Background Information:
The price of a portfolio at time t (t=0,1) is Vt(π)=ϕSt+ψBt The portfolio π is a perfect hedge for the claim X if V1(π)=X a.s. as random variables.
Given claim X, to have a perfect hedge then ϕ and ψ must satisfy ϕS1(u)+ψB1=X(u) ϕS1(d)+ψB2=X(d) We have solving for this that ϕ=X(u)−X(d)S1(u)−S1(d) ψ=B−11(X(u)−ϕS1(u))=B−11(X(d)−S1(d)) Thus the resulting value of X at t=0 is V0(X)=V0(π)=ϕS0+ψB0
Question:
Let β=B0B−11 be the discount factor. Show that
V0(X)=β[(β−1S0−S1(d)S1(u)−S1(d))X(u)+(S1(u)−β−1S0S1(u)−S1(d))X(d)]
I have tried this three times now and I still not getting the result what I do is this if you want to see my attempted work let me know. Otherwise it would be great if someone could give me a good start to this. There must be something that I am not seeing in regards to some algebra trick.
Answer
Using the values for ϕ and ψ that you have derived, V0(X)=ϕS0+ψB0=X(u)−X(d)S1(u)−S1(d)S0+B−11(X(u)−X(u)−X(d)S1(u)−S1(d)S1(u))B0=β(X(u)−X(d)S1(u)−S1(d)β−1S0+X(d)S1(u)−X(u)S1(d)S1(u)−S1(d))=β(β−1S0−S1(d)S1(u)−S1(d)X(u)+S1(u)−β−1S0S1(u)−S1(d)X(d)). What you need is to combine terms with X(u) together, and likewise for terms with X(d).
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