Upload
mohammad-imran-khan
View
247
Download
1
Embed Size (px)
Citation preview
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.
7/29/2019 binomial PPT.ppt
2/29
David Dubofsky and 17-2Thomas W. Miller, Jr.
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.
7/29/2019 binomial PPT.ppt
3/29
David Dubofsky and 17-3Thomas W. Miller, Jr.
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.
7/29/2019 binomial PPT.ppt
4/29
David Dubofsky and 17-4Thomas W. Miller, Jr.
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 = ______
7/29/2019 binomial PPT.ppt
5/29
David Dubofsky and 17-5Thomas W. Miller, Jr.
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
7/29/2019 binomial PPT.ppt
6/29
David Dubofsky and 17-6Thomas W. Miller, Jr.
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
7/29/2019 binomial PPT.ppt
7/29
David Dubofsky and 17-7Thomas W. Miller, Jr.
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!
7/29/2019 binomial PPT.ppt
8/29
David Dubofsky and 17-8Thomas W. Miller, Jr.
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?
7/29/2019 binomial PPT.ppt
9/29
David Dubofsky and 17-9Thomas W. Miller, Jr.
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:
7/29/2019 binomial PPT.ppt
10/29
David Dubofsky and 17-10Thomas W. Miller, Jr.
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
7/29/2019 binomial PPT.ppt
11/29
David Dubofsky and 17-11Thomas W. Miller, Jr.
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.
7/29/2019 binomial PPT.ppt
12/29
David Dubofsky and 17-12Thomas W. Miller, Jr.
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
7/29/2019 binomial PPT.ppt
13/29
David Dubofsky and 17-13Thomas W. Miller, Jr.
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
7/29/2019 binomial PPT.ppt
14/29
David Dubofsky and 17-14Thomas W. Miller, Jr.
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).
7/29/2019 binomial PPT.ppt
15/29
David Dubofsky and 17-15Thomas W. Miller, Jr.
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).
7/29/2019 binomial PPT.ppt
16/29
David Dubofsky and 17-16Thomas W. Miller, Jr.
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:
7/29/2019 binomial PPT.ppt
17/29
David Dubofsky and 17-17Thomas W. Miller, Jr.
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.
7/29/2019 binomial PPT.ppt
18/29
David Dubofsky and 17-18Thomas W. Miller, Jr.
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)
.
.
.
.
7/29/2019 binomial PPT.ppt
19/29
David Dubofsky and 17-19Thomas W. Miller, Jr.
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).
7/29/2019 binomial PPT.ppt
20/29
David Dubofsky and 17-20Thomas W. Miller, Jr.
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
7/29/2019 binomial PPT.ppt
21/29
David Dubofsky and 17-21Thomas W. Miller, Jr.
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.
7/29/2019 binomial PPT.ppt
22/29
David Dubofsky and 17-22Thomas W. Miller, Jr.
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
7/29/2019 binomial PPT.ppt
23/29
David Dubofsky and 17-23Thomas W. Miller, Jr.
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
7/29/2019 binomial PPT.ppt
24/29
David Dubofsky and 17-24Thomas W. Miller, Jr.
Binomial Put Pricing - III
26)(17
r)(1
p)P(1pPP du
durup)(1and
dudrp
Where:
7/29/2019 binomial PPT.ppt
25/29
David Dubofsky and 17-25Thomas W. Miller, Jr.
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.
7/29/2019 binomial PPT.ppt
26/29
David Dubofsky and 17-26Thomas W. Miller, Jr.
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
7/29/2019 binomial PPT.ppt
27/29
David Dubofsky and 17-27Thomas W. Miller, Jr.
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:
7/29/2019 binomial PPT.ppt
28/29
David Dubofsky and 17-28Thomas W. Miller, Jr.
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:
7/29/2019 binomial PPT.ppt
29/29
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.