NPV and Valuation

1
Chapters 14-16
Option Valuation The Black-Scholes- Merton Model

Mila Getmansky Sherman
2
Option Value and Asset Volatility

• Option value increases with the volatility of
  underlying asset.
• Example. Two firms, A and B, with the same
  current price of 100. B has higher volatility
  of future prices. Consider call options written
  on A and B, respectively, with the same exercise
  price 100.
• Clearly, call on stock B should be more valuable.

3
Determinants of Option Value

• Key factors in determining option value
• Price of underlying asset, S
• Strike price, X
• Time to maturity, T
• Interest rate, r
• Dividends, D
• Volatility of underlying asset,

4
Additional Factors

• Additional factors that can sometimes influence
  option value
• Expected return on the underlying asset
• Additional properties of stock price movements
• Investors attitude toward risk
• Characteristics of other assets

5
Binomial Option Pricing Model

• In order to have a complete option pricing model,
  we need to make additional assumptions about
• Pricing process of the underlying asset (stock)
• Other factors

6
Assumptions for Binomial Option Pricing Model

• Prices do not allow arbitrage
• Prices are reasonable
• A benchmark model Price follows a binomial
  process
• One-period borrowing/lending rate is r, and R1r
• No arbitrage requires ugtRgtd

7
Binomial Process
8
One-Period Option Pricing

• Example. Valuation of a European call on a
  stock.
• Current stock price is 50.
• There is one period to go.
• Stock price will either go up to 75 or go down
  to 25.
• There are no cash dividends
• The strike price is 50.
• One period borrowing and lending rate is 10.

9
Cont. One-Period Option Pricing

• The stock and bond present two investment
  opportunities
• The options payoff at expiration is
• Question What is C0, the value of the option
  today?

10
Calculation of C0

• Claim We can form a portfolio of stock and bond
  that gives identical payoffs as the call.
• Consider a portfolio (a,b) where
• a is the number of shares of the stock held
• b is the dollar amount invested in the riskless
  bond. Alternatively, you can think of b as the
  number of bonds.
• We want to find (a,b) so that
• 75a1.1b25
• 25a1.1b0
• There is a unique solution
• a0.5 and b-11.36.

11
Calculation of C0

• There is a unique solution
• a0.5 and b-11.36.
• That is
• Buy half of a share of stock and sell 11.36
  worth of bond
• Payoff of this portfolio is identical to that of
  the call
• Present value of the call must equal the current
  cost of this replicating portfolio which is
• 500.5-11.3613.64C0.

12
Two-Period Option Pricing

• Now we generalize the above example when there
  are two periods to go period 1 and period 2.
  The stock price process is (note, u1.5, and
  d0.5)
• The call price follows the following process
  (note, 62.5112.5-50)

13
Cont. Two-Period Option Pricing

• Where
• The terminal value of the call is known and
• Cu and Cd denote the option value next period
  when the stock price goes up and goes down,
  respectively.
• We derive current value of the call backwards
  first compute its value next period, and then its
  current value.

14
Step 1

• Start with Period 2
• 1. Suppose the stock price goes up to 75 in
  period 1
• Construct the replicating portfolio at node (t2,
  up)
• 112.5a1.21b62.5
• 37.5a1.21b0
• The unique solution is a0.834 and b-25.86
• The cost of this portfolio is 0.83475-25.8628.
  378 Cu

15
Cont. of Step 1

• 2. Suppose the stock price goes down to 25 in
  period 1. Repeat the above for node (t2, down)
• 37.5a1.21b0
• 12.5a1.21b0
• The unique solution is a0 and b0
• The cost of this portfolio is 0 Cd

16
Step 2

• Now go back one period, to Period 1
• The options value is either 34.075 or 0
  depending upon whether the stock price goes up or
  down
• If we can construct a portfolio of the stock and
  bond to replicate the value of the option next
  period, then the cost of this replicating
  portfolio must equal the options present value.

17
Cont. of Step 2

• Find a and b so that
• 75a1.1b28.378
• 25a1.1b0
• The unique solution is
• a0.5676
• b-12.8991
• The cost of this portfolio is 0.567650-12.8991
  15.48
• The present value of the option must be C015.48.

18
Play Forward This Strategy

• In period 0 spend 15.48 on option and borrow
  12.8991 at 10 interest rate to buy 0.5676 shares
  of the stock.
• In period 1
• When the stock price goes up, the option value
  becomes 28.378. Re-balance the portfolio to
  include 0.834 stock shares, financed by borrowing
  25.86 at 10.
• One period hence in period 2, the payoff of this
  portfolio exactly matches that of the call.
• When the stock price goes down, the portfolio
  becomes worthless. Close out the position.
• The portfolio payoff one period hence in period 2
  is zero.

19
Summary

• Replicating strategy gives payoffs identical to
  those of the call.
• Initial cost of the replicating strategy must
  equal the call price.

20
Lessons From the Binomial Model

• What have we used to calculate options value
• Current stock price
• Magnitude of possible future changes of stock
  price volatility
• Interest rate
• Strike price
• Time to maturity

21
Lessons From the Binomial Model

• What we have not used
• Probabilities of upward and downward movements
• Characteristics of securities other than the
  underlying stock and riskless bond
• Investors attitude towards risk.
• Note Investors may disagree on the
  probabilities of the upward and downward moves,
  but they agree on the option price!

22
Black-Scholes-Merton Option Pricing Formula (BSM)

• In the binomial model, if we let the
  period-length get smaller and smaller, we obtain
  the BSM formula for a call
• Where T is in units of a year
• R is the annual riskless interest rate
• C is the price of a call
• S is the price of the underlying stock
• X is the exercise price of the call
• is the standard deviation of the logarithm of
  the stocks return.
• N() denotes a value of the standard normal
  distribution.
• No dividends are paid before expiration

23
Example of BSM

• Consider a European call option on a stock with
  the following information
• S50, X50, T30 days, volatility 30/year,
  and current annual interest rate r 5.895.

24
BSM Put pricing

• Using the put-call parity, we get

25
BSM for Dividend Paying Stocks

• q is the continuous dividend yield per year.
  Note, that if dividends are not paid before T,
  then q 0, and we use the above BSM model.

26
Greeks Delta

• Delta represents the sensitivity of the price of
  an option (or portfolio) to changes in the price
  of the stock. The delta for European calls on
  dividend-paying stocks is
• For puts

27
Greeks Theta

• Theta measures how the price of an option changes
  with time. You can use it to estimate how fast
  an option will lose value with the passage of
  time. The theta for call options is
• For put options

28
Greeks Gamma

• Gamma is the sensitivity of a stocks delta to
  the stock price. It has the same value for both
  call and put options and is given by

29
Greeks Vega

• Vega is the rate of change of the value of an
  option with respect to the volatility of the
  stocks price. The vega for both call and put
  options is

30
Greeks Rho

• Rho measures the sensitivity of the price of an
  option with respect to the risk-free interest
  rate. For a call option, it is given by
• For a put option, it is