On Regret and Options - A Game Theoretic Approach for Option Pricing

Transcription

1 O Regret ad Optios - A Game Theoretic Approach for Optio Pricig Peter M. DeMarzo, Ila Kremer ad Yishay Masour Staford Graduate School of Busiess ad Tel Aviv Uiversity October, 005 This Revisio: 9/7/05 ABSTRACT. We study the lik betwee the game theoretic otio of regret miimizatio ad robust optio pricig. We demostrate how tradig strategies that miimize regret also imply robust upper bouds for the prices of Europea call optios. These bouds are based o o-arbitrage ad are robust i that they require oly miimal assumptios regardig the stock price process. We the focus o the optimal bouds ad demostrate that they ca be expressed as a value of a zero sum game. We solve for the optimal volatility-based bouds i closed-form, which i tur implies the optimal regret miimizig tradig strategy. We thak Sergiu Hart for helpful discussio ad semiar participats at Berkeley (IEOR), Tel Aviv uiversity, Hebrew Uiversity ad Staford for useful commets.

2 . Itroductio There is a growig literature i game theory o the strategic cocept of regret miimizatio for games uder ucertaity. Regret is defied as the differece betwee the outcome of a strategy ad that of the ex-post optimal strategy. This literature is based o earlier work by Haa (957) ad Blackwell (956) who studied dyamic robust optimizatio, ad is related to more recet work o calibratio ad the dyamic foudatios of correlated equilibria. I this paper we cosider a fiacial iterpretatio of regret miimizatio ad demostrate a lik to the robust pricig of fiacial assets. I particular we focus o fiacial optios, which we ca thik of as cotracts that allow ivestors to miimize their regret whe choosig a ivestmet portfolio. Usig the lik betwee regret miimizatio ad optio pricig, we the derive robust pricig bouds for fiacial optios. The classic, structural approach to optio pricig developed by Black ad Scholes (973) ad Merto (973), posits a specific stock price process (geometric Browia motio), ad the shows that the payoff of a optio ca be replicated usig a dyamic tradig strategy for the stock ad a risk-free bod. No arbitrage the implies that price of the optio must equal the cost of this tradig strategy. But because empirical stock prices do ot follow the process assumed by Black-Scholes- Merto, their argumet is ot a true arbitrage: the replicatio is perfect oly for a very restricted set of price paths. While our results are weaker (we provide bouds, rather tha exact prices), they are robust i that we do ot assume a specific price process. I sum, the goal of this paper is two-fold. First, we develop a fiace-based iterpretatio for the otio of regret miimizatio by showig the lik to robust (distributio-free) bouds for the value of fiacial optios. Secod, we look for the optimal such bouds. This fast growig literature examies the statistical otio of calibratio as well as dyamic foudatio for correlated or Nash equilibrium. See Hart (005), Foster Levie ad Vohra (999), ad Fudeberg ad Levie (998) for excellet surveys. Regret miimizatio is equivalet to the cocept of competitive aalysis that is used to evaluate the performace of a algorithm i computer sciece. The competitive ratio of a algorithm is the maximum, over all realizatios, of the ratio of the performace of the best ex-post algorithm to that of the give algorithm (see, e.g., Sleator ad Tara (985)).

3 The roots of regret miimizatio i game theory ca be traced to Haa (957) ad Blackwell s (956) work o dyamic optimizatio whe the decisio maker has very little iformatio about the eviromet. They cosidered a repeated decisio problem i which i each period the aget chooses a actio from some fixed fiite set. Although the set of actios is fixed, the payoffs to these actios vary i a potetially o-statioary maer, so that o learig is possible. They show that i the limit, there is a dyamic strategy that guaratees the aget a average payoff that is at least as high as that from the ex-post optimal static strategy i which the same actio is take repeatedly. Thus, i terms of the log ru average payoff, the aget suffers o regret with respect to ay static strategy. Haa (957) ad Blackwell s (956) provide foudatios to later work i egieerig, especially i computer sciece. Computer scietists are iterested i dyamic optimizatio methods (referred to as o-lie algorithms ) for eviromets i which a specific distributio of the ucertai variables is ukow. They have followed the view that i such eviromet oe should maximize a relative obective fuctio rather tha a absolute oe. I particular, they evaluate the worst-case loss relative to the optimal strategy if the ucertai variables were kow i advace. This differs from the more traditioal approach i ecoomics that cosiders a absolute obective (e.g. Gilboa ad Schmeidler (989)) i such a eviromet. It is importat to ote that i this paper we do ot take a stad o which is the right approach. Our results hold regardless of what oe believes is the right way to optimize or what best describes behavior observed i practice. We explore the lik to fiacial markets by examiig ivestmet decisios i a ucertai eviromet. Here we ca defie regret as the differece betwee the ivestor's wealth ad the wealth he could have obtaied had he followed alterative ivestmet strategies. By comparig the ivestor s payoff to that which could be attaied from a buy ad hold ivestmet of a the stock or a bod, we ca iterpret regret as the differece i payoff betwee a dyamic tradig strategy ad a call optio, allowig us to lik regret miimizatio to o arbitrage upper bouds for optio prices. We begi by adaptig the Haa-Blackwell results to a ivestmet settig. To do so, we eed to adust for the fact that i their settig, per period payoffs are additive ad draw from a fiite set. I a

