Multi-keyword multi-click advertisement option contracts for sponsored search
55Multi-Keyword Multi-Click Advertisement Option Contracts forSponsored Search
Bowei Chen , University College London
Jun Wang , University College London
Ingemar J. Cox , University of Copenhagen and University College London
Mohan S. Kankanhalli , National University of SingaporeIn sponsored search, advertisement (abbreviated ad) slots are usually sold by a search engine to an adver-tiser through an auction mechanism in which advertisers bid on keywords. In theory, auction mechanismshave many desirable economic properties. However, keyword auctions have a number of limitations includ-ing: the uncertainty in payment prices for advertisers; the volatility in the search engine’s revenue; and theweak loyalty between advertiser and search engine. In this paper we propose a special ad option that allevi-ates these problems. In our proposal, an advertiser can purchase an option from a search engine in advanceby paying an upfront fee, known as the option price. He then has the right, but no obligation, to purchaseamong the pre-specified set of keywords at the fixed cost-per-clicks (CPCs) for a specified number of clicksin a specified period of time. The proposed option is closely related to a special exotic option in finance thatcontains multiple underlying assets (multi-keyword) and is also multi-exercisable (multi-click). This novelstructure has many benefits: advertisers can have reduced uncertainty in advertising; the search engine canimprove the advertisers’ loyalty as well as obtain a stable and increased expected revenue over time. Sincethe proposed ad option can be implemented in conjunction with the existing keyword auctions, the optionprice and corresponding fixed CPCs must be set such that there is no arbitrage between the two markets.Option pricing methods are discussed and our experimental results validate the development. Compared tokeyword auctions, a search engine can have an increased expected revenue by selling an ad option.Categories and Subject Descriptors: J.4 [
Computer Applications ]: Social and Behaviour Science – Eco-nomicsGeneral Terms: Theory, Algorithms, ExperimentationAdditional Key Words and Phrases: Sponsored Search, Exotic Option, Pricing Model, Revenue Analysis
1. INTRODUCTION
Sponsored search has become an important online advertising format [PWC 2013],where a search engine sells ad slots in the search engine results pages (SERPs) gener-ated in response to a user’s search behaviour. An online user submits a term or phrasewithin the search box to the search engine. The term or phrase is collectively knownas the query . The SERP has two types of result listings in response to the submittedquery: organic results and paid results.
Organic results are the Web page listings that
Author’s addresses: Bowei Chen and Jun Wang, Department of Computer Science, University CollegeLondon, Gower Street, London, WC1E 6BT, United Kingdom; Ingemar J. Cox, 1). Department of ComputerScience, University of Copenhagen, Sigurdsgade 41, 2200 Copenhagen Ø, Denmark 2). Department ofComputer Science, University College London, Gower Street, London, WC1E 6BT, United Kingdom; MohanS. Kankanhalli, Department of Computer Science, School of Computing, National University of Singapore,Singapore 117417, Republic of Singapore.Permission to make digital or hard copies of part or all of this work for personal or classroom use is grantedwithout fee provided that copies are not made or distributed for profit or commercial advantage and thatcopies show this notice on the first page or initial screen of a display along with the full citation. Copyrightsfor components of this work owned by others than ACM must be honored. Abstracting with credit is per-mitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any componentof this work in other works requires prior specific permission and/or a fee. Permissions may be requestedfrom Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax + (cid:13) DOI: http://dx.doi.org/10.1145/0000000.0000000
ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. a r X i v : . [ c s . G T ] D ec :2 B. Chen et al. most closely match the user’s search query based on relevance [Jansen 2011]. Paid re-sults are online ads – the companies who have paid to have their Web pages displayedfor certain keywords, so such listings show up when an user submits a search querycontaining those keywords. The price of an ad slot is usually determined by a keywordauction such as the widely used generalized second price (GSP) auction [Edelman et al.2007; Varian 2007; Lahaie and Pennock 2007; B¨orgers et al. 2013; Qin et al. 2014]. Inthe GSP auction, advertisers bid on keywords present in the query, and the highestbidder pays the price associated with the second highest bid.Despite the success of keyword auctions, there are two major drawbacks. First, theuncertainty and volatility of bids make it difficult for advertisers to predict their cam-paign costs and thus complicate their business planning [Wang and Chen 2012]. Sec-ond, the “pay-as-you-go” nature of auction mechanisms does not encourage a stablerelationship between advertiser and search engine [Jank and Yahav 2010] – an adver-tiser can switch from one search engine to another in the next bidding at near-zerocost.To alleviate these problems, we propose a multi-keyword multi-click ad option . It isessentially a contract between an advertiser and a search engine. It consists of a non-refundable upfront fee, known as the option price , paid by the advertiser, in returnfor the right, but not the obligation, to subsequently purchase a fixed number of clicksfor particular keywords for pre-specified fixed cost per clicks (CPCs) during a spec-ified period of time. From the advertiser’s perspective, fixing the CPCs significantlyreduces the uncertainty in the cost of advertising campaigns. Moreover, for a keyword,if the spot CPC set by keyword auction falls below the fixed CPC of the option contract,the advertiser is not obligated to exercise the option, but can, instead, participate inkeyword auctions. Therefore, the option can be considered as an “insurance” that es-tablishes an upper limit on the cost of advertising campaigns. From the search engine’sperspective, the proposed option is not only an additional service provided for advertis-ers. We show that the search engine can, in fact, increase the expected revenue in theprocess of selling an ad option. Also, since the option covers a specific period of timeshould encourage a more stable relationship between advertiser and search engine.An important question for us is to determine the option price and the fixed CPCs as-sociated with candidate keywords in the advertiser’s request list. Clearly if the optionis priced too low, then significant loss in revenue to the search engine may ensure.Moreover, this may create an arbitrage opportunity where the buyer of the optionsells the clicks from their targeted keywords to gain extra profits. Conversely, if theoption is priced too high, then the advertiser will not purchase it. In this paper weconsider a risk-neutral environment and price the option under the no-arbitrage ob-jective [Wilmott 2006; Bj¨ork 2009]. We use the Monte Carlo method to price the optionwith multiple candidate keywords and show the closed-form pricing formulas for thecases of single and two keywords. Further, the effects of ad options on the search en-gine’s revenue is analysed.This paper has three major contributions. First, we propose a new way to pre-sell adslots in sponsored search which provides flexible guaranteed deliveries to advertisers.It naturally complements the current keyword auction mechanism and offers both ad-vertiser and search engine an effective risk mitigation tool to deal with fluctuations inthe bid price. Although the proposed ad option belongs to a family of exotic options, itdiffers from existing exotic options that we know from finance and other industries (seeTable I for detailed comparisons): it can be exercised not only once but also multipletimes during the contract period; it is not for a single keyword but multiple keywordsand each keyword has its own fixed CPC; it allows its buyer to choose which keywordto reserve and advertise at the corresponding fixed CPC later during the contract pe-riod. Second, we discuss a generalized pricing method for the proposed ad option (see
ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:3
Algorithm 1) to deal with the high dimensionality. Third, we demonstrate that, com-pared to keyword auctions, a search engine can have an increased expected revenueby selling an ad option.The rest of the paper is organised as follows. Section 2 reviews the related litera-ture. Section 3 introduces the design of proposed ad option, discusses the option pric-ing methods and analyses the option effects on the search engine’s revenue. Section 4presents our empirical evaluation and Section 5 concludes the paper. Several impor-tant mathematical results are provided in Appendices A-C.
2. RELATED WORK
The work presented in this paper touches upon several streams of literature. We firstreview the prior work on options in finance and other industries, and then discuss therelated literature in guaranteed advertising deliveries.
Options have been known and traded for many centuries and can be traced back tothe 17th century [Constantinides and Malliaris 2001]. A standard option is a contractin which the seller grants the buyer the right, but not the obligation, to enter into atransaction with the seller to either buy or sell an underlying asset at a fixed price onor prior to a fixed date. The fixed price is called the strike price and the fixed date iscalled the expiration date . The seller grants this right in exchange for a certain amountof money, called the option price . An option is called the call option or put option de-pending on whether the buyer is purchasing the right to buy or sell the underlyingasset. The simplest option is the European option [Wilmott 2006], which can be ex-ercised only on the expiration date. This differs from the
American option [Wilmott2006], which can be exercised at any time during the contract lifetime. Both Europeanand American options are called standard options .In the beginning of the 1980s, standard options became more widely understood andtheir trading volume increased dramatically. Financial institutions began to searchfor alternative forms of options, known as exotic options [Zhang 1998], to meet theirnew business needs. Among them, two types of options, multi-asset options and multi-exercise options, are particularly relevant to our research.
