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Black-Scholes 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)=μS(t)dt+σS(t)dW(t),
S(0)=s0
dB(t)=rB(t)dt,
B(0)=1,
where μ, σ and r are non-negative constants W a standard Brownian motion. Then the price ΠcE(t,K,T,s) of a European call option with maturity T, strike K and S(t)=s is given as
ΠcE(t,K,T,s)=sN(d1(t,s))-exp(-r(T-t))KN(d2(t,s))
d1(t,s)=(ln(s/K)+(r+σ2/2)(T-t))/(σ(T-t)1/2)
d2(t,s)= d1(t,s)-σ(T-t)1/2,
where N is the distribution function of the standard Gaussian distribution.

Black-Scholes prices

 

Questions: Magnus Wiktorsson
Last update: 2016 Sep 04 16:31:35. Validate: HTML CSS

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