4 ivestmet cotext, payoffs are multiplicative, ad bouds o the per-period returs will be required. While Haa-Blackwell focus o limitig results (similar to the traditioal work o regret miimizatio), to be useful i a ivestmet cotext we cosider miimizig regret over a fiite horizo. The optio price bouds we derive usig the Haa-Blackwell approach do ot deped o specific distributioal assumptios for the stock price path, ad so is i that sese robust. The most straightforward extesio of the Haa-Blackwell approach requires restrictig the magitude of the stock s retur each period. A more atural restrictio, ad oe which allows a direct compariso to the Black-Scholes-Merto framework, is to impose restrictios o the realized volatility, or quadratic variatio, of the stock price path. I subsectio 3., we show that simple mometum strategies (i which we ivest more i the stock whe its retur is positive) is effective at limitig regret whe the stock s quadratic variatio is bouded. These strategies thus lead to bouds for optio prices based o the stock s volatility. 3 It is importat to ote that the strategies metioed above are ot ecessarily optimal. There might be a lower upper boud correspodig to a hedgig strategy that is cheaper. Hece, a atural questio is what is the optimal boud/strategy? i sectio 4, we address this questio. We show that it ca be viewed as a solutio to a fiite horizo zero-sum game. Usig this approach we compute the boud usig dyamic programmig ad derive a simple closed-form solutio. We also derive the optimal robust tradig strategy, which is the lowest cost strategy with a payoff that exceeds the optio payoff for ay stock price path with a quadratic variatio below a give boud. These returs also provide the optimal strategies for miimizig regret i our settig. Fially we compare our price bouds to the Black-Scholes-Merto (BSM) model. Our optimal bouds exceed the BSM price we ca iterpret the bouds as the BSM price correspodig to a higher implied volatility. Ideed, the patter of implied volatility determied by our boud resembles the volatility smile that has bee documeted empirically i optios markets. We also compare our tradig strategy to the delta hedgig 3 Our results i this regard are related to work by Cover (9xx, 9xx) o the uiversal portfolio, a dyamic tradig strategy desiged to perform well compared to ay alterative fixed-weight portfolio. We discuss the relatioship of our results to Cover s i Sectio XX. 3

5 strategy of Black ad Scholes. We show that it is similar i ature but that the aget s stock positio is less sesitive to movemets i the uderlyig stock price. This isures him agaist umps that are ot cosidered by Black ad Scholes... Illustratio Before turig to the more techical part of the paper it is useful to demostrate some of the isights i this paper usig two simple examples. We begi with a example that demostrates the equivalece betwee regret miimizatio ad robust bouds for optio prices: EXAMPLE (Regret ad optio pricig): Suppose the curret stock price is $ ad the risk free iterest rate is zero. I a regret framework, we measure the performace of a strategy by its maximal loss relative to the ex-post optimal asset choice. For example, a T-period dyamic tradig strategy with a maximal loss of 0% implies that startig with $, our payoff at time T will exceed 0. 8max{, S T }, where S T is the fial stock price. To see how such a strategy ca be used to boud the value of a call optio o the stock, ote that by scalig the strategy by. 5= 08 /., we coclude that startig with $ 5. our strategy would have a payoff that exceeds max{, S T }. If we partly fiace our strategy by borrowig $ iitially, the after payig off our loa, our fial payoff exceeds max{0, S T }, which is the payoff of a T-period call optio o the stock with a strike price of $. Thus, to avoid arbitrage, the value of the call optio caot exceed the upfrot cost of $ 05.. The quality of our boud is determied by the maximal regret of our dyamic tradig strategy relative to best static decisio (which is either to buy bod or the stock). A loss of 0% traslated ito a upper boud of $ 05. for the call optio price, ad a lower loss would provide a tighter boud. We cotiue by presetig a simple example that demostrates the lik betwee the optimal bouds ad zero sum games; it is based o the classic oe period biomial model. 4

6 EXAMPLE (Optimal bouds ad zero sum games): A ivestor ad a adversary egage i a oe-period game. The adversary decides o the retur of the risky asset whose iitial price is ormalized to oe. We assume that he ca oly choose a retur of r =±σfor σ (0,), or radomize betwee the two alteratives. The ivestor starts with zero wealth ad decides o how may shares to buy, R. He fiaces this purchase by borrowig at a zero iterest rate; hece, his fial wealth is give by ( + r) = r. The game is a zero sum game where the adversary s payoff is the expected differece betwee the payoff of at-the-moey call optio ad the ivestor s wealth, E{max{0, r} r }. I sectio 4, we prove that a versio of the miimax theorem holds for this game (as well as the more geeral dyamic versio we will itroduce shortly). 4 Based o the miimax theorem we ca defie the value of this simultaeous-move game. I equilibrium the adversary radomizes while the ivestor does ot. I this simple example it is obvious that the adversary must choose r so that Er ( ) = 0. Otherwise the ivestor ca obtai a ifiite through a appropriate choice of. I the multi-period framework the coclusio would be that the adversary chooses the stochastic process to be a Martigale. If the adversary sets the expected retur o the risky asset to be zero the the ivestors expected wealth is zero regardless of the umber of shares he buys. Hece, the value of this game is simply the expected payoff of the optio, which i our case is σ /. I the more geeral setup this will lead us to a simple umerical procedure to compute the value of the game. Still, the umber of shares the ivestor buys caot be arbitrary. For the adversary to radomize he must be idifferet betwee a retur of σ ad σ; for this to happe the ivestor must buy = 05. shares. I the more geeral setup we argue that this leads to a differetial equatio. The umber of shares matches the derivative of the value with respect to the stock price. We have characterized the equilibrium of this game, the adversary sets Pr( r = x) = Pr( r = x) = 0. 5 while the ivestor buys = 05. shares; the value of the 4 We will eed to take ito accout the fact that the set of the ivestor s strategies is ot compact. 5

