3 The formula

The above lead to the following formula for the price of a call on a stock currently trading at price S, where the option has an exercise price of K, i.e. the right to buy a share at price K, at T years in the future. The constant interest rate is r and the constant stock volatility is v:

where

Here N is the cumulative normal distribution function.

The price of a put option may be computed from this by put-call parity and simplifies to:

The Greeks under the Black-Scholes model are also easy to calculate.

4 Extensions of the formula

The above option pricing formula is used for pricing European put and call options on non- dividend paying stocks. The Black-Scholes model may be easily extended to options on instruments paying dividends. For options on indexes (such as the FTSE) where each of 100 constituent companies may pay a dividend twice a year and so there is a payment nearly every business day, it is reasonable to assume that the dividends are paid continuously. The dividend payment paid over the time period is then modelled as

for some constant q. Under this formulation the arbitrage-free price the Black-Scholes model can be shown to be

where now

is the modified forward price that occurs in the terms d₁ and d₂.

Exactly the same formula is used to price options on foreign exchange rates, except now q plays the role of the foreign risk-free interest rate and S is the spot exchange rate. This is the Garman-Kohlhagen model (1983).

It is also possible to extend the Black-Scholes framework to options on instruments paying discrete dividends. This is useful when the option is struck on a single stock. A typical model is to assume that a proportion of the stock price is paid out at pre-determined times . The price of a stock is then modelled as

where n(t) is the number of dividends that have been paid at time t. The price of a call option on a such stock is again

where now

is the forward price for the dividend paying stock.

American options are more difficult to value, and a choice of models is available (for example Whaley, binomial options model).

5 Formula derivation

5.1 The Black-Scholes PDE

In this section we derive the partial differential equation (PDE) at the heart of the Black-Scholes model via a no-arbitrage or delta-hedging argument. The presentation given here is informal and we do not worry about the validity of moving between dt meaning an small increment in time and dt as a derivative.

As in the model assumptions above we assume that the underlying (typically the stock) follows a geometric Brownian motion. That is,

where W is Brownian. Now let V be some sort of option on S - mathematically V is a function of S and t. By Itô's lemma for two variables we have

Now consider a portfolio consisting of one unit of the option V and -dV/dS units of the underlying. The composition of this portfolio, called the delta-hedge portfolio, will vary from time-step to time-step. Now consider the change in value

of the portfolio by subbing in the equation above:

This equation contains no dW term. That is, it is entirely riskless. Thus, assuming no arbitrage (and also no transaction costs and infinite supply and demand) the rate of return on this portfolio must be equal to the rate of return on any other riskless instrument. Now assuming the risk-free rate of return is r we must have over the time period

If we now substitute in for and divide through by we obtain the Black-Scholes PDE

With the assumptions of the Black-Scholes model, this equation holds whenever V has two derivatives with respect to S and one with respect to t.

5.2 From the general Black-Scholes PDE to a specific valuation

We now show how to get from the general Black-Scholes PDE to a specific valuation for this option. Consider as an example the Black-Scholes price of a call on a stock currently trading at price S. The option has an exercise price of K, i.e. the right to buy a share at price K, at T years in the future. The constant interest rate is r and the constant stock volatility is v (all as at top). Now, for a call option the PDE above has boundary conditions:

for all t

as

In order to solve the PDE we transform the equation into a standard diffusion equation which may be solved using standard methods. To this end set

such that

such that

such that

Thus our Black-Scholes PDE becomes

where . The terminal condition now becomes an initial condition . If we now make a further transformation such that

then

a standard diffusion equation as desired. Our initial condition has translated to

Using the standard method for solving a diffusion equation we have:

where u₀ is the initial condition defined in the line above. This integral may be further transformed until we obtain:

where

and is identical to except that (c+1) is replaced by (c−1) everywhere.

Substituting v for u and the V for v, we finally obtain the value of a call option in terms of the Black-Scholes parameters:

where

.

As before, N is the cumulative normal distribution function.

The formula for the price of a put option, follows from this via put-call parity.

5.3 Other derivations

Above we used the method of arbitrage-free pricing (" delta-hedging") to derive a 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.

6 See also

• Binomial options model, which is able to handle a variety of conditions for which Black-Scholes cannot be applied.
• Black model a variant (and more general form) of the Black-Scholes option pricing model.
• Financial mathematics, which contains a list of related articles.

7 References

• Black, F. and M. Scholes, "The Pricing of Options and Corporate Liabilities" Journal of Political Economy 81, 1973, 637-654. Black and Scholes' original paper.
• Merton, Robert C., "Theory of rational option pricing", Bell Journal of Economics and Management Science 4 (1), 1973, 141-183.

8 External links

• The Black Scholes Option Pricing Model
  • option pricing theory links to more detailed articles on specific models, riskglossary.com
  • Options pricing using the Black-Scholes Model, The Investment Analysts Society of South Africa
  • The Black Scholes Option Pricing Model, optiontutor

• Variations on the model
  • options on non-dividend paying stocks (Black Scholes), riskglossary.com
  • options on stock indexes and continuous dividend-paying stocks, riskglossary.com
  • foreign exchange options, riskglossary.com
  • options on forwards (the Black model), riskglossary.com

• Derivations
  • The risk neutrality derivation of the Black-Scholes Equation, quantnotes.com
  • Arbitrage-free pricing derivation of the Black-Scholes Equation, quantnotes.com or an alternative treatment, Prof. Thayer Watkins
  • Solving the Black-Scholes Equation, quantnotes.com
  • Solution of the Black Scholes Equation Using the Green's Function, Prof. Dennis Silverman

• Online calculators
  • Black-Scholes Pricing Analysis- including dividends, hoadley.net

• Computer Programming Implementations
  • Black-Scholes in Multiple Languages, www.espenhaug.com

• Historical
  • The Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel for 1997
  • Trillion Dollar Bet - Companion Web site to a Nova episode originally broadcast on February 8, 2000. "The film tells the fascinating story of the invention of the Black-Scholes Formula, a mathematical Holy Grail that forever altered the world of finance and earned its creators the 1997 Nobel Prize in Economics."

financial mathematics Stock market

<Hide>