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# Riskneutral valuation formula

Assume that we have a market consisting of a risky asset S and a bank-account B, where the corresponding ℙ-dynamics are given as

dS(t)=S(t)μ(t,S(t))dt+S(t)σ(t,S(t))dW(t),

S(0)=s_{0}

dB(t)=rB(t)dt,

B(0)=1.

Assume that we have a simple claim with maturity T and pay-off Φ(S(T)). Using the Feynman-Kac representation theorem on the Black-Scholes equation we get that the price at time 0≤t≤T given that S(t)=s, F(t,s), is given by

F(t,s)=exp(-(T-t)r)E^{ℚ}[Φ(S(T))|S(t)=s],

where S(u) has the ℚ-dynamics

dS(u)=rS(u)ds+S(u)σ(t,S(u))dW^{ℚ}(u), for u>t,

S(t)=s.

The risk neutral valuation formula is valid also in more general settings than diffusion models e.g. in the framwork of stochastic jump diffusions such as exponential Lévy models, the Merton and Bates models etc.

dS(t)=S(t)μ(t,S(t))dt+S(t)σ(t,S(t))dW(t),

S(0)=s

dB(t)=rB(t)dt,

B(0)=1.

Assume that we have a simple claim with maturity T and pay-off Φ(S(T)). Using the Feynman-Kac representation theorem on the Black-Scholes equation we get that the price at time 0≤t≤T given that S(t)=s, F(t,s), is given by

F(t,s)=exp(-(T-t)r)E

where S(u) has the ℚ-dynamics

dS(u)=rS(u)ds+S(u)σ(t,S(u))dW

S(t)=s.

The risk neutral valuation formula is valid also in more general settings than diffusion models e.g. in the framwork of stochastic jump diffusions such as exponential Lévy models, the Merton and Bates models etc.

Questions: Magnus Wiktorsson

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