dstefan
03-22-2008, 01:32 PM
Chapter 8, Problem 5:
A three months at-the-money call on an underlying asset with spot price 30 paying dividends continuously at a 2% rate is worth $2.5. Assume that the risk free interest rate is constant at 6%.
(i) Compute the implied volatility with six decimal digits accuracy, using the bisection method on the interval [0.0001, 1], the secant method with initial guess 0.5, and Newton's method with initial guess 0.5.
(ii) Let \sigma_{imp} be the implied volatility previously computed using Newton's method. Use formula
\sigma_{imp} ~\approx~ \frac{\sqrt{2 \pi}}{S \sqrt{T}}~\frac{C ~-~ \frac{(r-q)T}{2}S}{1 - \frac{(r+q)T}{2}}
to compute an approximate value \sigma_{imp,approx} for the implied volatility, and compute the relative error \frac{|\sigma_{imp} - \sigma_{imp,approx}|}{\sigma_{imp}}.
Solution:
a
A three months at-the-money call on an underlying asset with spot price 30 paying dividends continuously at a 2% rate is worth $2.5. Assume that the risk free interest rate is constant at 6%.
(i) Compute the implied volatility with six decimal digits accuracy, using the bisection method on the interval [0.0001, 1], the secant method with initial guess 0.5, and Newton's method with initial guess 0.5.
(ii) Let \sigma_{imp} be the implied volatility previously computed using Newton's method. Use formula
\sigma_{imp} ~\approx~ \frac{\sqrt{2 \pi}}{S \sqrt{T}}~\frac{C ~-~ \frac{(r-q)T}{2}S}{1 - \frac{(r+q)T}{2}}
to compute an approximate value \sigma_{imp,approx} for the implied volatility, and compute the relative error \frac{|\sigma_{imp} - \sigma_{imp,approx}|}{\sigma_{imp}}.
Solution:
a