【導讀】andbonds.AtmaturityT,thefixedamount,F.firmandgetA.Theyownthefirm,A.AtmaturityT,whenA<F. AtmaturityT,whenA>F. exercisepriceofF.AtmaturityT,whenA<F. AtmaturityT,whenA>F. Valueofriskybond=Valueofdefault-freebond–Valueofputoption

　　

【正文】 ar beets are not as tied to the price of refined sugar as in the price of sugarcane because the peting firms currently do not use sugar beets as an input. The Production Decision of A Sugar Producer ?The numerical example ? In a good economy, the cost of producing refined sugar with sugarcane is $ per pound and the cost of using sugar beets is $ per pound. ? In a bad economy, the cost of producing refined sugar with sugarcane falls to $ per pound. The cost of producing with sugar beets falls to $ per pound. ? The price of refined sugar is always $ per pound greater than the cost of production using sugarcane. ? The risk neutral probabilities associated with each of these two states of the economy are assumed to be . ?Assuming that the fixed cost of building a sugarcane and sugar beet plant are the same, which method of producing refined sugar is better when (a) capacity is fixed, or (b) capacity is flexible in that the firm is mitted to producing at least 1 million pounds of refined sugar in the plant but, at a cost of $40,000 the firm can double capacity to 2 million pound on discovering the state of the economy. The Binomial Trees Good Economy Refined sugar = $ Sugar cane = $ Sugar beets = $ Bad Economy Refined sugar = $ Sugar cane = $ Sugar beets = $ Good Economy $140,000 = 2m($$)$40,000 Bad Economy $70,000 = 1m($$) Good Economy $30,000 = 1m($$) Bad Economy $30,000 = 1m($$) Sugar beets as input Sugarcane as input Sugarcane or Sugar Beets? (a) If the capacity is fixed ? Sugarcane: 1m()* + 1m()* = 30m ? Sugar beets: 1m()* + 1m()* = 10m ? The cost of producing the sugar using sugar beets is higher. (b) If the capacity can be expanded ? Sugarcane: 1m()* + 1m()* = 30m ? Sugar beets: [2m($$)$40,000]* + 1m($$)* = 35m ? Though using sugar beets is more costly, it introduces more variability which enhances the value of the flexibility (option to expand the capacity). Warrants ? Warrants are call options that give the holder the right, but not the obligation, to buy shares of mon stock directly from a pany at a fixed price for a given period of time. ? Warrants tend to have longer maturity periods than exchange traded options. ? Warrants are generally issued with privately placed bonds. ? Warrants are also bined with new issues of mon stock and preferred stock, given to investment bankers as pensation for underwriting services. ? In this case, they are often referred to as a Green Shoe Option. Warrants ?The same factors that affect call option value affect warrant value in the same ways. 1. Stock price + 2. Exercise price – 3. Risk free rate + 4. Volatility in the stock price + 5. Expiration date + 6. Dividends – The Difference Between Warrants and Call Options ?Call options are issued by individual investors or institutions, while warrants are issued by the firms. ?When a warrant is exercised, a firm must issue new shares of stock. But when the options are exercised there is no change in the total number of the shares. Only the ownership of the shares changes. ?The exercise of warrants can have the effect of diluting the claims of existing shareholders. Dilution Example ? Imagine that Mr. Armstrong and Mr. LeMond are shareholders in a firm whose only asset is 6 ounces of platinum. ? When they incorporated, each man contributed 3 ounces of platinum, then valued at $600 per ounce. They printed up two stock certificates, and named the firm LegStrong, Inc.. ? Suppose that Mr. Armstrong decides to sell Mr. Mercx a call option issued on Mr. Armstrong’s share. The call gives Mr. Mercx the option to buy Mr. Armstong’s share for $1,800. ? If the price of platinum rises above $600 per ounce, say $700 per ounce, Mr. Mercx will exercise and profit by $700*6/2$1,800 = $300 ? Nothing else changes except the names of the shareholders. Dilution Example ?Suppose that Mr. Armstrong and Mr. LeMond meet as the board of directors of LegStrong. The board decides to sell Mr. Mercx a warrant. The warrant gives Mr. Mercx the option to buy one share for $1,800. ?Suppose the warrant finishes inthemoney, (platinum increased to $700 per ounce). Mr. Mercx will exercise. The firm will print up one new share. Dilution Example ? The balance sheet of LegStrong Inc. when Legstrong is formed and the platinum price is $600 per ounces: Balance Sheet (platinum price is $600) Assets Liability and Equity Platinum (6 ounces) $3,000 Debt 0 Equity (2 shares) $3,000 Total $3,000 Total $3,000 Dilution Example ? The balance sheet of LegStrong Inc. when the platinum price is $700 per ounce and before the exercising of the warrant: Balance Sheet (platinum price is $700 before exercising of warrant) Assets Liability and Equity Platinum (6 ounces) $4,200 Debt 0 Equity (2 shares) $4,200 Total $4,200 Total $4,200 Dilution Example ? The balance sheet of LegStrong Inc. when the platinum price is $700 per ounce and after the exercising of the warrant: Balance Sheet (platinum price is $700 after exercising of warrant) Assets Liability and Equity Platinum (6 ounces) $4,200 Debt 0 Cash $1,800 Equity (3shares) $6,000 Total $6,000 Total $6,000 Mr. Armstrong’s claim falls in value from $2,100 = $4,200 / 2 to $2,000 = $6,000 / 3 Mr. Mercx profits by $2,000 $1,800 = $200 Warrant Pricing and the BlackScholes Model Compare the gains from exercising a call with the gains from exercising a warrant. The gain from exercising a call can be written as: Note that when n = the number of shares, share price is: Thus, the gain from exercising a call can be written as: Share price – Exercise price n Firm’s Value Net of Debt Exercise price Firm’s Value Net of Debt n