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Ken Shah FINC 580 Corning Option Valuation Steps: 1. Value Straight Bond 2. Valuing Conversion Option: Forecast up/down stock price using binomial model Forecast payoffs Use standard binomial valuation to value the call option (see handout) Obtain value of St. bond + conversion option value (per share – i.e. divide by # of shares in conversion) over the entire tree 3. Valuing Callable Feature: Forecast Bond price per period (to year 15) using the 2% yield. Price path over 15 years will grow by 2% every year. To the company, the value to calling (redeeming) bonds is: Value = Max[Stock price per share, redemption bond price per share] Therefore payoffs to calling (redeeming) are: Payoff = Max[Bond+Convert. Option - Redemption value, 0] Use binomial formula to value redemption option using payoffs above Obtain value of: Bond + conversion option – Redemption option 4. Repuchase Option: Assume that option is never exercised early Calculate the strike price per share = strike price at year 10 / # of shares Payoff = Max[Strike price per share – Bond value without repurchase option, 0] Use binomial model to value this option payoffs Alternatively, do the same assuming option is exercised early, and project payoffs upto year 5 only using exercise price in year 5. Repeat the binomial valuation. Seattle University – ASBE KEN SHAH FINC 580 Binomial Model: Need the following inputs: o Risk free rate per period o Path of underlying security values Charting the path of security value: If a security (e.g. underlying stock) o has an annual standard deviation of σ per year o and follows a binomial distribution (i.e., it can take 2 values next period) then the two values next period are: Up Value SU S e Down Value h S D S e h Where S = current security value h = length of period in years (e.g., if period = 6 months, h = .5) σ= annual standard deviation of stock returns For example, a stock has a current value of $100, and an annual standard deviation of returns of 20%, and h = 1 (period is 1 year) then SU = 100 x e.2 = $122.14 $122.14 $100 SD = 100 x e-.2 = $81.87 $81.87 From this stock value paths we can obtain the call option payoff path Payoff = Max [S – X, 0] Where S = current security value X = exercise price Suppose the X = 100 in the above example then the two payoffs next period are PU = Max [122.14 – 100, 0] = 22.14 PD = Max [81.87-100, 0 ] = 0 Now we can apply the binomial pricing formula for this one-period option: Binomial Option Pricing Formula for a Call Definitions: Up factor u e Down factor h d e h Call Value: Ce r f h q PU 1 q PD Where q e r f h d ud Note again: C = value of call option e = the napiers base (exponent) PU = call payoff when security is ‘up’ PD = call payoff when security is ‘down’ h = length of period in years rf = annual risk free rate u = up factor defined above d = down factor defined above In the numerical example PU = 22.14 PD = 0 h=1 rf = .05 (assume) u = e^.2 = 1.2214 d = e^-.2 = 0.8187 Therefore, e.051 .8187 = 0.5775 q= q 1.2214 .8187 C e .051 .5775 22.14 1 .5775 0 = $12.16 You can easily extend this model to multiple periods. Just work backwards from the last period at each node. Extending above example to two periods: 0 Stock Price 1 2 149.18 0 100 C 122.14 100 Call Payoff 1 2 49.18 22.14 81.87 0 0 67.03 0 First find the two call values C11 and C12. Then using C11and C12, find C 0 Call Value 1 2 C11 = 27.02 C = 14.84 C12 = 0 This method can be extended to any number of periods. Note- Original Issue Discount Assume a $100,000 three-year non-interest bearing note is purchased on January 1, 2000 for $86,384 (yielding an effective interest rate of 5%. Interest income is computed as follows: Year Jan 1 Carrying Value 2000 2001 2002 86,384 90,703 95,238 + Interest Income 4,319 (=86,384 x .05) 4,319 (=90,703 x .05) 4,319 (=95,238 x .05) = Dec. 31 Carrying Value 90,703 95,238 100,000