Options are popular instruments in the world of finance. A call option on a stock gives the owner the right to buy the stock at a predetermined price before the expiration date of the option. For example, suppose that call options are selling for Procter & Gamble stock that give the owner of the option the right to buy a share of stock for $60 within the next 21 days. The asking price on the option was $1.45 at the market close. How are options priced? A pricing formula for options was developed by Fischer Black and Myron Scholes and published in 1973. Scholes was later awarded the Nobel Prize for this work in 1997 (Black was deceased). The Black-Scholes pricing model is widely used today by hedge funds and traders. The Black-Scholes formula for the price of a call option is

C = S [PSN(Z)] - Xe-rT{PSNsZ - σ √T}

where

C = market price of the call option
X = strike or exercise price of the stock
S = current price of the stock
r =annual risk-free interest rate
T = time to maturity of the option
σ = yearly standard deviation

In the Black-Scholes formula,

and PSN(Z) is the probability of an observation of Z or less for a normal distribution with mean 0 and variance 1. The purpose of this exercise is to price a Procter & Gamble call option offered today that expires 21 days later. Use the yield on three-month Treasury bills as the risk-free interest rate, which you can assume is currently 0.0494. The strike price on the option is $60 today, the stock is currently trading at $60.87. In order to use the Black-Scholes formula, the yearly standard deviation, s is required. One way to obtain this number is to estimate the weekly variance of Procter & Gamble stock, multiply the weekly variance by 52, and then take the square root to get the annual standard deviation. For this problem, use a weekly variance of 0.000479376. Use these data to calculate the option price using the Black-Scholes formula.

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Z = [(r + o?/2)T + In(S/X)]/(oVT)