Chapter 4: Option Pricing Models: The Binomial Model

D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed.

Ch. 4: 1

Chapter 4: Option Pricing Models:The Binomial Model

You can think of a derivative as a mixture of its constituent You can think of a derivative as a mixture of its constituent underliers much as a cake is a mixture of eggs, flour, and underliers much as a cake is a mixture of eggs, flour, and milk in carefully specified proportions. The derivative’s milk in carefully specified proportions. The derivative’s model provides a recipe for the mixture, one whose model provides a recipe for the mixture, one whose ingredients’ quantities vary with time.ingredients’ quantities vary with time.

Emanuel DermanEmanuel Derman

RiskRisk, July, 2001, p. 48, July, 2001, p. 48


Ch. 4: 2

Important Concepts in Chapter 4

The concept of an option pricing modelThe concept of an option pricing model The one- and two-period binomial option pricing modelsThe one- and two-period binomial option pricing models Explanation of the establishment and maintenance of a Explanation of the establishment and maintenance of a

risk-free hedgerisk-free hedge Illustration of how early exercise can be capturedIllustration of how early exercise can be captured The extension of the binomial model to any number of The extension of the binomial model to any number of

time periodstime periods Alternative specifications of the binomial modelAlternative specifications of the binomial model


Ch. 4: 3

Definition of a modelDefinition of a model A simplified representation of reality that uses certain A simplified representation of reality that uses certain

inputs to produce an output or resultinputs to produce an output or result Definition of an option pricing modelDefinition of an option pricing model

A mathematical formula that uses the factors that A mathematical formula that uses the factors that determine an option’s price as inputs to produce the determine an option’s price as inputs to produce the theoretical fair value of an option.theoretical fair value of an option.


Ch. 4: 4

The One-Period Binomial Model Conditions and assumptionsConditions and assumptions

One period, two outcomes (states)One period, two outcomes (states) S = current stock priceS = current stock price u = 1 + return if stock goes upu = 1 + return if stock goes up d = 1 + return if stock goes downd = 1 + return if stock goes down r = risk-free rater = risk-free rate

Value of European call at expiration one period laterValue of European call at expiration one period later CCuu = Max(0,Su - X) or = Max(0,Su - X) or CCdd = Max(0,Sd - X) = Max(0,Sd - X)

See See Figure 4.1, p. 98Figure 4.1, p. 98


Ch. 4: 5

The One-Period Binomial Model (continued)

Important point: d < 1 + r < u to prevent arbitrageImportant point: d < 1 + r < u to prevent arbitrage We construct a hedge portfolio of h shares of stock and We construct a hedge portfolio of h shares of stock and

one short call. Current value of portfolio:one short call. Current value of portfolio: V = hS - CV = hS - C

At expiration the hedge portfolio will be worthAt expiration the hedge portfolio will be worth VVuu = hSu - C = hSu - Cuu

VVdd = hSd - C = hSd - Cdd

If we are hedged, these must be equal. Setting VIf we are hedged, these must be equal. Setting Vuu = V = Vdd

and solving for h givesand solving for h gives


Ch. 4: 6


These values are all known so h is easily computedThese values are all known so h is easily computed Since the portfolio is riskless, it should earn the risk-free Since the portfolio is riskless, it should earn the risk-free

rate. Thusrate. Thus V(1+r) = VV(1+r) = Vuu (or V (or Vdd))

Substituting for V and VSubstituting for V and Vuu

(hS - C)(1+r) = hSu - C(hS - C)(1+r) = hSu - Cuu

And the theoretical value of the option isAnd the theoretical value of the option is

SdSu

CCh du

−−

=


Ch. 4: 7


This is the theoretical value of the call as determined by This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and the stock price, exercise price, risk-free rate, and up and down factors.down factors.

