PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

APPLICATION OF THE OPTION MARKET PARADIGM 723justs the number of shares he holds (to n 1 ) to reflect any changein the stock price and the infinitesimal passage of time. He adjuststhe loan accordingly (to L 1 ). If n 0 and L 0 have been chosencorrectly and the time interval is short enough, the gain or loss inhis net position (i.e., the value of the net stock position less thevalue of the option) is effectively zero. The mean and variance ofhis net result is also zero. He repeats this adjustment procedurecontinuously until the option expires. In this way he ends upwith exactly the right amount of stock at expiry to generate thefunds to meet the option liability and repay the outstanding loan.Provided the sequences of n i and L i have been chosen correctly,the cumulative net result and its variance are both zero. Since thevariance is zero, there is no justification for a risk charge. Blackand Scholes proved that n 0 = N(d 1 )andL 0 = Se ¡rt ¢ N(d 2 )andthus thatcall 0 = P 0 ¢ N(d 1 ) ¡ Se ¡rt ¢ N(d 2 ), (1.1)where N(z) is the c.d.f. of the standard normal distribution andd 1 = ln(P 0 =S)+(r +0:5¾2 )t¾ p td 2 = ln(P 0=S)+(r ¡ 0:5¾ 2 )t¾ p tand= d 1 ¡ ¾ p t:Since (1.1) does not depend on ¹, the option seller engagingin the hedging strategy underlying the formula not only faces noprocess risk but also no ¹-related parameter risk. (There is stillparameter risk associated with ¾.) In our example, Formula (1.1)indicates a call premium of $2.4705.In the highly liquid, efficient market in which execution ofthis dynamic hedging strategy is possible, arbitrageurs will forcethe market’s “ask” price of the option to $2.4705. If the optionseller seeks a higher price, he will find no buyers, since anothertrader can and will undercut him without assuming any additionalrisk, simply by executing the hedging strategy. Note, however,