Topic 10: BlackScholes

1
Topic 10Black-Scholes
2
Introduction

• In this lecture (or series of lectures) we will
  lay the basic foundation that you need to begin
  working with the standard continuous-time option
  pricing model, the Black-Scholes model.
• This is really just an introduction to the
  mathematics of continuous-time finance, obviously
  in the Math 6202, 6203, and 6204 you will study
  this material in much greater detail.
• This lecture begins with a quick tour through
  Hulls Chapter 11, and then moves into the
  Black-Scholes analysis itself, which comes from
  Hull Ch. 12.

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Introduction

• Our first concern is to develop a statistical
  model of how stock prices behave over time.
• The term stochastic process simply means that the
  variable defined by it is random. 
• Stochastic processes can be classified in several
  ways.
• By time interval
• continuous time changes can occur at any point
  in time ex. temperature
• discrete time changes can only occur at specific
  points in time.
• By change increment
• Continuous variable can take on any value in
  range.
• Discrete variable can take on only specific
  values.
• Stocks are generally assumed to follow a
  continuous time, continuous variable stochastic
  process.

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The Markov Property of Stocks

• A Markov property is a particular type of
  stochastic process where only the present value
  of a variable is relevant for predicting the
  future. The past history of the variable and how
  it got to that point are irrelevant.
• There is overwhelming evidence that stock prices
  are Markov. If they were not, then chartists
  could make money above and beyond the risk
  adjusted rate of return. They cannot, and if fact
  we kind of define the markets as being weak-form
  efficient this says the stock prices fully
  incorporate all public information. This is
  caused by competition.

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Wiener Processes

• A Wiener process is a particularly useful type of
  Markov process it has a mean change of zero and
  variance
• The general format of a Wiener process is very
  adaptable. In fact it is its flexibility that
  makes it such a good candidate for describing
  many different phenomena, including stock prices.
• Lets begin by defining a variable, z, which
  follows a Weiner process. Define ?t as a small
  length of time (nearly instantaneous) and define
  ?z as the change in z during time ?t.

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Wiener Processes

• A Wiener process, it will have these properties
• Property 1 ?z is related to ?t by the equation
• where e is a random draw from a standard normal
  distribution (i.e. eN(1,0)).
• Property 2 The values of ?z for any two
  different short intervals are independent.
• It follows from property 1 that ?z itself is
  distributed normally with mean 0 and variance ?t.
  Its standard deviation is (?t).5

7
Weiner Processes

• What are the statistical properties of z during a
  non-short interval of time, T?
• Denote this as z(T)-z(0). It is simply the sum
  of N small time intervals of length ?t, where N
  T/?t.
• Thus,
• Thus, since the mean of a sum of random variables
  is simply the sum of the means,
• mean of z(T) - z(0) 0
• Further, since the draws from e are independent,
  the variance of the sum of those draws are
  additive (another way of saying this is that they
  are uncorrelated).

8
Weiner Processes

• Thus, s2 N?t T and hence s T.5.
• So far we have dealt with a discrete time case.
  We are interested in a continuous time case.
  Basically nothing changes except we use
  continuous time notation

9
Generalized Wiener Process

• So far we have dealt only with a simple Wiener
  process. It has a drift rate of 0 and a variance
  rate of 1.0. We can add non zero drift rates and
  non-unitary variance rates.
• To get to such a process we build off of the
  basic Weiner process we already have
• dx a dt b dz
• where a and b are constants, and dz is our
  previously defined Weiner process.
• To see this, consider the two components of this
  equation separately. Note that the first portion
  (a dt) is not random. It merely says that dx is
  generally drifting upward at some constant rate.

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Generalized Wiener Process

• Another way of looking at this is to ask the
  question, what if I divide through by dt? 
• dx/dt a.
• The second term, however, does contain a random
  component. What we have is the white noise
  process (dz) multiplied by some non-random
  constant.
• Recall that the mean of the product of a random
  variable times a constant is the product of the
  mean of that random variable times the constant
  amount.
• In this case, i.e. for
• b dz
• since dzN(0,1), the mean of this term is zero.
• Thus the drift of the entire process, is simply
  (adt).