The probabilities of the up and down moves were never The probabilities of the up and down moves were never specified. They are irrelevant to the option price.specified. They are irrelevant to the option price.

d)-d)/(u-r(1=p

wherer1

p)C(1pCC du

+

+−+

=


Ch. 4: 8


An Illustrative ExampleAn Illustrative Example S = 100, X = 100, u = 1.25, d = 0.80, r = .07S = 100, X = 100, u = 1.25, d = 0.80, r = .07 First find the values of CFirst find the values of Cuu,, C Cdd, h, and p:, h, and p:

CCuu = Max(0,100(1.25) - 100) = Max(0,125 - 100) = = Max(0,100(1.25) - 100) = Max(0,125 - 100) =

2525 CCdd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 h = (25 - 0)/(125 - 80) = .556h = (25 - 0)/(125 - 80) = .556 p = (1.07 - 0.80)/(1.25 - 0.80) = .6p = (1.07 - 0.80)/(1.25 - 0.80) = .6

Then insert into the formula for C: Then insert into the formula for C:

14.021.07

0.0).4((.6)25C =

+=


Ch. 4: 9


A Hedged Portfolio A Hedged Portfolio Short 1,000 calls and long 1000h = 1000(.556) = 556 Short 1,000 calls and long 1000h = 1000(.556) = 556

shares. See shares. See Figure 4.2, p. 101Figure 4.2, p. 101.. Value of investment: V = 556($100) - 1,000($14.02) = Value of investment: V = 556($100) - 1,000($14.02) =

$41,580. (This is how much money you must put up.)$41,580. (This is how much money you must put up.) Stock goes to $125Stock goes to $125

Value of investment = 556($125) - 1,000($25) = Value of investment = 556($125) - 1,000($25) = $44,500$44,500

Stock goes to $80Stock goes to $80 Value of investment = 556($80) - 1,000($0) = Value of investment = 556($80) - 1,000($0) =

$44,480$44,480


Ch. 4: 10


An Overpriced CallAn Overpriced Call Let the call be selling for $15.00Let the call be selling for $15.00 Your amount invested is 556($100) - 1,000($15.00) = Your amount invested is 556($100) - 1,000($15.00) =

$40,600$40,600 You will still end up with $44,500, which is a 9.6% return.You will still end up with $44,500, which is a 9.6% return. Everyone will take advantage of this, forcing the call price Everyone will take advantage of this, forcing the call price

to fall to $14.02to fall to $14.02

You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.


Ch. 4: 11

An Underpriced CallAn Underpriced Call Let the call be priced at $13Let the call be priced at $13 Sell short 556 shares at $100 and buy 1,000 calls at Sell short 556 shares at $100 and buy 1,000 calls at

$13. This will generate a cash inflow of $42,600.$13. This will generate a cash inflow of $42,600. At expiration, you will end up paying out $44,500.At expiration, you will end up paying out $44,500. This is like a loan in which you borrowed $42,600 and This is like a loan in which you borrowed $42,600 and

paid back $44,500, a rate of 4.46%, which beats the paid back $44,500, a rate of 4.46%, which beats the risk-free borrowing rate.risk-free borrowing rate.



Ch. 4: 12

The Two-Period Binomial Model

We now let the stock go up another period so that it ends We now let the stock go up another period so that it ends up Suup Su22, Sud or Sd, Sud or Sd22..

See See Figure 4.3, p. 105.. The option expires after two periods with three possible The option expires after two periods with three possible

values:values:

X]SdMax[0,C

X]SudMax[0,C

X]SuMax[0,C

2d

ud

2u

2

2

−=

−=

−=


Ch. 4: 13

After one period the call will have one period to go before expiration. Thus, it will worth either of the following two After one period the call will have one period to go before expiration. Thus, it will worth either of the following two valuesvalues

The price of the call today will beThe price of the call today will be

The Two-Period Binomial Model (continued)

r1

p)C(1pCC

or ,r1

p)C(1pCC

2

2

ddud

uduu

+−+

=

+−+

=


Ch. 4: 14


2d

2udu

2

du

r)(1

Cp)(1p)C2p(1CpC

as written be alsocan which r1

p)C(1pCC

22

+−+−+

=

+−+

=

•The hedge ratios are different in the different states:

2dud

d2

uduu

du

SdSud

CCh ,

SudSu

CCh ,

SdSu

CCh

22

−

−=

−

−=

−−

=


Ch. 4: 15


An Illustrative ExampleAn Illustrative Example SuSu22 = 100(1.25) = 100(1.25)22 = 156.25 = 156.25 Sud = 100(1.25)(.80) = 100Sud = 100(1.25)(.80) = 100 SdSd22 = 100(.80) = 100(.80)22 = 64 = 64 The call option prices are as followsThe call option prices are as follows

