Black-Scholes model

An option's premium depends on several things at once: how far the underlying is from the strike, how long is left, how volatile the underlying is, and interest rates. The Black-Scholes model turns those inputs into a single theoretical price.

It was published in 1973 by Fisher Black and Myron Scholes. The workbook calls it one of the most popular, relatively simple and fast methods of option pricing, because — unlike the binomial model — it does not rely on iteration. It is a formula you solve once.

Five inputs: spot price, strike price, volatility, time to expiry and the risk-free rate.

Outputs: the call and put premium, and the option Greeks — delta, gamma, vega, theta and rho.

For this paper the key fact is practical: Indian exchanges use the Black-Scholes model to compute the theoretical price of exchange-traded currency options.

Assumptions and limits (Chapter 4, section 4.6.2). The model was designed for options exercisable only at maturity (European) on assets that do not pay dividends, and assumes random market movements and zero transaction costs. In practice assets pay dividends and options are sometimes exercised early, so traders modify it.

Dividends. A dividend lowers the underlying price on the ex-date, making calls less valuable and puts more valuable. The workbook gives two fixes: for short-term options, subtract the present value of expected dividends from the spot price; for long-term options, adjust the model for a constant dividend yield.

Its relatives:

Implied volatility runs the model backwards: take the traded premium and the four observable inputs, and solve for the volatility that makes the model price equal the market price.

C = S·N(d1) − X·e^(−rt)·N(d2)
P = X·e^(−rt)·N(−d2) − S·N(−d1)

d1 = [ln(S/X) + (r + v²/2)·t] / (v·√t)
d2 = d1 − v·√t

S = spot, X = strike, t = years to expiry, r = continuously compounded risk-free rate,
v = annual volatility, N(·) = standard normal cumulative distribution

Illustrative inputs, chosen to show the arithmetic; they are not workbook figures. The workbook presents the formula in stock notation, so the example uses a hypothetical underlying priced at Rs 100.

S = Rs 100, X = Rs 100, t = 0.5 years, r = 7%, v = 20%.

v·√t      = 0.20 × 0.7071            = 0.14142
d1        = [ln(1) + (0.07 + 0.02) × 0.5] / 0.14142
          = 0.045 / 0.14142          = 0.3182
d2        = 0.3182 − 0.14142         = 0.1768
N(d1)     = 0.6248      N(d2) = 0.5702
X·e^(−rt) = 100 × e^(−0.035)         = 96.5605

Call = 100 × 0.6248 − 96.5605 × 0.5702 = Rs 7.43
Put  = 96.5605 × 0.4298 − 100 × 0.3752 = Rs 3.99

Check with put-call parity: C − P should equal S − X·e^(−rt):

7.43 − 3.99 = 3.44     and     100 − 96.5605 = 3.44  ✓

Reading it. Raise v to 25% and both premiums rise — which is why, as the workbook notes, rising implied volatility after you buy an option is good for the buyer and bad for the seller.

Chapter 4, section 4.6 (Option Pricing Methodology), covers the binomial model, Black-Scholes with its formula, inputs, outputs, assumptions and dividend adjustments, and Black (1976); section 4.7 explains implied volatility by running Black-Scholes in reverse. The standard questions: who developed it and when, its five inputs, which Indian products use Black-Scholes versus Black (1976), and how it differs from the binomial model.

• Black-Scholes uses spot prices; Black (1976) uses futures/forward prices. Indian exchanges use Black-Scholes for currency options and Black (1976) for interest rate options.
• 1973, Fisher Black and Myron Scholes. The binomial model is William Sharpe, 1978.
• Black-Scholes is not iterative; the binomial model is.
• Five inputs — spot, strike, volatility, time, risk-free rate. The option premium is an output, and the only unobservable input is volatility.
• Designed for European options on non-dividend-paying assets.
• Dividends make calls cheaper and puts dearer.