11
Generalized Wiener Process

• Similarly, the variance of the product of a
  random variable times a constant is the variance
  of the random variable times the square of the
  constant.
• So adding this all together we get dx N(a dt,
  b2 dt)
• and hence the st. deviation of dx is given by b
  (dt).5.
• Similarly over a non-small length of time T
• Mean of change in x aT
• variance of change in x is b2T
• standard deviation of change in x is b T.5
• If a and b are allowed to be functions of both x
  and t, then we have the most general Markov,
  stochastic process, the Ito process
• dx - a(x,t) dt b(x,t) dz

12
The Process for Stock Prices

• It is at first tempting to say that stocks follow
  a generalized Weiner process, that is, they have
  a constant drift and variance rate. However,
  this would not take into account that investors
  demand a rate of return that is independent of a
  stocks price.
• That is, investors want a return based on the
  risk of a stock, regardless if it is worth 10 or
  100.
• Because of this we cannot use the constant
  expected-drift rate, but rather have to have a
  constant proportional drift rate. This means
  that if S is the stock price, investors want a
  constant proportional return µS where µ is a
  constant.

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The Process for Stock Prices

• If a stock had a zero variance, this would imply
• ds µS dt,
• or, in another, more convenient, format
• dS/S µ dt.
• Thus,
• ST S0 e µt.
• Of course, stocks do exhibit volatility. A
  reasonable assumption (based on empirical
  observation) is that the variance is roughly
  constant over any relatively short time period.
  Define s2 as the variance rate of the
  proportional change in the stock price.
• This means that s2 ?t is the variance of the
  proportional change in the stock price over time
  ?t while the variance of the actual change is s2
  S2 ?t. Thus the instantaneous variance rate of S
  is therefore s2 S2 (i.e. as ?t -gt 0).

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The Process for Stock Prices

• This leads us to represent S by an Ito process of
  the form
• dS µS dt s S dz
• or, in the more generally seen form
• dS/S µ dt s dz
• Generally µ is referred to as the stocks
  expected return, and s as its volatility.
• It is really easy to build a simulation of this
  process.
• Lets assume that the at stock priced at 50 had
  an expected return ( µ) of 10 and that the
  volatility of that stock (i.e. standard deviation
  of the return, i.e. s) was 20. Let us divided
  the year into increments of 1 week (i.e. dt1/52)
  and chart the movements over 3 years

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The Process for Stocks
16
The Parameters

• Now notice that we have defined a process for
  stock prices that only involve two parameters µ
  and s2. Those parameters are usually expressed
  in time units of one year.
• What are the properties of those parameters?
• µ Generally, investors demand a higher value of
  µ for riskier stocks. Historically, the
  average stocks µ has been 8 percentage points
  higher than the risk-free rate of return.
  Fortunately, we dont have to worry too much
  about this parameter, since the expected rate of
  return does not matter for pricing derivatives.
  (Recall that we can price based on arbitrage
  arguments.)
• s This is an important parameter, and we will
  discuss how to estimate it from historical data
  later. It is important to note that s(T).5 will
  be approximately (but not exactly) the standard
  deviation of stock returns during the period T.

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Itos Lemma

• In the early 1950's Ito developed a method for
  describing how functions based on random
  variables evolve.
• Assume that G is a function of time, t, and a
  random variable x, i.e. G(x,t). Let x be a
  general Ito process
• dx a(x,t)dt b(x,t) dz
• then Itos lemma states that
• where dz is the typical Wiener process.

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Itos Lemma

• Thus G itself follows an Ito process with drift
  of
• and variance rate of
• Now, applying this to stocks, if a stock follows
  a proportional Ito process
• dS µS dt s S dz 

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Itos Lemma

• with constant µ and s, a function G of S (such
  as a derivative) will follow
• This is the key to pricing all derivatives.
  Notice that both S and G have one common (and
  sole) source of uncertainty dz. This is what
  allows us to price G in terms of S!

