Option Pricing for Inventory Management and Control

Transcription

1 29 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 1-12, 29 ThB7.3 Option Pricing for Inventory Management an Control Bryant Angelos, McKay Heasley, an Jeffrey Humpherys Abstract We explore the use of option contracts as a means of managing an controlling inventories in a retail market. Specifically, merchants can buy option contracts on unsol inventories of retail goos in an effort to hege, pool, or transfer risk. We propose a new kin of European put option on an inventory where the holer is allowe to freely ajust the original sale price of the unerlying goo throughout the contract perio as a means of controlling eman. Assuming the retailer will chose the profit maximizing pricing policy, we can price the option accoringly. I. INTRODUCTION The capital markets have been recently scrutinize in the press, congressional hearings, talk shows, etc., ue in part to the active traing of highly complex an yet poorly unerstoo over-the-couter erivatives an other investment vehicles use to transfer cash flows an risks in the consumer ebt an mortgage inustries. The recent financial crisis an global recession cause by large-scale holings of toxic paper involving collateralize ebt obligations, mortgagebacke securities, an subprime mortgages, emonstrates that we still have much to learn about financial engineering an the proper quantification of risk an rewar. Nonetheless, espite the ever-growing list of bluners an ebacles, businesses in the aggregate enjoy increasingly greater access to investment capital, less exposure to market risk, increase liquiity, an higher prouctivity as a result of erivative securities an investment banking [5], [7]. In the retail markets, merchants hol inventories of goos an services in a manner not terribly ifferent than investors holing portfolios of investment securities. Inee an investment portfolio can be viewe as an inventory an vice versa. Our goal is to ientify the commonalities between these two areas an help tie these two inustries together, at least mathematically, uner the common umbrella of systems theory. There are some obvious ifferences between the capital an retail markets. One notable example is that the former is highly instrumente an transactions are usually cleare centrally by financial intermeiaries on a traing floor, whereas the latter has willy varying traing practices an corresponing risks, thus resulting in relatively high transaction costs, bran effects, an geographically localize points of sale. Nonetheless as the information age continues to evolve an consumer purchases become more electronic, we may expect to see a similar commoitization in the retail markets over time. There are alreay web services such Bryant Angelos an McKay Heasley are research assistants uner the avisement of Jeffrey Humpherys, an assistant professor in the Department of Mathematics at Brigham Young University, Provo, UT, USA Corresponence shoul be sent to jeffh at math.byu.eu. as Paypal/eBay, PriceGrabber, an Amazon that centralize an/or clear numerous retail storefronts thus removing the nee for a irect relationship between the consumer an the retailer. Similar transaction clearinghouses likewise exist in the wholesale an shipping inustries. The iea of an option in retail can be mathematically motivate by the classical newsvenor problem, which we review in Section II. In this problem, the retailer seeks to etermine the optimal inventory neee to maximize expecte profit when the eman is uncertain, the sale price is fixe, an the inventory is perishable so that any remaining items at the en of the ay (or whatever the timeframe) are sol for scrap at some known salvage value; see for example [8]. Typical examples of perishable goos inclue bakery items in a grocery store an magazines on a newsstan, but electric power, theater tickets, an airplane seats also fit this escription. In fact one coul exten this to virtually any goo by using perishability as a proxy for epreciation, where the future epreciate value of the item is its salvage value uner the newsvenor paraigm. We remark that the solution of the newsvenor problem has many similarities to the Black-Scholes pricing of an option in the stock market. Recall that a put option contract gives the owner the right, but not the obligation, to sell a particular unerlying asset at a given price an at a specifie instance (or perio) of time. Hence, the value of a put option epens on the (expecte) future value of the unerlying asset. If the asset price rises beyon a certain level, calle the strike price, the put option becomes worthless. If it goes below the strike price, the put option can be exercise an the writer pays the buyer the ifference between the strike price an the asset price. In the case of a retail option, the merchant buys the option from the writer who then, at the en of the perio, pays the ifference between the