Haa ad Blackwell meet Black ad Scholes: Approachability ad Robust Optio Pricig Peter M. DeMarzo, Ila Kremer, ad Yishay Masour October, 2005 This Revisio: 4/20/09 ABSTRACT. We study the lik betwee the game theoretic otio of approachability or regret miimizatio ad robust optio pricig. This cocept was itroduced by Haa ad Blackwell ad is the basis behid a growig literature i game theory ad microecoomics. We demostrate how tradig strategies, based o approachability, that miimize regret also imply robust, ad empirically relevat, upper bouds for the prices of Europea call optios. We the argue that the gradiet strategy proposed by Haa-Blackwell is path depedet ad therefore suboptimal whe fiite horizos are cosidered. Based o path idepedece we solve for the optimal strategy ad boud. Staford Uiversity (DeMarzo ad Kremer) ad Tel Aviv Uiversity (Masour). We thak Sergiu Hart for helpful discussios ad semiar participats at Berkeley (IEOR), Tel Aviv Uiversity, Hebrew Uiversity, Staford, UCLA, Caltech, Duke, UIUC, Uiversity of Chicago, MIT ad Washigto Uiversity for useful commets.
1. Itroductio There is a growig literature i game theory that is based o approachability or regret miimizatio for games uder ucertaity. 1 Regret is defied as the differece betwee the outcome of a strategy ad that of the ex-post optimal strategy (withi a give class). This literature is based o earlier work by Haa (1957) ad Blackwell (1956) who studied robust dyamic optimizatio, ad is the basis for the more recet work o calibratio ad the dyamic foudatios of correlated equilibria; see Hart (2005), Foster, Levie, ad Vohra (1999), ad Fudeberg ad Levie (1998) for excellet surveys. I this paper we cosider a fiacial applicatio of these ideas ad demostrate a lik to the robust pricig of fiacial assets. I particular we focus o stadard Europea-style 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 (1973) ad Merto (1973), 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. o 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. This iterpretatio is iterestig as moetary payoffs provide a tagible way to measure the performace of regret-miimizig strategies (compared with stadard results based o the asymptotic average performace of these strategies). Secod, we look for the optimal such bouds. These bouds provide some measure of the performace of 1 We use the terms regret miimizatio ad approachability iterchageably. Formally, approachability is the more geeral cocept while regret miimizatio is a classic example of it. 1
kow, heuristic regret miimizig strategies (which despite beig asymptotically efficiet eed ot be optimal). The roots of approachability ad regret miimizatio i game theory ca be traced to Haa (1957) ad Blackwell s (1956) work o dyamic optimizatio whe the decisio maker has very little iformatio about the eviromet. They cosidered a ifiitely 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 ostatioary maer, so that learig is ot 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 expost 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. More geerally, approachability ad regret miimizatio ca be viewed as a alterative objective whe optimizig uder ucertaity to the traditioal approach i ecoomics that cosiders a absolute objective fuctio for the decisio-maker (e.g. Gilboa ad Schmeidler (1989)). 2 A regret-miimizig decisio-maker is ot cocered about the absolute performace of her strategy, but rather how well it performs compared to a defied set of alterative strategies. Of course, our purpose is ot to evaluate whether regret miimizatio is a appropriate objective, but rather to cosider the properties of the strategies that achieve it i a fiacial cotext. I a ucertai fiacial market, we ca defie regret as the ratio betwee the ivestor's wealth ad the wealth he could have obtaied had he followed a alterative ivestmet strategy. By comparig the ivestor s payoff to that which could be attaied from a buy ad hold ivestmet of a 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. To see why, suppose we have a strategy that ca guaratee a retur o worse tha 80% of the retur o ivestmet i a stock or i a bod (i.e., has a maximum regret 2 Several papers i ecoomics cosider regret miimizatio as a objective. Bergma ad Schlag (2005) examie a moopolist who miimizes regret. Boze, Ozdorze ad Pape (2004) examie optimal auctios i this cotext. Milor (1954) ad later Hayashi (2005) provide axiomatic foudatios for such prefereces. A geeral computer sciece settig where the performace of olie algorithms is compared to the optimal oe is competitive aalysis (see Borodi ad El-Yaiv (1998)). 2
