Question

Use the Black’s model to value a 1-year European put option on a 10-year bond. Assume that the current value of the bond is $112, the strike price is $113, the 1-year interest rate is 10% per annum, the bond’s forward price volatility is 15% per annum, and the present value of the coupons to be paid during the life of the option is $7.

Answer 1

Given:

Strike Price X = $113

Interest rate r = 10% per annum

Volatility σ = 15%

Maturity = 10 years

Time period t = 1 year

Current Value of Bond = $112

PV of Coupouns = $7

1year Forward Price = (112-7)*e ^(0.1*1)  = 116.043

S= $116.043

Black Scholes Formula

P= X*e^(-rt) * N(-d2) - S*e^(-rt)* N(-d1)

d1= (ln(S/X) + t(σ² /2 )) / σ√t

d2 =d1- σ√t

Now

d1= (ln(S/X) + t(r+σ² /2 )) / σ√t

d1= (ln(116.043/113) + 1*(0.15²/2)) / (0.15√1) = (0.0265725 + 0.01125) / 0.15 = 0.25215

d2 = 0.25215-0.15= 0.10215

N(-d1) =0.400462 (Norsm.s.dist(-0.25215,1)

N(-d2) = 0.459319 (Norsm.s.dist(-10215,1)

P= 113*e^(-0.1*1) * 0.459319 - 116.043*e^(-0.1*1) * 0.0.400462 = $4.9152