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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)=s_{0}

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 Π^{c}_{E}(t,K,T,s) of a European call option with maturity T, strike K and S(t)=s is given as

Π^{c}_{E}(t,K,T,s)=sN(d_{1}(t,s))-exp(-r(T-t))KN(d_{2}(t,s))

d_{1}(t,s)=(ln(s/K)+(r+σ^{2}/2)(T-t))/(σ(T-t)^{1/2})

d_{2}(t,s)= d_{1}(t,s)-σ(T-t)^{1/2},

where N is the distribution function of the standard Gaussian distribution.# Black-Scholes prices

dS(t)=μS(t)dt+σS(t)dW(t),

S(0)=s

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 Π

Π

d

d

where N is the distribution function of the standard Gaussian distribution.

Questions: Magnus Wiktorsson

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