Options Pricing and the Black–Scholes Model

An option gives its holder the right, but not the obligation, to buy or sell an asset at a fixed price on or before a future date. Valuing that right puzzled economists for decades until 1973, when Fischer Black and Myron Scholes, with key contributions from Robert Merton, published a formula that changed finance. This article explains what options are, builds the intuition with a simple binomial model, and then works through the Black–Scholes formula step by step.

Calls, Puts and Payoffs

There are two basic types of option:

• A call option gives the right to buy the underlying asset at the strike price \( K \).
• A put option gives the right to sell the underlying asset at the strike price \( K \).

A European option can be exercised only at expiry; an American option can be exercised at any time before expiry. Black–Scholes values European options.

If the share price at expiry is \( S_T \), the payoffs are:

$$ \text{Call payoff} = \max(S_T - K,\ 0), \qquad \text{Put payoff} = \max(K - S_T,\ 0) $$

The buyer pays a premium up front for this right. The pricing question is: what is a fair premium today?

What Affects an Option's Price?

Five inputs determine the value of a European option on a non-dividend-paying share:

1. Current share price \( S \). A higher price makes a call more valuable and a put less valuable.
2. Strike price \( K \). A higher strike lowers a call's value and raises a put's.
3. Time to expiry \( T \). More time usually means more chance of a favourable move.
4. Volatility \( \sigma \). More volatility raises both call and put values, because the holder benefits from big moves in one direction but is protected in the other.
5. Risk-free rate \( r \). A higher rate raises call values slightly and lowers put values.

Remarkably, the investor's expected return on the share is not on this list. That surprising fact is the key to the whole theory.

The Binomial Intuition: No-Arbitrage Pricing

Imagine a share now worth 100 that, in one year, will be either 120 or 90. A call with strike 100 will pay 20 if the price rises and 0 if it falls. The risk-free rate is 5% per year.

Step 1: Build a replicating portfolio. Buy \( \Delta \) shares and borrow some money so that the portfolio pays exactly what the option pays in both states. The share position must change by the same amount as the option:

$$ \Delta = \frac{20 - 0}{120 - 90} = \frac{2}{3} $$

Step 2: Find the borrowing. In the down state, \( \tfrac{2}{3} \) of a share is worth 60, but the option pays 0, so we must owe exactly 60 at year end. The amount borrowed today is

$$ \frac{60}{1.05} = 57.14 $$

Step 3: Cost of the portfolio.

$$ \tfrac{2}{3}(100) - 57.14 = 66.67 - 57.14 = 9.52 $$

Since this portfolio produces exactly the option's payoff in every state, the option must cost 9.52. Any other price would allow risk-free profit, called arbitrage.

The same answer comes from risk-neutral probabilities. Define

$$ q = \frac{(1+r) - d}{u - d} = \frac{1.05 - 0.9}{1.2 - 0.9} = 0.5 $$

Then the option price is the expected payoff under \( q \), discounted at the risk-free rate:

$$ C = \frac{0.5(20) + 0.5(0)}{1.05} = 9.52 $$

If we split the year into more and more tiny steps, this binomial model converges to the Black–Scholes formula.

The Black–Scholes Formula

For a European call on a non-dividend-paying share:

$$ C = S\,N(d_1) - K e^{-rT} N(d_2) $$

and for a European put:

$$ P = K e^{-rT} N(-d_2) - S\,N(-d_1) $$

where

$$ d_1 = \frac{\ln(S/K) + \left(r + \tfrac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T} $$

Here \( N(x) \) is the cumulative standard normal distribution function, \( r \) is the continuously compounded risk-free rate and \( T \) is in years.

A helpful interpretation: \( N(d_2) \) is the risk-neutral probability that the call ends in the money, and \( N(d_1) \) is the option's delta, the number of shares needed to hedge it.

Worked Example 1: An At-the-Money Call and Put

A share trades at \( S = 100 \). Find the price of a 6-month European call and put with strike \( K = 100 \), if \( r = 5\% \) and \( \sigma = 20\% \).