strike price an the salvage value on any remaining inventory, or alternatively the contract coul allow or require the writer to pay the strike price an take physical possession of the unsol inventory; see [2], [3]. One can price such options using the risk-neutral valuation, where the premium pai by the buyer at the beginning of the contract perio is the future value of the writer s expecte payout. In an effort to make retail options more useful, we relax the constraint on a fixe sale price u, thus allowing the retailer to freely ajust the sale price of the unerlying goo throughout the contract perio in orer to control eman, an thus maximize profit. In Section III, we formulate this by consiering eman as a non-homogenous Poisson process where the expecte arrival rate epens on the sale price of the unerlying goo. We moel our inventory state as /9/$ AACC 2831

2 a continuous Markov process, which can be escribe as a system of orinary ifferential equations (ODEs) that epen in part on the retailers pricing policy, which if known gives the risk-neutral valuation of the option contract. In Section IV, we etermine the optimal pricing policy by solving a ynamic program; see for example [1]. We consier specifically linear an log-linear eman rates, although our metho is quite general. We likewise reuce our ynamic program to a system of ODEs, which can be solve to provie the optimal pricing policy. We assume that the retailer will use the optimal pricing policy, an hence we set the option price accoringly. Finally, we perform some simulations in Section V an iscuss future irections an offer aitional ieas for using portfolio optimization tools in retail in Section VI. II. MOTIVATING EXAMPLE In this section we review the classic newsvenor problem. We consier the task of etermining the inventory policy (or prouction policy) that will maximize expecte profit, given that one has a goo forecast of future eman. More precisely, we want to know how much inventory I our retailer shoul buy from the wholesaler at cost C to maximize expecte profit, knowing that at the en of the sales perio [, T] the remaining inventory I T will be salvage at a price of C T per unit. We assume a fixe sale price u an that the eman X is a ranom variable with known statistics. Since the eman may excee the available inventory, we relate the quantity sol as the ranom variable Q = min(x,i ). Hence, the profit Π is the revenue minus the cost of inventory plus the salvage, or rather Π = Qu I C + I T C T. where I T is the remaining inventory, that is, the amount of prouct that oes not sell an is thus salvage at a price of C T at the en of the sale perio. This is given by I T = I Q = max(i X,) := (I X) +. (1) The profit Π is then expressible as the return one woul get from selling out, minus the lost revenue from excess inventory that is only partially recovere from salvage. This yiels Π = I (u C ) (u C T )I T. (2) Hence, the expecte profit is E[Π] = I (u C ) (u C T )E[I T ]. (3) We remark that in (2), the only ranom variable in computing the profit is I T. Hence it suffices in much of our analysis to compute E[I T ]. A. Optimal Inventory Policy In the classical newsvenor problem, we seek to etermine the optimal initial inventory I that maximizes profit, given a retail price u an marginal (or wholesale) cost C. For convenience, we assume X is a continuous ranom variable. Let F(x) be the (usually strictly monotone) cumulative istribution function (cf) an f(x) := F (x) be the probability ensity function (pf) for the eman X. To fin local extrema of E[Π], we take its erivative with respect to I an set it to zero. Hence E[Π] = u C (u C T ) E[I T ] =, I I or equivalently, I E[I T ] = u C u C T. (4) It is easy to show that the initial inventory I that satisfies (4) is profit maximizing. To compute E[I T ], we efine the inicator ranom variable { 1 X I J = (5) X > I. Then I T = J(I X) an E[I T ] = I E[J] E[JX]. (6) Hence, the expecte remaining inventory takes the general form E[I T ] = I f(x)x xf(x) x. (7) x I x I Taking the erivative yiels E[I T ] = E[J] + I E[J] E[JX]. I I I It is relatively straightforwar to show that E[J] = F(I ), Hence I E[J] = f(i ) an I E[I T ] = F(I ). I E[JX] = I f(i ). Combining with (4), the optimal inventory policy is ( ) u I = F 1 C. (8) u C T This is the solution to the classic newsvenor Problem. We remark that mathematically, the above analysis is very similar to the erivation of the Black-Scholes pricing of a European call option; see for example [1]. 2832

