Common use of Table I Clause in Contracts

Table I. Comparison of the proposed ad option and other options. The price of the ith underlying asset/keyword at time t is denoted by Ci(t), where t is a continuous time point in period [0, T ] and T is the contract expiration date; if there is only one underlying asset we denote its price by C(t). The strike/fixed payment price, of the ith underlying asset/keyword is denoted by Fi; if there is only one strike price we denote it by F . The weight of ith asset/keyword in a basket-type option is denoted by ωi. Note that in the n-keyword 1-click ad option, ωji represents the weight of the ith broad matched keyword for the jth candidate keyword, and kj represents the number of broad matched keywords. Detailed descriptions of notations are provided in Table II. Option contract Payoff function Underlying variable Exercise opportunity Early exercise Strike price Application area n-keyword 1-click ad option (keyword exact or broad match) max{C1(t) − F1, . . . , Cn(t) − Fn, 0} Multiple Single Yes Multiple Keywords n-keyword 1-click ad option Σk1 Σkn max ω1iC1i(t) − F1, · · · , ωniCni(t) − Fn, 0 i=1 i=1 Multiple Single Yes Multiple Keywords European standard call option [▇▇▇▇▇▇▇ 2006] max{C(T ) − F, 0} Single Single No Single Equity stock, or index American standard call option [▇▇▇▇▇▇▇ 2006] max{C(t) − F, 0} Single Single Yes Single Equity stock, or index Σn max ωiCi(T ) − F, 0 i=1 Index of Multiple Single No Single bonds or foreign currencies basket call option [▇▇▇▇▇▇ et al. 2006] European dual-strike call option [▇▇▇▇▇ 1998] max{C1(T ) − F1, C2(T ) − F2, 0} Double Single No Double Equity stocks, or indexes of equity stocks, or bonds, or foreign currencies European rainbow call on max option [Ouwehand and West 2006] max{max{C1(T ), . . . , Cn(T )} − F, 0} Multiple Single No Single European paying the best and cash option [▇▇▇▇▇▇▇ 1987] max{C1(T ), C2(T ), F} Double Single No Single European quotient call option [▇▇▇▇▇ 1998] max{C1(T )/C2(T ) − F, 0} Double Single No Single Multi-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 0:5 in search queries. Dual-strike options are options with two different strike prices for two different underlying assets [▇▇▇▇▇ 1998]. One simple version of our proposed ad options is a dual-strike call option, which allows an advertiser to switch between his targeted two keywords during the contract lifetime. However, in sponsored search, the number of candidate keywords to choose from is usually more than two, so the two key- words are extended to higher dimensions. In addition, as an advertiser usually needs more than a single click for guaranteed delivery, the dual-strike call option is extended to a multi-exercise option. Multi-exercise options are a generalisation of American options, which provide a buy- er with more than one exercise right and sometimes control over one or more other variables [▇▇▇▇▇▇▇▇▇ 2004], e.g., the amount of the underlying asset exercised in cer- tain time periods. Multi-exercise options have become more prevalent over the past decade, particularly, in the energy industry, such as electricity swing options and wa- ter options. Contributors to the multi-exercise options include ▇▇▇▇ [2000], ▇▇▇▇ and ▇▇▇▇ [2006], ▇▇▇▇▇▇▇ and ▇▇▇▇▇▇▇▇▇▇ [2000], ▇▇▇▇▇▇▇▇▇ [2004], ▇▇▇▇▇ [2006], ▇▇▇▇▇▇▇▇ et al. [2011] and ▇▇▇▇▇▇▇▇ [2012]. Their work is not further discussed here as our pro- posed ad option is a simple example of multi-exercise options. Compared to the energy industry, the multi-exercise opportunity in sponsored search is more flexible. Advertis- ers are allowed to exercise options at any time in the option lifetime, i.e. the exercise time is not pre-specified, and no minimum number of clicks is required for each exer- cise. Therefore, there is no penalty fee if the advertiser does not exercise the minimum clicks. In addition, there is no transaction fee for ad options in sponsored search. Motivated by an attempt to model the fluctuations of asset prices, Brownian motion (i.e., the continuous-time random walk process [▇▇▇▇▇▇ 2004]) was first introduced by ▇▇▇▇▇▇▇▇▇ [1900] to price an option. However, the impact of his work was not recog- nised by financial community for many years. Sixty five years later, ▇▇▇▇▇▇▇▇▇ [1965] replaced ▇▇▇▇▇▇▇▇▇’▇ assumptions on asset price with a geometric form, called the geo- metric Brownian