The Black–Scholes model, often simply called Black–Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. The Black–Scholes formula is a mathematical formula for the theoretical value of European put and call stock options that may be derived from the assumptions of the model. The equation was derived by Fischer Black and Myron Scholes; the paper that contains the result was published in 1973. They built on earlier research by Paul Samuelson and Robert Carhart Merton. The fundamental insight of Black and Scholes is that the call option is implicitly priced if the stock is traded. The use of the Black–Scholes model and formula is pervasive in financial markets.

Merton and Scholes received the 1997 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel for their work; Black died in 1995.

The model

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The key assumptions of the Black–Scholes model are:

• The price of the underlying instrument follows a geometric Brownian motion S_t, in particular with constant drift $\backslash mu$ (Expected gain) and volatility $\backslash sigma$:

$dS\_t\; =\; \backslash mu\; S\_t\backslash ,dt\; +\; \backslash sigma\; S\_t\backslash ,dW\_t\; \backslash ,$

• It is possible to short sell the underlying stock.
• There are no arbitrage opportunities.
• Trading in the stock is continuous.
• There are no transaction costs or taxes.
• All securities are perfectly divisible (e.g. it is possible to buy 1/100th of a share).
• The risk-free interest rate exists and is constant, and the same for all maturity dates.

The formula

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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 $\backslash sigma$.

$C(S,T)\; =\; SN(d\_1)\; -\; Ke^\{-rT\}N(d\_2)\; \backslash ,$

where

$d\_1\; =\; \backslash frac\{\backslash ln(S/K)\; +\; (r\; +\; \backslash sigma^2/2)T\}\{\backslash sigma\backslash sqrt\{T\}\}$

$d\_2\; =\; d\_1\; -\; \backslash sigma\backslash sqrt\{T\}.$

Here N is the standard normal cumulative distribution function.

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

$P(S,T)\; =\; Ke^\{-rT\}N(-d\_2)\; -\; SN(-d\_1).\; \backslash ,$

The Greeks under the Black–Scholes model are also easy to calculate :

	Calls	Puts
delta	$N(d\_1)\; \backslash ,$	$N(d\_1)\; -\; 1\; \backslash ,$
gamma	$\backslash frac\{\backslash phi(d\_1)\}\{S\backslash sigma\backslash sqrt\{T\}\}\; \backslash ,$
vega	$S\; \backslash phi(d\_1)\; \backslash sqrt\{T\}\; \backslash ,$
theta	$-\; \backslash frac\{S\; \backslash phi(d\_1)\; \backslash sigma\}\{2\; \backslash sqrt\{T\}\}\; -\; rKe^\{-rT\}N(d\_2)\; \backslash ,$	$-\; \backslash frac\{S\; \backslash phi(d\_1)\; \backslash sigma\}\{2\; \backslash sqrt\{T\}\}\; +\; rKe^\{-rT\}N(-d\_2)\; \backslash ,$
rho	$KTe^\{-rT\}N(d\_2)\backslash ,$	$-KTe^\{-rT\}N(-d\_2)\backslash ,$

Here, $\backslash phi$ is the normal probability density function. Note that the gamma and vega formulas are the same for calls and puts.

Extensions of the formula

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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 European options on instruments paying dividends. American options are more difficult to value, and a choice of models is available (for example Whaley, binomial options model, Monte Carlo Model).

Instruments paying continuous 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 simplify and make the assumption that the dividends are paid continuously.

The dividend payment paid over the time period $+\; dt$ is then modelled as

$qS\_t\backslash ,dt$

for some constant q (the dividend yield).

Under this formulation the arbitrage-free price implied by the Black–Scholes model can be shown to be

$C(S\_0,T)\; =\; S\_0\; e^\{-qT\}N(d\_1)\; -\; Ke^\{-rT\}N(d\_2)\; \backslash ,$

where now

$F\; =\; S\_0\; e^\{(r\; -\; q)T\}\; \backslash ,$

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

$d\_1\; =\; \backslash frac\{\backslash ln(F/K)\; +\; (\backslash sigma^2/2)T\}\{\backslash sigma\backslash sqrt\{T\}\}$

$d\_2\; =\; d\_1\; -\; \backslash sigma\backslash sqrt\{T\}.$

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).

Instruments paying discrete dividends

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 $\backslash delta$ of the stock price is paid out at pre-determined times $T\_1,T\_2,...$; this dividend forecast is often based on Fundamental analysis or company guidance.

