binomial ppt.ppt

7/29/2019 binomial PPT.ppt

1/29

David Dubofsky and 17-1Thomas W. Miller, Jr.

Chapter 17

The Binomial Option Pricing Model (BOPM) We begin with a single period.

Then, we stitch single periods together to form the Multi-Period

Binomial Option Pricing Model.

The Multi-Period Binomial Option Pricing Model is extremelyflexible, hence valuable; it can value American options (whichcan be exercised early), and most, if not all, exotic options.


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Assumptions of the BOPM

There are two (and only two) possible prices for the underlyingasset on the next date. The underlying price will either:

Increase by a factor of u% (an uptick) Decrease by a factor of d% (a downtick)

The uncertainty is that we do not know which of the two priceswill be realized.

No dividends.

The one-period interest rate, r, is constant over the life of theoption (r% per period).

Markets are perfect (no commissions, bid-ask spreads, taxes,price pressure, etc.)

, assumes a perfectly efficient market, and shortens the durationof the option.


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The Stock Pricing Process

ST,d = (1+d)ST-1

ST,u = (1+u)ST-1

ST-1

Suppose that ST-1 = 40, u = 25% and d = -10%. What are ST,u and ST,d?

40

ST,u = ______

ST,d = ______

Time T is the expiration day of a call option. Time T-1 is one periodprior to expiration.


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The Option Pricing Process

CT,d

= max(0, ST,d

-K) = max(0,(1+d)ST-1

-K)

CT,u = max(0, ST,u-K) = max(0,(1+u)ST-1-K)

CT-1

Suppose that K = 45. What are CT,u and CT,d?

CT-1

CT,u = ______

CT,d = ______


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The Equivalent Portfolio

(1+d)ST-1 + (1+r)B = ST,d + (1+r)B

(1+u)ST-1 + (1+r)B = ST,u + (1+r)B

ST-1+B

Set the payoffs of the equivalent portfolio equal to CT,u and CT,d, respectively.

(1+u)ST-1

+ (1+r)B = CT,u(1+d)ST-1 + (1+r)B = CT,d

These are two equations withtwo unknowns: and B

What are the two equations in the numerical example with ST-1 = 40, u= 25%, d = -10%, r = 5%, and K = 45?

Buy shares of stock and borrow $B.

NB: is not achange in S. It

defines the # ofshares to buy. For acall, 0 < < 1


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A Key Point

If two assets offer the same payoffs at time T, then they must bepriced the same at time T-1.

Here, we have set the problem up so that the equivalent portfolio

offers the same payoffs as the call.

Hence the calls value at time T-1 must equal the $ amountinvested in the equivalent portfolio.

CT-1 = ST-1 + B


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and B define the Equivalent Portfolio of a call

2)-(170B;r)d)(1(u

d)C(1u)C(1B

1)-(1710;SS

CC

d)S(u

CC

cuT,dT,

cdT,uT,

dT,uT,

1T

dT,uT,

Assume that the underlying asset can only rise by u% or decline by d%in the next period. Then in general, at any time:

4)-(17r)d)(1(u

d)C(1u)C(1B

3)-(17

SS

CC

d)S(u

CC

ud

du

dudu

CT-1 = ST-1 + B (17-5)

C = S + B (17-6)

NB: a negative sign

now denotes borrowing!


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So, in the Numerical Example.

ST-1 = 40, u = 25%, ST,u = 50, d = -10%, ST,d = 36, r = 5%, K = 45,

CT,u = 5 and CT,d = 0.

What are the values of, B, and CT-1?

What if CT-1 = 3?

What if CT-1 = 1?


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A Shortcut

du

rup)(1and

du

drp

where,

7)-(17r)(1

p)C(1pC

C

or,

r)(1

Cdu

ruC

du

dr

C

dT,uT,1T

dT,uT,

1T

8)-(17r)(1

p)C(1pCC du

In general:


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Interpreting p

p is the probability of an uptick in a risk-neutral world.