Step 1: Inputs. \( T = 0.5 \), \( \sigma\sqrt{T} = 0.2 \times \sqrt{0.5} = 0.2 \times 0.707107 = 0.141421 \).

Step 2: Compute \( d_1 \) and \( d_2 \). Since \( \ln(100/100) = 0 \):

$$ d_1 = \frac{0 + (0.05 + 0.02)(0.5)}{0.141421} = \frac{0.035}{0.141421} = 0.2475 $$

$$ d_2 = 0.2475 - 0.1414 = 0.1061 $$

Step 3: Normal probabilities (from tables or software).

$$ N(d_1) = N(0.2475) = 0.5977, \qquad N(d_2) = N(0.1061) = 0.5422 $$

Step 4: Discounted strike.

$$ K e^{-rT} = 100\,e^{-0.025} = 100 \times 0.975310 = 97.5310 $$

Step 5: Call price.

$$ C = 100(0.597734) - 97.5310(0.542235) = 59.7734 - 52.8847 = 6.89 $$

Step 6: Put price. Using \( N(-d_1) = 0.402266 \) and \( N(-d_2) = 0.457765 \):

$$ P = 97.5310(0.457765) - 100(0.402266) = 44.6463 - 40.2266 = 4.42 $$

Check with put–call parity. For European options:

$$ C - P = S - K e^{-rT} $$

Left side: \( 6.89 - 4.42 = 2.47 \). Right side: \( 100 - 97.53 = 2.47 \). The prices are consistent.

Worked Example 2: An Out-of-the-Money Call

A share trades at 50. Price a 3-month call with strike 55, with \( r = 6\% \) and \( \sigma = 30\% \).

Step 1: \( T = 0.25 \), \( \sigma\sqrt{T} = 0.3 \times 0.5 = 0.15 \), and \( \ln(50/55) = -0.095310 \).

Step 2:

$$ d_1 = \frac{-0.095310 + (0.06 + 0.045)(0.25)}{0.15} = \frac{-0.095310 + 0.02625}{0.15} = -0.4604 $$

$$ d_2 = -0.4604 - 0.15 = -0.6104 $$

Step 3: \( N(d_1) = 0.3226 \), \( N(d_2) = 0.2708 \).

Step 4: \( K e^{-rT} = 55\,e^{-0.015} = 55 \times 0.985112 = 54.1812 \).

Step 5:

$$ C = 50(0.322614) - 54.1812(0.270798) = 16.1307 - 14.6721 = 1.46 $$

Sanity check. The call is out of the money (the share is 5 below the strike) with only three months left, so a small premium makes sense. Its value must also lie between 0 and the share price of 50, which it does.

Assumptions and Limitations

The model assumes share prices follow geometric Brownian motion with constant volatility, no dividends, no transaction costs, continuous trading and a constant risk-free rate. Real markets break several of these. In particular, implied volatilities differ across strikes, a pattern called the volatility smile. Traders therefore treat Black–Scholes as a common language: they quote options by their implied volatility and adjust from there.

Islamic Finance and Options

Most Shariah scholars do not permit conventional options, mainly because of concerns about gharar (excessive uncertainty) and selling a right detached from an asset. Some Islamic structures, such as the urbun (a down-payment sale) and certain promises (wa'd), are discussed as alternatives. Students in Islamic finance programmes still study Black–Scholes to understand risk management in global markets.

Did You Know?

• Scholes and Merton received the 1997 Nobel Memorial Prize in Economic Sciences; Fischer Black had died in 1995, and the prize is not awarded posthumously.
• The Chicago Board Options Exchange opened in April 1973, the same year the Black–Scholes paper appeared, and traders were soon using the formula on calculators.
• The famous Black–Scholes equation is a partial differential equation closely related to the heat equation in physics.

Key Takeaways

• A call pays \( \max(S_T - K, 0) \); a put pays \( \max(K - S_T, 0) \).
• Options are priced by no-arbitrage replication, which leads to risk-neutral valuation.
• Black–Scholes: \( C = S N(d_1) - K e^{-rT} N(d_2) \).
• Higher volatility raises the value of both calls and puts.
• Always check answers with put–call parity: \( C - P = S - K e^{-rT} \).

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