Option Valuation

1
Chapter 15

• Option Valuation

2
Option Values

• Intrinsic value - profit that could be made if
  the option was immediately exercised
• Call stock price - exercise price
• Put exercise price - stock price
• Time value - the difference between the option
  price and the intrinsic value.
• Why is price gt intrinsic value?

3
Time/Volatility Value

• When a call option is out of the money there is
  still a chance that the stock price could go up.
  Willing to pay a positive price.
• When a call option is deep in the money the
  option will most likely be exercised at price X.
  What are you willing to pay, the present value of
  profit S PV(X).

4
Time Value of Options Call
Option value
Value of Call
Intrinsic Value
Time value
X
Stock Price
5
Summary of the effect of increasing one variable
while all others fixed
6
Methods to Price Options

• Binomial/Trinomial Trees modeling the path of
  stock prices and using risk neutral valuation of
  stock prices.
• Explicit solution to partial differential
  equation Black-Scholes
• Monte Carlo Simulation numerical method to
  evaluate option prices.

7
Binomial Option PricingOne step tree
200
100
50
Stock Price
8
Riskless Portfolio

• Must earn risk free rate.
• Consider a portfolio consisting of a long
  position in ? shares and short 1 call option.
• Calculate the value of ? that makes the portfolio
  riskless. (Same value at either end node)

9
Calculation of ?

• 200 ? - 75 50 ? ? .5
• A riskless portfolio consists of
• Long .5 shares
• Short 1 call option
• Portfolio value at either node is 25
• Present value of 25 at risk free rate is
  25/(1.08) 23.15, which is value of investment
  today. 100.5 c 23.15
• Thus c 26.85

10
?

• ? is sometimes called the hedge ratio, it is
  equal to
• ? (C-C-)/(S-S-) where C- indicate the calls
  value in the upstate and downstate, while S-
  indicate the value of the stock in the upstate
  and downstate.

11
Arbitrage Opportunity

• What if c 30 how could we profit?
• The call is too highly priced, buy portfolio ie
  (buy shares, sell option).
• Portfolio costs 20 to set up (100.5 30)
• Generates 25 at either node, ie 5 in profit
  which is greater than risk free rate.
• This will force call price to fall.

12
Two step tree and beyond
13
Black-Scholes Formula

• Fischer Black, Myron Scholes, and Robert Merton
  develop a model to price options.
• Explicit/exact solutions given restrictions
• Scholes/Merton received Nobel Prize in 1997.

14
Assumptions

• No taxes/transactions costs
• No arbitrage opportunities.
• Borrow and lend at risk free rate
• No early exercise
• No dividends
• Lognormal prices ie random walk
• Continuous trading

15
Black-Scholes Option Valuation E-Call

• Co Soe-dTN(d1) - Xe-rTN(d2)
• d1 ln(So/X) (r d s2/2)T / (s T1/2)
• d2 d1 - (s T1/2)
• where
• Co Current call option value.
• So Current stock price
• N(d) probability that a random draw from a
  normal dist. will be less than d.

16
Black-Scholes Option Valuation

• X Exercise price.
• d Annual dividend yield of underlying stock
• e 2.71828, the base of the nat. log.
• r Risk-free interest rate (annualizes
  continuously compounded with the same maturity as
  the option.
• T time to maturity of the option in years.
• ln Natural log function
• s Standard deviation of annualized cont.
  compounded rate of return on the stock

17
Call Option Example

• So 100 X 95
• r .10 T .25 (quarter)
• s .50 d 0
• d1 ln(100/95)(.10-0(.5 2/2))/(.5 .251/2)
• .43
• d2 .43 - ((.5)( .251/2)
• .18

18
Probabilities from Normal Dist.

• N (.43) .6664
• Table 17.2
• d N(d)
• .42 .6628
• .43 .6664 Interpolation
• .44 .6700
• .43 is between .42 and .43 so take average of
  N(d)s.

19
Probabilities from Normal Dist.

• N (.18) .5714
• Table 17.2
• d N(d)
• .16 .5636
• .18 .5714
• .20 .5793

20
Call Option Value

• Co Soe-dTN(d1) - Xe-rTN(d2)
• Co 100 X .6664 - 95 e- .10 X .25 X .5714
• Co 13.70
• Implied Volatility
• Using Black-Scholes and the actual price of the
  option, solve for volatility.
• Is the implied volatility consistent with the
  stock?

