The Black76 Options Pricing Formulas

The LME Black76 formula for calls is:

c = e -r(T+2/52) [FN(d1) - XN(d2)]

and for puts:

p = e -r(T+2/52) [XN(- d2) - FN(- d1)]

where N (.) stands for the cumulative normal distribution, T is the time to the option expiry, r is the continuously compounded interest rate, and

 d1 = ln(F/X) + σ2 T/2

σ √T

d2 = d1 – σ √T

Example

Today’s date is 5/7/07 and we want to price a 2100 call option on the August 2007 copper future. The prompt date for the August future is 19/8/07 i.e. in 45 days time. As the option expiry date is 14 days before this i.e. on 5/8/07, the number of days to the expiry of the option is 31.

Suppose input values to the formula are:

  • Futures price F = $2006
  • Strike price X = $2100
  • Volatility σ = 35%
  • Time to expiry T = 31/365 = 0.08493 years to 5 decimal places
  • Time to futures prompt date = 45/365 = 0.12329 to 5 dp
  • Annually compounded interest rate = 0.051342 to 6 dp1

First, calculate d1 and d2. Note that this uses T calculated from 31 days:

The logarithm to base e of F/X is ln(2006/2100) = ln(0.9552381) = -0.045795

σ 2 T/2 =0.35 * 0.35 * 0.5 * 0.08493 = 0.005202 

σ √T = 0.35 * √0.08493 = 0.1019996 

Therefore

d1 = -0.045795 + 0.005202
0.1019996
= -0.397972

and d2 = -0.397972 – 0.1019996 = -0.499972

Using EXCEL© function “=normsdist()” for the cumulative normal distribution, 
N (d1) = 0.345225 and N(d2) = 0.308547

Before discounting, this gives the terminal call price i.e. at expiry as 
(2006* 0.345225) - (2100* 0.308547) = 44.77308

The value of the discount factor is 0.993846. Its calculation is explained below.

Therefore the call price is

0.993846 * 44.77308 = $44.50 rounded to 2 dp.

How the Discount Factor is Calculated

Interest rates supplied by the LME are continuous compounded rates. You can convert from annually to continuous compounding by the formula

rcontinuous = ln(1 + rannualised

i.e rcontinuous = ln(1 + 0.051342) = 0.050067 to 6 dp

The time used in discounting the terminal call price to present value is increased by 14 days over the option expiry date i.e. to 45 days, as the option expires into the future on 5/8/07, but value is not obtained until the futures 
prompt date on 19/8/07.

To match this time, the maturity of the interest rate used should match the prompt date i.e. the LME supplies the 45-day rate.

The discount factor is therefore

e-rcontinuous (T+ 2/52) = e-0.050067(0.12329) = 0.993846

It is necessary to enter the annually compounded rate into LME Select to obtain the correct result. This is because LME Select “pre-processes” the annually compounded rate to convert it to continuous compounding before using it in Black’76. 

1 Note that the Black’76 formula requires a continuously compounded interest rate.

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