defd~1FT Thus c(0)=S2(0){S2(0)S1(0)N(d~1)N(d~2)}=S1(0)N(d~1)S2(0)N(d~2) Both stock prices grow at the same rate r, thus their ratio has no growth, and the option price is independent of r. The discounted call value process c(t)/exp(rt) is a martingale under probability P, so at time 0 exp(0T)c(0)=Ep[exp(rT)c(T)]=exp(rT)E[1[S(T)K]] where 1(.)is the indicator function which has value 1 when the condition in the curly brackets is satisfied, and 0 otherwise. As the expected value is taken of a random process that is a martingale, the payoff condition S(T)K is now expressed in terms of that martingale S as S(T) exp(rT)K, and c(0)=exp(rT)EP[1(S(T)exp(rT)K]]=exp(rT)P[S(T)exp(rT)K] as the expected value of an indicator function equals the probability of the indicator event. Under P S(T)=S(0)exp[212T+B(T)] Now determine the values of B(T) for which S(T)exp(rT)K, or equivalently ln[S(T)]rT+ln[K]ln[S(0)]212T+B(T)rT+ln[K]B(T)a{ln[KS(0)]+(r212)T}/B(T)a Then P[S(T)exp(rT)K]=P[B(T)a]=x=adensityofB(T)atxT21exp[21(Tx)2]dx The integrand is now transformed to the standard normal density by the change of variable y=x/T, lower integration limit x=a becomes [8.8.1] For the exchange option in Section 8.3, verify c(0) by evaluating E as in Section 6.5. [8.8.2] For the option on a bond in Section 8.3, verify c(0) by evaluating E as in Section 6.5. [8.8.3] For the European call under a stochastic interest rate in Section 8.3, verify c(0) by evaluating E as in Section 6.5. 8.3.1 Exchange Option S2(0)c(0)=EP{max[F(T)1,0]} Here S(T) becomes F(T)=S1(T)/S2(T),K becomes 1,2T becomes F2T where F2=12+22212. The expected value expression becomes S2(0)S1(0)N(d~1)N(d~2)whered~1=defFTln[S2(0)S1(0)]+21F2T 188 Brownian Motion Calculus and d~2=defd~1FT Thus c(0)=S2(0){S2(0)S1(0)N(d~1)N(d~2)}=S1(0)N(d~1)S2(0)N(d~2) Both stock prices grow at the same rate r, thus their ratio has no growth, and the option price is independent of r. The discounted call value process c(t)/exp(rt) is a martingale under probability P, so at time 0 exp(0T)c(0)=Ep[exp(rT)c(T)]=exp(rT)E[1[S(T)K]] where 1(.)is the indicator function which has value 1 when the condition in the curly brackets is satisfied, and 0 otherwise. As the expected value is taken of a random process that is a martingale, the payoff condition S(T)K is now expressed in terms of that martingale S as S(T) exp(rT)K, and c(0)=exp(rT)EP[1(S(T)exp(rT)K]]=exp(rT)P[S(T)exp(rT)K] as the expected value of an indicator function equals the probability of the indicator event. Under P S(T)=S(0)exp[212T+B(T)] Now determine the values of B(T) for which S(T)exp(rT)K, or equivalently ln[S(T)]rT+ln[K]ln[S(0)]212T+B(T)rT+ln[K]B(T)a{ln[KS(0)]+(r212)T}/B(T)a Then P[S(T)exp(rT)K]=P[B(T)a]=x=adensityofB(T)atxT21exp[21(Tx)2]dx The integrand is now transformed to the standard normal density by the change of variable y=x/T, lower integration limit x=a becomes