The price of a stock is then modelled as

$S\_t\; =\; S\_0(1\; -\; \backslash delta)^\{n(t)\}e^\{\backslash mu\; t\; +\; \backslash sigma\; W\_t\}$

where n(t) is the number of dividends that have been paid by time t.

The price of a call option on such a stock is again

$C(S\_0,T)\; =\; e^\{-rT\}(FN(d\_1)\; -\; KN(d\_2))\; \backslash ,$

where now

$F\; =\; S\_0(1\; -\; \backslash delta)^\{n(T)\}e^\{rT\}\; \backslash ,$

is the forward price for the dividend paying stock.

Black–Scholes in practice

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While in practice more advanced models are often used, many of the key insights provided by the Black–Scholes formula have become an integral part of market conventions. For instance, it is common practice for the implied volatility rather than the price of an instrument to be quoted. (All the parameters in the model other than the volatility—that is the time to maturity, the strike, the risk-free rate, and the current underlying price—are unequivocally observable. This means there is one-to-one relationship between the option price and the volatility.) Traders prefer to think in terms of volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on.

The volatility smile

However, the Black–Scholes model cannot model the real world exactly. If the Black–Scholes model held, then the implied volatility of an option on a particular stock would be constant, even as the strike and maturity varied, and roughly equal to the historic volatility. In practice, the volatility surface (the two-dimensional graph of implied volatility against strike and maturity) is not flat. In fact, in a typical market, the graph of strike against implied volatility for a fixed maturity is typically smile-shaped (see volatility smile). That is, at-the-money (the option for which the underlying price and strike coincide) the implied volatility is lowest; out-of-the-money or in-the-money the implied volatility tends to be different, usually higher on the put side (low strikes), and call side (high strikes).

In fact, the volatility surface of a given underlying instrument depends on, among other things, its historical distribution and is constantly changing as investors, market-makers, and arbitrageurs re-evaluate the probability of the underlying instrument reaching a given strike and the risk-reward (including factors related to liquidity) associated to it.

Valuing Bonds

Black–Scholes cannot be applied directly to bond securities because of the pull-to-par problem. As the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its volatility, and the simple Black–Scholes model does not reflect this process. A large number of extensions to Black–Scholes, beginning with the Black model, have been used to deal with this phenomenon.

Formula derivation

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Elementary derivation

Let S₀ be the current price of the underlying stock and S the price when the option matures at time T. Then S₀ is known, but S is a random variable. Assume that

$X\; =\; \backslash ln(S/S\_0)\; \backslash ,$

is a normal random variable with mean $\backslash mu\; T$ and variance $\backslash sigma^2\; T$. It follows that the mean of S is

$E(S)\; =\; S\_0\; e^\{rT\}\; \backslash ,$

for some constant r (independent of T), which may be readily identified with the interest rate. Define the value of the option as the present value of the expected payoff. For a call option with exercise price K this is

$C(S\_0,T)\; =\; e^\{-rT\}E(\backslash max(S\; -\; K,0)).\; \backslash ,$

The derivation of the formula for C is facilitated by the following lemma: Let Z be a standard normal random variable and let b be an extended real number. Define

If a is a positive real number, then

$E(e^\{aZ^+(b)\})\; =\; e^\{a^2/2\}N(-b\; +\; a)$

where N is the normal distribution function. As a special case, we have

$E(e^\{aZ\})\; =\; e^\{a^2/2\}.$

Now let

$Z\; =\; \backslash frac\{X\; -\; \backslash mu\; T\}\{\backslash sigma\backslash sqrt\{T\}\}$

and use the corollary to the lemma to verify the statement above about the mean of S. Define

$X^+\; =\; \backslash ln(S^+/S\_0)\; \backslash ,$

and observe that

$\backslash frac\{X^+\; -\; \backslash mu\; T\}\{\backslash sigma\backslash sqrt\{T\}\}\; =\; Z^+(b)$

for some b. Define

and observe that

$\backslash max(S\; -\; K,0)\; =\; S^+\; -\; K^+.\; \backslash ,$

The rest of the calculation is straightforward.

PDE based derivation

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; for more on the underlying logic, see the discussion at rational pricing.

The presentation given here is informal and we do not worry about the validity of moving between dt meaning a small increment in time and dt as a derivative.