0.0100]Max[0,64X]SdMax[0,C

0.0100]Max[0,100X]SudMax[0,C

56.25100]25Max[0,156.X]SuMax[0,C

2d

ud

2u

2

2

=−=−=

=−=−=

=−=−=


Ch. 4: 16


The two values of the call at the end of the first period areThe two values of the call at the end of the first period are

0.01.07

0.0).4((.6)0.0

r1

p)C(1pCCor

31.541.07

(.4)0.0+(.6)56.25=

r1

p)C(1pCC

2

2

ddud

uduu

=+

=+

−+=

=+

−+=


Ch. 4: 17


Therefore, the value of the call today isTherefore, the value of the call today is

17.691.07

0.0).4((.6)31.54r1

p)C(1pCC du

=+

=

+−+

=


Ch. 4: 18


A Hedge PortfolioA Hedge Portfolio See See Figure 4.4, p 109Figure 4.4, p 109.. Call trades at its theoretical value of $17.69.Call trades at its theoretical value of $17.69. Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = .701Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = .701 SoSo

Buy 701 shares at $100 for $70,100Buy 701 shares at $100 for $70,100 Sell 1,000 calls at $17.69 for $17,690Sell 1,000 calls at $17.69 for $17,690 Net investment: $52,410Net investment: $52,410


Ch. 4: 19


A Hedge Portfolio (continued)A Hedge Portfolio (continued) Note each of the possibilities:Note each of the possibilities:

Stock goes to 125, then 156.25Stock goes to 125, then 156.25 Stock goes to 125, then to 100Stock goes to 125, then to 100 Stock goes to 80, then to 100Stock goes to 80, then to 100 Stock goes to 80, then to 64Stock goes to 80, then to 64

In each case, you wealth grows by 7% at the end of the In each case, you wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.necessary for buying are borrowed at 7%.


Ch. 4: 20


A Hedge Portfolio (continued)A Hedge Portfolio (continued) Your wealth then grows by 7% from the end of the first Your wealth then grows by 7% from the end of the first

period to the end of the second.period to the end of the second. Conclusion: If the option is correctly priced and you Conclusion: If the option is correctly priced and you

maintain the appropriate mix of shares and calls as maintain the appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return indicated by the hedge ratio, you earn a risk-free return over both periods.over both periods.


Ch. 4: 21


A Mispriced Call in the Two-Period WorldA Mispriced Call in the Two-Period World If the call is underpriced, you buy it and short the stock, If the call is underpriced, you buy it and short the stock,

maintaining the correct hedge over both periods. You maintaining the correct hedge over both periods. You end up borrowing at less than the risk-free rate.end up borrowing at less than the risk-free rate.

If the call is overpriced, you sell it and buy the stock, If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.end up lending at more than the risk-free rate.

See See Table 4.1, p. 111Table 4.1, p. 111 for summary. for summary.


Ch. 4: 22

Extensions of the Binomial Model

Pricing Put OptionsPricing Put Options Same procedure as calls but use put payoff formula at Same procedure as calls but use put payoff formula at

expiration. In our example the put prices at expiration expiration. In our example the put prices at expiration areare

36.064]Max[0,100]SdMax[XP

0.0100]Max[0,100Sud]Max[XP

0.0156.25]Max[0,100]SuXMax[0,P

2d

ud

2u

2

2

=−=−=

=−=−=

=−=−=


Ch. 4: 23

Extensions of the Binomial Model (continued)

Pricing Put Options (continued)Pricing Put Options (continued) The two values of the put at the end of the first period The two values of the put at the end of the first period

areare

13.461.07

36).4((.6)0.0

r1

p)P(1pPPor

0.0,1.07

(.4)0.0+(.6)0.0=

r1

p)P(1pPP

2

2

ddud

uduu

=+

=+

−+=

=+

−+=


Ch. 4: 24


Pricing Put Options (continued)Pricing Put Options (continued) Therefore, the value of the put today isTherefore, the value of the put today is

5.031.07

13.46).4((.6)0.0r1

p)P(1pPP du

=+

=

+−+

=


Ch. 4: 25


Pricing Put Options (continued)Pricing Put Options (continued) Let us hedge a long position in stock by purchasing Let us hedge a long position in stock by purchasing

puts. The hedge ratio formula is the same except that puts. The hedge ratio formula is the same except that we ignore the negative sign:we ignore the negative sign:

Thus, we shall buy 299 shares and 1,000 puts. This Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.$5.03) for a total of $34,930.

