In their 1973 paper, The Pricing of Options and Corporate Liabilities, Fischer Black and Myron Scholes published an option valuation formula that today is known as the Black-Scholes model. It has become the standard method of pricing options.
The Black-Scholes formula calculates the price of a call option to be:
C = S N(d_{1}) – X e^{-rT} N(d_{2})
where
C = price of the call option | |
S = price of the underlying stock | |
X = option exercise price | |
r = risk-free interest rate | |
T = current time until expiration | |
N() = area under the normal curve | |
d_{1} = [ ln(S/X) + (r + σ^{2}/2) T ] / σ T^{1/2} | |
d_{2} = d_{1} – σ T^{1/2} |
Put-call parity requires that:
P = C – S + Xe^{-rT}
Then the price of a put option is:
P = Xe^{-rT} N(-d_{2}) – S N(-d_{1})
Assumptions
The Black-Scholes model assumes that the option can be exercised only at expiration. It requires that both the risk-free rate and the volatility of the underlying stock price remain constant over the period of analysis. The model also assumes that the underlying stock does not pay dividends; adjustments can be made to correct for such distributions. For example, the present value of estimated dividends can be deducted from the stock price in the model.
Warrant Pricing
Warrants are call options issued by a corporation. They tend to have longer durations than do exchange-traded call options. Warrants can be valued by the Black-Scholes model, but some modifications must be made to the parameters.
When warrants are exercised, the company typically issues new shares at the exercise price to fill the order. The resulting increase in shares outstanding dilutes the share value. If there were n shares outstanding, and m warrants are exercised, α represents the percentage of the value of the firm that is represented by the warrants, where
α = m / ( m + n )
When using the Black-Scholes model to value the warrants, it is worthwhile to use total amounts instead of per share amounts in order to better account for the dilution. The current share price S becomes the enterprise value (less debt) to be acquired by the warrant holders.
The exercise price is the total warrant exercise amount, adjusted for the fact that in paying cash to the firm to exercise the warrants,
the warrant holders in effect are paying a portion of the cash, α, to themselves.
The inputs to the Black-Scholes model for both option pricing and warrant pricing are outlined in the following table.
Black-Scholes Parameters for Pricing Options and Warrants
Input Parameter | Option Pricing | Warrant Pricing |
S | current share price | α V, where V is enterprise value minus debt. |
X | exercise price per share | total warrant exercise amount multiplied by (1 – α). |
T | current time to expiration | average T for warrants |
r | interest rate | interest rate |
σ | standard deviation of stock return | standard deviation for returns on enterprise value, including warrants |