The Black-Scholes Model

The Black-Scholes model is a pricing model used to determine the theoretical price of options contracts. It was developed by Fischer Black and Myron Scholes in 1973 and later refined by Robert Merton.

It revolutionized the options market by providing a fair and consistent framework to calculate the fair value and to manage the risk of an option contract, making options trading more transparent and accessible to investors.

The model has also played a crucial role in the development of more complex financial instruments and trading strategies that involve options.

Original Papers

The original paper, The Pricing of Options and Corporate Liabilities by Black and Scholes, was published in 1973 in the Journal of Political Economy. Almost at the same time, Merton wrote an academic paper “Theory of Rational Option Pricing” on the subject, he proposed some extensions to the model and also named it “Black-Scholes model”. The model led to a boom in the derivatives market, Merton and Scholes received the economic nobel prize in 1997 for their work. Black, who died 2 years before was mentioned as a contributor.

Assumptions

Here are the main assumptions of the model:

• Markets are random,market movements cannot be predicted, the underlying stock price follows a geometric brownian motion;

• Stocks are not paying any dividend;

• We can trade continuously, with no transaction costs;

• There is no limit on short selling, no borrowing constraints;

• The risk free interest rate is constant;

• And there is no opportunities for arbitrage in the market.

A Financial Market with Two Assets

It also assumes that the financial market has two assets:

• One risky asset such as a stock price;

• one risk-free asset such as a bank account.

Risky Asset

The stock price S_t  follows a Geometric Brownian motion, with the following stochastic differential equation:

W_t is a Wiener process, μ is the drift and σ the volatility of the stock price.

So S_t has the following expression, it follows a log normal distribution.

Simulations of the Stock Price

T = 5Y, 10000 simulations, nb intervals=100, r = 3%, sigma = 20%

Risk-Free Asset

We assume that there is one risk free asset  B_t, with r the continuously compounded risk free interest rate. There is no randomness here, one unit invested in the cash account at time 0 will be worth e^rt at time t.

Risk-Free Asset Price

T = 5Y, r = 3%

The Black-Scholes Equation

We will now derive the Black-Scholes equation which governs the dynamic of the price of European stock options.

For that we consider a third asset, a European option on the risky asset S. 

We know its final price at maturity T P(S, T), it is the final payoff of the option.

What is the price of the option P(S, t) at time t<T?

Self-Financing Portfolio

We consider the following self-financing portfolio Π:

It is short one option, the initial value is -P, it is long the underlying stock in a quantity equal the partial derivatives of P with respect to S and we borrow in cash the sum of the two in order to finance the strategy.

The initial value of the portfolio is zero by construction:

Here are the quantities we have on the three different assets:

Portfolio Variation

And we calculate the variation of the portfolio on a small interval dt assuming the quantities do not change.

Equation 1

We know that:

For the differential of the option price dP we apply the Itô lemma:

We know that:

The square of it is equal to:

So we get the following expression for the differential of the price of the option dP:

We replace dB and dP with the corresponding expressions in Equation 1 and we get the following expression for the variation of the value of the portfolio between t and t+dt:

Equation 2

We see that the terms in dS cancel each other. By purchasing a quantity equal to the partial derivative of the price with respect to S we are able to cancel the risk related to the risky asset in the portfolio.

The Black-Scholes Partial Differential Equation

We that that there is no randomness term in Equation 2 above, so it is a non-risky portfolio.

By no arbitrage condition its return is equal to the risk free interest rate.

As the initial value of the portfolio is equal to zero, the variation of the portfolio is also equal to zero.

And so we get the following expression which is the Black-Scholes Partial Differential Equation:

Black-Scholes Equation

We remark that there is no drift component.

Any european option satisfies this equation and we were able to find a hedging strategy offsetting the risk in the portfolio.

With boundary and final conditions, it can be solved numerically to determine the price of the derivatives, using methods such as numerical methods such as finite difference, Fourier, or Monte-Carlo. Closed-form solutions are available in some cases, which simplifies the computation.

The Black-Scholes Formula for European Call and Put Options

This is typically the case for European call and put options. The boundary and final conditions are determined by the payoff of the option.

European Call Option	European Put Option

And the equation can be solved analytically with closed-form solutions, these are the Black-Scholes formula for European call and put options.

Black-Scholes Formula

European Call Price: European Put Price: Where:

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