3 Fig. 1. option. Option Value p Option Value p (a) K = strike price p = option price K I Stock Price I = initial inventory p = option price (b) Deman Option value at expiration of (a) a stock option an (b) a retail III. A NEW KIND OF EUROPEAN PUT OPTION We present a new type of European put option for retail, which allows the option holer to ajust prices throughout the contract perio [, T] in an attempt to maximize profits. We present a metho for computing the risk-neutral valuation of this retail option, where the initial inventory is m units. We assume that the eman is a nonhomogeneous Poisson process, where customer purchases arrive at the rate λ(u(t)), which epens on the time-epenent sale price u(t) set by the merchant. Let I j (t) be the probability that the inventory level is j units at time t. Then the vector I(t) = (I (t),...,i m (t)) is a continuous-time Markov process, where the transition probabilities likewise epen on λ(u(t)). I i,j+1 λ(u i,j+1 ) t B. Fixe-Price Retail Option We now explore the newsvenor Problem from the point of view of an option contract; see also [2], [3]. Recall that a European put option gives the holer the right, but not the obligation to sell an unerlying goo at a specifie price an time. Consier a European put option on the remaining inventory I T with strike price K C T, initial inventory I, an fixe sale price u. In other wors, we have a erivative contract where the retailer pays p ollars to the option writer at the beginning of the contract perio an then receives payment of KI T in return for the remaining inventory. If the retailer sells out, the retailer pays nothing, an the net option value is p corresponing to the initial payment at the beginning of the contract perio; see Figure 1. Accoring to the risk-neutral valuation of the contract, the option is value at p = (K C T )E[I T ] (9) ollars (moulo interest). Moreover the profit for the retailer is given by Π = I (u C ) (u K)I T KE[I T ]. (1) We remark that the expecte profit in (1) is the same as that of (2). Inee options that are base on the risk-neutral valuation o not improve the expecte outcome, but they o reuce the uncertainty of the outcome. Specifically, the variance of (1) is given by (u K) 2 V ar(i T ), whereas the variance of (2) is larger an given by (u C T ) 2 V ar(i T ). The problem with fixe-price retail options is that retailers may want to change prices throughout the contract perio. If the unerlying prouct is selling faster than expecte, the retailer may be able to increase profits by raising prices. If the retailer is over stocke, he or she may be better off lowering prices to stimulate eman. In the next section, we introuce a new type of European put option that will allow for ynamic pricing I i,j 1 λ(u i,j ) t Ii+1,j Fig. 2. Single perio branch escribing (11). A. Discrete-Time Formulation We iscretize the time interval [, T]. Let u i,j enote the sales price when the inventory is at j units at time i t, where i =,...,n, j =,1,...,m, an t = T/n. For small intervals of time [t, t + t), we assume that an item sells with probability λ(u i,j ) t. Let I i,j = I j (i t) enote the probability that the inventory level is j units at time i t. Clearly m j= I i,j = 1, I i,j for all i,j. Hence, following Figure 2, the transition probabilities satisfy I i+1,j = λ(u i,j+1 ) ti i,j+1 + (1 λ(u i,j ) t)i i,j. (11) Hence, given a pricing policy {u i,j }, we can etermine the expecte remaining inventory by computing the finiteifference equations forwar to time t = T. Then the expecte remaining inventory is given by m E[I T ] = ji j (T). j=1 Then the option valuation comes from (9), yieling m p = (K C T ) ji j (T). (12) j=1 This is the risk-neutral valuation of the option. B. Continuous-Time Formulation We can now take the limit at t goes to zero, thus turning this problem into a system of ODEs. Simplifying (11), yiels I i+1,j I i,j t = λ(u i,j+1 )I i,j+1 λ(u i,j )I i,j,

4 which, as t, gives the ODE t I j(t) = λ(u j+1 (t))i j+1 (t) λ(u j (t))i j (t), (13) where I j () is the initial probability istribution. Hence, given the pricing policy {u j (t)} m j=, we can compute the state vector I(t) at time T, by integrating the system of ODEs in (13) on the interval [, T]. Figure 3 shows a sample solution to this ifferential equation for initial inventory I 1 = 1, an I 9 = I 8 = = I =. inventory. On each subinterval, the retailer chooses the price u that will maximize expecte remaining revenue on that subinterval, given the inventory level j. We procee by the principle of optimality. Let E i,j enote the expecte remaining revenue earne uring the i th subinterval of time when the inventory level is j units. If an arrival occurs, the expecte remaining revenue is E i+1,j 1 + u i,j, an if no arrival occurs, the expecte remaining revenue is E i+1,j ; see Figure 4. Thus at each point, we have E i,j = (1 λ i,j t)e i+1,j +λ i,j t(e i+1,j 1 +u i,j ), (14) I=1 I= where λ i,j = λ(u i,j ). By taking a erivative with respect to u, we fin that the expecte remaining revenue is maximize when u i,j satisfies = λ (u i,j)(e i+1,j 1 E i+1,j + u i,j) + λ(u i,j). (15) Probability I=7 I=3 Hence, by solving (15) for u i,j, we can compute the optimal prices an the expecte remaining revenues at time i t by using the optimal prices an expecte remaining revenues at time (i + 1) t; see Figure 5. This gives us a complete escription