7 game is σ /. This matches the solutio of the classic oe-period biomial model where oly oe price elimiates arbitrage. I this paper we cosider a more geeral setup i which returs i each period ca take more tha two values. As a result the markets are icomplete ad multiple prices are cosistet with No-arbitrage. As we shall discuss i subsectio 4., the value of the game i this case matches the highest price i this set or the optimal (lowest) upper boud. The ivestor s equilibrium strategy correspods to the robust hedgig strategy that esures agaist ay possible price path of the risky asset. Based o this, i sectio 4.3 we preset closed form solutio for the case whe the quadratic variatio is bouded... Literature Review While the Black-Scholes formula is oe of the most useful formulas developed i ecoomics, i recet years extesive empirical research has idetified several aomalies i the data. I geeral the formula seems to geerate prices for stock idex optios that are too low. Said aother way, the implied volatility of the stock idex computed based o the Black-Scholes formula is sigificatly higher o average tha the ex-post realized volatility. I additio, this effect is more proouced for call optios whose strike price is low. This effect is ofte referred to as the volatility smirk or smile. As a respose to these fidigs, there has bee a active research (e.g., Pa (00), Eraker, Johaes ad Polso (003), ad Eraker (004)) tryig to modify the Black ad Scholes formula to accout for these discrepacies. These papers examie differet stochastic processes for the idex, with modificatios that iclude ump processes ad stochastic volatility models. The result of our study will complemet this aalysis by offerig a ew perspective. Rather tha focusig o a specific formulatio for the stochastic process we rely o a geeric tradig strategy that works with ay evolutio for the risky asset as log as it satisfies some bouds o returs ad quadratic variatio. As a result of both academic ad practical iterest there are several papers that study the restrictios oe ca impose o the price of optios. These papers are similar i spirit to our work as the goal is to provide a robust boud by relaxig the specific assumptio made by Black ad Scholes. Myklad (000) cosiders a stochastic process that is more geeral tha Black ad Scholes assumptio that the stock has a costat volatility. He 6

8 models the stock price as a diffusio process, but allows the volatility to be stochastic. I this case the market eed ot be complete ad we might be uable to replicate a optio payoff. 5 Still he shows that oe ca use the Black-Scholes price as a upper boud if we take the volatility parameter to be the upper boud over all realizatios of the average stochastic volatility. The reaso for this ca be traced to Merto's argumet that i such a framework the Black ad Scholes formula holds if the average volatility is kow. While such bouds geeralize Black ad Scholes i a sigificat way they still impose sigificat restrictios o the stochastic process. For example, the price path is assumed to be cotiuous so the stochastic price has o umps; such umps were show to be empirically importat by Pa (00), Eraker, Johaes ad Polso (003), ad Eraker (004). For example, the Merto observatio fails i a discrete time versio of Black ad Scholes; the ability to trade cotiuously is critical. Still, Myklad s result, similar to our methodology, shows that a upper boud over the itegral or average volatility ca dramatically improve the bouds compared to the case i which oe assumes a upper boud over the istataeous volatility (see for example Shreve, El Karoui, ad Jeablac-Picque (998)). A alterative approach to that take here is developed by Berardo ad Lediot (000) ad Cochrae ad Saa-Requeo (000), who stregthe the o-arbitrage coditio by usig a equilibrium argumet. Specifically they assume bouds for the risk-reward ratio that should be achievable i the market. Based o these bouds ad existig market prices, they ca the determie upper ad the lower bouds for ew securities that may be itroduced ito the market. As metioed earlier, our research is also related to research i Computer Sciece/Statistics. I particular there is a literature that applies competitive algorithms i the cotext of ivestmets. Most of the literature follows the semial work by Cover (99). We follow the more covetioal paradigm i ecoomics of efficiet market ad hece provide a differet iterpretatio. We argue that oe should thik of these tradig algorithms as a way to super replicate a optio uder differet coditios. 5 A complete market is oe i which the existig assets allow all possible gambles o future outcomes. 7

9 . Model We cosider a discrete-time -period model which time is deoted by {0,,, }. There is a risky asset (e.g., stock) whose value (price) at time is give by S. We ormalize the iitial value to oe, S 0 =, ad assume that the asset does ot pay ay divideds. We deote by r the retur betwee ad so that S ( ) = S + r. Throughout this paper we assume that r > so that the stock price is always positive; we call r = r,, r the price path. I additio to the risky asset we have a risk free asset (e.g., bod). Uless otherwise stated, we assume that the risk free rate is zero. A dyamic tradig strategy startig with $c i cash has iitial value G0 period it distributes its curret value = c. At each G, betwee the assets, ivestig a fractio x i the risky asset ad i the risk free asset. Sice we assume zero iterest rate, at time x + its value is G+ = ( xg)( + r+ ) + ( x) G = G( + xr+ ) ; its fial value is G. Let CK ( ) be the value, at time = 0, of a Europea call optio whose strike price is K that matures at time. This is the preset value (at = 0 ) of the fial payoff at time T that is give by max{0, S K}. We cosider restrictios o the possible price path that are represeted by a subset of possible price paths, Ψ R ; we assume Ψ to be compact ad that 0 Ψ. For example, i some cases we assume bouded quadratic variatio so that { r r r } q = Ψ= >, <, i other cases we assume a boud o a sigle day retur so that Ψ= { r r >, r < m}, ad sometimes we assume both restrictios apply. Coditioal o Ψ, we assume that there is o arbitrage i prices. Namely, for ay two tradig strategies (or fiacial securities) A ad A, that start with cash $c ad $c, if for ay price path i Ψ the future payoff of A is always at least that of A, the c c. If this were ot true ad c < c, assumig that oe ca sell short assets (ad strategies), there would be a arbitrage opportuity: Ivestig i A ad shortig A would lead to a 8

