Valuing Options

1
Valuing Options

• The Binomial Method

2
Valuing Options

• Discounted cash flow method doesnt work
• Risk keeps changing over the lifetime of the
  option, so your discount rate would have to
  change
• Trick Owning the option is equivalent to owning
  some number of shares and borrowing money!
• Therefore the price of the option should be the
  same as the value of this replicating portfolio.

3
Example

• Lets assume a stock may be worth 140 or 80
  next year, with equal probabilities.
• The discount rate for the stock is 10.
• The stock has a beta of 1.
• T-bills are yielding 5.
• An option on the stock is traded on the NYSE.
  The strike price of this option is 110, and the
  maturity is one year.
• What is the option worth?

4
The Method

• Replicate the payoffs of the option using only
  the stock
• Step 1 Find the payoffs of the option in both
  good and bad states.
• Step 2 Choose the right number of shares to
  make the difference in outcomes equal to the
  option.
• An options delta or hedge ratio
• In general, the formula is
• Option Delta (Spread of possible option
  prices)
• (Spread of possible stock prices)
• Step 3 Now replicate the payoffs by adding the
  right size loan to your shares

5
Possibly more intuitive

• You might think of option deltas as solving
• (Option delta) (Spread in Share price)
• Spread in option price
• That is, the delta scales the amount of
  variation in the stock price so that the spread
  of outcomes in your replicating portfolio is the
  same as in the option.

6
Example cont.

• Options are riskier than the stock.
• Find the options expected return.
• Find the options beta.

7
The implicit loan in options

• Note that we just showed the option is like a
  levered version of stock.
• Its like buying the stock bundled with
  personal debt.

8
Example 2

• Stock is priced at 55. A call option expired in
  six months and has exercise price 55.
• Assume the stock will either rise to 73.33 or
  fall to 41.25.
• Let the six-month risk-free rate be 2.
• What is the call option worth?
• Note that you can answer this question without
  even knowing the probabilities of a rise or fall
  in the price!

9
Valuing the put option

• Method 1 Same as before.
• Find the options delta
• Option Delta (Spread of possible option
  prices)
• (Spread of possible stock prices)
• Then find the bank loan that equates the payoffs
  of (optionloan) and (stock position)
• Method 2 Since we already have the call option
  value, just use put-call parity to infer the put
  option value.

10
Expanding the binomial model to allow more
possible price changes
55
Now

• What is the price of a 6-month European call
  option with exercise price 55?
• Assume the 3- month risk-free rate is 1.

44.88
67.43
3 months
82.67
6 months
36.62
55
11
How to solve the two-period example

• Step 1 Find the option value 3 months from now,
  for each of the two cases.
• Same problem as before find a delta, then find
  the loan.
• Step 2 Now find the value at T0, given these
  two possible payoffs.
• Same problem as before find a delta, then find
  the loan.

12
Expanding the binomial model to allow more
possible price changes
Binomial to Black Scholes
1 step 2 steps 4 steps
(2 outcomes) (3 outcomes) (5
outcomes) etc. etc.
13
Binomial Option Pricing Model
The most important lesson (so far) from the
binomial option pricing model is

• the replicating portfolio intuition.

Many derivative securities can be valued by
valuing portfolios of primitive securities when
those portfolios have the same payoffs as the
derivative securities.