The Black-Scholes Formula

By Gretchen Browne

 

Abstract:  The mechanics and use of European put and call options are explained.  Using the no arbitrage assumption as a basis for understanding, the derivation of the Black-Scholes formula for option pricing is explained.  The relationship to the heat equation of physics is shown. An example is given of option pricing using the formula.

 

 

Most of us are familiar with traditional investments such as stocks, bonds or mutual funds.  Relatively recently, a new investment class has arisen.  These new securities are priced according to the value of an underlying asset.  The underlying asset may be any of number of things.  Some possibilities include stocks, bonds, interest rates, and currency values among many others.  Since the value of the contract depends upon the value of the underlying asset, the contract is considered to derive its value from that asset.  Hence, these contracts are often called “derivatives.”

            This paper shall focus on a specific type of derivative known as a European put or call option, with the underlying security being a stock.  Options are contracts giving the holder, or purchaser, of the contract “the right to buy or sell an underlying asset at a fixed price.” (Chriss)  While the holder has the right to buy or sell the underlying asset, he or she is not obligated to do so.  The fixed price at which the asset may be purchased or sold is known as the strike price or exercise price.  The right to buy the underlying security is known as holding a call option.  The right to sell the underlying security for the exercise price is called a put option.  If the holder of the option chooses to buy or sell the underlying asset, he is said to “exercise the option.”  European call options may only be exercised on a set date, known as the expiration date.  This date may also be simply termed “expiry.” 

            Naturally, having the right to buy or sell the underlying asset means someone must be selling the right to buy or sell that asset.  This seller of an option is said to “write a call.” In return for doing so, he or she charges the buyer a price.  This purchase price is known as the premium.  While the purchaser, or holder, of the option has the right to buy or sell at the strike price, the writer of the option is obligated to sell at the strike price if the option holder chooses to exercise his right

 

 

Call

Put

Option Buyer

The right to buy

the stock at the strike price

The right to sell

the stock at the strike price

Option Seller

The obligation to buy

the stock at the strike price

The obligation to sell

the stock at the strike price

 

            Let ST be the price of an underlying stock at the time of expiration.  If ST is greater than the strike price of the stock, a call holder will exercise his right to purchase the stock.  The payoff to the call holder will equal ST – X, with X equaling the strike price, or expiration price.  Should the current stock price be less than strike price, a rational call holder will choose not to exercise the option. In this case the option is said to expire worthless.

In summary, the payoff of a call option is:

                        ST – X  for ST > X       

                        0          for ST ≤ X

                        C= max(ST – X,0)       

The payoff of a call option, then, will be the greater of the two.

The payoff of an option should not be confused with the profit on the option.  To find the profit, the cost of purchasing the call must be subtracted from the option’s payoff.  Graphically, this can be depicted as a vertical translation of the call’s payoff, with the graph shifted down by the amount of the call’s premium.

            With this background in how call and put options work, our attention now turns to the question, “How does a prospective call buyer decide how much an option is worth?”  The answer to this question was discovered by Fisher Black, a financial professional, and Myron Scholes, a finance professor.  They developed a differential equation giving the value of a call or put options.  The solution to this equation, known as the Black-Scholes formula or Black-Scholes model, revolutionized the world of finance.   Fisher Black died in 1995.  In 1997 the Noble Prize in Economics was awarded to Myron Scholes and to Robert Merton for development of the Black-Scholes formula.  (Merton had adapted the original formula to derive a value for options on dividend-paying stocks.)

            The Black-Scholes differential equation is very complex.  It can be better appreciated by examining the assumptions behind the model. 

 

No arbitrage opportunities

            To better understand options, let us look at the simplest possible change in a stock price.  Suppose a stock currently sells for $100 at time T0.  At time T1 we know that the stock will either rise to $120 or fall to $80.  There is a ¾ probability the stock price will rise, and a ¼ probability it will fall.  If the stock rises, the call will have a payoff of $20.  A fall in the stock price will result in the call expiring worthless.  Thus we have the following potential payoffs:

 

 

A statistics student will quickly realize the expected value of buying the option is

20*¾ + 0*¼ = $15.  One’s natural inclination is to assume this must be the price one should pay for the call.  However, it can be shown by contradiction this is not the case.  In this illustration, the ability to borrow without paying interest is assumed.