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Application to a Forward Contract

• To see how this works, lets look at a forward
  contract. From earlier in the book we defined a
  forward price F to be F Ser(T-t).
• So we can define F as a function of S and t
  F(S,t), and then taking some standard
  derivatives

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Application to a Forward Contract

• Appling Itos lemma, F should evolve as dF,
• which in this case will be
• Since by definition F Ser(T-t), dF becomes
• Note that this is basis for the notion that the
  growth rate in a stock index futures price is the
  excess return over the risk free rate.

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Application to the Logarithm of a Stock Price

• Now it turns out that in many cases we will want
  to work with the log of stock price, not with the
  stock price itself.
• What process does ln S follow?
• Again, define a function G(S,t) ln S.
• Taking normal derivatives 
• Substituting into equation Itos lemma
•  

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Application to the Logarithm of a Stock Price

• So it has mean (over interval t,T)
• (µ - s2/2)(T-t) 
• and variance
• s2(T-t)
• Thus the change in the stock price is given by
• ln ST - ln S0
• and this is distributed
• ln ST - ln S0 N(µ - s2/2)(T-t), s(T-t).5
• or written slightly differently
•   ln ST Nln S0(µ - s2/2)(T-t), s(T-t).5
• The net effect is to show that ln ST is normally
  distributed. This means that a stocks price at
  time T, given its price today, is lognormally
  distributed, with standard deviation of s(T).5

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The Lognormal Property of Stock Prices

• What is neat (and convenient) is that our best
  evidence is that the lognormally distribution
  does a pretty decent job modeling real stock
  prices.
• Indeed, Hull has a nice quote our uncertainty
  about the logarithm of the stock price, as
  measured by its standard deviation, is
  proportional to the square root of how far ahead
  we are looking.

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The Distribution of the Rate of Return

• If the previous section tells us about the
  properties of the stock price, we are also
  concerned with the statistical properties of the
  rate of return on the stock.
• That is, if previously we were concerned with the
  way the price of the stock changed over time,
  here we are concerned with the way the rate of
  return of the stock changes over time.
• Define ? as the annual continuously compounded
  rate of return realized between t and T.
• By definition ST Se?(T-t) And ? 1/(T-t) ln
  (ST/S)
• and since
• ln(ST/S) f((µ-
  s2/2)(T-t),s(T-t).5)
• this implies
• ?f((µ- s2/2),s/(T-t).5).

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The Distribution of the Rate of Return

• Indeed, the last slide shows that our model
  assumes that stock returns are normally
  distributed.
• It might be nice to have some validation that
  this is indeed what we see in the real world.
• Short answer For most options pricing, it
  appears that stocks returns are normal-enough,
  although some researchers argue that real returns
  violate normality in two related ways
• The tales of the distributions are too fat to be
  a true normal, they are closer to a logistic
  distribution.
• There seems to be a higher probability of large
  downturns in price than upturns.

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The Distribution of the Rate of Return

• Let us examine the returns to two stocks to at
  least visually examine how they are distributed.
• General Electric (GE) and the parent company of
  American Airlines (AMR).
• Data collected from March 1994 through October
  2002.
• First, lets look at prices over that time

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The Distribution of the Rate of Return
29
The Distribution of the Rate of Return

• Now, lets look at a histogram of returns for
  each of these firms.