10 time 0 gai of c c without the possibility of loss i the future. As a result we have the followig defiitio: DEFINITION. We say that c= Cψ ( K) is a upper boud if there exists a dyamic tradig strategy that starts with $c ad for all possible price path i Ψ its fial payoff, G max 0, S K. G, satisfies: { } T Our goal i this paper is to show how this is related to regret ad what bouds ca we obtai. Before proceedig to the ext sectio we should discuss the importace of imposig certai restrictios o the price path give by Ψ. The first part of Merto (973) addresses this questio by askig what ca be said about the value of a call optio without makig ay additioal assumptio about the price path. The aswer is that with zero iterest rate oe ca oly say that for k > 0 : Ck ( ) [max(0, S k), S) 0 0 Hece, the optio is ot more valuable tha the uderlie asset. This is a very weak boud but it is tight as whe allowig for arbitrary price paths the value of the call optio ca be arbitrary close to it. 3. Regret As metioed i the itroductio, we first examie the cocept of regret miimizatio. This cocept is based o the semial work by Haa (957) ad Blackwell(956). I this sectio we describe the basic regret cocept ad demostrate how oe ca apply the origial method developed by Blackwell (956) i the cotext of fiacial market. We also show how this yields a robust upper boud for at the moey call optio; later we cosider differet setup ad improve o the bouds obtaied here. Cosider a setup i which we repeatedly choose a sigle actio amog I { I} =.. possible alteratives. Let π i, R deote the payoff of alterative i at time. We make almost o assumptios regardig these payoffs apart from assumig that differeces i payoffs are uiformly bouded so that π < for all ii, ', t for some m > 0. I particular i, πi', m the payoff may ot follow a statioary distributio or ay other distributio. We allow the 9

11 aget to radomize ad describe a radom strategy by { I} ξ.. so that ξ = i implies that at time we choose the ith alterative. Our payoff at time t is give by average payoff up to time is give by π ξt, = π ξ, ; our Give the few assumptios we have made, we do ot seek to maximize absolute performace. Istead, we cosider a relative bechmark, which is the regret measure. For each alterative i, we focus o the average payoff up to time, which is give by π = i,. The time regret of a give strategy measures how it compares to the best π t= i, static strategy ex-post, max { }. A corollary of Blackwell s approachability implies that: i COROLLARY (o asymptotic regret) There exists a radomized strategy so that for ay δ > 0 lim T π max{ π } δ ξ, i, a. s. i = = I additio oe ca boud the expected covergece rates. Give ay realizatio of payoffs, the expected distace coverges at a rate of m/. PROPOSITION. There exists a radomized strategy so that: E max{ } m ( I ) / π ξ π, i, i = = () We provide the proof i the appedix oly for the case whe there are two alteratives, I =. While this holds also for I >, the proof is somewhat more ivolved ad for our purpose the case I = is sufficiet. Specifically cosider two alteratives that are based o the two fiacial assets that we described i the previous sectio: a risky asset whose et retur at time t is give by r t ad a risk free asset with zero et retur. We defie payoffs by lookig at log-returs by lettig π 0, t = 0, π = l + r, ad assume that l( + r ) < m., 0

12 If at time we choose at radom a sigle alterative, i = 0. the we ca use Propositio to boud our expected regret. Istead, cosider ow a tradig strategy based o the radomized strategy described above. We costruct a determiistic strategy so that at time we ivest a fractio of x E ξ i the risky asset ad x i the risk free asset. Our retur at time is give by + x r, ad our fial payoff is give by Π +. Sice [ 0] ( ) x r = x, ad r > we have that: 6, l + xr xl( + r) = E π ξ Hece we ca coclude that: 7 π ξ, = = = l + x r E max{0, l + r } m which implies that our payoff always satisfies { } ( x r) exp m = max S Π +, () I the limit our geometric payoff coverges to the geometric average payoff of the best asset. For a fiite horizo we approximate the best asset ex-post to a factor of exp m our multiplicative regret is exp m. x, let f () r = + xr ad gr () = ( + r) x. Note that f(0) = g(0) =, f (0) = g (0) = x, ad g () r < 0 for ay r > &. Sice g is covex while f is liear i r we have that f () r g() r for r >. 6 For a give [ 0] 7 Oe eeds to be careful here. Cosider a two period model i which the stock price doubles itself i both periods with certaity. Suppose that a ivestor first chooses with equal probabilities whether to ivest his etire wealth i the stock or othig. He does ot chage his decisio i the secod period so i each period the expected fractio ivested i the stock equals a half. This radom strategy yields with probability 0.5 ad 4 with probability 0.5 so o average.5. Usig the procedure outlied i the text we trasform this strategy to a determiistic oe by ivestig half of our wealth i the stock i both periods; this strategy yields.5 with certaity. However, oce we look at log returs the radomized strategy yields o average 0. 5l(4) = l() while the determiistic oe yields l(. 5) = l(. 5). Hece, oly whe we look at logs, our determiistic portfolio performs better.

13 3.. Regret based Bouds I the previous sectio we have demostrated how to trasform the strategy i Blackwell to a ivestmet strategy i fiacial markets. As we shall demostrate this traslates to a upper boud for at-the-moey call optio. More geerally, to obtai bouds for differet strike prices we cosider a modified regret guaratee; we put differet weights o the assets: DEFINITION A dyamic tradig strategy has a ( α, β ) guaratee, if for ay price paths i Ψ its fial payoff, G, satisfies G max { αβs },. To gai some ituitio it is better to first examie a very simple tradig strategy. Suppose we decide to use a buy ad hold strategy i which we ivest a fractio β i the risky asset ad α = β i the risk free asset. The future payoff of this fixed portfolio, G, is G = α + βs α, βs max { } This implies that we implemeted a ( α, β ) guaratee for β = α. Compare the above to the payoff of a fixed portfolio of β call optios each with a strike price of K = combied with α ivested i the risk free asset. Such a fixed portfolio yields at time a payoff of exactly α β max{0 S ( α β)} max { α βs } +, / =,. By defiitio, the curret α price of this fixed portfolio is βck ( =, T) + α. Sice G max { αβ, S }, by the o arbitrage assumptio, we have, β C( α ) C( α β, + α, ) α = = S β β β 0 α β As metioed before, S 0 is a simple kow upper boud o the optio price. Our goal is to costruct a dyamic tradig strategy that starts with $ ad yields a future payoff that exceeds: max { αβs, }