            Suppose a call writer sells a call to a buyer for its expected value of $15.  However, the seller then borrows $40.  Combined with the proceeds from the sale of the call, the seller now has a total of $55.  $50 is used to buy ½ of a share of the stock underlying the call.  (Remember, this stock has a current price of $100.)  The remaining $5 is saved.

            Consider now the value of the seller’s position at time T1, with its two possible outcomes.

Outcome 1: 

            The stock price has risen to $120.  The call writer still has the $5 previously saved after selling the call and taking out the loan.  The call writer sells his half share of stock for $60, and repays the $40 loan, leaving the seller with $25.  The buyer of the call exercises his option to purchase the stock from the seller for $100.  Once the call seller has received the $100 from the call buyer, he purchases the stock at its current market price of $120 and turns over ownership of the stock to the call buyer.  The call seller still has $5 of profit remaining.

Outcome 2: 

            The stock price has fallen to $80.  The call writer still has the $5 previously saved after selling the call and taking out the loan.  The call buyer’s option expires worthless.  The call writer sells his half share of stock, receiving $40.  This $40 is used to pay off the loan.  The call writer still has $5 of profit remaining.  (Norstad)

 

Regardless of the change in stock price, the seller in this example made $5 without incurring any risk.  This ability to make a risk-free profit is referred to in the financial world as arbitrage.  Given the large number of highly trained, intelligent financial professionals in the world it is not hard to imagine what will happen next.  Seeing the opportunity to make a riskless profit, another call writer will come along and offer to sell the option to a prospective buyer for a mere $4 instead of the $5 charged by the original option seller.  A third caller writer will not be outdone by the second.  He will be willing to sell the option for only $3—after all, there is no risk incurred in the deal.  And so the process continues.  The price will be driven down by the process of arbitrage until the ability to make a risk-free profit is ended.  Thus, the value of the call is not the expected value of $15 predicted by statistics, but $10.  Because of this efficiency in the market, the price of the option must be $10.  If the option sells for more, there will only be willing call writers, but not call buyers.  If the option were to sell for less, market participants would only be willing to buy a call, with no one willing to sell a call.  Only when the option is priced at $10 will the opportunity to earn a riskless profit be eliminated.  The Black-Scholes model assumes there are no arbitrage opportunities available.  Rather, a call and a put are said to “put-call parity.”  The difference between the market price of the call and its expected value is called the “risk premium.”  Since a buyer is uncertain of the outcome of the change in stock price, he demands some compensation for investing in the call instead of investing in something with a guaranteed return.

            For simplicity the interest rate was assumed to be 0% in this example.  Of course, no rational lender will charge zero interest.  The interest rate used in financial modeling for such a riskless situation is that paid on what are considered to be the safest of investments, U.S. Government treasure bills or U.S. Government zero coupon bonds.

 

The Hedging Portfolio

 

            As shown above, in this two-state world selling a call option has the same value as borrowing at the risk-free interest rate and buying a fractional share of the underlying stock.  A portfolio can be constructed which will have the same payout as a call, regardless of whether the stock price moves up to the higher price or down to the lower price.  The amount of stock to purchase depends upon how much the call price changes relative to a change in stock price.  This is referred to as the hedge ratio:

                  

(Zhang)  (1)

A portfolio based upon this hedge ratio is called a “self-replicating portfolio” or a “synthetic call.”  Let the portfolio be represented by .  The price of this hedged portfolio is then given by:

(Zhang)  (2)

The Geometric Brownian Model

 

            Of course, this binomial model of stock price movement is extremely simplistic.  Never will a stock be found whose price will rise either up to a certain level or down to a certain level.  However at every instant in time at which a stock is being traded, the price can move either up or down.  This movement in stock price is analogous to the movement of a heavy particle surrounded by light particles.  As the light particles move about quickly, they will randomly run into the heavy particle, slightly changing its course.  The movement of each light particle is independent of the movement of the other light particles.  Likewise the force with which each light particle strikes is independent of the others.  The heavy particle is subject to independent, random forces of various directions and magnitudes which change its movement.  Over time, though, the various impacts upon the heavy particle form a normal distribution with a given mean and standard deviation.  This model of the heavy particle’s movement is called geometric Brownian motion.