Multi-asset options are the options written on at least two underlying assets [Zhang1998]. These underlying assets can be stocks, bonds, currencies and indices in eitherthe same category or different markets. Several types of multi-asset options are worthmentioning, such as basket options, dual-strike options, rainbow options, paying thebest and cash options, and quotient options. Table I provides a brief summary of thesemulti-asset options, and compares them to standard options and our proposed multi-keyword multi-click ad options (see Section 3) along the following seven dimensions:payoff function, underlying variable, exercise opportunity, early exercise opportunity,strike price and application area. The comparison indicates that our proposed ad op-tions is more complex than previous proposals.In Table I, it is worth emphasising basket options and dual-strike options.
Basketoptions are those options whose payoff is determined by the weighted sum of un-derlying asset prices [Wilmott 2006]. This structure can be extended to the keywordbroad match setting , where the weights are the probabilities that sub-phrases occur The keyword match type setting helps the search engine to control which searches can trigger an adver-tiser’s ad. Under the exact match setting , the advertiser’s ad may show on searches that are an exact termand close variations of that exact term; Under the broad match setting , the advertiser’s ad may show onsearches that include misspellings, synonyms, other relevant variations and related searches. For furtherdetails, see https://support.google.com/adwords/
ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :4 B. Chen et al. T ab l e I. C o m pa r i s ono ft hep r opo s edadop t i onando t he r op t i on s . T hep r i c eo ft he i t hunde r l y i nga ss e t/ k e y w o r da tt i m e t i s deno t edb y C i ( t ) , w he r e t i s a c on t i nuou s t i m epo i n t i npe r i od [ , T ] and T i s t he c on t r a c t e x p i r a t i onda t e ; i ft he r e i s on l y oneunde r l y i nga ss e t w edeno t e i t s p r i c eb y C ( t ) . T he s t r i k e / fix edpa y m en t p r i c e , o ft he i t hunde r l y i nga ss e t/ k e y w o r d i s deno t edb y F i ; i ft he r e i s on l y one s t r i k ep r i c e w edeno t e i t b y F . T he w e i gh t o f i t ha ss e t/ k e y w o r d i naba sk e t - t y peop t i on i s deno t edb y ω i . N o t e t ha t i n t he n - k e y w o r d - c li ck adop t i on , ω j i r ep r e s en t s t he w e i gh t o ft he i t hb r oad m a t c hed k e y w o r d f o r t he j t h c and i da t e k e y w o r d , and k j r ep r e s en t s t henu m be r o f b r oad m a t c hed k e y w o r d s . D e t a il edde sc r i p t i on s o f no t a t i on s a r ep r o v i ded i n T ab l e II. O p t i o n c o n t r a c t P a y o fff un c t i o n U n d e r l y i n g E x e rc i s e E a r l y S t r i k e A pp li c a t i o n v a r i ab l eo pp o r t un i t y e x e rc i s e p r i c e a r e a n - k e y w o r d - c li c k a d o p t i o n m a x { C ( t ) − F ,..., C n ( t ) − F n , } M u l t i p l e S i n g l e Y e s M u l t i p l e K e y w o r d s ( k e y w o r d e x a c t o r b r o a d m a t c h ) n - k e y w o r d - c li c k a d o p t i o n m a x (cid:26) k (cid:88) i = ω i C i ( t ) − F , ··· , k n (cid:88) i = ω n i C n i ( t ) − F n , (cid:27) M u l t i p l e S i n g l e Y e s M u l t i p l e K e y w o r d s ( k e y w o r d b r o a d m a t c h ) E u r o p e a n s t a n d a r d c a ll o p t i o n m a x { C ( T ) − F , } S i n g l e S i n g l e N o S i n g l e E q u i t y s t o c k , [ W il m o tt ] o r i n d e x A m e r i c a n s t a n d a r d c a ll o p t i o n m a x { C ( t ) − F , } S i n g l e S i n g l e Y e s S i n g l e E q u i t y s t o c k , [ W il m o tt ] o r i n d e x m a x (cid:26) n (cid:88) i = ω i C i ( T ) − F , (cid:27) M u l t i p l e S i n g l e N o S i n g l e I n d e x o f E u r o p e a n e q u i t y s t o c k s ba s k e t c a ll o p t i o n b o n d s o r [ K r e k e l e t a l . ]f o r e i g n c u rr e n c i e s E u r o p e a n m a x { C ( T ) − F , C ( T ) − F , } D o u b l e S i n g l e N o D o u b l e d u a l - s t r i k e c a ll o p t i o n [ Zh a n g ] E q u i t y s t o c k s , E u r o p e a n m a x { m a x { C ( T ) ,..., C n ( T ) } − F , } M u l t i p l e S i n g l e N o S i n g l eo r i n d e x e s o f r a i n b o w c a ll o n m a x o p t i o n e q u i t y s t o c k s , [ O u w e h a n d a n d W e s t ] o r b o n d s , o r E u r o p e a n m a x { C ( T ) , C ( T ) , F } D o u b l e S i n g l e N o S i n g l e f o r e i g n p a y i n g t h e b e s t a n d c a s h o p t i o n c u rr e n c i e s [ J o hn s o n ] E u r o p e a n q u o t i e n t c a ll o p t i o n m a x { C ( T ) / C ( T ) − F , } D o u b l e S i n g l e N o S i n g l e [ Zh a n g ] ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:5 in search queries.
Dual-strike options are options with two different strike prices fortwo different underlying assets [Zhang 1998]. One simple version of our proposed adoptions is a dual-strike call option, which allows an advertiser to switch between histargeted two keywords during the contract lifetime. However, in sponsored search, thenumber 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 needsmore than a single click for guaranteed delivery, the dual-strike call option is extendedto a multi-exercise option.
Multi-exercise options are a generalisation of American options, which provide abuyer with more than one exercise right and sometimes control over one or more othervariables [Villinski 2004], e.g., the amount of the underlying asset exercised in cer-tain time periods. Multi-exercise options have become more prevalent over the pastdecade, particularly, in the energy industry, such as electricity swing options and wa-ter options. Contributors to the multi-exercise options include Deng [2000], Deng andOren [2006], Clewlow and Strickland [2000], Villinski [2004], Weron [2006], Marshallet al. [2011] and Marshall [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 energyindustry, 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 exercisetime 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 minimumclicks. 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 [Shreve 2004]) was first introducedby Bachelier [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, Samuelson [1965]replaced Bachelier’s assumptions on asset price with a geometric form, called the geo-metric Brownian motion (GBM) . In the GBM model, the proportional price changes areexponentially 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 modelagainst which any alternative dynamics are judged.The research of Samuelson highly affected Black and Scholes [1973] and Merton[1973], who then examined the option pricing based on a GBM. They constructed aportfolio 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 Blackand Scholes [1973] and Merton [1973] were based on the assumption that investorson the market cannot obtain arbitrage. Therefore, the replicated portfolio is treatedas a self-adjusting process whose least expectation of returns increase at the samespeed as the constant bank interest rate. If considering the constant bank interestrate as a discount factor, the discounted value of the replicated portfolio would be amartingale [Bj¨ork 2009], whose probability measure is called the risk-neutral proba-bility measure . Since a closed-form pricing formula can be obtained from the settingsof Black and Scholes [1973] and Merton [1973], we normally call their work as the
Black-Scholes-Merton (BSM) option pricing formula . The BSM option pricing formulaspurred research in this field. Various numerical procedures then appeared, includinglattice methods, finite difference methods and Monte Carlo methods. These numericalprocedures are capable of evaluating more complex options when the closed-form so-lution does not exist. In this paper, the Monte Carlo method we discussed can quicklyprice an ad option where the number of candidate keywords is larger than two.