In a risk-neutral world, all assets (including the stock and theoption) will be priced to provide the same riskless rate of return, r.

In our example, if p is the probability of an uptick then

ST-1 = [(0.428571429)(50) + (0.571428571)(36)]/1.05 = 40

That is, the stock is priced to provide the same riskless rate ofreturn as the call option

du

drp


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Interpreting :

Delta, , is the riskless hedge ratio; 0 < c < 1.

Delta, , is the number of shares needed to hedge one call. I.e.,if you are long one call, you can hedge your risk by selling shares of stock.

Therefore, the number of calls to hedge one share is 1/. I.e., ifyou own 100 shares of stock, then sell 1/ calls to hedge yourposition. Equivalently, buy shares of stock and write one call.

Delta is the slope of the lines shown in Figures 14.3 and 14.4(where an options value is a function of the price of theunderlying asset).

In continuous time, = C/S = the change in the value of a callcaused by a (small) change in the price of the underlying asset.


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Two Period Binomial Model

ST,dd = (1+d)2ST-2

ST,uu = (1+u)2ST-2

ST-1,u = (1+u)ST-2

ST,ud = (1+u)(1+d)ST-2

ST-1,d = (1+d)ST-2ST-2

CT,dd = max[0,(1+d)2ST-2 - K]

CT,uu = max[0,(1+u)2ST-2 - K]

CT-1,uCT,ud = max[0,(1+u)(1+d)ST-2 - K]

CT-1,dCT-2


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Two Period Binomial Model: An Example

ST,dd = 36

ST,uu = 69.444

ST-1,u = 55.556

ST,ud = 50

ST-1,d = 40.00ST-2 = 44.444

CT,dd = 0

CT,uu = _______

CT-1,u = ____CT,ud = 5

CT-1,d = 2.0408CT-2


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Two Period Binomial Model:The Equivalent Portfolio

= 1B = -42.857143

= 0.357142857B = -12.24489796

= 0.6851312B = -24.1566014

T-2 T-1

Note that as S rises, also rises. As S declines, so does .

Note that the equivalent portfolio is self financing. This means that thecost of any purchase of shares (due to a rise in ) is accompanied by anequivalent increase in required borrowing (B becomes more negative).Any sale of shares (due to a decline in ) is accompanied by an

equivalent decrease in required borrowing (B becomes less negative).


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The Multi-Period BOPM

We can find binomial option prices forany number ofperiods by using the following five steps:(1) Build a price tree for the underlying.

(2) Calculate the possible option values in the last period (time T= expiration date)

(3) Set up ALL possible riskless portfolios in the penultimateperiod (next to last period).

(4) Calculate all possible option prices in the penultimate period.

(5) Keep working back through the tree to Today (Time T-n in ann-period, (n+1)-date, model).


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The n Period Binomial Formula:

15)-(17r)(1

Cp)(1Cp)3p(1p)C(13pCpC

3

dddT,3

uddT,2

uudT,2

uuuT,3

3T

If n = 3:

j)!(nj!

n!

j

n

The binomial coefficient computes the number of ways we can get j

upticks in n periods:

.K]Sd)(1u)(1max[0,p)(1pj

3

r)(1

1C

3

0j

3Tj3jj3j

33T

Thus, the 3-period model can be written as:


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The n Period Binomial Formula:

In general, the n-period model is:

17)(17.K]Sd)(1u)[(1p)(1pj

n

r)(11C

n

aj

nTjnjjnjn

Where a in the summation is the minimum number of

up-ticks so that the call finishes in-the-money.


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A Large Multi-period Lattice

Suppose that N = 100 days. Let u = 0.01 and d = -0.008. S0 = 50

135.241 = 50*(1.01^100)

132.830 = 50*(1.01^99)*(.992^1)

130.463 = 50*(1.01^98)*(.992^2)

50.00

50.50

51.00551.51505

49.60

49.203248.80957

50.096

50.59696

49.69523

T=0 T=1 T=2 T=3

T=100

23.214 = 50*(1.01^2)*(.992^98)22.801 = 50*(1.01^1)*(.992^99)

22.394 = 50*(.992^100)

.

