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Unformatted text preview: Math 238 Winter 2009 Problem Set 3  Solutions March 4, 2009 Problem 1 Consider a company with N outstanding shares and M outstanding European warrants. Each warrant entitles the holder to purchase shares from the company at time T at a price K per share. Find an implicit equation for the determination of the price W of the warrant. Show that this price at time t T is equal to the price of N/ ( N + M ) European call option with the stock price S replaced by S + ( M/N ) W , which gives an implicit equation for W . Interpret this formula in financial terms. (Hull Ch 12.10.) Solution Call V ( t ) the value of the company at time t. At time T if the warrants are exercised the companys value increases to V ( T ) + MK and it is divided in N + M shares. The price of a share at time T it is V ( T ) + MK N + M The warrant is exercised only if this new price of the share exceeds K, and the payoff is parenleftBig V ( T ) + MK N + M K parenrightBig + = N N + M parenleftBig V ( T ) N K parenrightBig + At any time t [0 ,T ] the value of the company is V=NS+MW, which gives that the payoff is N N + M parenleftBig ( S T + M N W ) K parenrightBig + which is the payoff of N/ ( N + M ) European call options on an asset with price process S+(M/N)W. If we assume that this process is a geometric Brownian motion, at any time t [0 ,T ] the price of the warrant satisfies W ( t ) = N N + M C BS ( t,S + M N W ( t ); K,T,r, ) (1) where represents the volatility of the price process S+(M/N)W. This is an implicit equation for W which has a unique solution for each t because the difference of the two sides (RHSLHS) of the equation is a monotone function of W that is positive for small W and negative for big W. To see the monotonicity observe that W ( C BS ( t,S + M N W ; K,T,r, ) W ) = N ( d 1 ) M N 1 < where N ( d 1 ) is as usual the hedge ratio and we assume that M < N . To interpret it financially we observe several things. First, for M N the price of the warrant becomes essentially the price of an ordinary European call option on the stock. Also, since S ( t ) = V ( t ) MW ( t ) N , issuing of warrants forces the stock price to go down. 1 If M is small compared to the number of the outstanding shares N, we can expand the equation for W and get W ( t,S ) = C BS bracketleftBig 1 N (1 C BS ( t,S ) S ) bracketrightBig (To see this, expand the RHS of (1) using Taylors theorem around the point S. Keep only the first two(To see this, expand the RHS of (1) using Taylors theorem around the point S....
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This note was uploaded on 11/08/2009 for the course MATH 238 taught by Professor Papanicolaou during the Winter '08 term at Stanford.
 Winter '08
 Papanicolaou
 Math

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