Black–Scholes

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For the financial art piece, see Black Shoals.

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 an equation which (in the model) the price of a derivative on the equity must satisfy.
  • The Black-Scholes formula is the result obtained by applying the Black-Scholes PDE to European put and call options.

The formula was derived by Fischer Black and Myron Scholes and published in 1973. They built on earlier research by Edward O. Thorp, Paul Samuelson, and Robert C. Merton. The fundamental insight of Black and Scholes is that the option is implicitly priced if the stock is traded.

Merton and Scholes received the 1997 Nobel Prize in Economics for this and related work; Black was ineligible, having died in 1995.

Contents

[edit] The model

The key assumptions of the Black–Scholes model are:

dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \,
  • 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).
  • It is possible to borrow and lend cash at a constant risk-free interest rate.

[edit] The PDE

Given the assumptions of the Black-Scholes model, the price Vt of a derivative written on a stock with price process St evolves according to the following partial differential equation (PDE):

\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0.

It is notable that the PDE does not contain μ, the drift of the stock. An informal derivation of the PDE, explaining this result, is given below.

[edit] The formula

The above lead to the following formula for the price of a call option with exercise price K on a stock currently trading at price S, i.e., the right to buy a share of the stock at price K after T years. The constant interest rate is r, and the constant stock volatility is σ.

C(S,T) = S\Phi(d_1) - Ke^{-rT}\Phi(d_2) \,

where

d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}
d_2 = d_1 - \sigma\sqrt{T}.

These are also more mnemonically notated

d_{\pm} = \frac{\ln(S/K) + (r \pm \sigma^2/2)T}{\sigma\sqrt{T}}

(so d + = d1 and d − = d2)

Here Φ 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}\Phi(-d_2) - S\Phi(-d_1). \,

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

Calls Puts
delta \Phi(d_1) \, \Phi(d_1) - 1 \,
gamma \frac{\phi(d_1)}{S\sigma\sqrt{T}} \,
vega S \phi(d_1) \sqrt{T} \,
theta - \frac{S \phi(d_1) \sigma}{2 \sqrt{T}} - rKe^{-rT}\Phi(d_2) \, - \frac{S \phi(d_1) \sigma}{2 \sqrt{T}} + rKe^{-rT}\Phi(-d_2) \,
rho KTe^{-rT}\Phi(d_2)\, -KTe^{-rT}\Phi(-d_2)\,

Here, φ is the standard normal probability density function. Note that the gamma and vega formulas are the same for calls and puts. This can be seen directly from put-call parity.

[edit] Extensions of the model

The above model can easily 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).

[edit] Instruments paying continuous yield 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 make the simplifying assumption that dividends are paid continuously, and that the dividend amount is proportional to the level of the index.

The dividend payment paid over the time period [t, t + dt] is then modelled as

qS_t\,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) = e^{-rT}(F\Phi(d_1) - K\Phi(d_2)) \,

where now

F = S_0 e^{(r - q)T} \,

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

d_1 = \frac{\ln(F/K) + (\sigma^2/2)T}{\sigma\sqrt{T}}
d_2 = d_1 - \sigma\sqrt{T}.

Exactly the same formula is used to price options on foreign exchange rates, except that 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).

[edit] Instruments paying discrete proportional dividends

It is also possible to extend the Black–Scholes framework to options on instruments paying discrete proportional dividends. This is useful when the option is struck on a single stock.

A typical model is to assume that a proportion δ of the stock price is paid out at pre-determined times t1, t2, .... The price of the stock is then modelled as

S_t = S_0(1 - \delta)^{n(t)}e^{ut + \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}(F\Phi(d_1) - K\Phi(d_2)) \,

where now

F = S_0(1 - \delta)^{n(T)}e^{rT} \,

is the forward price for the dividend paying stock.

