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Any deviations from parity can be used by investors to make riskless profitsApproach 1: Pseudo-American Valuation Step 1: Define when dividends will be paid and how much the dividends will be. Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described above, where the stock price is reduced by the present value of expected dividends. Step 3: Choose the maximum of the call values estimated for each ex-dividend day. 5.5: Using Pseudo-American option valuation to adjust for early exercise Consider an option, with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend Ex-Dividend Day The call option is first valued to just before the first ex-dividend date: The value of the call based upon these parameters is: Value of call = $4.7571 Pseudo-American value of the call = Maximum ($5.1312, $5.0732, $5.1285, $4.7571) = $5.1312 Approach 2: Using the binomial The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period rather than just the cash flows at expiration. The biggest limitation of the binomial is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved. Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes, convert these into inputs for the Binomial where u and d are the up and down movements per unit time for the binomial, T is the life of the option and m is the number of periods within that lifetime. Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period. Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment. 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. From Black-Scholes to Binomial The process of converting the continuous variance in a Black-Scholes modesl to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $ 30 currently and that you estimate the standard deviation in the asset value to be 40%; the riskless rate is 5%. For simplicity, let us assume that the option that you are valuing has a one-year life and that each period is a quarter. To estimate the prices at the end of each the four quarters, we begin by first estimating the up and down movements in the binomial: 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: 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 warrants value until there is convergence. 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 ?2 = 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: d1 = -0.1435N(d1) = 0.4430d2 = -0.5907N(d2) = 0.2774 Value of Call= S exp-(0.02) (5) (0.4430) - $60 exp-(0.10)(5) (0.2774) = $ 3.59 Value of warrant = Value of Call * ns /(nw + ns) = $ 3.59 *(1,000,000/1,500,000) = $2.39 5.7: Valuing a warrant on Avatek Corporation Avatek Corporation is a real estate firm with 19.637 million shares outstanding, $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 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% ?2 = Variance in value of stock = 0.932 y = Dividend yield on stock = 0.00% The results of the Black-Scholes valuation of this option are: d1 = 0.0418N(d1) = 0.5167d2 = -1.8182N(d2) = 0.0345 Value of Warrant= 0.3544 (0.5167) - 2.25 exp-(0.049)(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: 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: Thus, the replicating portfolio for a put is created by selling short (1-N(d1))) shares of stock and investing Ke-rt (1-N(d2)) in the riskless asset. 5.8: Valuing a put using put-call parity: Cisco& AT&T Consider the call that we valued on Cisco in Illustration 5.2. The call had a strike price of $15, 103 days left to expiration and was valued at $1.87. The stock was trading at $13.62 and the riskless rate was 4.63%. We could value the put as follows: Put Value = C - S + K e-rt = $1.87 - $13.62 + $15 e-(0.0463)(0.2822) = $3.06 The put was trading at $3.38. We also valued a long term call on AT&T in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: Put Value = C - S e-yt + K e-rt = $6.63- $20.5 e-(0.0251)(1.8333) + $20 e-(0.0485)(1.8333) = $5.35 The put was trading at $3.80. Jump Process Option Pricing Models If price changes remain larger as the time periods in the binomial are shortened, we can no longer assume that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downwards at a given rate. Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (?) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model . In this model, the value of an option is determined by the five variables specified in the Black Scholes model and the parameters of the jump process (?, k). Unfortunately, the estimates of the jump process parameters are so noisy for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice. |
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