of the optimal pricing policy an expecte remaining revenues for all subintervals of time an for all inventory levels Time Fig. 3. Inventory probabilities over time. j = 2 2K E i,j 1 λ(u i,j ) t E i+1,j λ(u i,j ) t E i+1,j 1 + u i,j Fig. 4. Single perio branch escribing (14). j = 1 j = { t K T Fig. 5. Multiple perios for computing the expecte remaining revenue. The rightmost column correspons to the final perio, where the option pays out K times the remaining inventory. By iterating backwars in time, we can etermine the rest of the gri values for both the optimal price u an the corresponing expecte remaining revenue values. IV. OPTIMAL PRICING In the previous section, we showe how to price an option for a given pricing policy. We now show how to compute the optimal pricing policy given a known eman rate λ(u). We first compute the optimal price by solving a ynamic program, an then, as in the section above, emonstrate that the problem can be compute as a solution to a system of ODEs by taking the subinterval length to zero. We assume that the retailer will choose the profit maximizing pricing policy, so that we can give the (Nash optimal) risk-neutral valuation of the option. A. Pricing as a Dynamic Program By partitioning the contract perio [, T] into n equal subintervals, as above, we can esign a ynamic program to etermine the optimal pricing policy for the unerlying B. Linear an Loglinear Deman Rates To more irectly connect the expecte remaining revenues an optimal prices, we compute optimal pricing policies for linear an log-linear eman rates. 1) Linear Deman: If the parameter of eman λ(u) is linear with respect to price, that is λ(u) = b au for some a,b >, then (15) gives u i,j = E i+1,j E i+1,j b 2a. 2) Loglinear Deman: We repeat the process escribe above but with a ifferent eman rate. We let λ(u) = au b, b > 1. Then (15) gives u i,j = b b 1 (E i+1,j E i+1,j 1 ). 2834

5 Sale Price Option Price Time (a) becomes very large, one coul treat inventory levels as a continuous variable, an resort to stochastic PDEs, as is one with option pricing in the stock market (espite the fact that the unerlying also takes on iscrete prices). As a check, we generate simulations to realize consumer purchases an examine how the price of both the unerlying an the option move over time; see Figure 6. We ranomly generate numbers on the unit interval at each time step, an then roppe the inventory by one if the number generate was less than λ(u i,j ). We then compute the option price at that moment. Notice that both images seem to behave like a ranom walk, an look similar to stock-market ata. Finally, we escribe how the option price varies as we toggle parameters. In Figure 7 we plot the option price against the initial inventory an the strike price, respectively. These vary in a monotonic, nonlinear fashion, as expecte, growing linearly in the infinite limit an converging to zero for small values. We remark that the option price for varying contract perios has an inverse relationship to how the option price varies with initial inventory, that is, small contract perios price as if the initial inventory were large, an large contract perios price as if the initial inventory were small Time (b) Fig. 6. Realization of (a) the optimal price of the unerlying asset throughout contract perio an (b) the option price throughout the contract perio. Option Price C. Pricing in Continuous Time We conclue this section by showing that the optimal prices an expecte remaining revenues can be compute by solving a certain system of ODEs much as we i in the previous section. By simplifying (14), we get E i+1,j E i,j = λ(u i,j )(E i+1,j E i+1,j 1 u i,j ), t which, by letting t, yiels the ODE t E j(t) = λ(u j (t))(e j (t) E j 1 (t) u j (t)), (16) where E (t) = an E j (T) = jk. V. SIMULATIONS We can coe up our two systems of ODEs in MATLAB in just a few lines of coe. For the integration, we use oe45. This variable-step routine allows for large steps when the errors are small, thus significantly speeing up computation time. Most of our runs execute in uner a secon. For large inventories or contract perios, the runtime increase proportionately. In the case that the inventory Option Price Fig. 7. price Initial Inventory (a) Strike Price (b) Option price as we vary (a) the initial inventory an (b) strike 2835

