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# Black-Scholes model

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.

Under the risk-neutral measure ℚ we have the dynamics

dS(t)=rS(t)dt+σS(t)dW(t),

S(0)=s_{0}

dB(t)=rB(t)dt,

B(0)=1,

where r and σ non-negative constants W a standard ℚ-Brownian motion.

The stock price here follows a geometric Brownian motion (both under ℙ and ℚ). The Black-Scholes market is free of arbitrage and complete. For prices of European put and call options see Black-Scholes formula.

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.

Under the risk-neutral measure ℚ we have the dynamics

dS(t)=rS(t)dt+σS(t)dW(t),

S(0)=s

dB(t)=rB(t)dt,

B(0)=1,

where r and σ non-negative constants W a standard ℚ-Brownian motion.

The stock price here follows a geometric Brownian motion (both under ℙ and ℚ). The Black-Scholes market is free of arbitrage and complete. For prices of European put and call options see Black-Scholes formula.

Questions: Magnus Wiktorsson

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