of 20%). Let the iterest rate o the bod be zero for simplicity. The by borrowig $100 ad ivestig $125 i this strategy, we ca attai a payoff that is o worse tha the payoff of a at the moey call optio o $100 worth of stock. 3 Thus, the value of the call optio caot exceed the iitial ivestmet of $25. Thus a regret guaratee of o more tha 20% is equivalet to a upper boud of $25 for a at-the-moey call optio. I Sectio 2 of the paper, we first review Haa-Blackwell s approachability. While Haa- Blackwell focus o limitig results (similar to the traditioal work o regret miimizatio), we cosider miimizig regret over a fiite horizo. We also costruct a simple but importat geeralizatio of their origial gradiet strategy. This geeralizatio will allow us to improve the fiite-horizo performace of the Haa-Blackwell strategy, ad will also prove extremely useful whe applyig the results to optios with differet strike prices. I Sectio 3 we apply the results to the pricig of call optios. To do so, we eed to adjust for the fact that i a ivestmet cotext, payoffs are multiplicative ad ot additive. We the apply the Haa-Blackwell gradiet strategy, with mixed strategies replaced by portfolio weights, to derive robust upper bouds for the prices of call optios. The bouds we derive are based o o arbitrage ad are robust i that they do ot deped o specific distributioal assumptios for the stock price path. For example, we ca allow both jumps i the stock price process ad tradig halts. The bouds we derive deped solely upo the total quadratic variatio of the stock price path. Because quadratic variatio is a equivalet measure to volatility i the Black-Scholes- Merto (BSM) framework, we ca directly compare our price bouds with those of the Black- Scholes formula. 4 Ulike Black-Scholes, however, our bouds do ot require the stock price paths to be cotiuous. The tradig strategies we develop are simple tred-followig or mometum strategies, ad ulike stadard optio hedgig strategies, these strategies are history depedet. While they are asymptotically optimal, they are ot ecessarily optimal for a fiite ivestmet horizo ad a give quadratic variatio. I Sectio 3 we also show that we ca improve upo the Haa- Blackwell strategy usig our geeralized gradiet approach. I Sectio 4, we further argue that the optimal regret miimizig strategy must be path idepedet; ad hece the Haa- 3 The future payoff of the optio is the max{$100,s 1 }-100 where S 1 is the future value of $100 ivested i the stock. 4 Our results i this regard are related to work by Cover (1991, 1996) o the uiversal portfolio, a dyamic tradig strategy desiged to perform well compared to ay alterative fixed-weight portfolio. 3
Blackwell strategy is suboptimal. This property suggests usig a recursive applicatio of our geeralized gradiet method. We use this idea to solve umerically for the optimal boud usig dyamic programmig ad derive the optimal robust tradig strategy. This strategy 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. This strategy is also the the optimal strategy for miimizig regret i our settig. I Sectio 4.5, we compare our optimal price bouds to the BSM model. Agai. our optimal bouds ecessarily exceed the BSM price thus, we ca iterpret the bouds as the BSM price correspodig to a higher implied volatility. We show that our bouds are sufficietly tight to be empirically relevat. 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 strategy of BSM. We show that it is similar i ature but that the stock positio is more sesitive overall to movemets i the uderlyig stock price. This strategy isures agaist jumps i the stock price that are ot cosidered by BSM, which we demostrate makes it much more robust tha the BSM hedgig strategy whe the assumptio of cotiuous price paths is dropped. 1.1. Literature Review The classic work of Foster ad Vohra (1998) reewed the iterest of game theorists i the importace of the approachability theorem. 5 While much of this literature is focused o abstract settigs a atural questio is whether these results ca be applied to fiacial markets. Our paper tries to build such a lik by examiig the implicatio to the classic Black-Scholes model. 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 (see Bolle ad Whaley (2004)). I additio, this effect is more proouced for call optios whose strike price is low. 5 Recet cotributios iclude Dekel, E. ad Y., Feiberg (2006), Al-ajjar, ad Weistei. (2008) ad Olszewski ad Sadroi (2008). 4
