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

APPLICATION OF THE OPTION MARKET PARADIGM 721the premium to fund the liability. We will assume that enoughinvestors or traders will find and execute the optimal strategy toforce the asking price 3 in the market to be no greater than thelevel indicated by this strategy. (This is the standard “no arbitrage”requirement.)This market premium is equal to the minimum expectedpresent value cost of acquiring sufficient assets to fund the expectedvalue liability at expiry and a risk charge related to theundiversifiable variability of the net result. If the variance of thenet result can be forced to zero, as it can be when Black-Scholesconditions are present, then the risk charge is zero and the premiumis simply equal to the minimum cost of acquiring the assetsto fund the liability.Case A–Underlying Asset is TradableThe traditional actuarial approach to valuing the liability, embodiedin Formula (1.3), is to assume the matching assets areinvested in risk-free Treasuries. 4 However, where the liabilityarises from an option on a traded stock, it is easy to improveon this approach. Since the expected value of the stock to betransferred to the option holder at expiry is P 0 e ¹t ¢ N(d (¹)1), theoption seller can match this expected liability by buying N(d (¹)1 )shares of stock at inception and holding them to expiry. He canfund most of the cost of the purchase, P 0 ¢ N(d (¹)1), by borrowingagainst his expected sale proceeds at expiry of S ¢ N(d (¹)2 ).Assuming he can borrow at the risk-free rate, he can raiseSe ¡rt ¢ N(d (¹)2 ) in this way. That leaves him short of the P 0 ¢ N(d(¹) 1 )he needs to buy the shares by P 0 ¢ N(d (¹)1 ) ¡ Se¡rt ¢ N(d (¹)2), whichis the amount he should ask for the option, before considerationof a risk charge. This indicates a formula for the premium before3 We will focus on the seller’s asking price. The question of whether there are buyers atthis asking price is beyond the scope of this discussion.4 Throughout this paper Treasuries are treated as risk-free assets and their yield as therisk free rate. If other assets meet that definition, they may be substituted for Treasuries.