            It’s easy to see how this model applies to stock prices.  According to the efficient market hypothesis, all known information about a company is already used by the market when buying or selling the stock.  At any time t, the stock price incorporates this known information.  The change in stock price, then, comes about with the addition of new information as time changes to t + 1.  By its definition of being “new,” it follows that this information cannot be predicted in advance.  Some of this information may apply to all stocks in general.  Say a new consumer confidence survey is released, showing a sharp upsurge in consumer sentiment.  All stocks will tend to rally as this new information is absorbed by the market.  Some information may be company specific and analogous to a high magnitude blow received by the heavy particle.  The pulling of a blockbuster drug from the market by the FDA deals a heavy blow to the stock price of its manufacturer.

            The movement of stock price is best understood by dealing in percentages.  Let’s say a stock’s value falls 15% at time t.  At time t+1, the stock rises in value by 15%.  Obviously the stock will not return to the original price.  Let x be the percentage change in stock price.  This cumulative change is described by the equation (1 – x)(1+x) = (1- x2).  For the 15% change described, we have 1-.152 = .9775, with the stock having a cumulative return of -2.25%. Because of this, the stock is said to have a geometric return.  The expression “geometric return” is synonymous with lognormal return, as the stock price will have a lognormal distribution. This stock price change is modeled as having two components: a deterministic component, equal to the average return of a stock in the past, μ, and a random component equal to standard deviation of a stock, σ. The change in stock price can be described by a stochastic differential equation:

 

            (Zahng) (3)

This equation is different from the functions familiar to beginning math students.  A function follows a deterministic behavior.  Given a particular input, a function will always yield the same output.  The stochastic differential equation describes a “process.”  There is uncertainty about what the outcome will be.

 

The Black-Scholes Differential Equation

 

The value of a call at time t, ct, is dependent upon two variables.  First, how much does the call price change relative to a change in the stock price?  This is the delta term used in hedging.  Second, how much time is there until the call expires?  The more time there is left for the stock price to rise, the more valuable that time is.  Thus we have the value of a call at time t based upon two variables, instead of the single variable characteristic of ordinary differential equations:

(Zahng) (4)

Using the Brownian stock price model, equation (3), and a technique from stochastic calculus known as Ito’s lemma, it can be shown that the change in value of the call is given by:

 

(Zahng) (5)

Recall that a perfectly hedged portfolio may be constructed in which the change in the price of the stock will be perfectly offset by a change in the price of the call, due to owning stock in proportion to this hedge, or delta, ratio.  A change in this portfolio’s value is given by:

 

Zhang)  (6)

 

 However, recall that this hedged portfolio has the same return regardless of the stock price movement.  Given a hedged portfolio, , and a risk-free interest rate of r, by the no arbitrage assumption, the return on the portfolio over a given change in time, dt, must be equal to the risk free rate:

(Zhang)  (7)

 

Setting the equations equal to each other and simplifying, we have:

 

(Zhang)  (8)

 

Setting this equal to zero yields the Black-Scholes partial differential equation:

 

                                          

(Zhang)  (9)

 

The writer hopes that by covering the reasoning which went into developing this differential equation, the reader will come to appreciate the insight which was required by Black and Scholes in order to develop this equation.  While developing this equation was truly a breakthrough, the challenge of solving the equation still remained!  This equation is a linear partial differential equation with boundary values since the call value at the time of expiration will equal the greater of zero or the stock price S minus the strike price K.

            To solve this differential equation, transformations must be made.  Just as the beginning O.D.E. student makes substitutions to solve a second order linear differential system, Black and Scholes rewrote their equation using transformations.  The first transformation made was:

Zhang ( 10)

Substituting these values into the original Black-Scholes differential equation now gives:

 

Zhang (11)

 

Another transformation is made:

Zhang (12)

Computing derivates using this transformation and substituting these derivatives back into the previous equations, it can eventually be shown that:

Zhang (13)

 

Amazingly enough, this series of transformation has shown that the Black-Scholes differential equation giving the value of a call option is exactly the same as a standard heat equation.  Such an equation would be used in physics to describe the temperature of a metal rod.  The solution to this heat equation is known.  By taking this known solution to the heat equation and using the same transformations in reverse, a solution to the Black-Scholes differential equation may be given.  The heat equation is solved using Green’s function:

Zhang (14)

This function describes the probability of a normal random number falling into a certain range given a specific initial condition.  The function is used to develop Green’s integral:

Zhang (15)

The solution to this integral is simply:

 

with:

 

Zhang (16)

 

There are two pieces to this solution, with each part having a normal distribution.  Simplifying this answer to Green’s heat equation, we now have:

 

Zhang (17)

Transforming these variables “backwards” using the same substitutions we can finally reach the solution to the original Black-Scholes differential equation:

 

Zhang (18)

d2 can be simplified to: d2 = d1-s√t

Rubash (1)

 

With  representing the time left until the option expires, the Black-Scholes formula shows that the call’s value at a particular time t comes from two sources. 

