Black–Scholes (from wikipedia.org)
Black–Scholes
From Wikipedia, the free encyclopedia
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 European put and call options.
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, by enhancing work that was published by Fischer Black and Myron Scholes. The paper was first published in 1973. The foundation for their research relied on work developed by scholars such as Louis Bachelier, A. James Boness, Sheen T. Kassouf, Edward O. Thorp, and Paul Samuelson. The fundamental insight of Black-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. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish academy.
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[edit] The model
The Black-Scholes model of the market for an equity makes the following assumptions:
- The price of the underlying instrument St follows a geometric Brownian motion defined by
- 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 (i.e. it is possible to buy any fraction of a share).
- It is possible to borrow and lend cash at a constant risk-free interest rate.
- The stock does not pay a dividend (see below for extensions to handle dividend payments).
[edit] The Equation
In the model, there is a unique price for any derivative of the stock. In particular, a European call option, which gives the right to buy one share at price K after T years, has the price C given by:
where
Where S is the current price of the stock, r is the continuously-compounded risk-free interest rate, and σ is the constant stock's volatility.
Here Φ is the standard normal cumulative distribution function, .
Interpretation: Φ(d1) and Φ(d2) 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.
Intuitively, Φ(d1) represents the non-discounted delta hedge ratio in the underlying asset; and Φ(d2) gives the unit price of the binary call.
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 calculated below:
Here, 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.
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).
[edit] 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).
[edit] Instruments paying continuous yield dividends
For options on indexes, 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
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
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 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
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
where now
is the forward price for the dividend paying stock.
[edit] Black–Scholes in practice
Results using the Black–Scholes model differ from real world prices due to simplifying assumptions of the model. One significant limitation is that in reality security prices do not follow a strict stationary log-normal process, nor is the risk-free interest actually known (and is not constant over time). The variance has been observed to be non-constant leading to models such as GARCH to model volatility changes. Pricing discrepancies between empirical and the Black-Scholes model have long been observed in options that are far out-of-the-money, and these correspond to the likelihood of extreme price changes. While these are very rare when price changes are normally distributed, they are observed in practice and can be modeled as a temporary increase in volatility.
Nevertheless, Black–Scholes pricing is widely used in practice [1], for it is easy to calculate and explicitly model the relationship of all the variables. It is a useful approximation, particularly when analyzing the directionality that prices move when crossing critical points. It is used both as a quoting convention and a basis for more refined models. Although volatility is not constant, results from the model are often useful in practice and helpful in setting up hedges in the correct proportions to minimize risk. Even when the results are not completely accurate, they serve as a first approximation to which adjustments can be made.
Additionally, rather than assuming a volatility a priori and computing prices from it, one can use the model to solve for volatility, which gives the implied volatility of an option at given prices, durations and exercise prices. Solving for volatility over a given set of durations and strike prices one can construct an implied volatility surface. In this application of the Black–Scholes model, a coordinate transformation from the price domain to the volatility domain is obtained. Rather than quoting option prices in terms of dollars per unit (which are hard to compare across strikes and tenors), option prices can thus be quoted in terms of implied volatility, which leads to trading of volatility in option markets.
[edit] The volatility smile
One of the attractive feature of the Black-Scholes model is that the parameters in the model (other than the volatility and the risk-free interest rate) — the time to maturity, the strike, and the current underlying price — are unequivocally observable. All other things being equal, an option's theoretical value is a monotonic increasing function of implied 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: compared to at-the-money, implied volatility is substantially 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."[2] 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] Interest rate curve
In practice, interest rates are not constant - they vary by tenor, giving an interest rate curve which may be interpolated to pick an appropriate rate to use in the Black-Scholes formula. Another consideration is that interest rates vary over time. This volatility may make a significant contribution to the price, especially of long-dated options.
[edit] Short stock rate
It is not free to take a short stock position. Similarly, it may be possible to lend out a long stock position for a small fee. In either case, this can be treated as a continuous dividend for the purposes of a Black-Scholes valuation.
[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
is a normal random variable with mean uT and variance σ2T. It follows that the mean of S is
for some constant q (independent of T). Now a simple no-arbitrage argument shows that the theoretical future value of a derivative paying one share of the stock at time T, and so with payoff S, is
where r is the risk-free interest rate. This suggests making the identification q = r for the purpose of pricing derivatives. Define 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 (using risk-neutral probabilities) is
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
where Φ is the standard normal cumulative distribution function. In the special case b = −∞, we have
Now let
and use the corollary to the lemma to verify the statement above about the mean of S. Define
and observe that
for some b. Define
and observe that
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 risk-free 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 risk-neutrality and the related theory is far from elementary. In elementary terms, the value of the option today is not the expectation of the value of the option at expiry, discounted with the risk-free rate. (So the basic capital asset pricing model (CAPM) results are not violated.) The value is instead computed using the expectation under another distribution of probability, of the value of the option at expiry, discounted with the risk-free rate. This other distribution of probability is called the "risk neutral" probability.
[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,
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
Now consider a trading strategy under which one holds one 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 [t, t + dt], 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). 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]
If we now substitute in for Π and divide through by dt we obtain the Black–Scholes PDE:
This is the law of evolution of the value of the option. 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.
[edit] Other derivations of the PDE
Above 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.
[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, or strike 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
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 V(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 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:
where
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:
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.