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

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Outline of these slides

• The option pricing problem
• An intuitive way to think about option pricing
• Two specific binomial option pricing approaches
• a. Option valuation using the replicating
  portfolio approach
• Option pricing using the risk-neutral approach
• Takeaways Its risk, not return that matters in
  option pricing

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1. The option pricing problem

• The value of a call option C f (S, X)
• At expiration CT max(0, ST X)

• Can we derive an exact pricing formula?

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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

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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)

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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.

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The binomial lattice for a four-week option
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How to value the optionusing 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 options expected payoff is (.0625)
  (10) (.25) (5) 1.875.
• The current option value is the present value of
  1.875.

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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
• Its risk, not return

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

• Key assumption No arbitrage opportunities
  exist
• If

Payoff of replicating portfolio
Payoff of option
They must cost the same today .
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(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
• The one-period risk-free rate is 6. A 1
  investment in a bond pays off

Su 110
S 100
Sd 90
1.06
B1
1.06
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(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
• ? the number of shares
• B the price of the bond (amount borrowed
  TODAY)

Value of replicating portfolio ? S ? B
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(3.a) Constructing the replicating portfolio

• Recall the path of the stock price
• Path of call price on above stock with X 100

Su 110
S 100
Sd 90
Cu max (0, Su X) 10
C ??
Cd max (0, Sd X) 0
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• Path of replicating portfolio consisting of ?
  shares of stock and B in bonds
• Path of call price
• Choose ? and B so that ? 110 ? 1.06 B 10
• ? 90 ? 1.06 B 0

? 110 ? 1.06 B
? 100 ? B
? 90 ? 1.06 B
Cu 10
C ??
Cd 0
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(3.a) Finding the exact ? and B

• Two equations, two unknowns
• ? 110 ? 1.06 B 10
• ? 90 ? 1.06 B 0
• Solving these two equations yields
• ? 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 t0

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(3.a) Check that we have replicated the option
payoff

• Path of stock price
• The value of the portfolio (at t 1)
• Up state Down state
• Portfoliou ? Su? 1.06 B
  Portfoliod ? Sd? 1.06 B
• .5 (110) ? (1.06)(42.45)
  .5 (90) ? (1.06)(42.45)
• 10 0

Su 110
S 100
Sd 90
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(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 ? S ? B
• .5 (100) ? 42.45
• 7.55
• The current value of the call is 7.55.
• Whew!

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(3.a) What is ? (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 ?, we only need to find B
  to find the calls price

C ? S B
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(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
• Path of a Hersheys call option with X 65

Su 85
S 68
Sd 54.40
Cu 20
C ??
Cd 0
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Hersheys 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

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Hersheys 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 t1
• Suppose we choose the t1 down state payoff
• ? 54.4 ? 1.025 B 0
• ? B 34.69
• Price of Hersheys call ? S ? B
• (0.6536) (68) ? 34.69
• 9.76

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(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 ? and B, calculate the
  price of call

? Sd ? (1r) B Cd
C ? S ? B
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(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!

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(3.a) Put option valuation with the replicating
portfolio approach

• Consider the original example
• Stock path
• What is the price path of a put (X 100)?

Su 110
S 100
Sd 90
Pu max(0, X Su) 0
P ??
Pd max(0, X Sd) 10
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(3.a) Calculating the put option value

• ?S - B p
• In the down state ?Sd - B(1r) 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

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(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
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3.b. The risk-neutral pricing method

• Two ways of pricing options
• 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

Stock and bond replication
Risk-neutral valuation
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(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
• Find the expected payoff of the call option at
  t1
• Discount the payoff to t0 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.)

Cu
p
C
Cd
1-p
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(3.b) The exact price of European call

• Definitions
• R 1 r
• u 1 percentage stock price increase
• d 1 percentage stock price decrease

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(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)

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(3.b) Back to the Hersheys example

• Hershey stock is currently priced at 68 and will
  rise by 25 or fall by 20 over the next period
• What are u and d?
• u 1 percentage of stock price increase
  1.25
• d 1 percentage of stock price decrease .80

Su 85
S 68
Sd 54.40
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(3.b) Hersheys example (slide 2 of 3)

• What are the risk-neutral probabilities?
• Recall the call price path
• The price of the option is just the discounted
  expected payoff

Cu 21.25
C ??
Cd 0
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(3.b) Hersheys example (slide 3 of 3)

• Price of call using risk-neutral valuation
• ? The replicating portfolio and risk-neutral
  pricing approaches yield the same value of the
  option.

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4. Takeaways Its 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
• Its risk, not return.