14 for some α + β >. Such a strategy yields a o-trivial boud, as stated i the followig claim, PROPOSITION. Assume that all price paths are ( q m), price paths. A dyamic tradig strategy with a ( α, β ) guaratee esures that for a call optio with strike price K = α β, u α C ( K q m ) β β K,,, =. Based o the above claim ad propositio we ca derive a upper boud for the value of at-the-moey call optio that is based o Blackwell: This boud is quite high; i fact if exp m m > l() the our upper boud is higher tha oe which is the iitial share price; hece to get a meaigful boud m caot be too high. I sectio 4.3 the optimal boud for this setup which will eable us to compare it to the above boud. A boud that depeds o restrictig the absolute per period retur suffers from the fact that eve for high frequecy a stock retur may be quite high. As a result tryig to impose a uiform boud o the absolute retur is likely to result i a boud which is too high. The other restrictio that we cosider that is based o boudig the quadratic variatio is much more useful as it relies o a boud of a global property over the etire price path. 3.. Bouds Based o Quadratic Variatio- Uiversal Portfolios Followig the discussio above we focus our attetio to tradig strategies that are based o the quadratic variatio. 8 We itroduce a mometum strategy that is useful i derivig upper bouds. We first describe a more geeral versio i which we trade I differet assets, ad its goal is to have its value approximate the value of the best asset. Later we shall see how a simple applicatio ideed yields the desired upper boud o the price of the optio. 8 These strategies are similar i spirit to the "Uiversal Portfolios approach by Cover (996). 3

15 Cosider I assets where we deote by V i, the price of asset i at time. We ormalize the iitial value of each asset to be oe, i.e., 0 =. The value at time satisfies V ( ) i, = Vi, + ri,, where ri, [ m, m] is the immediate retur of asset i at time. Our V i, tradig strategy is based o what we refer to as weights, w i,. We fix the iitial weights so that w i 0 i, =, ad the use the update rule w w i i ( r, + =, + η i, ), for some parameter η 0. At time t we forms a portfolio where the fractio of ivestmet i asset i is xi, = wi, / W where W = wi.. The value of tradig strategy is iitially, G 0 =, I I i= i, i, i= i, i,. ad G = ( x G )( + r ) = G ( + x r ) i The followig theorem, whose proof appears i the Appedix, summarizes the performace of our olie algorithm. ( ) PROPOSITION 3. Give parameters: η, m ( m), ad { w i, 0 }, where 0 the mometum tradig strategy described above, guaratees that for i i w, =, ay asset i, l( G ) l( V ) l ( η ) q i, i η wi, 0 where i t= i q = r,, ad r < m< / 03.. i, Cosider ow the applicatio to the special case we cosider of oly two assets. With a slight abuse of otatio we let w 0 deote the amout ivested i the risky asset ad assume that we ivest -w 0 i the risk free asset. Sice we assume a zero iterest rate we have q = 0 for the risk free asset ad coclude that: i ( ) COROLLARY Give parameters: η, m ( m), ad w0 (0, ), the mometum tradig strategy described above, whe applied to a risky asset ad a risk free asset, guaratees that 4

16 G S η q, l( ) max l( ) l ( ) l η w0 η w0 where q r = =, ad r < m< / 03.. From Corollary we have, where G, max { αβs } ( ) ( ) ( ) / ( ) q 0 w η / η η, = 0 ad 0, = 0 α w η β w η w e Now cosider the boud for a give strike price K. To fid the best boud we ca optimize over the two parameters η, w0, specifically we solve: ( K) max ( w ) β = β, η s. t. α β, w0 0 ( w0, η) = K ad η, ( w, η) m ( m) 0 η Oe ca simplify this problem by usig α( w0, η) β( w η) 0 = K to solve for w, 0 : w, = + Ke ( η K) 0 η η ( η ) q Hece, we eed to solve the followig maximizatio, / η ( ) η q β ( K) = max w0 ( η, K) e s. t. η, η m ( m) Let β ( K) be the solutio to the above optimizatio, our boud is the give by u CKT (, ) C( Kq,, m, ) K β ( K ) 5

17 4. Optimal Bouds I prior sectios we have looked at particular dyamic tradig strategies ad the optio price bouds that they imply. I this sectio we cosider optimal regret miimizig strategies, ad determie the tightest (lowest) possible optio price bouds. We focus o the followig costraits: (, ),max = Ψ qm = r q r m As metioed before this is the more relevat costrait ad it simplifies our expositio; still, most of our results hold for a more geeral Ψ. Let V(S,K,q,m,) be the miimal cost of a portfolio that super-replicates the optio for all -period stock price paths i Ψ ( qm, ). We ca defie V recursively as follows. For =0, V is equal to the optio payoff: V S K q m S K (,,,,0) = max{0, } For >0, V is the cost of the cheapest portfolio whose payoff ext period, after ay allowable retur, is sufficiet to super-replicate the optio from that poit oward. Because a portfolio with value V ad shares of the stock has payoff V + r S, we have (,,,, ) = mi, V V S K q m V s.t. V + rs V( S( + r), K, q r, m, ) for all r Ψ( q, m) () We first ote that sice the restrictios we cosider are o the returs, we ca coclude that V( S, K, q, m, ) = K V( S/ K,, q, m, ) (3) We also show i the appedix that: LEMMA. (i) For a give S, V( S, K, q, m, ) is covex i K. (ii) For a give K, V( S, K, q, m, ) is covex i S. 6