[edit] Black–Scholes in practice

[edit] The volatility smile

See Volatility smile for the complete article

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 a one-to-one relationship between the option price and the volatility. By computing the implied volatility for traded options with different strikes and maturities, we can test the Black-Scholes model. If the Black–Scholes model held, then the implied volatility for a particular stock would be the same for all strikes and maturities. In practice, the volatility surface (the three-dimensional graph of implied volatility against strike and maturity) is not flat. The typical shape of the implied volatility curve for a given maturity depends on the underlying instrument. Equities tend to have skewed curves: implied volatility is higher for low strikes, and slightly lower for high strikes. Currencies tend to have more symmetrical curves, with implied volatility lowest at-the-money, and higher volatilities in both wings. Commodities often have the reverse behaviour to equities, with higher implied volatility for higher strikes.

Despite the existence of the volatility smile (and the violation of all the other assumptions of the Black-Scholes model), the Black-Scholes PDE and Black-Scholes formula are still used extensively in practice. A typical approach is to regard the volatility surface as a fact about the market, and use an implied volatility from it in a Black-Scholes valuation model. This has been described as using "the wrong number in the wrong formula to get the right price" [Rebonato 1999]. This approach also gives usable values for the hedge ratios (the Greeks).

Even when more advanced models are used, traders prefer to think in terms of volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on.

[edit] Valuing bond options

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.

[edit] Formula derivation

[edit] 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 \equiv \ln(S/S_0) \,

is a normal random variable with mean uT and variance σ2T. It follows that the mean of S_T is

\mathbb{E}\left[ S_T \right] = S_0 e^{qT} \,

for some constant q (independent of T). Now a simple no-arbitrage argument shows that the value of a derivative paying one share of the stock at time T, and so with payoff S_T, has (future valued) theoretical value:

S_0 e^{rT} \,

Which suggests making the identification q = r for the purposes of pricing derivatives, and defining the theoretical value of a derivative as the present value of the expected payoff in this sense. For a call option with exercise price K this discounted expectation is:

C(S_0,T) = e^{-rT} \mathbb{E}\left[ \max(S - K,0) \right]. \,

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

Z^+(b) = \begin{cases} Z & \mbox{if }Z>b \\ -\infty & \mbox{otherwise} \end{cases}.

If a is a positive real number, then

\mathbb{E}\left[e^{aZ^+(b)}\right] = e^{a^2/2}\Phi(-b + a)

where Φ is the standard normal cumulative distribution function. In the special case b = −∞, we have

\mathbb{E}\left[e^{aZ}\right] = e^{a^2/2}.

Now let

Z = \frac{X - uT}{\sigma\sqrt{T}}

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

S^+ = \begin{cases} S & \mbox{if }S>K \\ 0 & \mbox{otherwise} \end{cases}
X^+ = \ln(S^+/S_0) \,

and observe that

\frac{X^+ - uT}{\sigma\sqrt{T}} = Z^+(b)

for some b. Define

K^+ = \begin{cases} K & \mbox{if }S>K \\ 0 & \mbox{otherwise} \end{cases}

and observe that

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

The rest of the calculation is straightforward.

Although the elementary derivation leads to the correct result, it is incomplete as it cannot explain, why the formula refers to the riskfree interest rate while a higher rate of return is expected from risky investments. This limitation can be overcome using the risk-neutral probability measure, but the concept of riskneutrality and the related theory is far from elementary.

[edit] 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.

[edit] 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 = \mu S_t\,dt + \sigma S_t\,dW_t \,

where Wt 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 = \left(\mu S \frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2}\right)dt + \sigma S \frac{\partial V}{\partial S}\,dW.

Now consider a trading strategy under which one holds one option and continuously trades in the stock in order to hold −∂V/∂S shares. At time t, the value of these holdings will be:

\Pi = V - S\frac{\partial V}{\partial S}

The composition of this portfolio, called the delta-hedge portfolio, will vary from time-step to time-step. Let R denote the accumulated profit and loss from following this strategy. Then over the time period [t, t + dt], the instantaneous profit or loss is:

dR = dV - \frac{\partial V}{\partial S}\,dS.

