MathFin2-4

Chapter 2
Binomial Trees, Replicating Portfolios, and Arbitrage
§1. Three Ways to Price a Derivative
Suppose that the price of a stock is $100 now and that in one year the stock price will be either $90 or $120. The interest rate is 5%, that is, one dollar will yields 1.05 dollars in one year. What is the fair price for the European call option with strike price $105 which expires in one year?
r S e S0 , i.e., q 1 2. u
Otherwise we short the stock as many as possible.
Expected value pricing approach
• Expected value
V Prob U q D 1-q
E[V] = qU + (1 – q)D.
– ቤተ መጻሕፍቲ ባይዱp:
aSu be U .
r
r aS be D. – Down: d
We have
U D a , Su S d r U D b U Su e Su S d
Since the portfolio and the option have the same value at time t =τ, they must have the same value at time t = 0, otherwise there are arbitrage opportunities, thus
• We can think of q as the probability of the stock goes up at time t =τ
V0 er E V
• The probability q is called the risk-neutral probability, or no-arbitrage pricing probability.
$130
$100 $ 90
$0
V0
$5
U D 5 a 0.125. Su Sd 40
V0 aS0 U aSu e r
0 0.125 130 0.125 100
1.05
$2.98
Homework
P.29 #1, #4.
§3. Replicating Portfolios
• This is just the short position of the portfolio at the beginning. • At the end of one year the value of the portfolio is -$90. (does not depend on the outcome!) • Thus we can make a risk-free profit 1.05x86 – 90 = $0.30.
Use
V0 e
r
qU (1 q)D
1. We have U=7, D=0, thus V=[7q+0(1-q)]/1.04=$5.38.
2. We have U=2, D-0, thus V=[2q+0(1-q)]/1.04=$1.54. 3. We have U=0, D=5, thus V=[0q+5(1-q)]/1.04=$0.96.
q
55
40
50
1-q
To find the pricing probability: 1.04 x 50 = q x 55 + (1-q) x 40 That is 52 = 15q + 40 Thus q = 12/15 = 0.8
•
Find the values for
1. Call option with X=$48. 2. Call option with X=$53. 3. Put option with X=$45.
Arbitrage
Suppose a dealer is willing to buy the option at $7.25: • Long one share of stock at $100, • Short 2 options • Cost: 100 – 7.25x2=$85.50, • Borrow $85.50 at the rate r = 0.04879 so that the yearly rate is
Let Π0 be the value of the portfolio at time t = 0. Then Π0 = aV + bS At time t = 1, Up state: S = 120, then Π1 = a(120 – 105) + b×120 Down state: S = 90, then Π1 = a ×0 + b×90
Suppose a dealer is offering the option at $7.00: • Short one share of stock at $100, • Long 2 options • Net: 100 – 7.00x2 = $86, • Invest $86 at the rate r = 0.04879,
– Down state: Πd = D – aSd.
Set Πu = Πd. Thus
U – aSu = D – aSd
We get
U D V a . Su Sd S
We have
e 0 .
r
V0 aS0 er U aSu .
Thus the derivative price formula
Let Then
e S0 S d q Su S d
r
Su e S0 1 q Su Sd
V0 e
r
r
so
qU (1 q)D
We show that 0≤q≤1: 1. er S S , i.e., q 0 0 d Otherwise we just borrow money and buy as many shares of stock a possible.
• Form a portfolio:
– a shares of stock; – b units of bond (risk free interest rate r).
• Initial value
Π0 = aS0+b.
• We want the portfolio to have the same value as the option at time t =τ, i.e., the portfolio replicates the option at t =τ. • At time t =τ:
120
100
90
Stock price
Three ways to price the option: • Game theory method; • Replicating portfolio method; • Probabilistic approach.
§2. Game Theory Method
V0 aS0 U aSu er .
Example
• A unit of stock is valued at $100. • In 1 year, the price will be either $130 or $90. • A put option with exercise price of $95. • One year interest rate is 5%. Find the put option price at t = 0.
V0 = aS0+b.
V0
aS0 U aSu e
r
r U D U D S0 U Su e Su S d Su S d S0 Su r r U e e Su S d Su S d S0 Su r D e Su S d Su S d r r r e S 0 S d r Su e S 0 Ue De Su S d Su S d
q
Su
Sd
q
U
D
S0
1-q Stock price
V0
1-q Option value
• How to remember the pricing probability?
e S0 qSu (1 q)Sd
• so
r
e S0 S d q Su S d
r
Example
• Current value of a stock is $50; • After 1 year, its value will be either $55 or $40; • One-year interest rate is 4%.
A General Formula
We assume that the stock price can take on only two values at timeτ. • Up state Su, option value U, • Down state Sd, option value D.