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Black-Scholes options pricing model

How can black-scholes options pricing model support strategic choice or positioning?

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Contents

In the world of financial markets, when you buy an ‘option’ on a corporate stock you are paying for the right, but not the obligation, to buy or sell an amount of that stock at a pre-specified price.

An equity option gives its owner the right, without the obligation, to buy or sell a specified quantity of shares at a predetermined price. The Black–Scholes model estimates what a European-style option should be worth under a defined set of market assumptions.

When to use it

To estimate a theoretical value for European options traded in financial markets.

To support hedging and risk-management decisions in a bank, investment firm or hedge fund.

To understand the factors affecting the value of employee stock options, while recognising that vesting and exercise restrictions may require a different model.

Origins

Fischer Black and Myron Scholes published “The Pricing of Options and Corporate Liabilities” in the Journal of Political Economy in 1973. Robert C. Merton independently developed and extended the continuous-time, no-arbitrage approach, which is now often called the Black–Scholes–Merton model. Myron Scholes and Robert Merton received the 1997 Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for the method. Fischer Black, who collaborated closely in its development, had died in 1995 and could not be awarded the prize posthumously.

What it is

The model produces theoretical values for a call—the right to buy the underlying share—and a put—the right to sell it. Investors use those values as reference points when trading options or constructing hedges.

The key insight is replication and no-arbitrage: a continuously adjusted portfolio of the underlying share and risk-free borrowing or lending can reproduce the option’s payoff. If two positions deliver the same future cash flows, competitive markets should not allow them to have different prices. The resulting option value depends on the current share price, strike price, time to expiry, risk-free rate and volatility of the underlying return.

Consider a non-dividend-paying share priced at $fifty, a ten-year option with a $50 strike and a continuously compounded risk-free rate of 2.75 per cent. Financing $50 for ten years at that rate would require about $thirty-eight point one two today, so the share price less the present value of the strike is approximately $11.88. This is a lower-bound relationship for the European call under the stated assumptions, not the complete Black–Scholes value; volatility adds time value because the holder participates in upside while losses are limited to the premium paid.

The relevant uncertainty is the volatility of the share’s own returns, not its beta relative to the market. Greater volatility generally increases both call and put values because it widens the range of possible terminal prices while the option holder can decline an unfavourable exercise. More time to expiry often increases this opportunity. With little volatility and expiry close at hand, an option’s value approaches its exercise payoff.

How to use it

Apply the model through a calculator, spreadsheet or financial system, but inspect every input and assumption rather than treating the output as fact. For a non-dividend-paying European call, the formula uses:

the current price of the underlying asset;

the option’s strike price;

time to expiry, expressed as a fraction of a year;

expected or implied volatility;

the continuously compounded risk-free interest rate.

The Black–Scholes call formula is:

*C = S N(d1) − K e−rt N(d*2)

where:

d1 = [ln(S / K) + (r + σ² / 2)t] / (σ√t)

d2 = d1 − σ√t

*C is the call value, S the current share price, K the strike, N the cumulative standard-normal distribution, e the exponential constant, σ the annualised standard deviation of returns, r the risk-free rate and t* the time to expiry. The first term represents the share component of the replicating portfolio after probability adjustment; the second represents the discounted strike component. The standard model relies on simplifying assumptions:

The option is European and can be exercised only at expiry.

The underlying share pays no dividend before expiry, unless the formula is adjusted for a known dividend yield.

Markets permit continuous trading and no riskless arbitrage.

There are no transaction costs or taxes, and assets can be divided and traded as needed.

The risk-free rate is known and remains constant over the option’s life.

The underlying return volatility is known and constant.

The share price follows a continuous lognormal process, with no sudden jumps.

Top practical tip

A calculator can perform the arithmetic, but useful application requires understanding how price, strike, time, rates and especially volatility affect the result. Compare the model value with market price and test several volatility and expiry scenarios.

Top pitfall

Do not confuse a model value with a guaranteed fair price. Volatility is not directly observable for the future, market-implied volatility varies by strike and expiry, and real prices can jump. Restructurings, mergers or major changes in competitive conditions can invalidate the smooth, constant-input assumptions.