The Black–Scholes PDE

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

$dS\_t\; =\; \backslash mu\; S\_t\backslash ,dt\; +\; \backslash sigma\; S\_t\backslash ,dW\_t\; \backslash ,$

where W_t is Brownian. Now let V be some sort of option on S - mathematically V is a function of S and t. $V(S,t)$ is the value of the option at time t if the price of the underlying stock at time t is S. The value of the option at the time that the option matures is known. To determine its value at an earlier time we need to know how the value evolves as we go backward in time. By Itō's lemma for two variables we have

$dV\; =\; \backslash left(\backslash mu\; S\; \backslash frac\{\backslash partial\; V\}\{\backslash partial\; S\}\; +\; \backslash frac\{\backslash partial\; V\}\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\backslash sigma^2\; S^2\backslash frac\{\backslash partial^2\; V\}\{\backslash partial\; S^2\}\backslash right)dt\; +\; \backslash sigma\; S\; \backslash frac\{\backslash partial\; V\}\{\backslash partial\; S\}\backslash ,dW.$

Now consider a portfolio

$\backslash Pi\; =\; V\; -\; S\backslash frac\{\backslash partial\; V\}\{\backslash partial\; S\}$

consisting of one unit of the option V and −∂V/∂S units of the underlying stock. The composition of this portfolio, called the delta-hedge portfolio, will vary from time-step to time-step. Now consider the change in return

$dR\; =\; dV\; -\; \backslash frac\{\backslash partial\; V\}\{\backslash partial\; S\}\backslash ,dS$

of the portfolio. By substituting in the equations above we get

$dR\; =\; \backslash left(\backslash frac\{\backslash partial\; V\}\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\backslash sigma^2\; S^2\backslash frac\{\backslash partial^2\; V\}\{\backslash partial\; S^2\}\backslash right)dt.$

This equation contains no dW term. That is, it is entirely riskless (delta neutral). 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 $+\; dt$

$r\backslash Pi\backslash ,dt\; =\; \backslash left(\backslash frac\{\backslash partial\; V\}\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\backslash sigma^2\; S^2\backslash frac\{\backslash partial^2\; V\}\{\backslash partial\; S^2\}\backslash right)dt.$

If we now substitute in for $\backslash Pi$ and divide through by dt we obtain the Black–Scholes PDE:

$\backslash frac\{\backslash partial\; V\}\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\backslash sigma^2\; S^2\backslash frac\{\backslash partial^2\; V\}\{\backslash partial\; S^2\}\; +\; r\; S\; \backslash frac\{\backslash partial\; V\}\{\backslash partial\; S\}\; -\; r\; V\; =\; 0.$

This is the law of evolution of the value of the option. 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.

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 $\backslash sigma$ (all as at top). Now, for a call option the PDE above has boundary conditions:

$V(0,t)\; =\; 0\; \backslash ,$ for all t

$V(S,t)\; \backslash rightarrow\; S\; \backslash ,$ as $S\; \backslash rightarrow\; \backslash infty\; \backslash ,$

$V(S,T)\; =\; \backslash max(S\; -\; K,0).\; \backslash ,$

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 standard diffusion equation which may be solved using standard methods. To this end set

$x$ such that $S\; =\; Ke^\{x\}\; \backslash ,$

$\backslash tau$ such that $t\; =\; T\; -\; \backslash frac\{2\backslash tau\}\{\backslash sigma^2\}\; \backslash ,$

$v(x,\backslash tau)$ such that $V\; =\; Kv(x,\backslash tau).\; \backslash ,$

Thus our Black–Scholes PDE becomes

$\backslash frac\{\backslash partial\; v\}\{\backslash partial\; \backslash tau\}\; =\; \backslash frac\{\backslash partial^2\; v\}\{\backslash partial\; x^2\}\; +\; (c\; -\; 1)\backslash frac\{\backslash partial\; v\}\{\backslash partial\; x\}\; -\; cv$

where $c\; =\; 2r/\backslash sigma^2$.

