Sunday, January 17, 2016

derivatives - How to understand the market price of risk


Consider the stochastic vol: dS=μSdt+σSdW1

dσ=p(σ,S,t)dt+q(σ,S,t)dW2
dW1dW2=ρdt
We want to obtain the price of option V(σ,S,t), we use the underlying asset S and another option V1(σ,S,t) to build the hedging portfolio: Π=VΔSΔ1V1
then make dΠ=rΠdt
eliminate the risk terms we have Vt+12σ2S22VS2+ρσSq2VSσ+12σ2q22Vσ2+rSVSrV=(pλq)Vσ.
Here λ is called market price of risk, since we can understand λ as dVrVdt=qVS(λdt+dW2)=qΔ(λdt+dW2)
this is the unit of extra return.


And we have another way to price V, the discounted value of V is martingale, namely dt term of d(ertV) is zero, then we find that, the PDE of V is exactly λ=0

in above PDE. So does that mean, the discounted value of V is martingale is equivalent to the market price of risk is zero?




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