18 Give (3), without loss of geerality we focus o the case K=; hece we suppress the secod argumet ad write (,,, ) V S q m Followig the example i the itroductio we begi by establishig the coectio to zero sum games. 4.. Optimal bouds as a zero sum game Cosider a zero sum game betwee a ivestor ad a adversary. The adversary chooses a price path ad the ivestor chooses a tradig strategy that starts with $W. The payoff to the adversary is the differece betwee the fial values of the optio ad the ivestor s portfolio; the ivestor s payoff is mius this amout. As we shall demostrate that the value of this game is the optimal optio price boud. While the game ca be described as a oe-shot game it is better to cosider a dyamic (extesive form) represetatio. I each period the adversary chooses the ext period retur, r, or more precisely a radom retur r%, the the ivestors decide how may shares to buy,. Formally, we cosider the followig recursive defiitio: For = 0 : For : f ( W, S, q, m,0) = max{ S, 0} W f( W, S, q, m, ) = sup% if Ef( W + rs %, S( + r% ), q r%, m, ) r Σ( q, m) where r% is the radom variable that represets the ext retur, ad Σ ( qm, ) is the set of radom variables whose magitude is bouded by mi(q,m). The above formulatio fits a setup i which the adversary moves first. The ivestor forms his portfolio after kowig the strategy of the adversary. The ivestor observes the distributio that the adversary has chose but ot the realized retur. Usig iductio we show i the appedix that: LEMMA. (i) f ( W, S, q, m, ) = f(0, S, q, m, ) W,(ii) f (0, Sq,, m, ) [0, S], (iii) Eif f( Sr%, S( + r% ), q r%, m, ) 0 if ad oly if Er % = 0,(iv) f ( W, S, q, m, ) is cotiuous i q, m ad S. 7

19 Part (iii) implies that the adversary must use a Martigale measure. It is iterestig to ote that the Martigale property arises from the fact that otherwise the ivestor could obtai a ifiite payoff. Usig part (i) i the above Lemma we ca write: { % % % } f W, S, q, m = E f, S + r, q r, m Sr W (, ) sup if (0 ( ), ) r% Σ( q, m) Usig part (iii) we ca focus o r% that satisfy Er % = 0, ad the choice of becomes irrelevat. Sice r% is chose from a compact set (based o C implies that topology), cotiuity f S q m Ef S r% q r% m st Er% (4) (0,,,, ) = max (0, ( + ),,, ).. = 0 r% Σ( q, m) Fially we argue that a versio of the miimax theorem holds. We eed to deal with the fact that is chose from a o-compact set ad hece rely o the versio of Sio (958). Applyig the miimax theorem ad otig that the optimum is obtaied with a fiite umber, we have LEMMA 3. From the miimax theorem: f Sq m E f S r% q m rs % (0,,,, ) = mi max { (0, ( + ), r%,, ) } r% Σ( q, m) Sice f(0, S, q, m,0) = max{ S,0}, comparig LEMMA 3 with () ad usig (4) we have prove that: PROPOSITION 4. The value of the above game matches the optimal upper boud for the value of the optio, that is, V( S, q, m, ) = f(0, S, q, m, ) (5) PROOF OF PROPOSITION 4 The proof immediately follows from the fact that () ca be writte as V( S, q, m, ) = mi max V( S( + r), K, q r, m, ) rs r Ψ ( q, m) 8

20 Due to the fact that i % % % the retur r is mi max E{ f(0, S( + r), q r, m, ) rs } r% Σ( q, m) chose after is kow LEMMA 3 ca be writte as: f, Sq,, m = f, S + r, q, m rs (0, ) mi max (0 ( ) r, ) r Ψ ( q, m) Fially, ote that V is bouded ad o-decreasig i. Hece, oe ca defie: f Sq m f Sq m = (0,,, ) lim (0,,,, ) V S q m V S q m (,, ) lim (,,, ) By removig the costrait o the umber of stock price movemets, this boud equals the maximal value of the optio whe the stock price evolves cotiuously with the oly costraits o the quadratic variatio ad the maximal ump size betwee tradig opportuities. 4.. Numerical Algorithm We have show that: r% V( S, q, m, ) = max E[ V( S( + r), q r, m, )] st.. Er% = 0, r% mi( q, m) (6) The above computatio ca be simplified by otig that we oly eed to cosider biary radom variables, amely: ru with probability π r% = rd with probability -π where r > 0, r < 0, π r + ( π ) r = 0. Hece, we ca write: u d u d r r V( S, q, m, ) = max { V( S( + r ), q r, m, ) + V( S( + r ), q r, m, ) d u ru, rd u u d d ru rd ru rd st.. r [0,mi( q, m)], r [ mi( q, m),0] u d This problem is straightforward to solve umerically. 9

21 4.3. Closed form solutio Case : Boudig the per-period retur If q > m, the oly relevat costrait is the boud m o the magitude of the stock s retur each period. I this case we suppress the argumet q, ad write V(S,, m) for the optio price boud. The followig result shows that i this case, the optimal boud is equivalet to the value of the optio computed usig a -period biomial model. PROPOSITION 5. The optio price boud V(S,, m) is equal to the value of the optio whe the stock has a biomial distributio with returs r t { m,m} each period. PROOF: Because V is covex i S, V(S(+r),, m) is covex i r. Therefore, (6) is solved with r% takig o the extreme values of m ad m with equal likelihood. Case : Boudig the quadratic variatio If q < m, the the boud o the per-period retur is ot bidig. We ca suppress the third argumet, we also look at the limitig case whe N ad write (, ) V S q. I this case we have: V( S, q ) = mi max V( S, q r ) rs (7) r [ q, q] ad V( S,0) = max{0, S k}. We argue that: ( ) / q q ( q ) S for S s0 = /( q ) q V( S, q ) = / q q (( q ) S) + S for S s0 = /( q ) + q (8) Oe ca umerically verify the above expressio usig the umerical procedure we described i subsectio 4.. We are uable to prove it aalytically ad istead provide few propositios that reveal how we foud the above expressio. 0

