BlackScholes

The BlackScholes model, often simply called BlackScholes, is a model of the varying price over time of financial instruments, and in particular stocks. The BlackScholes 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 BlackScholes 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 
The model
The key assumptions of the BlackScholes model are:
 The price of the underlying instrument is a geometric Brownian motion, in particular with constant drift and volatility.
 It is possible to short sell the underlying stock.
 There are no riskless 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 is constant, and the same for all maturity dates.
BlackScholes in practice
The use of the BlackScholes 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 riskfree rate and current underlying price—are unequivocally observable. This means there is onetoone 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.
However, the BlackScholes model can not be modelling the real world exactly. If the BlackScholes 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 twodimensional 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 smileshaped (see volatility smile). That is, atthemoney (the option for which the underlying price and strike coincide) the implied volatility is lowest; outofthemoney or inthemoney 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 among other things on its historical distribution, and is constantly reshaping as investors, marketmakers, and arbitrageurs reevaluate the probability of the underlying reaching a given strike and the riskreward ( including factors related to liquidity) associated to it.
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 v:
 <math>C(S,t)=S N(d_1)K e^{rT} N(d_2) \,<math>
where
 <math>d_1=\frac{\ln\frac{S}{K}+\left( r+v^{2}/2\right) T}{v\sqrt{T}}<math>
 <math>d_2=d_1v\sqrt{T}.<math>
Here N is the cumulative normal distribution function.
The price of a put option may be computed from this by putcall parity and simplifies to:
 <math> P(S,t) = Ke^{rT}N(d_2)SN(d_1). \, <math>
The Greeks under the BlackScholes model are also easy to calculate.
Extensions of the formula
The above option pricing formula is used for pricing European put and call options on nondividend paying stocks. The BlackScholes 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> q \, S_t \, dt <math>
for some constant q. Under this formulation the arbitragefree price the BlackScholes 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^{(rq)T}S_0 \,<math>
is the modified forward price that occurs in the terms d_{1} and d_{2}.
Exactly the same formula is used to price options on foreign exchange rates, except now q plays the role of the foreign riskfree interest rate and S is the spot exchange rate. This is the GarmanKohlhagen model (1983).
It is also possible to extend the BlackScholes 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 predetermined 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 such a 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).
Formula derivation
The BlackScholes PDE
In this section we derive the partial differential equation (PDE) at the heart of the BlackScholes model via a noarbitrage or deltahedging argument; for 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 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 is Brownian. Now let V be some sort of option on S  mathematically V is a function of S and t. By Itô's lemma for two variables we have
 <math> dV = \sigma S \frac{\partial V}{\partial S}dW + \left( \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}\right)dt. <math>
Now consider a portfolio <math>\Pi<math> consisting of one unit of the option V and dV/dS units of the underlying stock. The composition of this portfolio, called the deltahedge portfolio, will vary from timestep to timestep. Now consider the change in value
 <math>d\Pi = dV  \frac{\partial V}{\partial S} dS<math>
of the portfolio by subbing in the equations above:
 <math> d\Pi = \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. 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 riskfree rate of return is r we must have over the time period <math>[t,t+\delta t]<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 <math>dt<math> we obtain the BlackScholes 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 BlackScholes model, this equation holds whenever V has two derivatives with respect to S and one with respect to t.
From the general BlackScholes PDE to a specific valuation
We now show how to get from the general BlackScholes PDE to a specific valuation for this option. Consider as an example the BlackScholes 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) = \mbox{max}(SK,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 <math> such that <math>S = Ke^{x} \,<math>
 <math> \tau <math> such that <math>t=T\frac{\tau}{\frac{1}{2}\sigma^2} \,<math>
 <math> v(x,\tau) <math> such that <math>V = K.v(x,\tau) \,<math>
Thus our BlackScholes PDE becomes
 <math> \frac{\partial v}{\partial \tau}=\frac{\partial^2 v}{\partial x^2} + (c1)\frac{\partial v}{\partial x}  cv <math>
where <math>c=2r/\sigma^2<math>. The terminal condition <math>V(S,T)=\mbox{max}(SK,0)<math> now becomes an initial condition <math>v(x,0) = \mbox{max}(e^x1,0)<math>. If we now make a further transformation such that
 <math> v(x,\tau)=e^{\frac{1}{2}(c1)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}(c1)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{(xy)^2}{4\tau}}dy<math>
where u_{0} 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 value of a call option in terms of the BlackScholes 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_1v\sqrt{T}.<math>
As before, N is the cumulative normal distribution function.