The first is the gain (or loss) coming from ownership of the stock itself and is represented by SN(d1).  This is reached by multiplying the current stock price by the change in the call value relative to the change in the stock price.  In other words, given an expected return on the stock, the change in the call’s value is actually based upon the volatility of the stock price, not the average stock price.  This seems counterintuitive until one recalls the boundary values of a call option.  Upon expiration, the call can never have a payout of less than $0.  Consider two call options based on stocks with the same expected return, μ.  Both options specify a strike price of $30.  One is based upon a stock with historically high volatility, the other on a stock with low volatility.  The high volatility stock will have larger price swings: 

 

High Volatility Stock:

 

Stock Price

$10

$20

$30

$40

$50

Option Payoff

$0

$0

$0

$10

$20

 

 

Low Volatility Stock:

 

Stock Price

$20

$25

$30

$35

$40

Option Payoff

$0

$0

$0

$5

$10

 

Due to the “floor” or lower boundary in the option payout, the high volatility stock is more valuable to a call buyer.  It is more likely to end up “in the money” on the expiration date.

 

The second term in Black-Scholes formula gives the present value of paying the strike price on the expiration day.  Remember, the option holder will gain from the difference between the final stock price, ST less the strike price, K.  To give the current value of this strike price, the price is discounted at the continuously compounded risk free interest rate and multiplied times the probability the stock will exceed the strike price.  The term giving this value, Ke(-rt)N(d2) is subtracted from the first term to give the current value of purchasing the call option.

            Despite the challenge of solving the differential equation, once the formula was known the pricing of a call option becomes a simple process, quickly calculated using Matlab.  To figure the value the riskfree interest rate, standard deviation of the stock, time until expiration of the option, strike price and current stock price must be known.  Often the standard deviation of the stock is calculated for the most recent time period of the same length as the time until expiration.  So, for a 90 day option, the standard deviation of the stock over the past 90 days would be used as an input. 

 

 

Source: m.file written by Robert V. Kohn

 

 

 

 

 

 

 

 

 

 

Bibliography

 

Black, Fischer, “How We Came Up with the Option Formula,  Journal of Portfolio Management, v. 15, issue 2, 1989, p. 4-8.

 

Black, Fischer, “Fact and Fantasy in the Use of Options,  Financial Analysts Journal, v. 31, July-August, 1975, p. 36-41, 61-72.

 

Black, Fischer and Scholes, Myron, “The Pricing of Options and Corporate Liabilities,  Journal of Political Economy, v. 81, issue 3, 1973, p. 637-654.

 

Bodie, Zvi, Kane, Alex, and Marcus, Alan J., Essentials of Investments, 4th ed. Boston: McGraw-Hill, c2001.

 

Chriss, Neil A., Black-Scholes and Beyond: Option Pricing Models, Chicago: Irwin Publishing, 1997.

 

Chriss, Neil A., The Black-Scholes and Beyond Interactive Toolkit, Chicago: Irwin Publishing, 1997.

 

Grant, Dwight, Vora, Gautam, and Weeks, David, “Teaching Option Valuation: From Simple Discrete Distributions to Black/Scholes via Monte Carlo Simulation,” Financial Practice and Education, vol. 5, Fall/Winter 1996, p. 149-155.

 

Hull, John C.  Options, Futures and Other Derivatives, 5th ed., Saddle River, NJ: Prentice Hall, c2003.

 

Kohn, Robert V. “File call.m  http://www.math.nyu.edu/faculty/kohn/derivative.securities/2004/hw3_solution_addendum.pdf

 

Norstad, John, “Two-State Options,” April, 2005.  http://homepage.mac.com/j.norstad/finance/twostate.pdf

 

Rubash, Kevin, “A Study of Option Pricing Models,” http://bradley.bradley.edu/~arr/bsm/model.html

 

Zhang, Jin E.  “Mathematical Techniques of Finance I,” a series of pdf files available at:

http://hkusua.hku.hk/~jinzhang/hku/MathFin.html