21
Put Option Value Black-Scholes

• PXe-rT 1-N(d2) - S0e-dT 1-N(d1)
• Using the sample data
• P 95e(-.10X.25)(1-.5714) - 100 (1-.6664)
• P 6.35

22
Using Put-Call Parity

• P C PV (X) - So
• C Xe-rT - So
• Using the example data
• C 13.70 X 95 S 100
• r .10 T .25
• P 13.70 95 e -.10 X .25 - 100
• P 6.35

23
Practice - Call/Put

• So 42 X 40
• r .10 T .5
• s .20 d 0
• d1 N(d1)
• d2 N(d2)
• c 4.74
• p .79

24
Installing Derivagem software

• http//www.rotman.utoronto.ca/hull/software/
• Exel file DG150.xls
• Put .dll file in system directory
• Enable macros
• Black-Scholes is Analytic European
• Notice button to calculate implied volatility.

25
Put Call Parity Check

• P C PV (X) - So
• C Xe-rT - So
• ..794.74 40e-.1.5 42
• .79 .79
• Good Work!

26
Using the Black-Scholes Formula

• Hedging Hedge ratio or delta
• The number of stocks required to hedge against
  the price risk of holding one option
• Call N (d1)
• Put N (d1) - 1
• Option Elasticity
• Percentage change in the options value given a
  1 change in the value of the underlying stock

27
Determining the Inputs for Black-Scholes

• S current stock price, check the Lanterman
  ticker or internet.
• T time to maturity specified in contract
• X exercise price specified in contract
• D dividends paid during contract, estimate
  based on past dividend payments.

28
Finding r and s

• r T-bill rate (Continuous time)
• P PV(Face Value)
• Treasuries are zeros, ie only pay face value
• Need to discount at continuous rate.
• P exp(-r T)Face Value
• Solve for r.
• 99.7125 100exp(-r 90/365)
• To remove exp use ln on both sides
• ln (.997125) -r .2466 thus r 1.175

29
Volatility of stock returns

• Historical volatility based on past stock
  returns.
• return from stock S on day I we denote as ui
  ln(Si/Si-1)
• Mean return

30
Volatility

• S2 var(ui)
• S estimates sT.5
• s s/(T.5)
• Rule of thumb in choosing n use as many days
  past data as days to maturity.

31
Implied Volatility

• Given option price, x, s, r, T one can work out
  the implied volatility.
• This is the markets opinion of volatility.
• Volatility smile implied volatility as a
  function of its stock price
• Traders adjust B-S to account for this

32
Smile for fx options

• Volatility lower for at the money options.
• Implied distribution has fatter tails and is more
  peaked, which implies greater expected movement.
  
• Traders thus willing to pay higher prices for out
  of the money options, which results in higher
  implied vol.

33
Smile for Equity Options

• Volatility skew, ie volatility decreases as
  strike price increases.
• Implied distribution has fatter left tail/thinner
  right tail. More concerned over movement down.
• Crashaphobia has existed since 10/87.

34
Volatility Questions

• What pattern of implied volatility is likely to
  be observed when both tails are thinner than
  lognormal distribution?
• For a deep out of the money call, the probability
  of going above X is lower than lognormal
  assumption. Thus implied distribution should
  give low price, which corresponds to low implied
  volatility. Volatility Frown.

35
Put Call Parity Revisited

• pbs cbs Xe-rT S
• pmkt cmkt Xe-rT S
• pbs pmkt cbs cmkt
• Market price of c 3 and B-S says 3.50.
• If BS price of p 1, then market price .5

36
Monte Carlo Simulation

• Simulate Asset Price Movement
• Value of option is present value of expected
  payoff.
• Useful to evaluate options that are path
  dependent. Asian options, lookback options.

37
Steps

• Step 1 Simulate asset price movement dS rS dt
  sSdx
• Step 2 Evaluate payoff to option based on
  simulation.
• Step 3 Repeat 1 and 2 many times
• Step 4 Calculate average of payoffs and take the
  present value.
• Step 5 Finit

38
Example Asian Call

• Contract pays max (A X, 0) where A is the
  average asset price over the life of the
  contract.
• S 100, X 105, r .05, sigma .2, T 1year.

39
Derivatives Mishaps

• Barings Bank wiped out in 1995 due to
  arbitageur Nick Lesson who bet on future
  direction of Nikkei 225. Losses of 1 billion.
• O.C. - Treasurer Robert Citron in 1994 bet used
  derivatives and bet that interest rates would not
  rise. He was wrong, suffering losses of 2
  billion.

40

• LTCM run by 2 Nobel Prize winners among others.
  Try to take advantage of near arbitrage
  opportunities. Default of Russian bonds led to
  flight to quality and a divergence in
  opportunity. Fund loses 4 billion, is bailed out
  by FED.
• Sumitomo a single trader lost about 2 billion
  in copper futures and options market.

41
Derivatives and Regulation

• Derivatives can be used to speculate but they may
  also be used to reduce risk exposure.
• Regulators have a tough time evaluating their use