This effect is ofte referred to as the volatility smirk or smile. As a respose to these fidigs, there has bee a active research tryig to modify the Black ad Scholes formula to accout for these discrepacies. 6 These papers examie differet stochastic processes for the idex, with modificatios that iclude jump 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 quadratic variatio. As a result of both academic ad practical iterest there are several papers similar i spirit to our work as the goal is to provide a robust boud by relaxig the specific assumptio o the price process made by Black ad Scholes. For example, it has bee show that the Black-Scholes formula provides a price boud if a maximal istataeous volatility or maximal quadratic variatio is used i place of a costat volatility. 7 These results all maitai the assumptio of a cotiuous price path ad cotiuous tradig, however. These results therefore do ot hold i models with discrete tradig or price jumps (which are obviously relevat i practice), ulike the approach cosidered here. A alterative approach to that take here is developed by Berardo ad Ledoit (2000) ad Cochrae ad Saa-Requejo (2000), who stregthe the o-arbitrage coditio by usig a equilibrium argumet. They assume a specific stochastic process for the stock ad put bouds for the risk-reward ratio that should be achievable i the market. 8 Our research is also related to research i Computer Sciece ad Statistics. I particular there is a literature that applies regret miimizig algorithms (called competitive algorithms i this literature) i the cotext of ivestmets. 9 Most of the literature follows the semial work by Cover (1991), who cosider strategies to optimize the log-ru asymptotic performace of a portfolio relative to fixed-weight strategies. Cover ad Ordetlich (1998) cosider a fiite horizo settig, ad look at strategies that miimize regret agaist all fixed-weight strategies 6 E.g. Pa (2002), Eraker, Johaes ad Polso (2003), ad Eraker (2004). 7 See Shreve, El Karoui, ad Jeablac-Picque (1998), Grudy ad Wieer (1999), Myklad (2000), ad Shafer ad Vovk (2001). 8 See also Lo (1987) ad Bertsimas, Koga ad Lo (2001) who examie strategies that almost replicate the payoff of a optio give a stochastic process for the uderlie stock. 9 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 Tarja (1985)). 5
over this horizo. They the iterpret the result i terms of the price of the exotic derivative that pays ex-post the best costat rebalaced portfolio. While their results are mathematically very elegat, they are ot useful for stadard optios such as call optios that are traded i the market, ad the bouds are much too weak eve weaker tha stadard, static bouds to be of practical or theoretical importace for stadard optios. 2. Haa ad Blackwell s Gradiet Strategy We first examie the origial strategy proposed by Blackwell (1956) for adversarial games. While Blackwell focuses o the ifiite horizo properties of this strategy, we provide a characterizatio of the strategy s fiite horizo performace. We also costruct a simple geeralizatio of the strategy cosidered by Blackwell that will be useful i our aalysis. 2.1. The Multi-Actio Game Blackwell (1956) cosiders a settig i which a aget repeatedly chooses a sigle actio amog I possible alteratives. The realized payoffs of the actios at time are give by π 1 I, with π j,i the payoff of alterative i at time. At this poit we make o assumptios regardig the uderlyig distributio of these payoffs. I particular the payoffs eed ot be statioary. The aget s actio choice at time is give by the radom variable { 1 I} time is π, ξ, ad the aget s aggregate payoff up to time is π, ξ. ξ... The aget s payoff at = 1 At each date, the aget observes the payoff of every actio choice. Give the history up to date 1 of prior payoffs ad actio choices, the aget s strategy at date specifies a distributio for the actio choice ξ, which we deote by the vector I 1 ρ, where i, Pr( i) ρ = ξ =. Give the few assumptios we have made, the aget does ot seek to maximize absolute performace. Istead, Blackwell cosidered a relative bechmark: How does the aget s aggregate payoff compare to the payoff of a simple static strategy of choosig the same alterative each period? Specifically, we compute the regret with the best static strategy ex-post as: max π π (1) i, i, ξ = 1 = 1 6