6 VI. INVENTORY VS. PORTFOLIO MANAGEMENT From an inventory manager s point of view, it can be ifficult, when looking at raw transactions ata, to know whether low sales are ue to low eman (e.g., noboy is buying) or low supply (e.g., retailer is out of stock). A retail option allows one to easily ifferentiate between low sales an low supply. Inee the price of the option goes to zero when the inventory runs out whereas low sales woul result in the option value being high. By using options an other financial instruments in a retail environment, merchants can better unerstan their markets by treating their inventories as money managers o portfolios. However in retail, there is the ae avantage of being able to ynamically control the price of the unerlying goos in an attempt to maximize profits. There are many irections to take this portfolio view of inventory theory an control. Several recent results exist for managing portfolios of stock options over multiple perios; see for example [9]. We coul also explore risk-management strategies ranging from a myria of techniques incluing value-at-risk, cash-flow-at-risk, etc. Another irection is the further use of more exotic options. Variations on floors, caps, Asian options, Bermua options, etc., shoul all be consiere to see if there are retail possibilities. Another area of interest is how to price an option when ealing with an incompetent retailer. Note that the price of the option propose here requires that the retailer continually ajust prices to maximize profits. However, if the retailer isn t ajusting them optimally, then the writer will be force to buy more unsol inventory than necessary at the en of the contract perio. While the existence of a Nash equilibrium shoul preclue such events theoretically assuming symmetric information, market efficiency, etc., in practice prices may be set baly. Room for such incompetence may nee to be price in the option. We remark that options have been aroun for centuries in the financial markets [11] (most notably over the past 3 years), an they have only recently been suggeste for use in a retail environment [2], [3]. There is, however, an extensive supply chain contracts literature going back for ecaes; see [12] for a review, as well as a real options literature in business strategy; see [6] an references within. It woul be worthwhile to pull all of this literature together uner the common umbrella of systems theory. Inee recent events eman better analysis of financial moeling an a eeper unerstaning of the ynamics of the markets. Finally, it is unclear whether retail option contracts coul ever materialize into exchange-trae instruments like options on stocks, futures, etc. Surely some coul fin a natural home in certain niche retail markets, but overall the lack of stanars, trust, thir-party or government regulation, an/or ifficulties in instrumentation may make it ifficult to reliably create an active market for retail options in a general setting. We remark, however, that the use of retail options as a means of centralize inventory management an control in a retail chain, or more generally an organize network of manufactures, wholesalers, istributors, retailers, speculators, an financiers, may prove to be valuable. Inee, retailers can essentially operate as portfolio managers. Retail chains can buy an sell options internally, between their istribution centers, iniviual stores, an their own corporate finance epartment can act as the hege fun for the company by pooling risks much like an insurance company. Moreover, by using the put option propose in this paper, where the prices can be ajuste ynamically, the retailers can then use the price to control the returns in their portfolio, both for an iniviual store an in the aggregate. VII. FUTURE WORK We are currently consiering variations on this work. To make our retail option viable, we nee to further explore the egree to which the option price is sensitive to input parameters. We also nee to flesh out the etails about the variance in the option payoff. We are also intereste in seeing if a set of options can be written on an unerlying collection of inventory, whether is is reasonable to have variable payoffs epening on how much inventory is remaining, an how the option might work with multiple perios an multiple echelons. Other issues inclue state estimation of the arrival rate, the effects of substitutes an complements, an the effects of coupons an other manufacturer-base promotions on the option price. VIII. ACKNOWLEDGMENTS This work was supporte in part by the National Science Founation, Grant No. DMS , BYU mentoring funs, an a gift from Park City Group, Inc. The authors thank PCG Presient an CEO Ranall Fiels an his associates for useful an stimulating conversations throughout this work. REFERENCES [1] D. Bertsekas. Dynamic Programming an Optimal Control, (Volumes 1 an 2). Athena Scientific, 25. [2] A. Burnetas an P. Ritchken. Option pricing with ownwar-sloping eman curves: The case of supply chain options. Management Science, 51(4):566 58, 25. [3] F. Chen an M. Parlar. Value of a put option to the risk-averse newsvenor. IIE Transactions, 39(5):481 5, 27. [4] J. Cox, S. Ross, an M. Rubinstein. Option Pricing: A Simplifie Approach. International Library of Critical Writings in Economics, 143: , 22. [5] A. Demirguc-Kunt an R. Levine. Stock markets, corporate finance, an economic growth: An overview. Worl Bank Economic Review, 1(2):223 39, May [6] A. Dixit an R. Pinyck. 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