A futures contract is an agreement to buy or sell at a future date a specified amount of a particular asset at a pre-agreed price. An option on a futures contract gives the holder the right to purchase (via a call) or sell (via a put) a futures contract. The calculation of a futures option is similar to that of a stock option, except that futures don’t pay dividends. A mathematical function known as the Black-Scholes model is used to calculate futures option prices.
Identify values for the five inputs: 1) F = futures contract price. 2) K = strike price of the option. 3) T = time to expiration of the option contract, in years. 4) r = the annual risk free rate, in percent. 5) V= annual futures contract price volatility, in percent.
The first three are specified by the given futures and options contracts. You can use the current yield on 3-month Treasury bills as the risk-free rate. The annual futures volatility can be estimated using the “rule of 16”: estimate the daily percent volatility of the futures contract from the last week of trading and then multiply by 16 to get the annual volatility. For instance, if the futures contract has been experiencing approximately 2 percent daily price moves over the last week, V would equal 32 percent.
Calculate the two normal distributions d1 and d2. The formulas are:
d1 equals [(natural log(F divided by K)) plus( 0.5 (V squared))T] divided by [V multiplied by (square root of T)]
d2 equals d1 minus [V multiplied by (square root of T)]
Work out the two terms X1 and X2 of the Black-Sholes equation. You will be using the constant “e”, the base for natural logarithms, in the calculations. The approximate value of e is 2.71828. You will need the cumulative normal distribution N of d1 and d2 – see Tips.
X1 equals F multiplied by [e exponentially raised by (negative r multiplied by T)] multiplied by N(d1).
X2 equals K multiplied by [e exponentially raised by (negative r multiplied by T)] multiplied by N(d2).
Compute the price of the futures call option C, which is equal to X1 minus X2.
The price of a futures put option P is equal to X2 minus X1 with one proviso: the values of d1 and d2 must first be multiplied by minus 1 when calculating the cumulative normal distributions. In other words, use N(-d1) and N (-d2) instead of N(d1) and N(d2), respectively.
Excel has built-in functions to ease the calculation process: (1) The function LN returns the natural log of a given number. (2) The function EXP returns e raised to the power of a given number. (3) The function SQRT returns the square root of a given number. (4) The function NORMSDIST returns the cumulative normal distribution N of a given number.
The Black-Sholes equation provides an estimate of a theoretical price. It is an estimate because the volatility is estimated. The price is theoretical because the actual prices are determined by supply and demand. This equation assumes European-style options, which can only be exercised on expiration. The calculation of American style – can be exercised at any time -- option prices is considerably more complex.
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