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Executive Stock Options: Value of At-The-Money Options for ARB

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Executive stock options. It is March 1, 2004 and you have received a job offer from a publicly traded fund manager (stock symbol: ARB) to start in six months. The company needs good people, so they are willing to leave the offer open until then. The compensation package includes 1,000 call options on ARB whose features are discussed below. (Ignore any issues with non-transferability, etc. Assume these are ordinary European options).

Denote by t = 0; 1; 2; 3 the dates March 1, 2004; Sept 1, 2004; March 1, 2005; Sept 1, 2005.

Assume that the company stock price today (t = 0) is S0 = 100, and that every six months the stock moves up or down by 25% with equal probability. Assume the simple semi-annual interest rate is 4%. ARB will not pay any dividends.

(a) If the options are issued at-the-money on the day the offer expires (t = 1), so that K will be S1, and expire one year later (t = 3), what is the value today of their offer.

(b) If, instead, the strike price were fixed today, K = S0, expiring at t = 2, What is the value of the package?