The formula for the price of a put option, follows from this via putcall parity.
Other derivations
Above we used the method of arbitragefree pricing ("deltahedging") to derive a PDE governing option prices given the BlackScholes 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 riskneutral measure, which differs from the real world measure.
See also
 Binomial options model, which is able to handle a variety of conditions for which BlackScholes cannot be applied.
 Black model a variant (and more general form) of the BlackScholes option pricing model.
 Financial mathematics, which contains a list of related articles.
References
 Black, F. and M. Scholes, "The Pricing of Options and Corporate Liabilities" Journal of Political Economy 81, 1973, 637654. Black and Scholes' original paper.
 Merton, Robert C., "Theory of rational option pricing", Bell Journal of Economics and Management Science 4 (1), 1973, 141183.
External links
 The Black Scholes Option Pricing Model
 Option pricing theory (http://www.riskglossary.com/link/option_pricing_theory.htm) links to more detailed articles on specific models, riskglossary.com
 The Black & Scholes Model (http://www.globalderivatives.com/options/blackscholes.php), globalderivatives.com
 Options pricing using the BlackScholes Model (http://www.moneymax.co.za/articles/displayarticlewide.asp?ArticleID=271869), The Investment Analysts Society of South Africa
 The Black Scholes Option Pricing Model (http://www.ftsmodules.com/public/texts/optiontutor/eappch6.htm), optiontutor
 Variations on the model
 options on nondividend paying stocks (Black Scholes) (http://www.riskglossary.com/articles/black_scholes_1973.htm), riskglossary.com
 options on stock indexes and continuous dividendpaying stocks (http://www.riskglossary.com/articles/merton_1973.htm), riskglossary.com
 foreign exchange options (http://www.riskglossary.com/articles/garman_kohlhagen_1983.htm), riskglossary.com
 options on forwards (the Black model) (http://www.riskglossary.com/articles/black_1976.htm), riskglossary.com
 Derivations
 The risk neutrality derivation of the BlackScholes Equation (http://www.quantnotes.com/fundamentals/options/riskneutrality.htm), quantnotes.com
 Arbitragefree pricing derivation of the BlackScholes Equation (http://www.quantnotes.com/fundamentals/options/blackscholes.htm), quantnotes.com or an alternative treatment (http://www.sjsu.edu/faculty/watkins/blacksch.htm), Prof. Thayer Watkins
 Solving the BlackScholes Equation (http://www.quantnotes.com/fundamentals/options/solvingbs.htm), quantnotes.com
 Solution of the Black Scholes Equation Using the Green's Function (http://www.physics.uci.edu/%7Esilverma/bseqn/bs/bs.html), Prof. Dennis Silverman
 Tests of the Model
 Anomalies in option pricing: the BlackScholes model revisited (http://www.findarticles.com/p/articles/mi_m3937/is_1996_MarchApril/ai_18367627), New England Economic Review, MarchApril, 1996
 Online calculators
 BlackScholes Pricing Analysis (http://www.hoadley.net/options/optiongraphs.aspx) including dividends (http://www.hoadley.net/options/optiongraphs.aspx?divs=Y), hoadley.net
 Computer Programming Implementations
 BlackScholes in Multiple Languages (http://www.espenhaug.com/black_scholes.html), www.espenhaug.com
 Historical
 The Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel for 1997 (http://www.nobel.se/economics/laureates/1997/press.html)
 Trillion Dollar Bet (http://www.pbs.org/wgbh/nova/stockmarket/)  Companion Web site to a Nova episode originally broadcast on February 8, 2000. "The film tells the fascinating story of the invention of the BlackScholes Formula, a mathematical Holy Grail that forever altered the world of finance and earned its creators the 1997 Nobel Prize in Economics."de:BlackScholesModellfr:Modèle BlackScholes