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Guaranteed contracts appeared in the early stages of online advertising (particularlyin display advertising). They were mostly negotiated by advertisers and publishers privately [Edelman et al. 2007]. Each negotiation contains an amount of needed dis-play impressions over a certain period of time and a pre-specified guaranteed price.Hence, in discussing guaranteed deliveries, the following issues must be considered:allocation and pricing. Many studies discussed these two issues separately. Allocationmodels is reviewed first, then the pricing models.Feldman et al. [2009] studied an ad selection algorithm for a publisher whose ob-jective is not only to fulfil the guaranteed contracts but also to deliver the well-targeted display impressions to advertisers. This research was more relevant to aservice matching problem. The allocation of impressions between the guaranteed andnon-guaranteed channels was first discussed by Ghosh et al. [2009], where a publisherwas considered to act as a bidder who bids for guaranteed contracts. This modellingsetting was reasonably good as the publisher acts as a bidder who would allocate im-pressions to online auctions only when other winning bids are high enough. Balseiroet al. [2011] investigated the same allocation problem but used some stochastic controlmodels. In their model, for a given price of an impression, the publisher can decidewhether to send it to ad exchanges or assign it to an advertiser with a fixed reserveprice. The decision making process aims to maximise the expected total revenue. Roelsand Fridgeirsdottir [2009] proposed a similar allocation framework to Balseiro et al.[2011], where the publisher can dynamically select which guaranteed buy requeststo accept and to deliver the guaranteed impressions accordingly. However, comparedto Balseiro et al. [2011], the uncertainty in advertisers’ buy requests and the traffic of awebsite were explicitly modelled under the revenue maximisation objective. Recently, alightweight allocation framework was proposed by Bharadwaj et al. [2012]. They useda simple greedy algorithm to simplify the computations of revenue maximisation.Two algorithms for pricing the guaranteed display contracts were discussedby Bharadwaj et al. [2010]. Each contract has a large number of impressions and theproposed algorithms solved the revenue optimisation problem for the given number ofuser visits (i.e., the demand level). However, their work did not consider the auctioneffects on the contract pricing, and the developed algorithms were purely based on thestatistics of users’ visits.Consider the case where the online advertising market is bouyant (i.e., the winningpayment prices for specific ad slots from online auctions increase) and non-guaranteedselling becomes more profitable for publishers. In this case, they may want to cancelthe sold guaranteed contracts before the time that the targeted impressions will becreated. Online auctions with cancellations were recently discussed by Babaioff et al.[2009] and Constantin et al. [2009]. They both considered a design where a publishercan cancel the sold guaranteed contracts but needs to pay a penalty to the advertisers.The proposed auctions with cancellations exhibit interesting economic properties, suchas allocative efficiency and equilibrium solution. However, there may exist speculatorswho pursue the cancellation penalty only. In fact, the discussed cancellation penalty isvery similar to over-selling of flight tickets [Talluri and van Ryzin 2005].Salomatin et al. [2012] studied a framework of guaranteed deliveries for sponsoredsearch, under which advertisers are able to send their guaranteed requests to a searchengine. Each guaranteed request includes the needed number of clicks and the ad bud-get. The search engine then decides guaranteed deliveries according to search queriesand available positions. Since the allocation decision is based on the joint revenue max- Publishers are sellers in display advertising.
ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:7 imisation from guaranteed deliveries and keyword auctions, some advertisers may notreceive all their demanded clicks. In such cases, the search engine pays a penalty. How-ever, advertisers still have less control of the ad exposure time and the position of thead. In addition, with the number of guaranteed advertisers increasing, it is less likelythat advertisers can meet their business needs with such a mechanism.The concept of ad option was initially introduced by Moon and Kwon [2010] (eventhough Meinl and Blau [2009] discussed the possibility of Web service derivatives,their proposal was not intended for online advertising). Moon and Kwon [2010] pro-posed that the ad option buyer can be guaranteed the right to choose the minimumpayment between cost-per-mille (CPM) and CPC once click-through rate (CTR) is re-alized. This option structure was similar to a paying the worst and cash option [Zhang1998]. In addition, Moon and Kwon [2010] suggested option pricing under the frame-work of a Nash bargaining game. Simply, they considered two utility functions: one forthe advertiser and one for the publisher. The objective function is the product of thesetwo utilities and each utility function is restricted by a negotiation power. Therefore,the option price is the optimal solution which maximises the negotiated join utility.Another ad option was discussed by Wang and Chen [2012] (and later Chen and Wang[2014]) for display advertising. The option allows its buyer to select his preferred pay-ment scheme (either CPM or CPC) for the fixed payment. For example, an advertisercan choose to pay a fixed CPC for targeted display impressions. They discussed the lat-tice methods for option pricing and investigated the stochastic volatility (SV) model forthe cases where the GBM assumption is not valid empirically. However, their work waslimited to an univariate case as the SV model cannot be easily extended to multiplevariables based on the lattice framework.
3. MULTI-KEYWORD MULTI-CLICK AD OPTIONS
We first introduce how a multi-keyword multi-click ad option works, then discuss theoption pricing methods, and finally provide an analysis of the search engine’s revenue.
We use the following example to illustrate our idea. Suppose that a computer sciencedepartment creates a new master degree programme on ‘Web Science and Big DataAnalytics’ and is interested in an advertising campaign based around relevant searchterms such as ‘MSc Web Science’, ‘MSc Big Data Analytics’ and ‘Data Mining’, etc. Thecampaign is to start immediately and last for three months and the goal is to generateat least 1000 clicks on the ad which directs users to the homepage of this new masterprogramme. The department (i.e., advertiser) does not know how the clicks will bedistributed among the candidate keywords, nor how much the campaign will cost ifbased on keyword auctions. However, with the ad option, the advertiser can submita request to the search engine to lock-in the advertising cost. The request consists ofthe candidate keywords, the overall number of clicks needed, and the duration of thecontract. The search engine responds with a price table for the option, as shown inFigure 1. It contains the option price and the fixed CPC for each keyword. The CPCsare fixed yet different across the candidate keywords. The contract is entered intowhen the advertiser pays the option price.During the contract period [0 , T ] , where T represents the contract expiration date(and is three months in this example), the advertiser has the right, at any time, toexercise portions of the contract, for example, to buy a requested number of clicks fora specific keyword. This right expires after time T or when the total number of clickshave been purchased, whichever is sooner. For example, at time t ≤ T the advertisermay exercise the right for 100 clicks on the keyword ‘MSc Web Science’. After receiv-ing the exercise request, the search engine immediately reserves an ad slot for the ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :8 B. Chen et al.
Fig. 1 . Schematic view of buying, selling and exercising a multi-keyword multi-click ad option in sponsoredsearch.
ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:9 keyword for the advertiser until the ad is clicked on 100 times. In our current design,the search engine decides which rank position the ad should be displayed as long asthe required number of clicks is fulfilled - we assume there are adequate search im-pressions within the period. It is also possible to generalise the study in this paperand define a rank specific option where all the parameters (CPCs, option prices etc.)become rank specific. The advertiser can switch among the candidate keywords andalso monitor the keyword auction market. If, for example, the CPC for the keyword‘MSc Web Science’ drops below the fixed CPC, then the advertiser may choose to par-ticipate in the auction rather than exercise the option for the keyword. If later in thecampaign, the spot price for the keyword ‘MSc Web Science‘’ exceeds the fixed CPC,the advertiser can then exercise the option.Figure 1 illustrates the flexibility of the proposed ad option. Specifically, (i) the ad-vertiser does not have to use the option and can participate in keyword auctions aswell, (ii) the advertiser can exercise the option at any time during the contract period,(iii) the advertiser can exercise the option up to the maximum number of clicks, (iv) theadvertiser can request any number of clicks in each exercise provided the accumulatednumber of exercised clicks does not exceed the maximum number, and (v) the adver-tiser can switch among keywords at each exercise at no additional cost. Of course, thisflexibility complicates the pricing of the option, which is discussed next.
The proposed multi-keyword multi-click ad option enables an advertiser to fix his ad-vertising cost and construct a set of candidate keywords beforehand, yet leave thedecision of selecting suitable keywords for matching and the exact timing to place thead to later. Since the advertiser enjoys great flexibility in sponsored search, there is anintrinsic value associated with an ad option and the buyer needs to pay an upfront op-tion price first. In the following discussion, we focus on calculating a fair upfront optionprice for the given option candidate keywords, the current winning payment prices, thevolatility of these keywords, the length of contract period, the risk-less bank interestrate, and the fixed CPCs for candidate keywords. Note that the fixed CPCs are con-sidered as given variables as they can be set by the search engine after receiving theadvertiser’s request or be proposed by the advertiser in his request. Either case willnot affect our valuation of the option. We follow the scenario of the motivating examplepresented in Figure 1 and consider the search engine sets the fixed CPCs.Recall that Table I presents two different payoff functions for the proposed ad option.The first payoff function can be used to price an ad option with either the keyword exactor broad match setting, which is determined by what is the match type of the winningpayment prices used. However, if only having the exact match winning payment pricesfrom keyword auctions and the advertiser wants to have an ad option with keywordbroad match setting, the second payoff function can be used for option pricing. In thefollowing, we discuss the option pricing based on the first payoff function. Same methodcan be applied to the second payoff function, for further details see Section 3.2.4.
The winning payment CPC of the candidate keyword K i (for a specific slot/position) at time t is denoted by C i ( t ) . Its movement can be de-scribed by a multivariate geometric Brownian motion (GBM) [Samuelson 1965]: dC i ( t ) = µ i C i ( t ) dt + σ i C i ( t ) dW i ( t ) , i = 1 , . . . , n, (1) ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :10 B. Chen et al.
