Option Valuation

1
Chapter 18
Option Valuation
2
Chapter Summary

• Objective To discuss factors that affect option
  prices and to present quantitative option pricing
  models.
• Factors influencing option values
• Black-Scholes option valuation
• Using the Black-Scholes formula
• Binomial Option Pricing

3
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

4
Time Value of Options Call
5
Factors Influencing Option Values Calls

• Factor Effect on value
• Stock price increases
• Exercise price decreases
• Volatility of stock price increases
• Time to expiration increases
• Interest rate increases
• Dividend Rate decreases

6
Restrictions on Option Value Call

• Value cannot be negative
• Value cannot exceed the stock value
• Value of the call must be greater than the value
  of levered equity
• C gt S0 - ( X D ) / ( 1 Rf )T
• C gt S0 - PV ( X ) - PV ( D )

7
Allowable Range for Call
8
Summary Reminder

• Objective To discuss factors that affect option
  prices and to present quantitative option pricing
  models.
• Factors influencing option values
• Black-Scholes option valuation
• Using the Black-Scholes formula
• Binomial Option Pricing

9
Black-Scholes Option Valuation

• Co SoN(d1) - Xe-rTN(d2)
• d1 ln(So/X) (r ?2/2)T / (??T1/2)
• d2 d1 (??T1/2)
• where,
• Co Current call option value
• So Current stock price
• N(d) probability that a random draw from a
  normal distribution will be less than d

10
Black-Scholes Option Valuation (contd)

• X Exercise price
• e 2.71828, the base of the natural 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
• ????Standard deviation of annualized continuously
  compounded rate of return on the stock

11
Call Option Example

• So 100 X 95
• r .10 T .25 (quarter)
• ??? .50

12
Probabilities from Normal Distribution

• N (.43) .6664
• Table 18.2
• d N(d)
• .42 .6628
• .43 .6664 Interpolation
• .44 .6700

13
Probabilities from Normal Distribution

• N (.18) .5714
• Table 18.2
• d N(d)
• .16 .5636
• .18 .5714
• .20 .5793

14
Call Option Value

• Co SoN(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?

15
Put Value using Black-Scholes

• P Xe-rT 1-N(d2) - S0 1-N(d1)
• Using the sample call data
• S 100 r .10 X 95
• g .5 T .25
• P 95e-10x.25(1-.5714)-100(1-.6664)6.35

16
Put Option Valuation 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

17
Adjusting the Black-Scholes Model for Dividends

• The call option formula applies to stocks that
  pay dividends
• One approach is to replace the stock price with a
  dividend adjusted stock price
• Replace S0 with S0 - PV (Dividends)

18
Summary Reminder

• Objective To discuss factors that affect option
  prices and to present quantitative option pricing
  models.
• Factors influencing option values
• Black-Scholes option valuation
• Using the Black-Scholes formula
• Binomial Option Pricing

19
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

20
Portfolio Insurance - Protecting Against
Declines in Stock Value

• Buying Puts - results in downside protection with
  unlimited upside potential
• Limitations
• Tracking errors if indexes are used for the puts
• Maturity of puts may be too short
• Hedge ratios or deltas change as stock values
  change

21
Hedging Bets on Mispriced Options

• Option value is positively related to volatility
• If an investor believes that the volatility that
  is implied in an options price is too low, a
  profitable trade is possible
• Profit must be hedged against a decline in the
  value of the stock
• Performance depends on option price relative to
  the implied volatility

22
Hedging and Delta

• The appropriate hedge will depend on the delta.
• Recall the delta is the change in the value of
  the option relative to the change in the value of
  the stock.

23
Mispriced Option Text Example

• Implied volatility 33
• Investor believes volatility should 35
• Option maturity 60 days
• Put price P 4.495
• Exercise price and stock price 90
• Risk-free rate r 4
• Delta -.453

24
Hedged Put Portfolio

• Cost to establish the hedged position
• 1000 put options at 4.495 / option 4,495
• 453 shares at 90 / share 40,770
• Total outlay 45,265

25
Profit Position on Hedged Put Portfolio

• Value of put as function of stock price
• implied volatility 35
• Stock Price 89 90 91
• Put Price 5.254 4.785 4.347
• Profit/loss per put .759 .290
  (.148)
• Value of and profit on hedged portfolio
• Stock Price 89 90 91
• Value of 1,000 puts 5,254 4,785
  4,347
• Value of 453 shares 40,317 40,770
  41,223
• Total 45,571 45,555 45,570
• Profit 306 290 305

26
Summary Reminder

• Objective To discuss factors that affect option
  prices and to present quantitative option pricing
  models.
• Factors influencing option values
• Black-Scholes option valuation
• Using the Black-Scholes formula
• Binomial Option Pricing

27
Binomial Option PricingText Example
28
Binomial Option PricingText Example
Alternative Portfolio Buy 1 share of stock at
100 Borrow 46.30 (8 Rate) Net outlay
53.70 Payoff Value of Stock 50 200 Repay
loan - 50 -50 Net Payoff 0
150
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Binomial Option PricingText Example
30
Another View of Replicationof Payoffs and Option
Values

• Alternative Portfolio - one share of stock and 2
  calls written (X 125)
• Portfolio is perfectly hedged
• Stock Value 50 200
• Call Obligation 0 -150
• Net payoff 50 50
• Hence 100 - 2C 46.30 or C 26.85

31
Generalizing the Two-State Approach

• Assume that we can break the year into two
  six-month segments
• In each six-month segment the stock could
  increase by 10 or decrease by 5
• Assume the stock is initially selling at 100
• Possible outcomes
• Increase by 10 twice
• Decrease by 5 twice
• Increase once and decrease once (2 paths)

32
Generalizing the Two-State Approach
33
Expanding to Consider Three Intervals

• Assume that we can break the year into three
  intervals
• For each interval the stock could increase by 5
  or decrease by 3
• Assume the stock is initially selling at 100

34
Expanding to Consider Three Intervals
35
Possible Outcomes with Three Intervals
Event Probability Stock Price 3 up
1/8 100 (1.05)3 115.76 2 up 1 down 3/8 100
(1.05)2 (.97) 106.94 1 up 2 down 3/8 100
(1.05) (.97)2 98.79 3 down 1/8 100
(.97)3 91.27
36
Multinomial Option Pricing

• Incomplete markets
• If the stock return has more than two possible
  outcomes it is not possible to replicate the
  option with a portfolio containing the stock and
  the riskless asset
• Markets are incomplete when there are fewer
  assets than there are states of the world (here
  possible stock outcomes)
• No single option price can be then derived by
  arbitrage methods alone
• Only upper and lower bounds exist on option
  prices, within which the true option price lies
• An appropriate pair of such bounds converges to
  the Black-Scholes price at the limit