motion (GBM). In the GBM model, the proportional price changes are exponentially generated by a Brownian motion. While the GBM model is not appropri- ate for all financial assets in all market conditions, it remains the reference model against which any alternative dynamics are judged. The research of ▇▇▇▇▇▇▇▇▇ highly affected Black and ▇▇▇▇▇▇▇ [1973] and ▇▇▇▇▇▇ [1973], who then examined the option pricing based on a GBM. They constructed a portfolio from risky and risk-less underlying assets to replicate the value of an Euro- pean option. Risky assets can be stocks, foreign currencies, indices, and so on; risk-less assets can be bonds. Once the risky part of the replicated portfolio is estimated, the option value can be obtained accordingly. The pricing methods proposed by ▇▇▇▇▇ and ▇▇▇▇▇▇▇ [1973] and ▇▇▇▇▇▇ [1973] were based on the assumption that investors on the market cannot obtain arbitrage. Therefore, the replicated portfolio is treated as a self- adjusting process whose least expectation of returns increase at the same speed as the constant bank interest rate. If considering the constant bank interest rate as a discoun- t factor, the discounted value of the replicated portfolio would be a martingale [Bjo¨rk 2009], whose probability measure is called the risk-neutral probability measure. Since a closed-form pricing formula can be obtained from the settings of Black and Scholes [1973] and Merton [1973], we normally call their work as the Black-▇▇▇▇▇▇▇-▇▇▇▇▇▇ (B- SM) option pricing formula. The BSM option pricing formula spurred research in this field. Various numerical procedures then appeared, including lattice methods, finite d- ifference methods and Monte Carlo methods. These numerical procedures are capable of evaluating more complex options when the closed-form solution does not exist. In this paper, the Monte Carlo method we discussed can quickly price an ad option where the number of candidate keywords is larger than two. ACM Transactions on Intelligent Systems and Technology, Vol. 0, No. 0, Article 0, Publication date: March 2015. 0:6 ▇. ▇▇▇▇ et al.

Table I. Comparison of the proposed ad option and other options. The price of the ith underlying asset/keyword at time t is denoted by Ci(t), where t is a continuous time point in period [0, T ] and T is the contract expiration date; if there is only one underlying asset we denote its price by C(t). The strike/fixed payment price, of the ith underlying asset/keyword is denoted by Fi; if there is only one strike price we denote it by F . The weight of ith asset/keyword in a basket-type option is denoted by ωi. Note that in the n-keyword 1-click ad option, ωji represents the weight of the ith broad matched keyword for the jth candidate keyword, and kj represents the number of broad matched keywords. Detailed descriptions of notations are provided in Table II. Option contract Payoff function Underlying variable Exercise opportunity Early exercise Strike price Application area n-keyword 1-click ad option (keyword exact or broad match) max{C1(t) − F1, . . . , Cn(t) − Fn, 0} Multiple Single Yes Multiple Keywords n-keyword 1-click ad option (keyword broad match) . Σk1 Σkn Σ max ω1iC1i(t) − F1, · · · , ωniCni(t) − Fn, 0 i=1 i=1 Multiple Single Yes Multiple Keywords European standard call option [▇▇▇▇▇▇▇ 2006] max{C(T ) − F, 0} Single Single No Single Equity stock, or index American standard call option [▇▇▇▇▇▇▇ 2006] max{C(t) − F, 0} Single Single Yes Single Equity stock, or index Σn max ωiCi(T ) − F, 0 i=1 Index of Multiple Single No Single bonds or foreign currencies European basket call option [▇▇▇▇▇▇ et al. 2006] . Σn Σ max ωiCi(T ) − F, 0 i=1 Multiple Single No Single Index of equity stocks bonds or foreign currencies European dual-strike call option [▇▇▇▇▇ 1998] max{C1(T ) − F1, C2(T ) − F2, 0} Double Single No Double Equity stocks, or indexes of equity stocks, or bonds, or foreign currencies European rainbow call on max option [Ouwehand and West 2006] max{max{C1(T ), . . . , Cn(T )} − F, 0} Multiple Single No Single European paying the best and cash option [▇▇▇▇▇▇▇ 1987] max{C1(T ), C2(T ), F} Double Single No Single European quotient call option [▇▇▇▇▇ 1998] max{C1(T )/C2(T ) − F, 0} Double Single No Single Multi-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 0:5 5:5 in search queries. Dual-strike options are options with two different strike