# Option Pricing Using Binomial Trees

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```					            Option Pricing Using Binomial Trees

FIN 509: Foundations of Asset Valuation
Class session 4
Professor Jonathan M. Karpoff
Outline of these slides

1. The option pricing problem
2. An intuitive way to think about option pricing
3. Two specific binomial option pricing approaches
a. Option valuation using the replicating portfolio approach
b. Option pricing using the risk-neutral approach
4. Takeaways: It’s risk, not return that matters in option pricing
1. The option pricing problem

• The value of a call option:   C = f (S, X)

• At expiration:                CT = max(0, ST – X)
Option pricing notation (repeat)

S = The underlying stock (or asset) price
X = The exercise, or strike, price, at which the asset can be
purchased or sold
t = The time to expiration, expressed in years
s = The volatility of the underlying asset, equal to the square root of
the variance of the asset's rate of return over a very short time
interval
r =The risk-free rate of interest over the life of the option
More option pricing notation

• c is the value of a call option.
c is a function of S, X, t, s, and r, and is expressed as c(S,X,t,s,r).

• p is the value of a put option.
p is a function of S, X, t, s, and r, and is expressed as p(S,X,t,s,r)
2. An intuitive way to think about option pricing

• Suppose there is a call option on a stock. The current stock price
is \$40, and the option exercise price is \$40. The option expires in
4 weeks.
• Each week, the stock price has equal probability of going up
\$2.50, or down \$2.50.
• The lattice on the next slide shows how the stock price can move
during the four weeks, and the five possible final outcomes.
• If the stock price ends up above \$40, the option will be worth
something. At or below a \$40 stock price, the option is worthless.
The binomial lattice for a four-week
option
How to value the option
using the binomial lattice

• The option pays off only when the stock price upon expiration of
the option is \$50 or \$45.
• A \$50 value occurs with 6.25% probability, and a \$45 value
occurs with a 25% probability.
• So the option’s expected payoff is: (.0625) (\$10) + (.25) (\$5) =
\$1.875.
• The current option value is the present value of \$1.875.
3. Two specific binomial option pricing
approaches

• The replicating portfolio approach
– Price the option by pricing a portfolio of stock and bonds with identical
cash flows
• The risk-neutral approach
– Take advantage of the result that the actual probabilities of up and
down movements do NOT enter into the option pricing formula
– “It’s risk, not return”
Overview of binomial option pricing

•   Goal: Find exact formula for value of the option before expiration

•   Binomial tree: Diagram that represents different possible paths a stock
price might follow over the life of an option

t=0               t=1
Cu
C
Cd
3.a. Replicating portfolio approach

• Key assumption:   No arbitrage opportunities exist

Payoff of
Payoff of
If      replicating          =
option
portfolio

They must cost the same today .
(3.a) Numerical example

•   Consider a stock that is currently priced at \$100 and will either be \$110 or
\$90 at the end of one year

Su =110
S =100
Sd =90
•   The one-period risk-free rate is 6%. A \$1 investment in a bond pays off:

1.06

B=1
1.06
(3.a) Stock and bond replication

• We can replicate the payoffs of the call option by buying shares of
the stock and selling the risk-free bond
• Step 1: Find the replicating portfolio
D: the number of shares
B: the price of the bond (amount borrowed TODAY)

Value of replicating portfolio = D S - B
(3.a) Constructing the replicating portfolio

•   Recall the path of the stock price:

Su =110
S =100
•   Path of call price on above stock with X = Sd =90
\$100:

Cu = max (0, Su – X) = 10

C = ??
Cd = max (0, Sd – X) = 0
• Path of replicating portfolio consisting of D shares of stock and B in bonds:

D 110 - 1.06 B
D 100 - B
• Path of call price:                       D 90 - 1.06 B

Cu = 10

C = ??
• Choose D and B so that:                     D 110 - 1.06 B = 10
Cd = 0
D 90 - 1.06 B = 0
(3.a) Finding the exact ∆ and B

•   Two equations, two unknowns:
D 110 - 1.06 B = 10
D 90 - 1.06 B = 0
•   Solving these two equations yields:
D = 0.5
B = 42.45
•   Therefore, the replicating portfolio consists of:
– Buying 1/2 share of stock
– Selling the risk-free bond (that is, borrowing) in the amount of \$42.45 at t=0
(3.a) Check that we have replicated the
option payoff

Path of stock price:
Su =110
S =100
The value of the portfolio (at t = 1):                         Sd =90
Up state:                                Down state:
Portfoliou = D Su- 1.06 B                           Portfoliod = D Sd- 1.06 B
= .5 (110) - (1.06)(42.45)                           = .5 (90) - (1.06)(42.45)
= 10                                                  =0
(3.a) Find the option value

•   Because the replicating portfolio and the call option have identical
payoffs, by the no-arbitrage principle, they must have the same cost
today
•   The current value of the portfolio (at t = 0) is:
Portfolio = D S - B
= .5 (100) - 42.45
= 7.55
•   The current value of the call is \$7.55.
•   Whew!
(3.a) What is D (“delta”)?