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The Distribution of the Rate of Return
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Estimation of Volatility from Historical Data

• Sometimes we want to estimate volatility from
  historical data. When we do this, we typically
  examine a series of closing prices. Lets adopt
  the following notation
• n1 the number of observations of closing
  prices.
• Si the stock price at the end of the ith
  interval.
• t length of time interval in years.
• Define µi ln (Si/Si-1) for i1,2,...,n.
• By definition we have constructed a series of
  continuously compounded return values. The usual
  estimator that we get in stats class look like

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Estimation of Volatility from Historical Data

• Or
• Of course we know from equation 11.1 that the
  standard deviation of µi is s(t.5), and so we
  must adjust our estimate of the standard
  deviation, we do this by dividing s by (t.5),
• s s/(t.5)

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Assumptions Behind the Black-Scholes Differential
Equation

• Although the mathematics appear a bit more
  complex and a bit more intimidating, the basic
  idea behind the Black Scholes analysis is the
  same as in the binomial.
• That is, if you can set up a portfolio that
  consists of the option and the stock, and that
  portfolio is perfectly risk-free, you can
  back-out the price of the option.
• To get to this point we make a number of
  assumptions. Most of these assumption are not
  very binding.
• The stock price follows the process developed in
  Chapter 10 with µ and s constant.
• The short selling of securities is allowed.
• There are not transactions costs or taxes, all
  securities are perfectly divisible.
• There are no dividends during the life of the
  option.
• There are not arbitrage opportunities.
• Securities trade continuously.
• The risk-free rate is constant.

34
Derivation of the Black-Scholes Differential
Equation

• Recall that we assume a stock price S which
  follows the process
• dS µS dt sS dz
• working with a discrete time version of this
• ?S µS ?t sS ?z.
• Let f represent the value of any derivative
  written on S. From Itos lemma, it must be the
  case that

35
Derivation of the Black-Scholes Differential
Equation

• Now clearly the only risk from this derivative
  security comes from the ?z term.
• If I construct a portfolio consisting of -1
  shares of the derivative and (?f/?s) shares of
  the stock, will have an instantaneously riskless
  portfolio. To see this note that the value of
  the portfolio (?) is
• and so the change in this portfolio is
• which by substitution gives us

36
Derivation of the Black-Scholes Differential
Equation

• Notice that we have managed to remove all of the
  ?zs from the equation. As a result we know that
  it must grow at the risk-free rate. It follows,
  therefore, that
• ?? r ? ?t
• where r is the risk-free rate. Substituting back
  into previous equations gives
• which leads us to the general equation

37
Derivation of the Black-Scholes Differential
Equation

• Notice that f is ANY derivative based on S, not
  just a call or a put. In fact, to price any
  particular derivative, you must first impose upon
  this equation appropriate boundary conditions.
• These literally specify the boundary values of S
  and t, and how the derivative behaves at those
  boundaries.
• Fortunately, these boundary conditions are
  basically our payoff conditions
• call option f max(S-X,0) when tT
• put option f max(X-S,0) when tT

38
Risk-Neutral Valuation

• Notice in the PDE that the drift rate of the
  stock does not appear. It has fallen out in all
  of the algebra.
• Thus, how investors feel about µ are irrelevant
  when it comes to pricing the option f. We can
  choose to work in a world with any set of
  investor attitudes about µ that we want to, and
  values obtained in those worlds are valid in any
  other world.
• This allows us to make a hugely simplifying
  assumption. Just assume that investors are
  risk-neutral. In such a world, all securities
  would grow at the risk-free rate because
  investors do not demand a premium to induce them
  to take risk.

39
Risk-Neutral Valuation

• This provides us with a rather simple way of
  valuing a derivative in this world simply assume
  that the stock will grow at the risk free rate r
  until time T.
• At that point see what would be the payoffs to
  the derivatives, and then discount those payoffs
  at the risk-free rate back to today. This is the
  value of the derivative in both the risk-neutral
  and risk-averse world.

40
The Black-Scholes Pricing Formulas

• Lets start with the call option. As you know,
  the value of a call option at maturity in a
  risk-neutral world is Emax(ST-X,0)
• where E represents the risk-neutral world. This
  means that the value of this option at time (T-t)
  is simply that payoff discounted back to time
  (T-t) at the risk free rate (just like any other
  discounted cash flow)
• c e-r(T-t)Emax(ST-X,0)
• So now, all we have to deal with is determining
  this expected value.