prices for two different underlying assets [▇▇▇▇▇ 1998]. One simple version of our proposed ad options is a dual-strike call option, which allows an advertiser to switch between his targeted two keywords during the contract lifetime. However, in sponsored search, the number of candidate keywords to choose from is usually more than two, so the two key- words are extended to higher dimensions. In addition, as an advertiser usually needs more than a single click for guaranteed delivery, the dual-strike call option is extended to a multi-exercise option. Multi-exercise options are a generalisation of American options, which provide a buy- er buyer with more than one exercise right and sometimes control over one or more other variables [▇▇▇▇▇▇▇▇▇ 2004], e.g., the amount of the underlying asset exercised in cer- tain time periods. Multi-exercise options have become more prevalent over the past decade, particularly, in the energy industry, such as electricity swing options and wa- ter options. Contributors to the multi-exercise options include ▇▇▇▇ [2000], ▇▇▇▇ and ▇▇▇▇ [2006], ▇▇▇▇▇▇▇ and ▇▇▇▇▇▇▇▇▇▇ [2000], ▇▇▇▇▇▇▇▇▇ [2004], ▇▇▇▇▇ [2006], ▇▇▇▇▇▇▇▇ et al. [2011] and ▇▇▇▇▇▇▇▇ [2012]. Their work is not further discussed here as our pro- posed ad option is a simple example of multi-exercise options. Compared to the energy industry, the multi-exercise opportunity in sponsored search is more flexible. Advertis- ers are allowed to exercise options at any time in the option lifetime, i.e. the exercise time is not pre-specified, and no minimum number of clicks is required for each exer- cise. Therefore, there is no penalty fee if the advertiser does not exercise the minimum clicks. In addition, there is no transaction fee for ad options in sponsored search. Motivated by an attempt to model the fluctuations of asset prices, Brownian motion (i.e., the continuous-time random walk process [▇▇▇▇▇▇ 2004]) was first introduced by ▇▇▇▇▇▇▇▇▇ [1900] to price an option. However, the impact of his work was not recog- nised by financial community for many years. Sixty five years later, ▇▇▇▇▇▇▇▇▇ [1965] replaced ▇▇▇▇▇▇▇▇▇’▇ assumptions on asset price with a geometric form, called the geo- metric Brownian motion (GBM). In the GBM model, the proportional price changes are exponentially generated by a Brownian motion. While the GBM model is not appropri- ate for all financial assets in all market conditions, it remains the reference model against which any alternative dynamics are judged. The research of ▇▇▇▇▇▇▇▇▇ highly affected Black and ▇▇▇▇▇▇▇ [1973] and ▇▇▇▇▇▇ [1973], who then examined the option pricing based on a GBM. They constructed a portfolio from risky and risk-less underlying assets to replicate the value of an Euro- pean option. Risky assets can be stocks, foreign currencies, indices, and so on; risk-risk- less assets can be bonds. Once the risky part of the replicated portfolio is estimated, the option value can be obtained accordingly. The pricing methods proposed by ▇▇▇▇▇ and ▇▇▇▇▇▇▇ [1973] and ▇▇▇▇▇▇ [1973] were based on the assumption that investors on the market cannot obtain arbitrage. Therefore, the replicated portfolio is treated as a self- self-adjusting process whose least expectation of returns increase at the same speed as the constant bank interest rate. If considering the constant bank interest rate as a discoun- t discount factor, the discounted value of the replicated portfolio would be a martingale [Bjo¨rk 2009], whose probability measure is called the risk-neutral probability proba- bility measure. Since a closed-form pricing formula can be obtained from the settings of Black and Scholes [1973] and Merton [1973], we normally call their work as the Black-▇▇▇▇▇▇▇-▇▇▇▇▇▇ (B- SMBSM) option pricing formula. The BSM option pricing formula spurred research in this field. Various numerical procedures then appeared, including lattice methods, finite d- ifference difference methods and Monte Carlo methods. These numerical procedures are capable of evaluating more complex options when the closed-form solution so- lution does not exist. In this paper, the Monte Carlo method we discussed can quickly price an ad option where the number of candidate keywords is larger than two. ACM Transactions on Intelligent Systems and Technology, Vol. 07, No. 01, Article 05, Publication date: March October 2015. 0:6 5:6 ▇. ▇▇▇▇ et al.