Tipology : Encyclopedia

The most famous and most general option pricing model was developed in the early 1970s by Fisher Black and Myron Scholes (1973). Originally, this model was developed to price European-style financial options (i.e. those that cannot be settled before expiry) and since the first version it has contributed to and influenced all subsequent pricing models. An important contribution to the definite development of the Black and Scholes model undoubtedly goes to Merton who made modifications and improvements on the basis of the 1973 version.
In the Black & Scholes model, as in the binomial model, the basic assumption is that an option-equivalent portfolio can be created, consisting partly of units of the underlying and partly of risk-free bonds. The main difference to the binomial model is that in this case the assumption is that returns are distributed among infinite states of nature according to a normal statistical law. The Black and Scholes model represents the limit in the continuum of the binomial model (which is discrete).

The Black and Scholes model makes it possible to define and evaluate an option from knowledge of six fundamental variables, which are:
S = Value of the underlying asset
K= "strike" price of the option
t = option maturity
r= risk-free interest rate corresponding to the life of the option
      σ= volatility of the underlying
Given these values, Black and Scholes show that, in the presence of a geometric Brownian stochastic process (the stochastic process that corresponds to the assumption of lognormality of the instantaneous distributions of the reference variable), the following result is obtained:

Value of call option

with:

N represents a standardized normal distribution, i.e. a normal distribution that has mean equal to 0 and standard deviation equal to 1.
The Black and Scholes (B&S) method is based on the idea of the replicating portfolio. Since in an efficient market there is no possibility of arbitrage, this portfolio must have the same value as the option given by a risk-free combination of credit and debt assets and the underlying.
The first term in the equation can be interpreted as the value of the shares to be bought and the second term as that of the bonds to be issued. The value of the option thus expresses the difference between a given value of the underlying asset and the present value of the debt .

the number of shares of the underlying to create the replicating portfolio is called the option delta. The and values denote the probabilities that the option will expire with a value of the underlying greater than the strike value (i.e. "in the money" as they say in financial jargon). Note, that, as in the binomial model, these probabilities are 'pseudo-probabilities' and do not correspond to actual probabilities except in the particular hypothesis that traders are risk-neutral. Similarly to the value of the European call, one can derive the value of the European option which is equal to

The Black and Scholes model assumes: that the price trend of the underlying asset can be approximated by a log-normal process; that there is a perfectly efficient and frictionless market (including the absence of taxes and transaction costs); that the market interest rate is the same for loans and borrowings and is constant over the life of the option; that the variance of the underlying asset is constant over the life of the option. If the market meets these characteristics, the model under consideration provides a rigorous basis for calculating the value and risk characteristics of an option. The key factor for this calculation is the changes in the price of the security. The sensitivity of the value of an option with respect to the price factor is indicated by the delta coefficient (the first derivative of C with respect to S), which measures the ratio between the changes in C and those in S in the constant of the other factors. The delta coefficient makes it possible to estimate the impact on the ex ante calculated value C of a given change in the price of the underlying asset (change in C = delta * change in S). Mathematically, the delta is derived from the Black and Scholes formula: delta = N(d¹ ). The delta of a call varies in a range between 0 and 1. Its value is lowest when S is much lower than K and the option's maturity is close. In such a case, the probability of price increases such that the call option is in the money at expiration is very remote: the market expects the option to expire worthless and therefore the link to the stock price is very weak. The delta tends to unity for prices (S) much higher than K, as it is very likely that the option will be exercised. Other coefficients that can be derived from the Black and Scholes formula are gamma, theta, vega and rho (q.v.) which measure, respectively, the sensitivity: of delta to small changes in the stock price; of the value of the option to small changes in the time to expiration; of the value of the option to small changes in volatility: of the value of the option to changes in the interest rate. Put options. In European-style put options, the theoretical price P is given by the formula ? = Ke ¯ rt N(-d²)-SN -(d¹).
The delta is then equal to the 1's complement of the delta of a call option written under the same conditions, i.e. it is : =|N(d¹) - 1|

Adaptations of the Black and Scholes formula. Financial theory has made several refinements to the Black and Scholes formula. In 1973 Robert Merton relaxed the assumption of non-distribution of dividends during the option period. In 1976, Jonathan Ingerson relaxed the no-tax and transaction cost constraint and Rober erton removed the constraint of a constant interest rate. Several adaptations have also been developed to extend the formula to options on currencies, bonds, futures and options on interest rates (caps; floors etc.). Empirical adaptations have also been made to evaluate American options (which, unlike European options, give the holder the option of exercising early before maturity). Applications to market risk assessment. Delta is a measure of the sensitivity of the calculated option value to small changes in the share price and is therefore an index of position risk. Delta is the coefficient used to calculate capital requirements for outstanding market risks on options, according to one (delta-plus method) of the three procedures prescribed by the Bank of Italy. The Black and Scholes formula is one of the methods prescribed by the Bank of Italy for calculating the delta (see Supervisory Instructions, Title IV, Chapter 3, Section IX, and Annex D).

Bibliography
BLACK F. and SCHOLES M. (1973), “The Pricing of Options and Corporate Liabilities”, in Journal of Political Economy, 81 (May-June), 1973, pp. 637-659
HULL J.C. (1997), Options, Futures and Other Derivative Securities, Prentice Hall International; translated by Emilio Barone, Il Sole 24 Ore Libri, Milan
MERTON R. (1973), Theory of Rational Option Pricing, Bell Journal of Economics and Management Science 4 (1), 141-183
PENNISI G. and SCANDIZZO P. L. (2006) "Economic Evaluation in an Age of Uncertainty", Evaluation, Vol. 12, No. 1, pp. 77-94
PENNISI G. and SCANDIZZO P. L. (2003) "Evaluating Uncertainty: Cost Benefit Analysis in the 21st Century" G. Giappichelli Editore