0.29980125

13.460h −=

−−

=


Ch. 4: 26


Pricing Put Options (continued)Pricing Put Options (continued) Stock goes from 100 to 125. We now haveStock goes from 100 to 125. We now have

299 shares at $125 + 1,000 puts at $0.0 = $37,375299 shares at $125 + 1,000 puts at $0.0 = $37,375 This is a 7% gain over $34,930. The new hedge This is a 7% gain over $34,930. The new hedge

ratio isratio is

So sell 299 shares, receiving 299($125) = $37,375, So sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.which is invested in risk-free bonds.

0.00064100

0.00.0h =

−−

=


Ch. 4: 27



299 shares at $80 + 1,000 puts at $13.46 = $37,380299 shares at $80 + 1,000 puts at $13.46 = $37,380 This is a 7% gain over $34,930. The new hedge This is a 7% gain over $34,930. The new hedge

ratio isratio is

So buy 701 shares, paying 701($80) = $56,080, by So buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.borrowing at the risk-free rate.

1.00064100

360h −=

−−

=


Ch. 4: 28


Pricing Put Options (continued)Pricing Put Options (continued) Stock goes from 125 to 156.25. We now haveStock goes from 125 to 156.25. We now have

Bond worth $37,375(1.07) = $39,991Bond worth $37,375(1.07) = $39,991 This is a 7% gain.This is a 7% gain.

Stock goes from 125 to 100. We now haveStock goes from 125 to 100. We now have Bond worth $37,375(1.07) = $39,991Bond worth $37,375(1.07) = $39,991 This is a 7% gain.This is a 7% gain.


Ch. 4: 29



1,000 shares worth $100 each, 1,000 puts worth $0 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan in which we owe $56,080(1.07) = each, plus a loan in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain$60,006 for a total of $39,994, a 7% gain

Stock goes from 80 to 64. We now haveStock goes from 80 to 64. We now have 1,000 shares worth $64 each, 1,000 puts worth $36 1,000 shares worth $64 each, 1,000 puts worth $36

each, plus a loan in which we owe $56,080(1.07) = each, plus a loan in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain$60,006 for a total of $39,994, a 7% gain


Ch. 4: 30


American Puts and Early ExerciseAmerican Puts and Early Exercise Now we must consider the possibility of exercising the Now we must consider the possibility of exercising the

put early. At time 1 the European put values wereput early. At time 1 the European put values were PPuu = 0.00 when the stock is at 125 = 0.00 when the stock is at 125

PPdd = 13.46 when the stock is at 80 = 13.46 when the stock is at 80

When the stock is at 80, the put is in-the-money by $20 When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pso exercise it early. Replace Puu = 13.46 with P = 13.46 with Puu = 20. = 20.

The value of the put today is higher atThe value of the put today is higher at

7.481.07

20).4((.6)0.0P =

+=


Ch. 4: 31


Dividends, European Calls, American Calls, and Early Dividends, European Calls, American Calls, and Early ExerciseExercise One way to incorporate dividends is to assume a One way to incorporate dividends is to assume a

constant yield, constant yield, , per period. The stock moves up, then , per period. The stock moves up, then drops by the rate drops by the rate ..

See See Figure 4.5, p. 114Figure 4.5, p. 114 for example with a 10% yield for example with a 10% yield The call prices at expiration areThe call prices at expiration are

0100)0Max(0,57.6C

0.0100)Max(0,90CC

40.625100)625Max(0,140.C

2

2

d

duud

u

=−==−==

=−=


Ch. 4: 32


Dividends, European Calls, American Calls, and Early Dividends, European Calls, American Calls, and Early Exercise (continued)Exercise (continued) The European call prices after one period areThe European call prices after one period are