By substituting in the equations above we get

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

This equation contains no dW term. That is, it is entirely riskless (delta neutral). Thus, given that there is no arbitrage, 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 [t, t + dt]

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

If we now substitute in for Π and divide through by dt we obtain the Black–Scholes PDE:

\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 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.

[edit] 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.

[edit] Solution of the Black–Scholes PDE

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 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 σ. Now, for a call option the PDE above has boundary conditions

V(0,t) = 0 \, for all t
V(S,t) \sim S \, as S \rightarrow \infty \,
V(S,T) = \max(S - K,0). \,

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

x = \ln(S/K) + (r - \sigma^2/2)(T - t) \,
\tau = T - t \,
u = Ve^{r(T - t)}. \,

Then the Black–Scholes PDE becomes a diffusion equation

\frac{\partial u}{\partial \tau} = \frac{\sigma^2}{2} \frac{\partial^2 u}{\partial x^2}.

The terminal condition V(S,T) = max(S − K,0) now becomes an initial condition

u(x,0) = u_0(x) \equiv K\max(e^x - 1,0). \,

Using the standard method for solving a diffusion equation we have

u(x,\tau) = \frac{1}{\sigma\sqrt{2\pi\tau}}\int_{-\infty}^{\infty}u_0(y)e^{-(x - y)^2/(2\sigma^2\tau)}\,dy.

After some algebra we obtain

u(x,\tau) = Ke^{x + \sigma^2\tau/2}\Phi(d_1) - K\Phi(d_2)

where

d_1 = \frac{x + \sigma^2\tau}{\sigma\sqrt{\tau}}
d_2 = \frac{x}{\sigma\sqrt{\tau}}

and Φ is the standard normal cumulative distribution function.

Substituting for u, x, and τ, we obtain the value of a call option in terms of the Black–Scholes parameters:

V(S,t) = S\Phi(d_1) - Ke^{-r(T - t)}\Phi(d_2) \,

where

d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T - t)}{\sigma\sqrt{T - t}}
d_2 = d_1 - \sigma\sqrt{T - t}.

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

[edit] Remarks on notation

The reader is warned of the inconsistent notation that appears in this article. Thus the letter S is used as:

(1) a constant denoting the current price of the stock
(2) a real variable denoting the price at an arbitrary time
(3) a random variable denoting the price at maturity
(4) a stochastic process denoting the price at an arbitrary time

It is also used in the meaning of (4) with a subscript denoting time, but here the subscript is merely a mnemonic.

In the partial derivatives, the letters in the numerators and denominators are, of course, real variables, and the partial derivatives themselves are, initially, real functions of real variables. But after the substitution of a stochastic process for one of the arguments they become stochastic processes.

The Black–Scholes PDE is, initially, a statement about the stochastic process S, but when S is reinterpreted as a real variable, it becomes an ordinary PDE. It is only then that we can ask about its solution.

The parameter u that appears in the discrete-dividend model and the elementary derivation is not the same as the parameter μ that appears elsewhere in the article. For the relationship between them see Geometric Brownian motion.

[edit] See also

[edit] References

[edit] Primary references

  • Black, Fischer, Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): 637-654. [1] (Black and Scholes' original paper.)
  • Merton, Robert C. (1973). "Theory of Rational Option Pricing". Bell Journal of Economics and Management Science 4 (1): 141-183. [2]

[edit] Historical and sociological aspects

  • Bernstein, Peter. Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press. ISBN 0-02-903012-9.
  • MacKenzie, Donald (2003). "An Equation and its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics". Social Studies of Science 33 (6): 831-868. [3]
  • MacKenzie, Donald, Yuval Millo (2003). "Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange". American Journal of Sociology 109 (1): 107-145. [4]
  • MacKenzie, Donald. An Engine, not a Camera: How Financial Models Shape Markets. MIT Press. ISBN 0-262-13460-8.

[edit] External links

[edit] Discussion of the model

[edit] Variations on the model

[edit] Derivation and solution

[edit] Tests of the model

[edit] Online calculators

[edit] Computer implementations

[edit] Historical

[edit] Trivia


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