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=V−ΔS−Δ1V1 then make dΠ=rΠdt
eliminate the
risk terms
we have ∂V∂t+12σ2S2∂2V∂S2+ρσSq∂2V∂S∂σ+12σ2q2∂2V∂σ2+rS∂V∂S−rV=−(p−λq)∂V∂σ. Here λ is called
market price of risk
, since we can understand λ as dV−rVdt=q∂V∂S(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(e−rtV) 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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