Black-Scholes

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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 was ineligible, having died in 1995.

Contents

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The model

The key assumptions of the Black–Scholes model are:

<math> dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \, </math>
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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 <math> \sigma </math>.

<math> C(S,T) = SN(d_1) - Ke^{-rT}N(d_2) \, </math>

where

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

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

<math> P(S,T) = Ke^{-rT}N(-d_2) - SN(-d_1). \, </math>

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

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

Here, <math> \phi </math> is the normal probability density function. Note that the gamma and vega formulas are the same for calls and puts.

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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 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).

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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 <math> [t,t + dt] </math> is then modelled as

<math> qS_t\,dt </math>

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

<math> C(S_0,T) = S_0 e^{-qT}N(d_1) - Ke^{-rT}N(d_2) \, </math>

where now

<math> F = S_0 e^{(r - q)T} \, </math>

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

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

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).

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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 <math> \delta </math> of the stock price is paid out at pre-determined times <math> T_1,T_2,... </math>; this dividend forecast is often based on Fundamental analysis or company guidance.

The price of a stock is then modelled as

<math> S_t = S_0(1 - \delta)^{n(t)}e^{\mu t + \sigma W_t} </math>

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

<math> C(S_0,T) = e^{-rT}(FN(d_1) - KN(d_2)) \, </math>

where now

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

is the forward price for the dividend paying stock.

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Black–Scholes in practice

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.

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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). Furthermore, all implied volatilities on the surface tend to be higher than historic volatility.

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.

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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.

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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

<math> X = \ln(S/S_0) \, </math>

is a normal random variable with mean <math> \mu T </math> and variance <math> \sigma^2 T </math>. It follows that the mean of S is

<math> E(S) = S_0 e^{rT} \, </math>

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

<math> C(S_0,T) = e^{-rT}E(\max(S - K,0)). \, </math>

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

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

If a is a positive real number, then

<math> E(e^{aZ^+(b)}) = e^{a^2/2}N(-b + a) </math>

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

<math> E(e^{aZ}) = e^{a^2/2}. </math>

Now let

<math> Z = \frac{X - \mu T}{\sigma\sqrt{T}} </math>

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

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

and observe that

<math> \frac{X^+ - \mu T}{\sigma\sqrt{T}} = Z^+(b) </math>

for some b. Define

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

and observe that

<math> \max(S - K,0) = S^+ - K^+. \, </math>

The rest of the calculation is straightforward.

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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.

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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,

<math> dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \, </math>

where Wt is Brownian. Now let V be some sort of option on S - mathematically V is a function of S and t. <math> V(S,t) </math> 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

<math> 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. </math>

Now consider a portfolio

<math> \Pi = V - S\frac{\partial V}{\partial S} </math>

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

<math> dR = dV - \frac{\partial V}{\partial S}\,dS </math>

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

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

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 <math> [t,t + dt] </math>

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

If we now substitute in for <math> \Pi </math> and divide through by dt we obtain the Black–Scholes PDE:

<math> \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0. </math>

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.

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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 <math> \sigma </math> (all as at top). Now, for a call option the PDE above has boundary conditions:

<math> V(0,t) = 0 \, </math> for all t
<math> V(S,t) \rightarrow S \, </math> as <math> S \rightarrow \infty \, </math>
<math> V(S,T) = \max(S - K,0). \, </math>

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

<math> x </math> such that <math> S = Ke^{x} \, </math>
<math> \tau </math> such that <math> t = T - \frac{2\tau}{\sigma^2} \, </math>
<math> v(x,\tau) </math> such that <math> V = Kv(x,\tau). \, </math>

Thus our Black–Scholes PDE becomes

<math> \frac{\partial v}{\partial \tau} = \frac{\partial^2 v}{\partial x^2} + (c - 1)\frac{\partial v}{\partial x} - cv </math>

where <math> c = 2r/\sigma^2 </math>.

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

<math> v(x,\tau) = e^{-\frac{1}{2}(c - 1)x - \frac{1}{4}(c + 1)^2\tau}u(x,\tau) </math>

then

<math> \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2} </math>

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

<math> u(x,0) = \mbox{max}(e^{\frac{1}{2}(c + 1)x} - e^{\frac{1}{2}(c - 1)x},0). </math>

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

<math> u(x,\tau) = \frac{1}{2\sqrt{\pi\tau}}\int_{-\infty}^{\infty}u_0(y)e^{-\frac{(x - y)^2}{4\tau}}\,dy </math>

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

<math> u(x,\tau) = I_1 - I_2 \, </math>

where

<math> I_1 = e^{\frac{1}{2}(c + 1)x + \frac{1}{4}(c + 1)^2\tau}N(d_1) </math>
<math> d_1 = \frac{x}{\sqrt{2\tau}} + \frac{1}{2}(c + 1)\sqrt{2\tau} </math>

and <math> I_2 </math> is identical to <math> I_1 </math> except that c + 1 is replaced by c − 1 everywhere.

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

<math> V(S,t) = SN(d_1) - Ke^{-r(T - t)}N(d_2) \, </math>

where

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

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.

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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.

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See also

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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.
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External links

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Discussion of the model

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Variations on the model

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Derivation and solution

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Tests of the model

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Online calculators

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Computer implementations

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Historical

ja:ブラック-ショールズ方程式

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