The European call value at time 0 isThe European call value at time 0 is

00.01.070.0).4( (.6)0.0

C

22.781.07

0.0).4((.6)40.625C

u

u

=+=

=+

=

12.771.07

0.0).4((.6)22.78Cu =

+=


Ch. 4: 33


Dividends, European Calls, American Calls, and Early Dividends, European Calls, American Calls, and Early Exercise (continued)Exercise (continued) If the call is American, when the stock is at 125, it pays If the call is American, when the stock is at 125, it pays

a dividend of $12.50 and then falls to $112.50. We can a dividend of $12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of option is worth $25. We replace the binomial value of CCuu = $22.78 with C = $22.78 with Cuu = $25. At time 0 the value is = $25. At time 0 the value is

14.021.07

0.0).4((.6)25Cu =

+=


Ch. 4: 34


Dividends, European Calls, American Calls, and Early Dividends, European Calls, American Calls, and Early Exercise (continued)Exercise (continued) Alternatively, we can specify that the stock pays a Alternatively, we can specify that the stock pays a

specific dollar dividend at time 1. Assume $12. specific dollar dividend at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Unfortunately, the tree no longer recombines, as in Figure 4.6, p. 115Figure 4.6, p. 115. We can still calculate the option . We can still calculate the option value but the tree grows large very fast. See value but the tree grows large very fast. See Figure 4.7, p. 116Figure 4.7, p. 116..

Because of the reduction in the number of Because of the reduction in the number of computations, trees that recombine are preferred over computations, trees that recombine are preferred over trees that do not recombine.trees that do not recombine.


Ch. 4: 35


Dividends, European Calls, American Calls, and Early Dividends, European Calls, American Calls, and Early Exercise (continued)Exercise (continued) Yet another alternative (and preferred) specification is Yet another alternative (and preferred) specification is

to subtract the present value of the dividends from the to subtract the present value of the dividends from the stock price (as we did in Chapter 3) and let the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down adjusted stock price follow the binomial up and down factors. For this problem, see factors. For this problem, see Figure 4.8, p. 117Figure 4.8, p. 117..

The tree now recombines and we can easily calculate The tree now recombines and we can easily calculate the option values following the same procedure as the option values following the same procedure as before.before.


Ch. 4: 36


Dividends, European Calls, American Calls, and Early Dividends, European Calls, American Calls, and Early Exercise (continued)Exercise (continued) The option prices at expiration areThe option prices at expiration are

0.0100)2Max(0,56.8C

0.0100)9Max(0,88.7C

38.74100)74Max(0,138.C

2

2

d

ud

u

=−==−=

=−=


Ch. 4: 37

Extensions of the Binomial Model (continued) Dividends, European Calls, American Calls, and Early Dividends, European Calls, American Calls, and Early

Exercise (continued)Exercise (continued) At time 1 the option prices areAt time 1 the option prices are

We exercise at time 1 so that CWe exercise at time 1 so that Cuu is now 22.99. At time 0 is now 22.99. At time 0

The European option value would be 12.18.The European option value would be 12.18.

0.01.07

0.0).4((.6)0.0C

21.721.07

0.0).4((.6)38.74C

d

u

=+

=

=+

=

12.891.07

0.0).4((.6)22.99C =

+=


Ch. 4: 38

Extensions of the Binomial Model (continued) Extending the Binomial Model to n PeriodsExtending the Binomial Model to n Periods

With n periods to go, the binomial model can be easily With n periods to go, the binomial model can be easily extended. There is a long and somewhat complex extended. There is a long and somewhat complex looking formula in the book. The basic procedure, looking formula in the book. The basic procedure, however, is the same. See however, is the same. See Figure 4.9, p. 119Figure 4.9, p. 119 in which in which we see below the stock prices the prices of European we see below the stock prices the prices of European and American puts. This illustrates the early exercise and American puts. This illustrates the early exercise possibilities for American puts, which can occur in possibilities for American puts, which can occur in multiple time periods.multiple time periods.

At each step, we must check for early exercise by At each step, we must check for early exercise by comparing the value if exercised to the value if not comparing the value if exercised to the value if not exercised and use the higher value of the two.exercised and use the higher value of the two.


Ch. 4: 39

Extensions of the Binomial Model (continued) The Behavior of the Binomial Model for Large n and a The Behavior of the Binomial Model for Large n and a

Fixed Option LifeFixed Option Life The risk-free rate is adjusted to (1 + r)The risk-free rate is adjusted to (1 + r)T/nT/n

The up and down parameters are adjusted toThe up and down parameters are adjusted to

where where is the volatility. Let us price the AOL June is the volatility. Let us price the AOL June 125 call with one period.125 call with one period.