22 LEMMA 4. Let V be defied by (7), ad let be the umber of shares i the optimal portfolio. If V is differetiable with respect to the secod argumet the V (S, q ) =. Based o the above propositio we coclude that if V is differetiable with respect to the secod argumet the (7) implies that: V( S, q ) + V ( S, q ) Sr V( S( + r), q r ) for all r q (9) The boudary coditio is give by: V( S,0) = max{0, S K} (0) Whe r = ± q, (9) combied with the boudary coditio yields: V( S, q ) V ( S, q ) Sq 0 () V( S, q ) + V ( S, q ) Sq S( + q) () We derive the expressio i (8) by coecturig that at each poit oe of the costraits i () ad () bids. Hece, LEMMA 5. Let V be defied by (9) ad (0), the V(S, q ) V (S, q ), where V * is defied by (8) Compariso to Black ad Scholes Formally the Black ad Scholes model is ot ested i our model. This is due to the fact that Black ad Scholes is a cotiuous time model while we cosider a discrete time model. Nevertheless oe ca overcome this techical differece usig the fact the Black ad Scholes model is a limit of discrete time models. Black ad Scholes model with volatility σ ca be expressed as the limit of biomial trees where the quadratic variatio is q =σ². If we let m deote the daily retur ad N the umber of periods the we keep m²=σ² as N goes to ifiity. Sice the bouds we derive do ot deped o the umber of periods we ca coclude that the Black ad Scholes price

23 with volatility σ² is ot higher the our upper bouds whe σ² is a boud o the quadratic variatio. Oe may woder about the restrictio of costat quadratic variatio i the cotext of Black ad Scholes. Whe we look at a geometric Browia motio at discrete itervals, the icremets are ormally distributed which may suggest that we allow for ubouded quadratic variatio. However, i such a case the discrete Black ad Scoles tradig strategy fails to replicate the optio payoff; moreover, the loss is ubouded. A differet way of sayig this is that while we ca defie as limit of differet sequeces of discrete time processes oly particular sequeces yield the Black ad Scholes equatio: i.e. biomial trees. At the cotiuous time limit almost surely a paths is cotiuous ad has a fixed quadratic variatio The above graph illustrates the relatio betwee the Black ad Scholes price ad the expressio we derive i (8). The bottom lie is the itrisic value of the optio where K = as a fuctio of S ; hece, it is max{0, S }. The middle lie represets the value of a optio as accordig to Black ad Scholes whe σ = 0., while the top lie represet the expressio i (8) whe q = 0. A alterative way of represetig this relatio is by lookig at the implied volatility. That is we use the Black ad Scholes formula to solve for the volatility:

24 BS Implied Volatility at Boud Stock Price It is iterestig to see that the boud we derived has a smile feature as the implied volatility is higher for optios that are deep out of the moey or i the moey. Fially we plot the umber of shares i our hedgig strategy,, ad compare it to the hedgig strategy i Black ad Scholes Delta Stock Price 3

25 5. Coclusio To be added 6. Appedix PROOF OF PROPOSITION. To apply Blackwell theorem oe eeds to describe the setup as a game with vector i payoffs. We let a, deote the vector payoff of alterative i at time whe we defie. Our payoff at time is give by a ξ,, which measures our regret i, a π i, π i', relative to the idividual strategies. Followig Hartt (003) we let a = a Usig this formulatio our goal is to coverge as fast as possible to the positive quadrat. I I particular if we let δ = dist( a, R ) the, + δ = max 0, E π ξ max{ } π, i, i = t= = ξ, Claim 5. Whe I = there exists a radomized strategy so that δ m/ Proof. Our strategy is give by: If a 0 ad a 0 : we chose the first alterative so that ξ = with certaity. We are already i the positive quadrat ad hece our actio is arbitrary. If a < 0 ad a < 0 : we radomize ad choose the first actio with a probability pr( ξ = ) =. a + a If a 0 ad a < 0 : we chose the risky asset which is the secod alterative so that ξ = with certaity. If a < 0 ad a 0 : we chose the risk free asset which is the first alterative so that ξ = with certaity. We argue by that, E δ ( ) δ + m, this implies the result by iductio. The claim the otig that δ 0 = 0. We cosider the differet cases:. a 0, a 0 : I this case δ is zero ad hece ξ,, δ E a = a m. 4

26 . a < 0 ad a < 0 I this case that ξ,, a a, i, a + a a + a E a ξ = ( π π )(, ). Note E a m ad that E a ξ, is orthogoal to a. As a result whe we measure our distace from ( 00, ) we obtai ( ) ( ) δ E a = a + E a ( ) δ + m ξ, 3. a < 0 ad a 0 or a 0 ad a < 0 : I this case we are gettig closer to the positive quadrat compared to the case where we are already o oe of the axis, that is, a = 0 or a = 0 ; this is essetially covered by the previous case. PROOF OF PROPOSITION 3 We first establish the followig techical lemma. ( ) LEMMA 6. Assumig that m (0, ), r > m, ad η, m ( m) the: ηl( + r) l( + ηr) ηl( + r) η( η ) r Proof. For the first iequality defie a fuctio f () r = η l( + r ) l( + ηr ) We have f (0) = 0, ad η η η η () r = = f ( ) r + r + ηr ( + r)( + ηr) Hece, for r > 0 the f () r > 0 ad for r < 0 we have f () r < 0. Therefore 0 is a miimum poit of f. For the secod iequality we have: Agai, f (0) = 0 ad f () r = l( + ηr ) η l( + r ) + η ( η ) r η η f () r = + ( ηη ) r = ηη ( ) r + ηr + r ( + r)( + ηr) 5

27 We use a similar argumet as before ad claim that for r > 0 the f () r > 0ad for r < 0 we have f () r 0 <. To show this we oly eed to verify that: ( + r)( + ηr) For r > 0 this clearly holds so we focus o r < 0. I this case, sice the miimum of the expressio is whe r = m, it is sufficiet to guaratee that ( m)( ηm) /. Solvig for η we get, / m η = m( m) m ( m) ad i additio we eed that m <. We ow ca prove the propositio. For each i =,, N we get W w + i, + l l = l wi, 0 + l ( + ηri, ) W W = l w + l( + ηr ) ( ) = i, 0 i, t= i, 0 η i η( η ) i,, t= l w + l( + r ) r = lw + ηl( V) η η Q i, 0 i i where V i, is the value of asset i at time, ad l( + ηz) η l( + z), W W i = i I + + l = l = l ( + ηri, ) xi, W = W = i= I = l η ri xi +,, = l ηrg, + = i = = ηl + rg, = ηl( G ) = Combiig the two iequalities ad dividig by η, we get l wi, 0 l( G) + l( Vi, ) ( η ) Qi η Q = r,. O the other had, usig PROOF OF LEMMA (i) We eed to show that: 6