The terminal condition $V(S,T)\; =\; \backslash mbox\{max\}(S\; -\; K,0)$ now becomes an initial condition $v(x,0)\; =\; \backslash mbox\{max\}(e^x\; -\; 1,0)$. If we now make a further transformation such that

$v(x,\backslash tau)\; =\; e^\{-\backslash frac\{1\}\{2\}(c\; -\; 1)x\; -\; \backslash frac\{1\}\{4\}(c\; +\; 1)^2\backslash tau\}u(x,\backslash tau)$

then

$\backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash tau\}\; =\; \backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; x^2\}$

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

$u(x,0)\; =\; \backslash mbox\{max\}(e^\{\backslash frac\{1\}\{2\}(c\; +\; 1)x\}\; -\; e^\{\backslash frac\{1\}\{2\}(c\; -\; 1)x\},0).$

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

$u(x,\backslash tau)\; =\; \backslash frac\{1\}\{2\backslash sqrt\{\backslash pi\backslash tau\}\}\backslash int\_\{-\backslash infty\}^\{\backslash infty\}u\_0(y)e^\{-\backslash frac\{(x\; -\; y)^2\}\{4\backslash tau\}\}\backslash ,dy$

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

$u(x,\backslash tau)\; =\; I\_1\; -\; I\_2\; \backslash ,$

where

$I\_1\; =\; e^\{\backslash frac\{1\}\{2\}(c\; +\; 1)x\; +\; \backslash frac\{1\}\{4\}(c\; +\; 1)^2\backslash tau\}N(d\_1)$

$d\_1\; =\; \backslash frac\{x\}\{\backslash sqrt\{2\backslash tau\}\}\; +\; \backslash frac\{1\}\{2\}(c\; +\; 1)\backslash sqrt\{2\backslash tau\}$

and $I\_2$ is identical to $I\_1$ except that c + 1 is replaced by c − 1 everywhere.

Substituting for u, v, x, and $\backslash tau$, we finally obtain the value of a call option in terms of the Black–Scholes parameters:

$V(S,t)\; =\; SN(d\_1)\; -\; Ke^\{-r(T\; -\; t)\}N(d\_2)\; \backslash ,$

where

$d\_1\; =\; \backslash frac\{\backslash ln(S/K)\; +\; (r\; +\; \backslash sigma^2/2)(T\; -\; t)\}\{\backslash sigma\backslash sqrt\{T\; -\; t\}\}$

$d\_2\; =\; d\_1\; -\; \backslash sigma\backslash sqrt\{T\; -\; t\}.$

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.

Other derivations of the PDE

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.

See also

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• Black model, a variant of the Black–Scholes option pricing model.
• Binomial options model, which is a discrete numerical method for calculating option prices.
• Monte Carlo option model, using simulation in the valuation of options with complicated features.
• Financial mathematics, which contains a list of related articles.
• Heat equation, to which the Black–Scholes PDE can be transformed.
• Mathematical Finance Programming in TI-Basic, which contains some TI-Basic programs to perform stochastic calculus with Texas Instruments Calculators
• Black Shoals, an art installation

References

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• 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.

External links

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Discussion of the model

• Option pricing theory, Overview and links to more detailed articles on specific models, riskglossary.com
• The Black–Scholes Model, global-derivatives.com
• Options pricing using the Black-Scholes Model, The Investment Analysts Society of South Africa
• The Black–Scholes Option Pricing Model, optiontutor
• Black, Merton, and Scholes: Their work and its consequences, by Ajay Shah
• A Study of Option Pricing Models, Prof. Kevin Rubash
• Black-Scholes in English, risklatte.com
• The Black-Scholes Option Pricing Formula, greekshares.com

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
• Employee Stock Option Valuation, esomanager.com

Derivation and solution

• 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
• Solution via risk neutral pricing or via the PDE approach using Fourier transforms (includes discussion of other option types), Simon Leger
• Step-by-step derivation of delta from the Black-Scholes equation (site also contains step-by-step derivations of some of the other greeks), quantnotes.com

Tests of the model

• Anomalies in option pricing: the Black–Scholes model revisited, New England Economic Review, March-April, 1996

Online calculators

• Graphical Online Option Calculator, www.OptionsToolbox.com
• Online Option Calculator with Simulation Table, www.option-price.com
• Black–Scholes Option Calculator, tradingtoday.com
• Option Valuation and Calculators, DerivativeOne.com
• Black–Scholes Pricing Analysis- including dividends, hoadley.net
• Java mobile edition phone calculator, www.getjar.com

Computer implementations

• Black–Scholes in Multiple Languages, espenhaug.com
• Option Pricing Spreadsheet with documented VBA, OptionTradingTips.com
• Excel spreadsheet with VBA sourcecode, quantnotes.com
• Excel format (.xls) Option Position Calculators, financialhub.info

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."
• BBC Horizon A TV-programme on the so-called Midas formula and the bankruptcy of Long Term Capital Management (LTCM)

Mathematical finance | Stock market | Equations | Finance theories | Options

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