Let the interest rate r and the volatility o > 0 be constant. Let 1 St = So exp t+ oWt be a geometric Brownian motion with mean rate of return u, where the initial stock price So is positive. Let K be a positive constant. Show that, for T > 0, E[e-r" (ST – K)*] = S,N(d4) – Ke-rTN(d_), where So log K 1 1 o² = +p σνΤ + and ry Nw) = L 1 N(y) e-z²/2 dz.
Let the interest rate r and the volatility o > 0 be constant. Let 1 St = So exp t+ oWt be a geometric Brownian motion with mean rate of return u, where the initial stock price So is positive. Let K be a positive constant. Show that, for T > 0, E[e-r" (ST – K)*] = S,N(d4) – Ke-rTN(d_), where So log K 1 1 o² = +p σνΤ + and ry Nw) = L 1 N(y) e-z²/2 dz.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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Let the interest rate r and the volatility σ > 0 be constant. Let
St = S0exp((µ − σ2/2)t + σWt )
be a geometric Brownian motion with mean rate of return µ, where the initial stock price S 0 is positive. Let K be a positive constant. Show that, for T > 0, satisfy the following equation shown in the below picture:
Text Book: STOCHASTIC CALCULUS FOR FINANCE, Shreve vol. II
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