Investment Analysis and Portfolio Management

1
Investment Analysis and Portfolio Management

• Lecture 10
• Gareth Myles

2
Put-Call Parity

• The prices of puts and calls are related
• Consider the following portfolio
• Hold one unit of the underlying asset
• Hold one put option
• Sell one call option
• The value of the portfolio is
• P S Vp Vc
• At the expiration date
• P S maxE S, 0 maxS E, 0

3
Put-Call Parity

• If S lt E at expiration the put is exercised so
• P S E S E
• If S gt E at expiration the call is exercised so
• P S S E E
• Hence for all S
• P E
• This makes the portfolio riskfree so
• S Vp Vc (1/(1r)t)E

4
Valuation of Options

• At the expiration date
• Vc maxS E, 0
• Vp maxE S, 0
• The problem is to place a value on the options
  before expiration
• What is not known is the value of the underlying
  at the expiration date
• This makes the value of Vc and Vp uncertain
• An arbitrage argument can be applied to value the
  options

5
Valuation of Options

• The unknown value of S at expiration is replaced
  by a probability distribution for S
• This is (ultimately) derived from observed data
• A simple process is assumed here to show how the
  method works
• Assume there is a single time period until
  expiration of the option
• The binomial model assumes the price of the
  underlying asset must have one of two values at
  expiration

6
Valuation of Options

• Let the initial price of the underlying asset be
  S
• The binomial assumption is that the price on the
  expiration date is
• uS with probability p up state
• dS with probability 1- p down state
• These satisfy u gt d
• Assume there is a riskfree asset with gross
  return R 1 r
• It must be that u gt R gt d

7
Valuation of Options

• The value of the option in the up state is Vu
• maxuS E, 0 for a call
• maxE uS, 0 for a put
• The value of the option in the down state is Vd
• maxdS E, 0 for a call
• maxE dS, 0 for a put
• Denote the initial value of the option (to be
  determined) by V0
• This information is summarized in a binomial tree
  diagram

8
Valuation of Options
9
Valuation of Options

• There are three assets
• Underlying asset
• Option
• Riskfree asset
• The returns on these assets have to related to
  prevent arbitrage
• Consider a portfolio of one option and D units
  of the underlying stock
• The cost of the portfolio at time 0 is
• P0 V0 DS

10
Valuation of Options

• At the expiration date the value of the portfolio
  is either
• Pu Vu - DuS
• or
• Pd Vd - DdS
• The key step is to choose D so that these are
  equal (the hedging step)
• If D (Vu Vd)/S(u d) then
• Pu Pd (uVd dVu)/(u d)

11
Valuation of Options

• Now apply the arbitrage argument
• The portfolio has the same value whether the up
  state or down state is realised
• It is therefore risk-free so must pay the
  risk-free return
• Hence Pu Pd RP0
• This gives
• RV0 DS (uVd dVu)/S(u
  d)

12
Valuation of Options

• Solving gives
• This formula applies to both calls and puts by
  choosing Vu and Vd
• These are the boundary values
• The result provides the equilibrium price for the
  option which ensures no arbitrage
• If the price were to deviate from this then
  risk-free excess returns could be earned

13
Valuation of Options

• Consider a call with E 50 written on a stock
  with S 40
• Let u 1.5, d 1.125, and R 1.15

14
Valuation of Options

• This gives the value
• For a put option the end point values are
• Vu max50 60, 0 0
• Vd max50 45, 0 5
• So the value of a put is

15
Valuation of Options

• Observe that
• 40 4.058 0.58 43.478
• And that
• (1/1.15) 50 43.478
• So the values satisfy put-call parity
• S Vp Vc (1/R)E

16
Valuation of Options

• The pricing formula is
• Notice that
• So define

17
Valuation of Options

• The pricing formula can then be written
• The terms q and 1 q are known as risk neutral
  probabilities
• They provide probabilities that reflect the risk
  of the option
• Calculating the expected payoff using these
  probabilities allows discounting at the risk-free
  rate

18
Valuation of Options

• The use of risk neutral probabilities allows the
  method to be generalized

19
Valuation of Options

• u and d are defined as the changes of a single
  interval
• R is defined as the gross return on the risk-free
  asset over a single interval
• For a binomial tree with two intervals the value
  of an option is

20
Valuation of Options

• With three intervals
• Increasing the number of intervals raises the
  number of possible final prices
• The parameters p, u, d can be chosen to match
  observed mean and variance of the asset price
• Increasing the number of periods without limit
  gives the Black-Scholes model