Table II. Summary of notations.
Notation Description r Constant continuous (risk-less) interest rate. T Option expiration date. t Continuous time point in [0 , T ] . m Number of total clicks specified by an ad option. n Number of total number of keywords specified by an ad option. K Keywords specified by an ad option, K = { K , . . . , K n } . F Pre-specified fixed CPCs for keywords K . C ( t ) Winning payment CPCs for keywords K from auctions at time t . V ( t, C ( t ); T, F , m ) Value of an n -keyword m -click ad option at time t . µ i Constant drift of CPC for keyword K i , i = 1 , . . . , n . σ i Constant volatility of CPC for keyword K i , i = 1 , . . . , n . W ( t ) Standard Brownian motion at time t . Σ Price correlation matrix, in which ρ ij is the correlation coefficient betweenkeywords K i and K j , such that ρ ii = 1 and ρ ij = ρ ji . M Σ M Price covariance matrix, where M is the matrix with σ i along the diagonaland zeros everywhere else. Φ( C ( t )) Payoff function of an ad option at time t . π Option price (i.e., upfront fee) of an ad option. N ( µ, σ ) Normal distribution with mean µ and variance σ . MV N ( µ , M Σ M ) Multivariate normal distribution with mean µ and variance M Σ M . N [ · ] Cumulative probability distribution of a standard normal distribution. where µ i and σ i are constants representing the drift and volatility of the CPC respec-tively, and W i ( t ) is a standard Brownian motion satisfying the conditions: E ( dW i ( t )) = 0 , var( dW i ( t )) = E ( dW i ( t ) dW i ( t )) = dt, cov( dW i ( t ) , dW j ( t )) = E ( dW i ( t ) dW j ( t )) = ρ ij dt, where ρ ij is the correlation coefficient between keywords K i and K j , such that ρ ii = 1 and ρ ij = ρ ji . The correlation matrix is denoted by Σ , so that the covariance matrixis simply M Σ M , where M is the matrix with the σ i along the diagonal and zeroseverywhere else. For the reader’s convenience, detailed descriptions of notations areprovided in Table II.Since the GBM assumption lays the foundation of pricing the proposed ad option, weprovide several discussions and investigations of it. In Section 3.2.4, we explain whythe GBM assumption is suitable for pricing an ad option in sponsored search, and alsohighlight its limitations. In Section 4.2, we discuss the estimation of GBM parameters.In Section 4.3, we conduct goodness-of-fit tests with real datasets and track the “errors”of the calculated option price when the GBM assumption is not valid empirically. To simplify the discussion and without loss of generality,the value of an n -keyword m -click ad option can be decomposed as the sum of m inde-pendent n -keyword -click ad options. If an advertiser buys an ad option at time , theoption price π can be expressed as follows π = V (0 , C (0); T, F , m ) = mV (0 , C (0); T, F , , (2)where V (0 , C (0); T, F , m ) represents the option value at time .Our focus now centres on the n -keyword -click ad option. Adopting the basic eco-nomic setting [Narahari et al. 2009], we assume that an advertiser is risk-neutral. Inother words, he has no preference across the candidate keywords and exercises the op-tion for the keyword which has the maximum difference between its winning paymentprice and the pre-specified fixed price. This difference shows the value of the optionbecause the advertiser is offered the right to move from the auction market to theguaranteed market. ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:11
Let us first consider if the advertiser exercises the option at the contract expirationdate T , the option payoff can be defined as follows Φ( C ( T )) = max { C ( T ) − F , . . . , C n ( T ) − F n , } . (3)Note that the option payoff in sponsored search does not mean the direct reward but itmeasures the difference of advertising cost between the auction market and the guar-anteed market. By having Eq. (3), we can see if the advertiser would like to exercisethe option early by using the backward deduction method. The option value at timetime t < T is then V ( t, C ( t ); T, F ,
1) = (cid:40) Φ( C ( t )) , if early exercise , E Q t (cid:2) e − r ( T − t ) Φ( C ( T )) (cid:3) , if not early exercise , where r is the constant risk-less bank interest rate and E Q t [ · ] is the conditional expec-tation with respect to time t under the probability measure Q . As we use the risk-lessbank interest rate as the discounted factor, the probability measure Q is also called the risk-neutral probability measure [Bj¨ork 2009]. Appendix B discusses the rationale forusing the risk-less bank interest rate and introduces an alternative method of optionpricing.Let us now return to the decision making problem. If the ad option is exercisedearly at time t , the option value is equal to its payoff Φ( C ( t )) . However, if the ad op-tion is not exercised, the option value at time t is equal to the discounted value ofthe expected payoff at the expiration date T . The comparison between Φ( C ( t )) and E Q t (cid:2) e − r ( T − t ) Φ( C ( T )) (cid:3) informs the optimal decision for the advertiser. Since the payofffunction defined is convex, we then obtain the following inequality (see Appendix A): Φ( C ( t )) ≤ E Q t (cid:2) e − r ( T − t ) Φ( C ( T )) (cid:3) . (4)Eq. (4) illustrates, to gain the maximum option value, the advertiser will not exercisethe option until its expiration date. Hence, the option price should be computed atthe discounted value of the expected payoff from the expiration date T . Together withEq. (2), we can obtain the option pricing formula for the n -keyword m -click ad option: π = me − rT E Q (cid:2) Φ( C ( T )) (cid:3) . (5)It is worth noting that we rule out arbitrage [Varian 1987] between the auctionmarket and the guaranteed market in option pricing. The concept of arbitrage canbe understood as the “free lunch”. As a market designer, we need to make sure thateveryone obtains something by paying something so that it is fair to both the buyand sell sides. Since we assume that an advertiser is risk-neutral, the risk-less bankinterest rate can be employed as the benchmark rate to rule out arbitrage. Eq. (5) canalso be obtained by constructing an advertising strategy for the advertiser as discussedin Appendix B. Eq. (5) can be expanded in integral form as follows π = me − rT (cid:0) πT (cid:1) − n | Σ | − (cid:32) n (cid:89) i =1 σ i (cid:33) − × (cid:90) ∞ · · · (cid:90) ∞ Φ( (cid:101) C ) (cid:81) ni =1 (cid:102) C i exp (cid:26) − ζ T Σ − ζ (cid:27) d (cid:101) C , (6)where ζ = ( ζ , . . . , ζ n ) (cid:48) , ζ i = σ i √ T (cid:0) ln { (cid:101) C i /C i (0) } − ( r − σ i ) T (cid:1) , and other notations aredescribed in Table II. ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :12 B. Chen et al.
