DERIVATION OF THE AD OPTION PRICING FORMULA. Since the proposed ad option complements the existing keyword auctions, there may exist a situation that some advertisers only want to make guaranteed profits from the difference of costs between option and auction markets without taking any risk. This situation is called arbitrage [Varian 1987; Bjo¨rk 2009]. Hence, we must fairly evaluate the option so that arbitrage is eliminated. ∈ In the context of sponsored search, we consider that an advertiser buys a n-keyword m-click ad option at time 0. Then at time t, t [0, T ], the difference between the option value and the market value of candidate keywords can be expressed as Σ Π(t) = V (t, C(t); F , T, m) − ψi(t)Ci(t), (19)
DERIVATION OF THE AD OPTION PRICING FORMULA. Since the proposed ad option complements the existing keyword auctions, there may exist a situation that some advertisers want to make guaranteed profits only from the difference of costs between option and auction markets without taking any risk. This situation is called arbitrage [Varian 1987; Bjo¨rk 2009]. Hence, we must fairly evaluate the option so that arbitrage is eliminated. ∈ In the context of sponsored search, we consider that an advertiser buys an n-keyword m-click ad option at time 0. Then, at time t, t [0, T ], the difference between the option value and the market value of candidate keywords can be expressed as n П(t) = V (t, C(t); F, T, m) — ψi(t)Ci(t), (19) . = where ψi(t) represents the number of clicks needed for the keyword Ki such that i ψi(t) m. Here, we call П(t) the value difference process. Recall that, in Equation (3), we consider the value of an n-keyword m-click option as the sum of m independent n- keyword 1-click options; for mathematical convenience, Equation (19) can be rewritten as follows: П(t) =m V (t, C(t); F, T, 1) — i=1 ∆iCi(t) , (20) . where ∆i represents the probability that a single click goes for the keyword Ki and n i=1 ∆i = 1. The changes of П over a sufficient small period of time dt is then n n n n dП(t) = m⎝ ∂t dt + 2 σiσjρijCiCj ∂Ci∂Cj dt + ∂Ci dCi — ∆idCi⎠ . (21) ⎛ ∂V ∂2V ∂V ⎞ i=1 i=1 = i=1 j=1 The uncertain components in dП(t) can be removed if ∆i ∂V /∂Ci. This is called delta hedging in option pricing theory [Xxxxxxx 2006]. Hence, П(t) now becomes a riskless process over time: n n dП(t) = m⎝ ∂t + 2 σiσjρijCiCj ∂Ci∂Cj ⎠ dt. (22) ⎛ ∂V 1 ∂2V ⎞ i=1 j=1 ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015.