.

.

.


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Suppose the Number of PeriodsApproachs Infinity

S

TIn the limit, that is, as N gets large, and if u and d are consistentwith generating a lognormal distribution for ST, then the BOPMconverges to the Black-Scholes Option Pricing Model (theBSOPM is the subject of Chapter 18).


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Stocks Paying a Dollar Dividend Amount

Figure 17.4: The stock trades ex-

dividend ($1) at time T-2.

Figure 17.5: The stock trades ex-

dividend ($1) at time T-1.

25.410

23.100

22 => 21 21.945

19.950

20.000 18.9525

21.780

19.800

19 => 18 18.810

17.100

16.245

T-3 T-2 T-1 T

25.520

24.20 => 23.20

20.040

22.000

21.890

20.000 20.90 => 19.90

18.905

19.000

18.755

18.05 => 17.05

16.1975

T-3 T-2 T-1 T


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American Calls on Dividend Paying Stocks

The key is that at each node of the lattice, the value of an

American call is:

19)(17.KS,r)(1

p)C(1pCmax du

If the first term in the brackets is less than the calls intrinsic value,

then you must instead value it as equal to its intrinsic value. Moreover,if the dividend amount paid in the next period exceeds K-PV(K), thenthe American call should be exercised early at that node.


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Binomial Put Pricing - I

ST,u = (1+u)ST-1

ST-1

ST,d = (1+d)ST-1

PT,u = max(0,K-ST,u) = max(0,K-(1+u)ST-1)

PT-1

PT,d = max(0,K-ST,d) = max(0,K-(1+d)ST-1)

(1+u)ST-1 + (1+r)B = ST,u + (1+r)B = PT,uST-1+B

(1+d)ST-1 + (1+r)B = ST,d + (1+r)B = PT,d


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Binomial Put Pricing - II

PT-1 = ST-1 + B (17-24)

22)(17SSPP

d)S(uPP

du

dudu

23)(17r)d)(1(u

d)P(1u)P(1B ud

Where:

-1 < p < 0

A put is can be replicated by selling shares of stock short, andlending $B. and B change as time passes and as S changes.Thus, the equivalent portfolio must be adjusted as time passes.

B > 0


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Binomial Put Pricing - III

26)(17

r)(1

p)P(1pPP du

durup)(1and

dudrp

Where:


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Binomial American Put Pricing

27)(17r)(1

p)P(1pPS,KmaxP du

At any node, if the 2nd term in the brackets is less than the Americanputs intrinsic value, then value the put to equal its intrinsic value

instead. American puts cannot sell for less than their intrinsic value.The American put will be exercised early at that node.


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Binomial Put Pricing Example - I

79.86

72.6

66 68.97

60 62.757 59.565

54.13

51.4425

T-3 T-2 T-1 T

The StockPricingProcess:

u = 10%d = -5%r = 2%K = 65p = 0.466667


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Binomial Put Pricing Example - II

0

0

1.485924 03.9776 2.84183

6.306976 5.435

9.57549

13.5575

T-3 T-2 T-1 T

European Put Values:


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Binomial Put Pricing Example - III

= 0.0B = 0.0

= -0.2870535

B = 20.431458

= -0.5356724 = -0.5778841

B = 36.117946 B = 39.075163

= -0.7875626

B = 51.198042

= -1.0

B = 63.72549

T-3 T-2 T-1

Composition of the

equivalent portfolioto the European put:


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David Dubofsky and 17-29

Thomas W. Miller, Jr.

Binomial Put Pricing Example - IV

00

1.485924 0

4.86284 2.84183

5

6.97339 5.435

8

9.57549

10

13.5575

T-3 T-2 T-1 T

American put pricing: Ifeqn. 17.25 yields anamount less than theputs intrinsic value, then

the Americans put value

is K S (shown in bold),

and it should beexercised early.

binomial ppt.ppt

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