## D exp2 06505

Based upon these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first quarter): Up price = $ 30 (1.4477) = $ 43.43

3. The Impact of Exercise On The Value Of The Underlying Asset

The Black-Scholes model is based upon the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants compared to otherwise similar call options. The adjustment for dilution in the Black-Scholes to the stock price is fairly simple. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance:

Dilution-adjusted S =

where

S = Current value of the stock nS + nW

3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity. We will discuss them in chapter 16.

nw = Number of warrants outstanding W = Value of warrants outstanding ns = Number of shares outstanding

When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.

There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (say, the exercise value or the current market price of the warrant). This will yield a value for the warrant and this estimated value can then be used as an input to re-estimate the warrant's value until there is convergence.

Illustration 5.6: Valuing a warrant with dilution

MN Corporation has 1 million shares of stock trading at $50, and it is considering an issue of 500,000 warrants with an exercise price of $60 to raise fresh equity for the firm. The warrants will have a five-year lifetime. The standard deviation in the value of equity has been 20%, and the five-year riskless bond rate is 10%. The stock is expected to pay $1 in dividends per share this year, and is expected to maintain this dividend yield for the next five years.

The inputs to the warrant valuation model are as follows: S = (1,000,000 * $50 + 500,000 * $ W )/(1,000,000+500,000) K = Exercise price on warrant = $60 t = Time to expiration on warrant = 5 years a2 = Variance in value of equity = 0.22 = 0.04 y = Dividend yield on stock = $1 / $50 = 2%

Since the value of the warrant is needed as an input to the process, there is an element of circularity in reasoning. After a series of iterations where the warrant value was used to re-estimate S, the results of the Black-Scholes valuation of this option are:

Value of Call= S exp-(0 02) (5) (0.4430) - $60 exp-(010)(5) (0.2774) = $ 3.59 Value of warrant = Value of Call * ns /(nw + ns) = $ 3.59 *(1,000,000/1,500,000) = $2.39

Illustration 5.7: Valuing a warrant on Avatek Corporation

Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March, 2001, the company had 1.8 million options outstanding, with four years to expiration, with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) is 93%. The four-year treasury bond rate was 4.90%. (The warrants were trading at $0.12 apiece at the time of this analysis)

The inputs to the warrant valuation model are as follows:

S = (0.38 * 19.637 + 0.12* 1.8 )/(19.637+1.8) = 0.3582

K = Exercise price on warrant = 2.25

t = Time to expiration on warrant = 4 years r = Riskless rate corresponding to life of option = 4.90%

a2 = Variance in value of stock = 0.932

The results of the Black-Scholes valuation of this option are:

Value of Warrant= 0.3544 (0.5167) - 2.25 exp-(0049)(4) (0.0345) = $0.12 The warrant was trading at $0.25.

^ warrant.xls: This spreadsheet allows you to estimate the value of an option, when there is a potential dilution from exercise.

The Black-Scholes Model for Valuing Puts

The value of a put can be derived from the value of a call with the same strike price and the same expiration date.

C - P = S - K e-rt where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration t. To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows:

Position |
Payoffs at t if S*>K |
Payoffs at t if S*<K |

Sell call |
-(S*-K) |
0 |

Buy put |
0 |
K-S* |

Buy stock |
S* |
S* |

Total |
K |
K |

Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e-rt). S+P-C = K e-rt C - P = S - K e-rt

Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get:

where

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