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

720 APPLICATION OF THE OPTION MARKET PARADIGMwhere N(z) is the cumulative distribution function (c.d.f.) of thestandard normal distribution, andd (¹)1= ln(P 0 =S)+(¹ +0:5¾2 )t¾ p andtd (¹)2= ln(P 0=S)+(¹ ¡ 0:5¾ 2 )t¾ p = d (¹)1¡ ¾ p t:tThe first term in Formula (1) is the expected market valueof the assets to be sold by the call option grantor to the optionholder at expiry. The second term is the expected value of thesale proceeds from that transaction.The variance of the call payoff obligation at expiry is givenbyZ 1Var(call t )= (x ¡ S) 2 f(x)dx ¡ E(call t ) 2S=E(x 2 ) ¢ N(d (¹)0) ¡ 2S ¢ E(x) ¢ N(d(¹)1 )+ S 2 ¢ N(d (¹)2 ) ¡ E(call t )2 , (2)where N(z) is the c.d.f. of the standard normal distribution, andd (¹)1and d (¹)2are defined as in Formula (1) and d (¹)0= d (¹)1+ ¾ p t.Returning to the example of the 20-day call option withP 0 = S = $100, ¹ = 13%, r =5%and¾ = 25%, the expected payoffliability at expiry associated with that option is $2.7174. Thatamount is the difference between the expected market value ofthe stock the grantor of the option will sell to the option holder($56.4009), given by the first term of Formula (1), and the expectedvalue of his sale proceeds ($53.6835), which is given bythe second term of Formula (1). The variance, given by Formula(2), is $14.4456, implying a standard deviation of $3.8007.We will illustrate the pricing of this expected payoff liabilityof $2.7174 in various available asset scenarios. The premiumthat the market can be expected to ask for assuming this liabilitydepends on the optimal strategy available for investment of