f b e559f2 binomial pricing
DESCRIPTION
TRANSCRIPT
Binomial Pricing
Pricing of American Options
Introduction
• It is also based in an arbitrage argument.• The option can be replicated with the
underlying stock and bonds.• Objective:1 Find the price of the option.2 Derive the replicating portfolio: basis of
hedging and (therefore) investment banking.
Binomial Setting
• The price of the stock only can go up to a given value or down to a given value.
S
uS
dS
• Besides, there is a bond (bank account) that will pay interest of r.
Binomial Setting (cont.)
• We assume u (up) > d (down).
• For Black and Scholes we will need d = 1/u
• For consistency we also need u > (1+r) > d.
• Example: u = 1.25; d = 0.80; r = 10%.
S=100
S = 125
S = 80
Binomial Setting (cont.)
• Basic model that describes a simple world.
• As the number of steps increases, it becomes more realistic.
• We will price and hedge an option: it applies to any other derivative security.
• Key: we have the same number of states and securities (complete markets)
Basis for arbitrage pricing.
Option Pricing
• Introduce an European call option:– X = 110.– It matures at the end of the period.
S=100
uS = 125
dS = 80
S C (X=110)
Cu = 15
Cd = 0
Option Pricing (cont.)
• We can replicate the option with the stock and the bond.
• Construct a portfolio that pays Cu in state u and Cd in state d.
• The price of that portfolio has to be the same as the price of the option.
• Otherwise there will be an arbitrage opportunity.
Option Pricing (cont.)
• We buy shares and invest B in the bank.
• They can be positive (buy or deposit) or negative (shortsell or borrow).
• We want then,
d
u
CrBdS
CrBuS
)1(
)1(
• With solution,
)1)((;
)( rdu
CdCuB
duS
CC uddu
Option Pricing (cont.)
• In our example, we get for stock:
3
1
80251100
015
)..(d)S(u
CCΔ du
• And, for bonds:
24.24)1.1()8.025.1(
158.0025.1
)1)((
rdu
CdCuB ud
• The cost of the portfolio is,
09.924.241003
1 BS
Option Pricing (cont.)
• The price of the European call must be 9.09.• Otherwise, there is an arbitrage opportunity.• If the price is lower than 9.09 we would buy
the call and shortsell the portfolio.• If higher, the opposite.• We have computed the price and the hedge
simultaneously:– We can construct a call by buying the stock and
borrowing.– Short call: the opposite.
Risk Neutral Pricing
• Remember that
)1)((;
)( rdu
CdCuB
duS
CC uddu
• And
BSC • Substituting,
)1)(()( rdu
CdCu
du
CCC uddu
Risk Neutral Pricing (cont.)• After some algebra,
du Cdu
ruC
du
dr
rC
)(
)1(
)(
1
1
1
• Observe the coefficients,
)(
)1(,
)(
1
du
ru
du
dr
• Positive.
• Smaller than one.
• Add up to one.Like a probability.
Risk Neutral Pricing (cont.)
• Rewrite
du CpCpr
C
)1(1
1
• Where
)(
)1(1,
)(
1
du
rup
du
drp
• This would be the pricing of:– A risk neutral investor– With subjective probabilities p and (1-p)
Multiperiod Setting
• Suppose the following economy,
S
uS
dS
u2S
udS
d2S
• We introduce an European call with strike price X that matures in the second period.
Multiperiod Setting (cont.)
• The price of the option will be:
)],0max()1(2
),0max()1(
),0max([)1(
1
22
222
XudSpp
XSdp
XSupr
C
• There are “two paths” that lead to the intermediate state (that explains the “2”).
Multiperiod Setting (cont.)
• Consider now n periods.
)],0max()1(
)!(!
Binomial Pricing• The price of the European call becomes,
),,()1(
)',,( pnar
XpnaSC
n
• Where,
)1('
r
upp
• And (a,n,p’) is the complementary binomial distribution.
• Probability of getting at least a “up” changes after n tosses with p’ the probability of “up” at each toss.
Binomial Pricing (cont.)
• The binomial distribution is tabulated.(a,n,p’) is (approximately) the “delta” of the
call:Number of shares (smaller than one) we need to
replicate the European call.
• Suppose we know the volatility and the time to maturity t.
• We can retrieve u and d:
udeu nt /1;/
American Options
• Objective: we want to value an American call that matures in two periods.
• Strike price, X = 100.
• Interest rate: 5% (each period).
• The underlying will pay a dividend of 8 after the first period.
• Problem: should we exercise after the first period or wait until maturity?
American Options (cont.)
100
Call Payoff
90
(82)
90.2
73.8
0
0
110
(102)
112.2
91.8
12.2
0
I
II
III
In parenthesis: ex-dividend
American Options (cont.)
• Price of option at node II, 0: regardless of what happens afterwards the call pays zero.
• Price of call option at node I:
a If we exercise it (before the dividend is paid), 110-100 = 10.
b Unexercised: we compute the value of the replicating portfolio.
Then, we compare.
American Options (cont.)
110
(102)
112.2
91.8
12.2
0
I
29.52;6.00)05.1(8.91
2.12)05.1(2.112
BB
B
Value of call:91.829.52102)6.0( C
Solution: exercise and get 10.
Replicating portfolio:
American Options (cont.)• At node III:
100
110
90
10
0
III
• Value of call:
86.42;5.00)05.1(90
10)05.1(110
BB
B
• Replicating portfolio:
14.786.42100)5.0( C