The term Black–Scholes refers to three closely related concepts:
- The Black–Scholes model is a mathematical model of the market for an equity, in which the equity's price is a stochastic process.
- The Black–Scholes PDE is a partial differential equation which (in the model) must be satisfied by the price of a derivative on the equity.
- The Black–Scholes formula is the result obtained by solving the Black–Scholes PDE for a European call option.
Fischer Black and Myron Scholes first articulated the Black–Scholes formula in their 1973 paper, "The Pricing of Options and Corporate Liabilities." The foundation for their research relied on work developed by scholars such as Jack L. Treynor, Paul Samuelson, A. James Boness, Sheen T. Kassouf, and Edward O. Thorp. The fundamental insight of Black–Scholes is that the option is implicitly priced if the stock is traded.
Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model and coined the term "Black-Scholes" options pricing model.
Merton and Scholes received the 1997 The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for this and related work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish academy .
Black–Scholes model
The Black–Scholes model of the market for a particular equity makes the following explicit assumptions:
- It is possible to borrow and lend cash at a known constant risk-free interest rate.
- The price follows a geometric Brownian motion with constant drift and volatility.
- There are no transaction costs.
- The stock does not pay a dividend (see below for extensions to handle dividend payments).
- All securities are perfectly divisible ( i.e. it is possible to buy any fraction of a share).
- There are no restrictions on short selling.
- There is no arbitrage opportunity
From these ideal conditions in the market for an equity (and for an option on the equity), the authors show that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in , whose value will not depend on the price of the stock." "
Notation
Define
N ( x ) denotes the standard normal cumulative distribution function,
.
N '( x ) denotes the standard normal probability density function,
.
Black–Scholes PDE
As per the model assumptions above, we assume that the underlying asset (typically the stock) follows a geometric Brownian motion. That is,
where W t is Brownian—the dW term here stands in for any and all sources of uncertainty in the price history of a stock.
The payoff of an option V ( S , T ) at maturity is known. To find its value at an earlier time we need to know how V evolves as a function of S and T . By Itō's lemma for two variables we have
Now consider a trading strategy under which one holds a single option and continuously trades in the stock in order to hold
shares. At time t , the value of these holdings will be
The composition of this portfolio, called the delta-hedge portfolio, will vary from time-step to time-step. Let R denote the accumulated profit or loss from following this strategy. Then over the time period , the instantaneous profit or loss is
By substituting in the equations above we get
This equation contains no dW term. That is, it is entirely riskless (delta neutral). Black and Scholes reason that under their ideal conditions, the rate of return on this portfolio must be equal at all times to the rate of return on any other riskless instrument; otherwise, there would be opportunities for arbitrage. Now assuming the risk-free rate of return is r we must have over the time period
If we now substitute in for Π and divide through by dt we obtain the Black–Scholes PDE :
With the assumptions of the Black–Scholes model, this partial differential equation holds whenever V is twice differentiable with respect to S and once with respect to t .
Other derivations of the PDE
See also: Martingale pricingAbove we used the method of arbitrage-free pricing ("delta-hedging") to derive some PDE governing option prices given the Black–Scholes model. It is also possible to use a risk-neutrality argument. This latter method gives the price as the expectation of the option payoff under a particular probability measure, called the risk-neutral measure, which differs from the real world measure.
Black–Scholes formula
The Black Scholes formula is used for obtaining the price of European put and call options. It is obtained by solving the Black–Scholes PDE as discussed - see derivation below.
The value of a call option in terms of the Black–Scholes parameters:
The price of a put option is:
For both, as above:
- N(•) is the standard normal or cumulative distribution function
- T - t is the time to maturity
- S is the spot price of the underlying asset
- K is the strike price
- r is the risk free rate (annual rate, expressed in terms of continuous compounding)
- σ is the volatility in the log-returns of the underlying
Interpretation
N ( d 1 ) and N ( d 2 ) are the probabilities of the option expiring in-the-money under the equivalent exponential martingale probability measure (numéraire = stock) and the equivalent martingale probability measure (numéraire = risk free asset), respectively. The equivalent martingale probability measure is also called the risk-neutral probability measure. Note that both of these are probabilities in a measure theoretic sense, and neither of these is the true probability of expiring in-the-money under the real probability measure.
Derivation
We now show how to get from the general Black–Scholes PDE to a specific valuation for an option. Consider as an example the Black–Scholes price of a call option, for which the PDE above has boundary conditions
The last condition gives the value of the option at the time that the option matures. The solution of the PDE gives the value of the option at any earlier time,
. In order to solve the PDE we transform the equation into a diffusion equation which may be solved using standard methods. To this end we introduce the change-of-variable transformation
Then the Black–Scholes PDE becomes a diffusion equation
The terminal condition C ( S , T ) = max( S − K ,0) now becomes an initial condition
Using the standard method for solving a diffusion equation we have
After some algebra we obtain
where
and
Substituting for u , x , and τ , we obtain the value of a call option in terms of the Black–Scholes parameters:
where
The price of a put option may be computed from this by put-call parity and simplifies to
Greeks
The Greeks under Black–Scholes are given in closed form, below:
Note that the gamma and vega formulas are the same for calls and puts. This can be seen directly from put-call parity.
In practice, some sensitivities are usually quoted in scaled-down terms, to match the scale of likely changes in the parameters. For example, rho is often reported divided by 10,000 (1bp rate change), vega by 100 (1 vol point change), and theta by 365 or 252 (1 day decay based on either calendar days or trading days per year).
Extensions of the model
The above model can be extended to have non-constant (but deterministic) rates and volatilities. The model may also be used to value European options on instruments paying dividends. In this case, closed-form solutions are available if the dividend is a known proportion of the stock price. American options and options on stocks paying a known cash dividend (in the short term, more realistic than a proportional dividend) are more difficult to value, and a choice of solution techniques is available (for example lattices and grids).
Instruments paying continuous yield dividends
For options on indexes, it is reasonable to make the simplifying assumption that dividends are paid continuously, and tha
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