Option pricing: Black, Scholes and Merton
The formula recognised by the 1997 Nobel prize is the foundation of the derivatives desks on a trading floor, and its central idea reaches well beyond options - to the way a company's debt and equity are valued.
- The share
- A higher share price today
- Higher volatility
- The contract
- A lower strike price
- A longer time to maturity
- The market
- A higher risk-free rate
The comparative statics of the Black-Scholes formula as the Nobel committee's 1997 press release states them.
01the formula
- A call's payoff at expiry
max(ST − K, 0)ST is the share price at expiry, K the strike.- A put's payoff at expiry
max(K − ST, 0)- Black-Scholes value of a call
C = S·N(d1) − K·e−rT·N(d2)S share price today, K strike, r risk-free rate, T years to expiry, N the standard normal distribution.- The two arguments
d1 = [ln(S/K) + (r + σ2/2)·T] / (σ·√T), d2 = d1 − σ·√Tσ is the volatility of the share's return.
What an option is1
An option gives the right, but not the obligation, to buy or sell a security at a set price in the future. That one-sidedness is its use: it insures against a loss beyond a level, or secures a gain above one.
1973, and three names1
Fischer Black and Myron Scholes published the formula in 1973, and Robert Merton derived it another way that generalised widely. Merton and Scholes received the 1997 prize; Black had died in 1995.
No risk premium needed1
The central insight is that valuing an option does not require an estimate of the stock's expected return. A position in the stock and the option can be combined to be riskless, so it must earn the risk-free rate; the risk premium is already inside the stock price.
The hedge has to be rebalanced1
In the committee's example, if the option moves $1 for every $2 move in the stock, writing two options per share owned removes the risk. As the stock moves and maturity approaches that ratio changes, so the position has to be adjusted gradually to stay riskless.
The formula as the committee printed it1
The press release gives the value of a call as the share price times one probability term, less the present value of the strike times another. Its inputs are the share price, the strike, the time to expiry, the risk-free rate and the share's volatility - the same five things the figure above lists, and nothing about the share's expected return.
A worked price (computed)1
Share at 100, strike 100, one year to expiry, a 5% risk-free rate and 20% volatility. Then d1 = (ln(100/100) + (0.05 + 0.2²/2) × 1) / (0.2 × 1) = 0.07 / 0.2 = 0.35, and d2 = 0.35 − 0.2 = 0.15. From the normal table N(0.35) = 0.6368 and N(0.15) = 0.5596, and the strike's present value is 100 × e^(−0.05) = 95.12. C = 100 × 0.6368 − 95.12 × 0.5596 = 63.68 − 53.23 = 10.45. Raise the volatility to 30% and the same call is worth 14.23: more swing raises the value because the payoff below the strike is floored at zero.
Payoffs at expiry (computed)1
A call struck at 100 pays 30 if the share ends at 130 and nothing if it ends at 80: the holder simply does not exercise, which is the right-but-not-obligation at work. A put struck at 100 pays 20 at 80 and nothing at 130.
next: beyond options →
02beyond options
Equity is an option on the firm1
A company's shares, preferred stock and debt depend on the value of the whole firm in essentially the way an option depends on its stock. The 1973 papers already observed this, laying the foundation for valuing corporate liabilities as options.
Guarantees, insurance and real options1
A guarantee or an insurance contract is a kind of option, and so is flexibility in an investment - closing a mine when the metal price is low, or switching energy sources. The same method values all three.