28 λv( S, K, q, m, ) + ( λ) V( S, K, q, m, ) V( S, K, q, m, ) For all λ [0,] where K = λk + ( λ) K The above holds sice a portfolio of λ optios with strike K ad (-λ) optios with strike K domiates the payoff of a sigle optio with a strike of λ K +(-λ) K. (ii) For a fix K we ote that V(, K / S, q, m, ) is covex i S as give (i) it is a compositio of two covex fuctios; we also ote that it is icreasig i S. The proof the follows as if f(x) is icreasig ad covex i x the so is xf(x) PROOF OF LEMMA. (i) Follows from a simple argumet based o iductio that reveals that f ( W, S, q, m, ) = f(0, S, q, m, ) + W (ii) The fact that f( W, S, q, m, ) 0 follows by iductio sice we ca set r % = 0 The fact that f ( W, S, q, m, ) S follows by iductio usig (i) sice we ca set = (iii) If Er% 0 the sice f (0, Sq,, m, ) [0, S] usig the fact that is ubouded we cam choose so that Ef ( W + rs %, S( + r% ), q r%, m, ) > 0 which is a cotradictio to (ii). (iv)the proof i both cases follows by iductio usig the fact that r% ad V are bouded. PROOF OF LEMMA 3. Sice ErS % + Ef (0, S( + r% ), q r%, m, ) is liear i the distributio of r% ad ad sice the space of r% is compact uder a appropriate topology (C ) eables us to use Sio(958) ad coclude that: {% % r% } f W Sq m W E rs f S r q m (,,,, ) = + supif + (0, ( + ),,, ) r% The fact that the space of σ is compact ad cotiuity i q eables to use miimizatio over q. Sice the fuctio is liear i we eed to show that we ca restrict the domai of to a compact subset of R. As metioed before whe cosiderig ifr% sup E Sr% + Ef(0, S( + r% ), σ r%, ) we ca assume to be zero. Hece we focus o sup if E { Sr f (0 S r ( r ) +, +, σ r, ) } f (0 S σ ) S % % %. Sice we kow that,,, > we ca restrict so that S/ σ. PROOF OF LEMMA 4 Covexity of V i S implies that it is sufficiet to show that: ad ( ( + ε), σ ) (, σ ) limsup V S V S S (3) ε 0 ε 7

29 (, σ ) ( ( ε), σ ) limif V S V S ε 0 ε S (4) Usig (7) we kow that by lettig r = ε : Thus for ε > 0 : V S S V S (, σ ) + ε ( ( + ε), σ ε ) V( S( + ε), σ ε ) V( S, σ ) S ε Hece for (3) oe eeds to show that V( S( + ε), σ ε ) V( S( + ε), σ ) lim = 0 ε 0 ε which holds uder our differetiability assumptio. The proof for (4) is similar whe we take r = ε. PROOF OF LEMMA 5. Note that () is equivalet to V( S, q ) V ( S, q ) Sq which limits the rate of declie of V to the left of (s 0, v 0 ). The steepest declie occurs if () holds with equality. The resultig differetial equatio has the solutio l / q = l V ( S, q ) c S where c l is chose so that V(s 0, q ) = v 0 ; that is, c l = v 0 s 0 /q. Similarly, to the right of (s 0, v 0 ), () determies the miimal rate of icrease of V. The slowest rate of icrease occurs whe () bids, or r / q r V ( S, q ) = c S + S with c r = (v 0 s 0 + ) s 0 /q. Both V r ad V l are icreasig i c r ad c l, ad so are icreasig i v 0. For a give s 0, what is the lowest possible value of v 0? Addig () ad (), we fid that V(S, q ) ½(S (+q) ). Therefore, if we set 8

30 v 0 = ½(s 0 (+q) ) the the true V exceeds V r to the right of s 0 ad V l to the left of s 0. We the fid V by choosig s 0 to maximize V r ad V l by maximizig c r ad c l. I both cases, this occurs with s 0 = /( q ) The result the follows by solvig for c r ad c l give (s 0, v 0 ). 7. Refereces Berardo A. E. ad Ledoit O. (000) Gai, loss ad asset pricig, Joural of Political Ecoomy, 08:44 7. Black F. ad M. Scholes (973) The pricig of optios ad corporate liabilities, Joural of Political Ecoomy, 8: Blackwell D. (956), A aalog of the MiiMax theorem for vector payoffs, Pacific Joural of Mathematics, 6: 8. Borodi A. ad R. El-Yaiv. (998) Olie Computatio ad Competitive Aalysis, Cambridge Uiversity Press. Cochrae J.H. ad J. Saa-Requeo (000) Beyod arbitrage: Good-deal asset price bouds i icomplete markets, Joural of Political Ecoomy, 08:79 9. Cover T. (99), Uiversal portfolios, Mathematical Fiace, :-9. Cover T. (996), Behavior of sequetial predictors of biary sequeces, Trasactios of the Fourth Prague Coferece o Iformatio Theory.. Cover T. ad E. Ordetlich.(996) Uiversal portfolios with side iformatio, IEEE Trasactios o Iformatio Theory, 4: Cover T. ad E. Ordetlich.(998) The cost of achievig the best portfolio i hidsight, Mathematics of Operatios Research, , Darrell Duffie D. (00). Dyamic Asset Pricig Theory, Priceto Uiversity Press, 00. 9

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