Algorithm 1
Pricing a multi-keyword multi-click ad option via Monte Carlo simula-tions. Detailed description of notations is provided in Table II. function
OptionPricingMC ( K , C (0) , Σ , M , m, r, T ) for k ← to (cid:101) n do (cid:101) n is the number of simulations; [ z ,k , . . . , z n,k ] ← GeneratingMultivariateNoise ( M V N [0 , M Σ M ]) for i ← to n do C i,k ← C i (0) exp (cid:110) ( r − σ i ) T + σ i z i,k √ T (cid:111) . end for G k ← Φ([ C ,k , . . . , C n,k ]) . end for π ← me − rT E [Φ( C ( T ))] ≈ me − rT (cid:16) (cid:101) n (cid:80) (cid:101) nk =1 G k (cid:17) . return π end function Closed form solutions to Eq. (6) can be derived if n ≤ . If n = 1 , Eq. (6) is equivalentto the Black-Scholes-Merton (BSM) pricing formula for an European call option [Blackand Scholes 1973; Merton 1973]. If n = 2 , Eq. (6) contains a bivariate normal distri-bution and the option price can be obtained by employing the pricing formula for adual-strike European call option [Zhang 1998]. The closed form solutions are providedin Appendix C.For n ≥ , taking integrals in Eq. (6) is computationally difficult. In such a case,we resort to numerical techniques to approximate the option price. Algorithm 1 illus-trates our Monte Carlo method. For (cid:101) n number of simulations, for each simulation, wegenerate a vector of multinormal noise and then calculate the CPCs at time T . Eq. (4)shows that there is no need to generate the whole paths in each simulation as we onlyconsider the CPCs on the expiration date in the calculation of option payoff. Hence, byhaving (cid:101) n payoffs at time T , the option price π can be then approximated numerically.We refer to this as Algorithm 1. The candidate keywords’ prices may not follow the GBM assump-tion empirically because some time series features, such as jumps and volatility clus-tering, cannot be captured effectively by a GBM [Marathe and Ryan 2005]. However,the GBM model is still a good choice for pricing ad options in sponsored search. First,in our data analysis (see Section 4.3.1), we find that 15.73% keywords’ CPCs satisfythe GBM assumption. Second, for the cases where the GBM assumption is not validempirically (see Section 4.3.2), we find that the pricing model is reasonably robust asthe identified arbitrage values in many experimental groups are small. Of course, ourdataset might be biased. However, other previous research in keyword auctions sup-port the GBM assumption: Lahaie and Pennock [2007] tested the log-normality of bidson Yahoo! search advertising data and gave the estimated distribution parameters;Ostrovsky and Schwarz [2011] performed experiments based on the log-normal bidson Yahoo! search advertising platform; Pin and Key [2011] observed random bids fromMicrosoft Bing search platform and simulated similar bids based on the log-normaldistribution. Since in these research the advertisers’ bids are tested across auctions,the winning payment prices (i.e., the second-highest bids from auctions) over timealso satisfy the log-normal distribution. Recall that in the GBM model, the differencebetween two logarithms of winning payment prices follows a time dependent normaldistribution. If we consider the average daily winning payment price as the underlyingvariable, these previous work can provide the distribution hypothesis tests to supportthe GBM assumption in sponsored search. However, for display advertising, the GBM
ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:13 assumption is usually not valid empirically, and this has been recently investigatedby Chen et al. [2014] and Yuan et al. [2014].Table I shows that if only having the exactly matched C ( T ) , we can still construct abroad match structure for the option. Similar to Eq. (3), the option payoff function ontime T can be defined as follows Φ( C ( T )) = max (cid:26) k (cid:88) i =1 ω i C i ( T ) − F , · · · , k n (cid:88) i =1 ω ni C ni ( T ) − F n , (cid:27) . (7)where ω ji is the probability that the i th broad matched keyword (i.e., the sub-phraseoccurs in search queries) for the keyword K j , and k j represents the number of broadmatched keywords. Eq. (1) can be still used to model the underlying CPCs’ movementand the option price π can be directly calculated by Algorithm 1. The proposed ad option can be considered as an “insurance” for an advertiser. Theadvertiser needs to pay the upfront option price, which contributes to the search en-gine’s revenue. In the following discussion, we analyse the effect of an ad option on thesearch engine’s revenue. We provide a functional analysis for the -keyword -click adoption in this section and leave the empirical investigation of the n -keyword cases toSection 4.Let D ( F ) be the difference between the expected revenue from an ad option and theexpected revenue from only keyword auctions, we then have D ( F ) = (cid:18) C (0) N [ ζ ] − e − rT F N [ ζ ] + e − rT F (cid:19) P ( E Q [ C ( T )] ≥ F ) (cid:124) (cid:123)(cid:122) (cid:125) = Discounted value of expected revenue from option if E Q [ C ( T )] ≥ F + (cid:18) C (0) N [ ζ ] − e − rT F N [ ζ ] + e − rT E Q [ C ( T )] (cid:19) P ( E Q [ C ( T )] < F ) (cid:124) (cid:123)(cid:122) (cid:125) = Discounted value of expected revenue from option if E Q [ C ( T )] 12 ( ζ − ζ ) (cid:27) = C (0) e rT F . (12)Taking the derivative of ζ and ζ with respect to F gives ∂ζ ∂F = ∂ σ √ T (cid:18) ln { C (0) /F } + ( r + σ ) T (cid:19) ∂F = − F σ √ T , (13) ∂ζ ∂F = ∂ζ ∂F − ∂σ √ T∂F = − F σ √ T . (14)and D ( F ) achieves its maximum or minimum value at F = E Q [ C ( T )] . Further, takingthe second derivative of D ( F ) with respect to F = E Q [ C ( T )] gives ∂ D ( F ) ∂F = ∂ P ( E Q [ C ( T )] ≥ F ) ∂F = ∂ N [ ζ ] ∂ζ ∂ζ ∂F = − √ π e − ζ F σ √ T < . Hence, if the fixed CPC is set the same as the estimated spot CPC on the contractexpiration date (i.e., F = E Q [ C ( T )] ), the search engine can increase its profit. ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:15 Table III. Overview of experimental settings of data. Market Group Training set (31 days) Deve&test set (31 days)US 1 25/01/2012-24/02/2012 24/02/2012-25/03/20122 30/03/2012-29/04/2012 29/04/2012-31/05/20123 10/06/2012-12/07/2012 12/07/2012-17/08/20124 10/11/2012-11/12/2012 11/12/2012-10/01/2013UK 1 25/01/2012-24/02/2012 24/02/2012-25/03/20122 30/03/2012-29/04/2012 29/04/2012-31/05/20123 12/06/2012-13/07/2012 13/07/2012-19/08/20124 18/10/2012-22/11/2012 22/11/2012-24/12/2012 4. EXPERIMENTS In this section, we describe our data and experimental settings, conduct assumptionand fairness tests, and investigate the option’s effects on the search engine’s revenue. The data used in the experiments is collected from Google AdWords by using its TrafficEstimation Service [Yuan and Wang 2012]: when an advertiser submits his targetedkeywords, budget, and other settings to Google, the Traffic Estimation Service will re-turn a list of data values, including the estimated CPCs, clicks, global impressions, lo-cal impressions and position. These values are recorded for the period from 26/11/2011to 14/01/2013, for a total of 557 keywords in the US and UK markets. Note that in thedata 21 keywords have missing values and 115 keywords’s CPCs are all 0.For each market, as illustrated in Table III, we split the data into 4 experimentalgroups and each group has one training, one development, and one test set. The train-ing set is used to: (i) select the keywords with non-zero CPCs; (ii) test the statisticalproperties of the underlying dynamic and estimate the model parameters. We thenprice ad options and simulate the corresponding buying and selling transactions inthe development set. Finally, the test set is used as the baseline to examine the pricedad options. The GBM parameters are estimated by using the method suggested by Wilmott [2006].Specifically, for the keyword K i , the volatility σ i is the sample standard deviation ofchange rates of log CPCs and the correlation ρ ij is given by ρ ij = (cid:80) (cid:101) mk =1 (cid:0) y i ( k ) − ¯ y i (cid:1)(cid:0) y j ( k ) − ¯ y j (cid:1)(cid:113)(cid:80) (cid:101) mk =1 (cid:0) y i ( k ) − ¯ y i (cid:1) (cid:80) (cid:101) mk =1 (cid:0) y j ( k ) − ¯ y j (cid:1) , (15)where (cid:101) m is the size of training data and y i ( t k ) is the k th change rate of log CPCs.Figure 2 illustrates an empirical example, where the candidate keywords are K = (cid:40) K K K (cid:41) = (cid:40) ‘canon cameras’‘nikon camera’‘yahoo web hosting’ (cid:41) , and the estimated model parameters are σ = (cid:32) . . . (cid:33) , Σ = (cid:32) . . . . . − . . − . . (cid:33) . Note that a high contextual relevance of keywords normally means that they have ahigh substitutional degree to each other, such as ‘canon cameras’ and ‘nikon camera’,whose CPCs move in the same direction with correlation . . The other keyword ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :16 B. Chen et al. ‘yahoo web hosting’ is contextually less relevant to the formers and also has very lowprice correlations to them. The example also shows that the contextual relevance ofkeywords has an impact on their CPCs movement.Based on the estimated parameters, we draw a sample of simulated paths of a 3-dimensional GBM in Figure 2(a) for 31 days (where the x-axis is expressed in termsof year value). Recall that the option payoff at any time t in the contract lifetime is max { C ( t ) − F , . . . , C n ( t ) − F n , } . In Figure 2(b), we plot the price difference betweenthe spot CPC and the fixed CPC of each candidate keyword (i.e., C i ( t ) − F i , i = 1 , . . . , n )and also indicate the corresponding option daily payoffs (shown by the cyan curve). Itsuggests that switching among keywords would help the advertiser to maximise thebenefits of the ad option. Repeating the above simulations 50 times generates 50 sim-ulated vales of each keyword for each day, as shown in Figure 2(c). We then calculate50 option payoffs and their daily mean values to obtain the final option price, as shownin Figure 2(d).To examine the fairness (i.e., no-arbitrage) of the calculated option price, we can con-struct a risk-less value difference process by delta hedging ∂V /∂C j (see Appendix B)and check if any arbitrage exists [Wilmott 2006]. The hedging delta of the 1-keyword1-click ad option can be calculated as follows ∂V∂C = N (cid:20) σ √ T (cid:18) ln (cid:26) C (0) F (cid:27) + ( r + σ T (cid:19)(cid:21) . (16)For the n -keyword -click option, the hedging delta of each keyword can be com-puted by the Monte Carlo method, i.e., ∂V /∂C i = E Q [ ∂V ( T, C ( T )) /∂C i ( T )] . Accordingto Appendix B, we can define the 31-day growth rate of the value difference process as (cid:101) γ = (cid:16) Π( t ) − Π( t ) (cid:17) / Π( t ) , and compare (cid:101) γ to the risk-less bank interest rate r = 5% (equivalent to (cid:101) r = 4 . per 31 days return ). The arbitrage detection criteria is | (cid:101) γ − (cid:101) r | ≤ ε ? arbitrage does not exist : arbitrage exists , (17)where the notation ε is the model variation threshold (and we set ε = 5% in experi-ments). Hence, a positive (cid:101) γ − (cid:101) r means that the advertiser buys an option can obtainarbitrage while a negative (cid:101) γ − (cid:101) r indicates the case of making arbitrage by selling anoption. Then the identified arbitrage α is defined as the excess return, that is α = (cid:26) (cid:101) γ − ( (cid:101) r − ε ) , if (cid:101) γ < (cid:101) r − ε, (cid:101) γ − ( (cid:101) r + ε ) , if (cid:101) γ > (cid:101) r + ε. (18)Table IV presents the overall results of our arbitrage test based on the GBM model.We generate paths for candidate keywords with 100 simulations and examine the op-tions price using delta hedging. There are 99.76% (1-keyword), 93.06% (2-keyword)and 92.71% (3-keyword) options fairly priced. Only a small number of options exhibitsarbitrage and most of the mean arbitrage values lie within 5%, such as shown in Fig-ure 3. The existence of small arbitrage may be due to two reasons. First, the stabilityof process simulations in both option pricing and arbitrage test. Second, the candi-date keywords are randomly selected for the 2-keyword and 3-keyword options. Thesignificant differences on the absolute prices of these keywords can generates a largevariation of calculated option payoffs, which then trigger arbitrage. The relationship between the continuous compounding r and the return per 31 days (cid:101) r is: (cid:101) r = e r × / [Hull 2009]. ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:17 W i nn i ng p a y m e n t C P C (c) Paths from 50 simulations ’canon cameras’’nikon camera’’yahoo web hosting’0 0.02 0.04 0.06 0.08456789 Time W i nn i ng p a y m e n t C P C (a) Path from 1 simulation ’canon cameras’’nikon camera’’yahoo web hosting’ 0 0.02 0.04 0.06 0.08−0.500.511.522.533.5 Time P r i ce d i ff e r e n ce (d) Option payoffs from 50 simulations Early exerciseHolding valueOption payoff0 0.02 0.04 0.06 0.08−0.500.511.522.533.5 Time P r i ce d i ff e r e n ce (b) Option payoff from 1 simulation ’canon cameras’’nikon camera’’yahoo web hosting’Option payoff Fig. 2 . Empirical example of pricing a 3-keyword 1-click ad option via Monte Carlo method, where K = ‘canon cameras’, K = ‘nikon camera’, K = ‘yahoo web hosting’, F = 3 . , F = 4 . and F =6 . . ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :18 B. Chen et al. Table IV. Test of arbitrage for ad options based on a GBM: n is the numberof candidate keywords, N is the number of options priced in a group, P ( α ) is percentage of options in a group with identified arbitrage, and the E [ α ] is the average arbitrage value of the options, where the arbitrage α isdefined by Eq. (18) and the risk-less bank interest rate r = 5% . n Group US market UK market N P ( α ) E [ α ] N P ( α ) E [ α ] −0.1 0 0.1−0.0500.050.1 Invest return A r b it r a g e (a) n=2 & group 2 −0.05 0 0.05 0.1−0.0500.05 Invest return A r b it r a g e (b) n=2 & group 3 −0.1 0 0.1 0.2 0.3−0.0500.050.10.150.2 Invest return A r b it r a g e (c) n=2 & group 40 0.1 0.200.050.1 Invest return A r b it r a g e (d) n=3 & group 4 −0.05 0 0.05−0.0500.05 Invest return A r b it r a g e (e) n=3 & group 2 Priced optionsRisk−less returnArb threshold Fig. 3 . Empirical example of arbitrage analysis based on GBM for the US market. We now examine the GBM assumption and investigate if arbitrage exists when thecandidate keywords in an option do not follow a GBM. Two validation conditions of the GBMmodel are tested [Marathe and Ryan 2005]: (i) the normality of change rates oflog CPCs; and (ii) the independence from previous data. Normality can be eitherchecked graphically by histogram/Q-Q plot or verified statistically by the Shapiro-Wilktest [Shapiro and Wilk 1965]. To examine independence, we employ the autocorrela-tion function (ACF) [Tsay 2005] and the Ljung-Box statistic [Ljung and Box 1978]. ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:19 Fig. 4 . Empirical example of checking the GBM assumption for the keyword ‘canon 5d’, where the Shapiro-Wilk test is with p -value 0.3712 and the Ljung-Box test is with p -value 0.4555. N u m b e r o f k e y w o r d s Non−GBMGBM 1 2 3 4050100150 (b) UK marketExperimental group ID N u m b e r o f k e y w o r d s Non−GBMGBM Fig. 5 . Overview of checking the GBM assumption for all keywords of experimental groups. Figure 4 provides an empirical example of the keyword ‘canon 5d’. Figure 4 (a)-(b) ex-hibit the movement of CPCs and log change rates while Figure 4 (c)-(d) show that thestated two conditions are satisfied in this case.We check the discussed two conditions with the training data. As shown in Figure 5,there are 14.25% and 17.20% of keywords in US and UK markets that satisfy theGBM assumption, respectively. Thus 15.73% of keywords can be effectively priced intoan option based on a GBM. It is worth mentioning that not all keywords follow a GBM.Next, we examine the robustness of pricing model and investigate the arbitrage basedon non-GMB models. ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :20 B. Chen et al. WilcoxonA−B K−S020406080100120 (a) US marketTesting model S i m il a r it y ( % ) WilcoxonA−B K−S020406080100120 (b) UK marketTesting model S i m il a r it y ( % ) CEVMRDCIRHWV Fig. 6 . Overview of model similarity tests: Wilcoxon test, Ansari-Bradley (A-B) test and Two-sampleKolmogorov-Smirnov (K-S) test. n=1 n=2 n=3020406080100 Number of keywords in ad option F a i r n e ss ( % ) GBMCEVMRDCIRHWV Fig. 7 . Overview of pricing model robust tests. Several popular stochastic processes(together with the real data) are tested to check the arbitrage in option pricing. Ta-ble V shows the candidate models and each model can capture certain features of timeseries data, such as mean-reversion, constant volatility and square root volatility [Hull2009]. The arbitrage tests here are slightly different from that of the GBM model. Weestimate the model parameters from the actual data in the test sets instead of thelearning sets and treat the actual data as one single path of each model. Hence, thesimulated data has the same drift, volatility and correlations as the test data. We arenow able to examine the arbitrage multiple times when the real-world environmentdoes not follow a GBM. Also, for the candidate models, hypothesis tests are used tocheck if the simulated path and actual data come from a same distribution. Thesetests include the Wilcoxon test [Wilcoxon 1945], Ansari-Bradley test [Mood et al. 1974]and Two-sample Kolmogorov-Smirnov test [Justel et al. 1997]. Figure 6 summarisesthe results of models’ goodness-of-fit tests, where the y-axis represents the mean per-centage of simulated paths not rejected by the hypothesis tests. Even though the threetests give different absolute percentages, the dynamics’ performance is similar andconsistent: the CEV model has the best simulations for the actual data, followed bythe MRD; the CIR and HWV models are very close.Table VI presents the results of arbitrage tests for the non-GBM dynamics, wheremost of experimental groups exhibit arbitrage. The CEV model gives the best no-arbitrage performance, showing that 78.65% of CEV-based keywords can be properlypriced by using the GBM-based option pricing model. About 53.05% of the CIR model ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:21 R e v e nu e o r r e v e d i ff e r e n ce (a) GBM 3.8 4 4.2 4.4 4.600.20.40.6 Fixed CPC R e v e nu e o r r e v e d i ff e r e n ce (b) CEV 3.8 4 4.2 4.4 4.600.20.40.6 Fixed CPC R e v e nu e o r r e v e d i ff e r e n ce (c) MRD3.8 4 4.2 4.4 4.600.20.40.6 Fixed CPC R e v e nu e o r r e v e d i ff e r e n ce (d) CIR 3.8 4 4.2 4.4 4.600.20.40.6 Fixed CPC R e v e nu e o r r e v e d i ff e r e n ce (e) HWV Reve differenceOption priceExpected winningpayment CPC Fig. 8 . Empirical example of analysing the search engine’s revenue for the keyword ‘canon cameras’. and about 43% of the MRD or HWV models based options have no arbitrage. For 1-keyword options, the fairness percentage is more than 85% across all experimentalgroups. However, this number drops to around 38% for multi-keyword options (36.27%for 2-keyword options and 42% for 3-keyword options). For the identified arbitrage,many groups (especially 1-keyword options) show small arbitrage values (around 10%)while arbitrage explodes in some groups.In summary, Tables IV and VI illustrate that our option pricing methods are effectiveand reasonably robust for the real sponsored search data. As shown in Figure 7, whenthe keywords’s price follow a GBM (15.73%), the pricing model ensures that 95.17%of ad options are fairly priced under the 5% arbitrage precision. For the non-GBMkeywords, the CEV model is the best performance model, giving 78.65% of fairness; theCIR model is worst performance model and is with only 31.97% of fairness. Overall,the best expected fairness for all keywords is 81.25% while the worst is 41.91%. Wefind that the increase of the number of candidate keywords in an ad option increasesthe likelihood of arbitrage. This is confirmed by the fact that expected fairness dropsfrom 86.83% (99.76% GBM and 83.60% non-GBM for 1-keyword options) to 43.69%(2-keyword options) and 53.39% (3-keyword options), respectively. Let us start with the case of 1-keyword options. The example of keyword ‘canoncameras’ in Figure 8(a) illustrates (other keywords exhibit the similar pattern) theconclusions from our theoretical analysis in Section 3.3 that (i) the revenue differ-ence between option and auction is always positive and (ii) that when the fixed CPC F = E Q t [ C ( T )] , the revenue difference D ( F ) achieves its maximum and the two bound-ary values are approximately zero. ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :22 B. Chen et al. T ab l e V . T e s t ednon - G B M d y na m i cs : k i = . w h il eo t he r pa r a m e t e r s a r e l ea r ned f r o m t he t r a i n i ngda t a . D y n a m i c S t o c h a s t i c d i ff e r e n t i a l e q u a t i o n ( S D E ) C o n s t a n t e l a s t i c i t y o f v a r i a n c e ( CEV ) m o d e l [ C o x a n d R o ss ] d C i ( t ) = µ i C i ( t ) d t + σ i ( C i ( t )) / d W i ( t ) M e a n - r e v e r t i n g d r i f t ( M R D ) m o d e l [ W il m o tt ] d C i ( t ) = k i ( µ i − C i ( t )) d t + σ i ( C i ( t )) / d W i ( t ) C o x - I n g e r s o ll - R o ss ( C I R ) m o d e l [ C o x e t a l . ] d C i ( t ) = k i ( µ i − C i ( t )) d t + ( σ i ) / C i ( t ) d W i ( t ) H u ll - W h i t e / V a s i c e k ( H W V ) m o d e l [ H u ll a n d W h i t e ] d C i ( t ) = k i ( µ i − C i ( t )) d t + σ i d W i ( t ) T ab l e V I. O v e r v i e w o f de l t ahedg i nga r b i t r age t e s t i ng f o r non - G B M d y na m i cs : s a m eno t a t i on s a s i n T ab l e I V . M a r k e t n G r o u p N r e a l d a t a + CEV s i m u r e a l d a t a + M R D s i m u r e a l d a t a + C I R s i m u r e a l d a t a + H W V s i m u P ( a r b ) M e a n a r b P ( a r b ) M e a n a r b P ( a r b ) M e a n a r b P ( a r b ) M e a n a r b U S . % - . % . % - . % . % . % . % - . % . % . % . % . % . % . % . % - . % . % - . % . % - . % . % . % . % - . % . % . % . % . % . % . % . % . % . % . % . % - . % . % - . % . % - . % . % . % . % . % . % . % . % . % . % . % . % - . % . % - . % . % - . % . % . % . % . % . % . % . % . % . % - . % . % - . % . % - . % . % - . % . % - . % . % - . % . % - . % . % - . % . % . % . % - . % . % - . % . % - . % . % . % . % . % . % . % . % . % U K . % . % . % . % . % . % . % - . % . % . % . % . % . % . % . % . % . % - . % . % - . % . % . % . % - . % . % . % . % . % . % . % . % . % . % - . % . % - . % . % - . % . % - . % . % . % . % . % . % . % . % . % . % . % . % - . % . % - . % . % - . % . % . % . % . % . % . % . % . % . % - . % . % - . % . % - . % . % - . % . % . % . % - . % . % - . % . % - . % . % . % . % - . % . % - . % . % - . % . % . % . % . % . % . % . % . % ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:23 (a)Fixed CPC F O p ti on p r i ce (b)Fixed CPC F R e v e nu e d i ff e r e n ce Fig. 9 . Empirical example of analysing the search engine’s revenue for the keywords ‘non profit debt con-solidation’ and ‘canon 5d’, where ρ = 0 . . The non-GBM cases are further examined in Figure 8(b)-(e), which show that whenthe fixed CPC is close to zero, the revenue difference D ( F ) → . This is because whenthe fixed CPC approximates zero, it is almost certain that the option will be used inthe contract period. As such, the only income for the keyword is from the option price,which in this case is close to the CPC in the auction market (discounted back to t =0).On the other hand, if the fixed CPC is very high, it is almost certain that the optionwon’t be used. In this case, the option price π → and the probability of exercisingthe option P ( E Q t [ C ( T )] ≥ F ) → . Hence, D ( F ) is zero. However, under the non-GBMdynamics, the point F = E Q t [ C ( T )] is not the optimal value that gives the maximum D ( F ) , which indicates that arbitrage may occur.Next, Figure 9 illustrates an empirical example a 2-keyword ad option. The candi-date keywords are ‘non profit debt consolidation’ and ‘canon 5d’. Figure 9(a) tells thatthe higher the fixed CPCs the lower is the option price (even though the option priceis less sensitive to the keyword ‘canon 5d’) and it achieves the maximum when all thefixed CPCs are zeros. These monotone results are as same as the 1-keyword options.Figure 9(b) then shows the revenue difference curve of the search engine, where thered star represents the value where F = E Q t [ C ( T )] and F = E Q t [ C ( T )] . The expectedrevenue differences are all non-negative, showing that this 2-keyword ad option is ben-eficial to the search engine’s revenue. However, the red star point is not the maximumdifference revenue. This is different to the 1-keyword ad options.For higher dimensional ad options (i.e., n ≥ ), it is not possible to graphically exam-ine the revenue difference. However, based on the earlier discussions, two findings canbe summarised. First, there are boundary values of the revenue differences. If every F i → , D ( F ) → ; and if every F i → ∞ , D ( F ) → . Second, there exists a maximumrevenue difference value even though this may not at the point where F i = E Q t [ C i ( T )] .Hence, compared to only keyword auctions, proper setting the fixed CPCs can increasethe search engine’s expected revenue. ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :24 B. Chen et al. 5. CONCLUDING REMARKS In this paper, we proposed a novel framework to provide flexible guaranteed deliveriesfor sponsored search, from which both buy and sell sides can benefit. On the buy side,advertisers are able to secure a certain number of clicks from their targeted keywordsin the future and can decide how to advertise later. They can be released from auctioncampaigns and can manage price risk under the given budgets. On the sell side, thesearch engine can sell the future clicks in advance and can receive a more stable andincreased expected revenue over time. In addition, advertisers would be more loyal toa search engine due to the contractual relationships, which has the potential to boostthe search engine’s revenue on the long run.We also believe that the proposed ad options will soon be welcomed by the sponsoredsearch market. Several similar but different developments appeared in the displaydigital markets are able to support our point of view. They are: AOL’s Programmatic Upfront . OpenX Programmatic Guarantee [OpenX 2013]. Adslot Media’s Programmatic Direct Media Buying . Shiny Ads Programmatic Direct Advertising Platform . iSOCKET’s Programmatic Direct .Our work differs to the above developments in many aspects. First, we focus on spon-sored search while they are for display advertising. Second, the proposed ad optionsprovide flexible guaranteed deliveries (e.g., multi-keyword targeting, multi-click exer-cise, early exercise, no obligation of exercise) while other recent developments do notprovide such features.Our work leaves several directions for future research. First, to address the limita-tions of GBM, other stochastic processes tailored to some specific keywords are worthstudying, such as the jump-diffusion model [Kou 2002] and the stochastic volatilitymodel [Chen and Wang 2014]. 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PROOF OF THE NO-EARLY EXERCISE PROPERTY FOR THE PROPOSED AD OPTION Eq. (3) can be rewritten as Φ( x ) = max { x − f , } , where x (cid:48) = [ x , . . . , x n ] and f (cid:48) =[ f , . . . , f n ] . It is not difficult to find that Φ( x ) is multivariate convex. Let ≤ λ ≤ andlet y (cid:48) = [ y , . . . , y n ] , if the elements of vector a = y − x are all non-negative, then Φ (cid:0) λ x + (1 − λ ) y (cid:1) ≤ λ Φ( x ) + (1 − λ )Φ( y ) . ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :28 B. Chen et al. If taking y (cid:48) = (0 , . . . , , and using the fact that Φ( ) = 0 , we obtain Φ( λ x ) ≤ λ Φ (cid:0) x (cid:1) , for all x i ≥ , ≤ λ ≤ . For ≤ s ≤ t ≤ T , since ≤ e − r ( t − s ) ≤ , we then have E Q s (cid:2) e − r ( t − s ) Φ (cid:0) X ( t ) (cid:1)(cid:3) ≥ E Q s (cid:2) Φ (cid:0) e − r ( t − s ) X ( t ) (cid:1)(cid:3) ≥ Φ (cid:0) E Q s (cid:2) e − r ( t − s ) X ( t ) (cid:3)(cid:1) ( by the Jenen’s Inequality )= Φ (cid:0) e rs E Q s (cid:2) e − rt X ( t ) (cid:3)(cid:1) , where E Q s [ · ] is the conditional expectation with respect to time s under the risk-neutralprobability measure Q . Since e − rt X ( t ) is a martingale under Q [Bj¨ork 2009], then Φ (cid:0) e rs E Q s (cid:2) e − rt X ( t ) (cid:3)(cid:1) = Φ (cid:0) e rs e − rs X ( s ) (cid:1) = Φ (cid:0) X ( s ) (cid:1) . Hence, E Q s (cid:2) e − r ( t − s ) Φ (cid:0) X ( t ) (cid:1)(cid:3) ≥ Φ (cid:0) X ( s ) (cid:1) , showing that e − rt Φ (cid:0) X ( t ) (cid:1) is a sub-martingaleunder Q . This tells that the proposed ad option can be priced as same as its Europeanstructure, focusing on the payoff on the contract expiration date. For further detaileddiscussions about martingale and sub-martingale, please see [Bj¨ork 2009]. B. DERIVATION OF THE AD OPTION PRICING FORMULA Since the proposed ad option complements the existing keyword auctions, there mayexist a situation that some advertisers only want to make guaranteed profits from thedifference of costs between option and auction markets without taking any risk. Thissituation is called arbitrage [Varian 1987; Bj¨ork 2009]. Hence, we must fairly evaluatethe 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 . Then at time t , t ∈ [0 , T ] , the difference between the optionvalue and the market value of candidate keywords can be expressed as Π( t ) = V ( t, C ( t ); F , T, m ) − n (cid:88) i =1 ψ i ( t ) C i ( t ) , (19)where ψ i ( t ) represents the number of clicks needed for the keyword K i such that (cid:80) i ψ i ( t ) = m . Here we call Π( t ) as the value difference process . Recall that in Eq. (3),we consider the value of an n -keyword m -click option as the sum of m independent n -keyword -click options, for the mathematical convenience, Eq. (19) can be rewrittenas follows Π( t ) = m (cid:32) V ( t, C ( t ); F , T, − n (cid:88) i =1 ∆ i C i ( t ) (cid:33) , (20)where ∆ i represents the probability that a single click goes for the keyword K i and (cid:80) ni =1 ∆ i = 1 . The changes of Π over a sufficient small period of time dt is then d Π( t ) = m (cid:32) ∂V∂t dt + 12 n (cid:88) i =1 n (cid:88) j =1 σ i σ j ρ ij C i C j ∂ V∂C i ∂C j dt + n (cid:88) i =1 ∂V∂C i dC i − n (cid:88) i =1 ∆ i dC i (cid:33) . (21)The uncertain components in d Π( t ) can be removed if ∆ i = ∂V /∂C i . This is calledthe delta hedging in option pricing theory [Wilmott 2006]. Hence, Π( t ) now becomes arisk-less process over time d Π( t ) = m (cid:32) ∂V∂t + 12 n (cid:88) i =1 n (cid:88) j =1 σ i σ j ρ ij C i C j ∂ V∂C i ∂C j (cid:33) dt. (22) ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. ulti-Keyword Multi-Click Advertisement Option Contracts for Sponsored Search 5:29 We assume that the advertiser has no initial fund and he borrows the money fromothers at the risk-less bank interest rate r , so the interest of this borrowing is d Π( t ) = r Π( t ) dt = rm (cid:32) V − n (cid:88) i =1 ∂V∂C i C i (cid:33) dt. (23)Eqs. (22)-(23) need to be equal otherwise arbitrage exists. If the risk-less growthrate of the value difference process is larger than the risk-less bank interest rate, theadvertiser can obtain arbitrage by: (i) borrowing the money from bank at interest rate r to buy an ad option first; (ii) selling the ad option later to repay the bank interest.In the case where the risk-less growth rate of the value difference process is smallerthan the risk-less bank interest rate, the advertiser can obtain the risk-less surplusby: (i) selling short an ad option first and saving the revenue in a bank account; (ii)using the deposit money to buy the clicks of underlying keywords later. In either case,the advertiser can finally receive a risk-less surplus; therefore, arbitrage exists.Solving Eqs. (22)-(23) can give a parabolic partial differential equation (PDE) for theno-arbitrage equilibrium: ∂V∂t + r n (cid:88) i =1 ∂V∂C i C i + 12 n (cid:88) i =1 n (cid:88) j =1 ∂ V∂C i ∂C j σ i σ j ρ ij C i C j − rV = 0 . The above PDE satisfies the boundary condition in Eq. (3). By employing the multidi-mensional Feynman-Ka˘c stochastic representation [Bj¨ork 2009], we obtain the solution V ( t, C ( t ); F , T, 1) = e − r ( T − t ) E Q t [Φ( C ( T ))] , where E Q t [ · ] is the conditional expectation with respect to time t under the risk-neutralprobability Q . The process C i ( t ) can be rewritten as dC i ( t ) = rC i ( t ) dt + σ i C i ( t ) dW Q i ( t ) , where W Q i ( t ) is the standard Brownian motion under Q . Therefore, the option price π can be calculated by the following formula: π = V (0 , C (0); F , T, m ) = mV (0 , C (0); F , T, 1) = me − rT E Q [Φ( C ( T ))] . C. OPTION PRICING FORMULAS FOR SPECIAL CASES If n = 1 , Eq. (6) is equivalent to the Black-Scholes-Merton (BSM) pricing formula foran European call option [Black and Scholes 1973; Merton 1973]. Then we have π = mC (0) N [ ζ ] − mF e − rT N [ ζ ] , (24)where ζ = σ √ T (cid:0) ln { C (0) /F } + ( r + σ ) T (cid:1) and ζ = ζ − σ √ T .If n = 2 , Eq. (6) contains a bivariate normal distribution. Hence, we can calculatethe option price as same as the dual-strike European call option [Zhang 1998]: π = mC (0) (cid:90) ζ + σ √ T −∞ f ( u ) N (cid:20) q ( u + σ √ T ) − ρσ √ T + ρu (cid:112) − ρ (cid:21) du + mC (0) (cid:90) ζ + σ √ T −∞ f ( v ) N (cid:20) q ( u + σ √ T ) − ρσ √ T + ρv (cid:112) − ρ (cid:21) dv − me − rT (cid:32) F (cid:90) ζ −∞ f ( u ) N (cid:20) q ( u ) + ρu (cid:112) − ρ (cid:21) du + F (cid:90) ζ −∞ f ( v ) N (cid:20) q ( v ) + ρv (cid:112) − ρ (cid:21) dv (cid:33) , (25) ACM Transactions on Intelligent Systems and Technology, Vol. 7, No. 1, Article 5, Publication date: October 2015. :30 B. Chen et al. where q ( u ) = 1 σ √ T (cid:32) ln (cid:26) F − F + C (0) e ( r − σ ) T − uσ √ T C (0) (cid:27) − ( r − σ ) T (cid:33) ,q ( u ) = 1 σ √ T (cid:32) ln (cid:26) F − F + C (0) e ( r − σ ) T − vσ √ T C (0) (cid:27) − ( r − σ ) T (cid:33) ,ζ = 1 σ √ T (cid:32) ln { C (0) /F } + ( r − σ ) T (cid:33) ,ζ = 1 σ √ T (cid:32) ln { C (0) /F } + ( r − σ ) T (cid:33) ..