1/ud

eu T/n

==


Ch. 4: 40


Fixed Option Life (continued)Fixed Option Life (continued) The parameters are nowThe parameters are now

The new stock prices areThe new stock prices are Su = 125.9375(1.293087) = 162.8481Su = 125.9375(1.293087) = 162.8481 Sd = 125.9375(0.773343) = 97.3929Sd = 125.9375(0.773343) = 97.3929

0.7733431/1.293087d

1.293087eu

.0042851(1.0456)r0.0959/10.83

0.0959/1

====

=−=


Ch. 4: 41


Fixed Option Life (continued)Fixed Option Life (continued) The new option prices would beThe new option prices would be

• CCuu = Max(0,162.8481-125) = 37.85 = Max(0,162.8481-125) = 37.85

• CCdd = Max(0,97.3929 - 125) = 0.0 = Max(0,97.3929 - 125) = 0.0 p would be (1.004285 - 0.773343)/(1.293087 - p would be (1.004285 - 0.773343)/(1.293087 -

0.773343) = .444; 1 - p = .556.0.773343) = .444; 1 - p = .556. The price of the option at time 0 is, therefore,The price of the option at time 0 is, therefore,

16.741.004285

0.00).556(5(.444)37.8C =

+=


Ch. 4: 42


Fixed Option Life (continued)Fixed Option Life (continued) The actual price of the option is 13.50, but The actual price of the option is 13.50, but

obviously one binomial period is not enough.obviously one binomial period is not enough. Table 4.2, p. 121Table 4.2, p. 121 shows what happens as we shows what happens as we

increase the number of binomial periods. The price increase the number of binomial periods. The price converges to around 13.56. In Chapter 5, we shall converges to around 13.56. In Chapter 5, we shall see that this is approximately the Black-Scholes see that this is approximately the Black-Scholes price.price.


Ch. 4: 43


Alternative Specifications of the Binomial ModelAlternative Specifications of the Binomial Model We can use a different specification of u, d and pWe can use a different specification of u, d and p

where ln(1 + r) is the continuously compounded interest where ln(1 + r) is the continuously compounded interest rate. Here p will converge to .5 as n increases.rate. Here p will converge to .5 as n increases.

T/nT/n

T/n(T/n)/2

T/n/2)(T/n)r)(ln(1

T/n/2)(T/n)r)(ln(1

ee

eep

ed

eu

2

2

2

−

−

−−+

+−+

−−

=

=

=


Ch. 4: 44


Alternative Specifications of the Binomial Model Alternative Specifications of the Binomial Model (continued)(continued) Now let us price the AOL June 125 call but use two Now let us price the AOL June 125 call but use two

periods. We have r = (1.0456)periods. We have r = (1.0456)0.0959/20.0959/2 - 1 = .0021. Using - 1 = .0021. Using our previous formulas,our previous formulas,

.46050.83381.1993

0.83381.0021p

0.83381/1.1993d

1.1993eu 0.0959/20.83

=−−

=

====


Ch. 4: 45


Alternative Specifications of the Binomial Model (continued)Alternative Specifications of the Binomial Model (continued) Now let us use these new formulas:Now let us use these new formulas:

We can use .5 for p. See We can use .5 for p. See Figure 4.10, p. 123Figure 4.10, p. 123. The prices are close . The prices are close and will converge when n is large.and will converge when n is large.

See bsbin3.xls and bsbwin2.2 for software to calculate the binomial See bsbin3.xls and bsbwin2.2 for software to calculate the binomial model.model.

.500251ee

eep

0.8219ed

1.1822eu

0.0959/20.830.0959/20.83

0.0959/20.83/2(0.0959/2)0.83

0.0959/20.83/2)/2)(0.09590.83-)(ln(1.0456

0.0959/20.83/2)/2)(0.09590.83-)(ln(1.0456

2

2

2

=−−

=

==

==

−

−

−

+


Ch. 4: 46

Summary


Ch. 4: 47

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Chapter 4: Option Pricing Models: The Binomial Model

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Transcript of Chapter 4: Option Pricing Models: The Binomial Model