Black-Scholes
The Black-Scholes model, often simply 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 Fisher Black and Myron Scholes; the paper that contains the result was published in 1973. They built on earlier research by Paul Samuelson and Robert Merton. The fundamental insight of Black and Scholes was 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.
The key assumptions of the Black-Scholes model are:
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:
- <math>V(S,t)=SN(d_1)-K e^{-rT} N(d_2)</math>
where
- <math>d_1=\frac{\log \frac{S}{K}+\left( r+v^{2}/2\right) T}{v\sqrt{T}}</math>
- <math>d_2=d_1-v\sqrt{T}</math>.
- 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:
- <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.
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 <math>[t,t+\delta t]</math> is then modelled as
- <math> qS_t dt</math>
for some constant q. Under this formulation the arbitrage-free price under the Black-Scholes model can be shown to be
- <math> C(S,T)= e^{-qT}S_0 N(d_1) - e^{-rT}KN(d_2)</math>
where now
- <math> F = e^{(r-q)T}S_0</math>
is the modified forward price that occurs in the <math>d_.</math> terms.
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 <math>\delta</math> of the stock price is paid out at pre-determined times <math>T_1,T_2,...</math>. The price of a stock is then modelled as
- <math> S_t = S_0(1-\delta)^{n(t)}e^{\sigma W_t + \mu t}</math>
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
- <math> C(S,T) = FN(d_1)-Ke^{-rT}N(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.
American options are more difficult to value, and a choice of models is available (for example Whaley, binomial options model).
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,
- <math>dS_t = \mu S dt + \sigma S dW_t</math>
where W Brownian. Now let V be some sort of option on S - mathematically V is a function of S and t. By Ito's Lemma for two variables we have
- <math> dV = \sigma S \frac{\partial V}{\partial S}dW + ( \mu S \frac{\partial V}{\partial S}+ \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + \frac{\partial V}{\partial t})dt </math>
Now consider a portfolio <math>\Pi</math> 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
- <math>d\Pi = dV - \frac{\partial V}{\partial S} dS</math>
of the portfolio by subbing in the equation above:
- <math> d\Pi = ( \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2})dt </math>
This equation contains no <math>dW</math> 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 <math>r</math> we must have over the time period <math>[t,t+\delta t]</math>
- <math> r\Pi dt = ( \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2})dt </math>
If we now substitute in for <math>\Pi</math> and divide through by <math>dt</math> 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>
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.
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:
- <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>
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\ s.t.\ S = Ke^{x}</math>
- <math> \tau\ s.t. \ t=T-\frac{\tau}{\frac{1}{2}\sigma^2}</math>
- <math> v(x,\tau)\ s.t.\ V = K.v(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 = 0</math>
where <math>c=2r/\sigma^2</math>. The terminal condition <math>V(S,T)=max(S-K,0)</math> now becomes an initial condition <math>v(x,0) = 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) = 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 v for u and the V for v, we finally obtain the the value of a call option in terms of the Black-Scholes parameters:
- <math>V(S,t)=SN(d_1)-K e^{-rT} N(d_2)</math>
where
- <math>d_1=\frac{\log \frac{S}{K}+\left( r+v^{2}/2\right) T}{v\sqrt{T}}</math>
- <math>d_2=d_1-v\sqrt{T}</math>.
- N is the cumulative Normal distribution function.
The formula for the price of a put option, follows from this via put-call parity.
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.
The use of the Black-Scholes formula is pervasive in the markets. In fact the model has become such an integral part of market conventions that 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 expiry, the strike, the risk-free rate and current underlying price - are unequivocably observable. This means there is one-to-one relationship between the option price and the volatility.) Traders prefer to think in terms of volatility.
However, the Black-Scholes model can not be modelling the real world completely accurately. 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. 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 co-incide) the implied volatility is lowest; out-of-the-money or in-the-money the implied volatility tends to be higher. The reason for this smile is still the subject of much speculation and research. A prominent proposed explanation is that the market in options away from the money is less liquid than at-the-money: traders demand a premium for these options because they know it may be more difficult to reverse an option position in illiquid markets. This view is consistent with the fact the smile was first observed shortly after the stock market crash of 1987. Before this crash, the first and most severe since the introduction of options, the Black-Scholes was more widely trusted.
- 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.
- 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."
- The Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel for 1997
- Options pricing using the Black-Scholes Model, Investment Basics: XLII., The Investment Analysts Society of South Africa
- The Black Scholes Option Pricing Model, optiontutor
- Generalized Black-Scholes Calculator, Written by Espen Gaarder Haug (himself) 1998
- Further information on pricing options on continuous dividend-paying stocks
- Further information on pricing options on foreign exchange options.
Related Finance and Investing Topics
|
|
|
|
|
The Original Article used for this page is available for download under the terms of the GNU Free Documentation License.
| 