• ∆ is the number of shares needed to replicate the call option

• It is the spread of option prices relative to the spread of stock
prices

• Also known as the hedge ratio:

• With this definition of D, we only need to find B to find the call’s
price:
C=DS–B
(3.a) Another example: Hershey Foods
option

• Hershey Foods, Inc. stock is currently priced at \$68 and will rise
by 25% or fall by 20% over the next period:
Su = 85
S =68
Sd = 54.40

• Path of a Hershey’s call option with X = \$65:

Cu = 20

C = ??
Cd = 0
Hershey’s example (slide 2 of 3)

• Solve for option D:

• Next step: Find B, using the assumption that the interest rate is 2.5% over
the next period
Hershey’s example (slide 3 of 3)

• To find how much is needed in bonds:
– Set the payoff of the replicating portfolio equal to option payoff at
t=1
– Suppose we choose the t=1 down state payoff:
D 54.4 - 1.025 B = 0
Þ B = 34.69
• Price of Hershey’s call = D S - B
= (0.6536) (68) - 34.69
= \$9.76
(3.a) Summary of replicating portfolio
approach

Step 1: Set up binomial trees for the stock and option path prices.
Step 2: Using terminal stock and option prices, find option delta:

Step 3: Find B (the amount borrowed) by setting the terminal payoff of the
replicating portfolio equal to the terminal payoff of the option:

Step 4: Using option D and B, calculate the price of call:
D Sd - (1+r) B = Cd

C=DS-B
(3.a) Does this approach work with put
options?

• Only the terminal payoffs are different

• With a call option, the terminal node payoff = max (0, ST – X)

• With a put option, the terminal node payoff = max (0, X – ST)

• All else is the same!
(3.a) Put option valuation with the
replicating portfolio approach

Consider the original example
•   Stock path                                    Su =110
S =100
Sd =90
•   What is the price path of a put (X = \$100)?

Pu = max(0, X – Su) = 0

P = ??
Pd = max(0, X – Sd) = 10
(3.a) Calculating the put option value

•   ∆S - B = p
•   In the down state: ∆Sd - B(1+r) = pd
•   ∆ = (pu - pd)/(Su - Sd) = (0-10)/(110-90) = -0.5
•   Using the down state to calculate B:
-0.5 (90) - (1.06) B = 10
B = -51.89
•   Now calculate p:
p = -0.5 (100) - (-51.89) = 1.89
(3.a) Put-call parity

• Verify that put-call parity holds

C = P + S – PV(X)
P = C – S + PV(X)
= 7.55 – 100 + (100 / 1.06)
= 1.89
3.b. The risk-neutral pricing method

Two ways of pricing options

Stock and bond replication                  Risk-neutral valuation
In the risk-neutral world:
– Investors are indifferent to risk
– The expected return of a stock is equal to the risk-free rate
– We can find the risk-neutral probabilities of up or down
movements in stock prices
(3.b) Overview of risk-neutral valuation

•   Find risk-neutral probabilities of up and down movements
p is the risk-neutral probability of an up move in the stock price
(1 - p) is the risk-neutral probability of a down move in the stock price

p         Cu
C
•   Find the expected payoff of the call option at t=1
•                                        1-p          Cd
Discount the payoff to t=0 at the risk-free rate to find the price of the call
(This is similar to the intuitive example at the beginning of this lecture.)
(3.b) The exact price of European call

Definitions:
R=1+r
u = 1 + percentage stock price increase
d = 1 – percentage stock price decrease
(3.b) What determines p ?

p = probability of Su
•   The current stock price is the PV of the expected future price:
•   S = (1/R) (p u S + (1- p) d S)
•   Divide by S:            1 = (1/R) (p u + (1- p) d)
•   Multiply by R and expand:        R = (p u + d - p d)
•   Rearrange terms:                  p = (R-d)/(u-d)
(3.b) Back to the Hershey’s example

•   Hershey stock is currently priced at \$68 and will rise by 25% or fall by
20% over the next period:

Su = 85
S = 68
Sd = 54.40
•   What are u and d?
u = 1 + percentage of stock price increase = 1.25
d = 1 – percentage of stock price decrease = .80
(3.b) Hershey’s example (slide 2 of 3)

•   What are the risk-neutral probabilities?

•   Recall the call price path:

Cu = 21.25
C = ??
•   The price of the option is just the discounted expected payoff
Cd = 0
(3.b) Hershey’s example (slide 3 of 3)

•   Price of call using risk-neutral valuation:

Þ The replicating portfolio and risk-neutral pricing approaches yield the
same value of the option.
4. Takeaways: It’s risk not return….

• Know the replicating portfolio and risk-neutral approaches to
valuation
• Probabilities of future up or down movements of the stock do not
affect option prices (such probabilities already are incorporated
into the price of the stock)
• Rather, option prices depend on the volatility of stock
• It’s risk, not return….

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