41
The Black-Scholes Pricing Formulas

• As we showed earlier, the rate of return on S in
  a risk-neutral world is r, not µ, therefore the
  distribution of ST is given by
• we will avoid what Hull describes as tedious
  algebra and cut to the equation 
• c SN(d1) - Xe-r(T-t)N(d2)
• where

42
The Black-Scholes Pricing Formulas

• And
• N(x) is the cumulative probability distribution
  function for a variable that is normally
  distributed with mean zero and a standard
  deviation of 1.
• Note that we can re-write this equation as
• c e-r(T-t)Ser(T-t) N(d1) - XN(d2).
• Expressed this way you can see what this is
  really getting at, lets break it down

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The Black-Scholes Pricing Formulas

• First, notice that Ser(T-t) is the expected value
  of ST in a risk-neutral world. Thus you can think
  of the term Ser(T-t)N(d1) as the expected payoff
  to a security which is worth ST is STgtX and zero
  otherwise.
• So this is the first half of your option how
  much the asset on which you hold the option is
  worth at the end, and XN(d2) is the expected
  amount you have to pay for it times the
  probability of your making that payout.
• So essentially this has two parts the value of
  the asset after the payment, and the cost of
  exercising the option. the e-r(T-t) is simply
  discounting it back to todays price.
• Note I will provide you with tables for N(x) on
  the exam.
• You can get a similar result for puts, but it is
  somewhat easier to use put-call parity
• p Xe-r(T-t)N(-d2) - SN(-d1).

44
The Black-Scholes Pricing Formulas

• Lets work an example. Lets say that you hold a
  European option with these parameters
• S 42, K 40, r .10, s.2, and T-t.5.
• What is the value of that option today?
• First, lets calculate d1 and d2

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The Black-Scholes Pricing Formulas

• Obviously the next step is to calculate N(d1) and
  N(d2), and there are several ways you could do
  this
• In Excel, use the function normsdist(),
• Use a polynomial approximation, or
• Use a table of normal values.
• Lets look at each of these
• In Excel simply enter
• normsdist(0.7693) 0.7791normsdist(0.6278)
  0.7349
• Hull presents a polynomial approximation on page
  248 of his book. This approximation is as
  follows

46
The Black-Scholes Pricing Formulas

• Hulls approximation
• For d1 0.7693, the approximate value for
  N(0.7693) works out to be 0.779142, and for
  d20.6278 the approximate value for N(0.6278)
  works out to be 0.734933.

47
The Black-Scholes Pricing Formulas

• If you look in the back of Hulls book you can
  see an example of a standard normal table similar
  to the one I have created here
• To use it find the row/column combination that
  most closely equals your value of (x), and then
  interpolate between that value and the next
  closest one.

48
The Black-Scholes Pricing Formulas

• All three method give you approximately the same
  answer for N(d1) and N(d2)
• The next step is to calculate Ke-r(T-t).
• Ke-r(T-t) 40 e-.1(.5) 38.049.
• If this is a call option that you hold, its value
  is given by
• And if it is a put, its value is given by

49
The Black-Scholes Pricing Formulas

• Notice that if you wanted to solve for the put
  value, you could have used put-call parity as
  well
• p c Ke-r(T-t) S0
• or
• p 4.76 40e-.10(.5)- 42
• p 4.76 38.049 42 0.81

50
The Black-Scholes Pricing Formulas

• McDonald presents the basic Black-Scholes
  equations in a slightly different format
• In this case the parameter d is a continuous
  dividend yield. This is just a minor modification
  to Hulls Black-Scholes model. Note that this is
  in keeping with the idea of setting the total
  return to the risk-free rate as we have done in
  previous settings.

51
The Black-Scholes Pricing Formulas

• For a put the equations change to
• Where again d is the continuous dividend yield.

52
The Black-Scholes Pricing Formulas

• Clearly the two previous equations would be
  practical for use with stock index, but would
  not be particularly useful for most stocks that
  pay discrete dividends.
• We can incorporate discrete dividends into the
  Black-Scholes equation in much the same way that
  we did for the binomial model remove the present
  value of the future dividends from the time 0
  stock price and then model that as the current
  stock price. (This obviously only applies to
  European options)
• Technically we would have to estimate the
  volatility of the risky portion of the stock
  price, but this is not really an issue provided
  we are using implied volatilities they would be
  based upon this value.

