dstefan
07-08-2008, 02:04 PM
Chapter 5, Problem 6
(i) What is the approximate value P_{approx,r=0,q=0} of an at-the-money put option on a non-dividend-paying underlying asset with spot price S = 60, volatility \sigma = 0.25, and maturity T=1 year, if the constant risk--free interest rate is r=0?
(ii) Compute the Black--Scholes value P_{BS,r=0,q=0} of the put option, and estimate the relative approximate error
\frac{|P_{BS,r=0,q=0}-P_{approx,r=0,q=0}|}{P_{BS,r=0,q=0}}
(iii) Assume that r=0.06 and q=0.03. Compute the approximate value P_{approx,r=0.06,q=0.03} of an ATM put option and estimate the relative approximate error
\frac{|P_{BS,r=0.06,q=0.03}-P_{approx,r=0.06,q=0.03}|}{P_{BS,r=0.06,q=0.03}}
where P_{BS,r=0.06,q=0.03} is the Black-Scholes value of the put option.
Solution:
(i) P_{approx,r=0,q=0} ~=~ \sigma S \sqrt{\frac{T}{2 \pi}} ~=~ 5.984134
(ii) From the Black-Scholes formula, we find that P_{BS,r=0,q=0} ~=~ 5.968592, and therefore
\frac{|P_{BS,r=0,q=0}-P_{approx,r=0,q=0}|}{P_{BS,r=0,q=0}} ~=~ 0.002604 ~=~ 0.26\%
(iii) P_{approx,r \neq 0,q \neq 0} ~=~ \sigma S \sqrt{\frac{T}{2 \pi}}~\left( 1 - \frac{(r+q)T}{2} \right) ~-~ \frac{(r-q) T}{2} S ~=~ 4.814848
From the Black--Scholes formula, we find that P_{BS,r=0.06,q=0.03} ~=~ 4.886985, and therefore
\frac{|P_{BS,r=0.06,q=0.03}-P_{approx,r=0.06,q=0.03}|}{P_{BS,r=0.06,q=0.03}} ~=~ 0.014761 ~=~ 1.4761\%
(i) What is the approximate value P_{approx,r=0,q=0} of an at-the-money put option on a non-dividend-paying underlying asset with spot price S = 60, volatility \sigma = 0.25, and maturity T=1 year, if the constant risk--free interest rate is r=0?
(ii) Compute the Black--Scholes value P_{BS,r=0,q=0} of the put option, and estimate the relative approximate error
\frac{|P_{BS,r=0,q=0}-P_{approx,r=0,q=0}|}{P_{BS,r=0,q=0}}
(iii) Assume that r=0.06 and q=0.03. Compute the approximate value P_{approx,r=0.06,q=0.03} of an ATM put option and estimate the relative approximate error
\frac{|P_{BS,r=0.06,q=0.03}-P_{approx,r=0.06,q=0.03}|}{P_{BS,r=0.06,q=0.03}}
where P_{BS,r=0.06,q=0.03} is the Black-Scholes value of the put option.
Solution:
(i) P_{approx,r=0,q=0} ~=~ \sigma S \sqrt{\frac{T}{2 \pi}} ~=~ 5.984134
(ii) From the Black-Scholes formula, we find that P_{BS,r=0,q=0} ~=~ 5.968592, and therefore
\frac{|P_{BS,r=0,q=0}-P_{approx,r=0,q=0}|}{P_{BS,r=0,q=0}} ~=~ 0.002604 ~=~ 0.26\%
(iii) P_{approx,r \neq 0,q \neq 0} ~=~ \sigma S \sqrt{\frac{T}{2 \pi}}~\left( 1 - \frac{(r+q)T}{2} \right) ~-~ \frac{(r-q) T}{2} S ~=~ 4.814848
From the Black--Scholes formula, we find that P_{BS,r=0.06,q=0.03} ~=~ 4.886985, and therefore
\frac{|P_{BS,r=0.06,q=0.03}-P_{approx,r=0.06,q=0.03}|}{P_{BS,r=0.06,q=0.03}} ~=~ 0.014761 ~=~ 1.4761\%