53
The Black-Scholes Pricing Formulas

• Example Suppose that you have a call option on a
  stock that will pay a 3 dividend in 1 month.
  This stock is currently priced at 41, and has a
  volatility of 30. The risk free rate is 8, and
  the option will expire in 3 months. The strike
  price is 40.
• Begin by determining the present value of the
  dividend
• I 3e-.08(.25) 2.98
• Reset the stock price (S0) to (41-2.98)
  38.02, and then plug it into the Black-Scholes,
  first calculate d1 and d2

54
The Black-Scholes Pricing Formulas

• Next, calculate N(d1) and N(d2)
• N(-0.13011) 0.4482
• N(-0.28011) 0.3897
• Then determine Ke-rT
• 40e-.08(.25)39.21
• Finally, calculate the call value
• c S0N(d1) Ke-rTN(d2)
• c 38.02(0.4482) 39.21(0.3897)
• c 1.76

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The Black-Scholes Pricing Formulas

• But what about American options?
• We know that in the presence of dividends it is
  sometimes optimal to exercise an American call
  option early. It was relatively easy to handle
  this within the binomial model, but how do we
  handle this within the Black-Scholes model?
• Fortunately, Hull demonstrates that normally it
  is the case that only the final dividend date
  matters for the purposes of pricing (this may be
  violated if, for example there were an
  extraordinarily large early dividend followed by
  one or more much smaller dividends.) Thus, we can
  behave, in most cases, as if there were only one
  dividend, even when there are many.
• Generally there are a couple of methods that are
  commonly used. The first is an approximation
  proposed by Black, and the second is an exact
  pricing formula known as the Roll, Geske, and
  Whaley model. We will briefly examine each of
  these.
• The Black approximation is widely used in
  practice, and has the benefit of being extremely
  easy to implement.

56
The Black-Scholes Pricing Formulas

• The Black approximation is widely used in
  practice, and has the benefit of being extremely
  easy to implement.
• Assume that a stock will pay a dividend at time
  tn, and that tngtT.
• Price two calls using the Black-Scholes formula
  one with a maturity date of tn and one with T.
  Then assume that the American options value is
  the greater of the two. (Adjusting S0 for the
  intermediate dividends.)
• The Roll, Geske, and Whaley model is more
  precise, but also significantly more complex to
  implement.

57
The Black-Scholes Pricing Formulas

• The RGW model is given by
• Note that M(a,b?) is the cumulative probability
  in a standardized bivariate normal distribution
  that the first variable is less than a and the
  second variable is less than b, and the
  correlation coefficient between the two is ?.
  Also,

58
The Black-Scholes Pricing Formulas

• S is defined to be that value of the initial
  stock price that sets the option value exactly
  equal to the initial stock price plus the
  dividend less the strike price
• c(S) S-D1-K
• Technically the RGW model allows for only one
  dividend, although in most cases you can treat
  the last dividend as being the only dividend
  (adjusting S0 by the present value of all but the
  last dividend amount, of course) and then solving
  using this procedure.

59
The Black-Scholes Pricing Formulas

• Finally, what about options on futures contracts?
• The model for pricing futures options is normally
  referred to as Blacks model.
• McDonald does a nice job presenting this from the
  standpoint of prepaid futures contracts. Recall
  that the time 0 value of a prepaid futures
  contract in a risk-neutral world is just the
  present value of the futures price Fe-rT.
• The formula is given by

60
The Black-Scholes Pricing Formulas

• Consider a European put futures option on crude
  oil, with the options maturity date in four
  months. The current futures price is 20, the
  exercise price is 20, the risk free rate is 9,
  and the volatility of the futures price is 25.
• F020, K20, r0.09, T4/12, s.25.