2.2. A Gradiet Strategy I his semial paper Blackwell (1956) costructs a simple radomized strategy that miimizes the asymptotic regret i (1). Let A i, deote the aget s relative performace compared to alterative i after periods, A, let A max, {,0 i Ai, } π π i, ', i ', ξ ' ' = 1 + = ad let A deote the vector of relative performace after periods where we suppress i. The radomized strategy is give by: with Pr( ξ = i) = 1/ I for all i if 1, i 0. A Pr( ξ = i) =ρ =, (2) i A+ =, i i' + 1, i + A 1, i' This strategy has a ice geometric iterpretatio. The vector A + represets the loss of the aget at time j relative to the differet alteratives. The aget starts at the origi ad tries to slow his drift away from the o-regret regio, which is the egative orthat show i Figure 1. He does so by choosig a strategy that moves him orthogoally relative to the closest poit of the o regret regio. Specifically, defie the vector ΔA A A 1 to be the chage i the aggregate loss o date j. The give the strategy defied i (2), the expected chage ΔA is coditioally orthogoal to A + 1 :10 E + + + ΔA A 1 π, A 1 =π A 1 π ρ A 1, i 0 i = (3) 10 We use the coditioal expectatio loosely here, as we have ot defied a probability distributio for the payoff vectors. The expectatio i (3) is over the aget s actio choice. 7
Regret vs. Actio #2 A j-1 E[ΔA j ] E[A j ] o Regret Regio Regret vs. Actio #1 Figure 1 Updatig the Loss Vector usig the Blackwell Gradiet Strategy Figure 1 illustrates the updatig whe there are two actio choices. Because of the orthogoal updatig of the loss vector, the squared distace from the o regret regio accumulates i a additive fashio. This fact leads to the followig key result: PROPOSITIO 1. For ay payoff history ( ) above satisfies: π = π,..., 1 π, the gradiet strategy described + 2 i, i π Δ ' π ' = 1 (4) E max A E A Proof: ote that + 2 2 2 2 + + 1 1 2 + +Δ = + Δ + Δ 1 A A A A A A A Therefore, we ca coclude usig (3) that: 2 + 2 π Δ ' π ' = 1 E A E A + + + 2 The result the follows from the fact that E( max i A, i ) E( A ) E( A ). 8
For the special case of two actios, because 2 2 A (,1,2 ) Δ π π we have as a immediate corollary: Corollary A. For I = 2, Eq. (4) is equivalet to 2 max i, i π ( π,1 π,2) (5) j 1 E A + = Hece, 2 ( π,1 π,2 ) is the maximum expected loss of this strategy relative to the best static j= 1 alterative ex-post. ote that this expectatio is with regard to the aget s strategy choice ad does ot reflect ay radomess i the payoff structure as it holds for ay payoff history. evertheless there is ucertaity regardig the actual realizatio of the loss. A Geeralized Gradiet strategy Cosider a simple yet useful geeralizatio of the Blackwell gradiet strategy. As we shall later see this geeralizatio is useful for (i) improvig the boud for at the moey call optios, ad (ii) bouds for optios with differet strike prices We describe it for the special case of just two alteratives I = 2 ; this is the relevat case for our applicatio. Suppose that i the strategy we have just described, istead of startig with a loss vector A 0 = 0, we start at some arbitrary poit A0 = ( x, y ) > 0. Let A ( x, y ) deote our locatio after j rouds. I this case, followig the same logic as above, we have: + 2 2 2 max, (, ) i i π ( π,1 π,2) + + (6) = 1 E A x y x y 2.3. Approachability ad Calibratio While i this paper we focus o fiite horizos, we ote that PROPOSITIO 1 implies the famous Approachability result of Blackwell (1956). Specifically, if the payoffs are appropriately bouded, the the gradiet strategy implies that the aget will have o asymptotic regret: Corollary B. (Approachability) If π ji, π ji, ' is uiformly bouded for all (i, i', j), the with the geeralized gradiet strategy the aget has o log ru average regret: 9
1 1 lim if max j, i j, 0 π π j a s i 1 ξ.. = = 1 (7) The proof for this follows the fact that Eq. (4) implies that the average regret goes to zero at a rate of -1/2 ad usig strog law of large umbers. To appreciate this remarkable result, cosider the followig example: EXAMPLE (Approachability ad o Regret): Suppose a aget must repeatedly predict the outcome of a coi toss, where the coi eed ot be a fair oe. We are iterested i the success rate he ca guaratee i the log ru, that is i the limit as the umber of coi tosses goes to ifiity. If the aget repeatedly predicts heads or tails radomly with equal probability, the i the log ru his success rate is 50% with probability oe. He could do better if he kew the parameter of the coi. If the probability of heads, p, is more tha 0.5, the he could always aouce heads ad achieve a success rate of p. Coversely if p < 0.5 he could achieve a success rate of 1 p by always predictig tails. ote that if the same coi is used over ad over agai the fact that p is ukow is ot a issue i the log ru. The aget ca first focus o estimatig p, as failure i the first rouds does ot affect his log term success rate. Oce he lears p he ca guaratee a success rate of max{p,1 p}. We ca describe this i terms of regret miimizatio by cosiderig two simple strategies: (i) always predict heads, ad (ii) always predict tails. Thus, we ca obtai zero asymptotic regret provided that the sequece is statioary; we use the same coi over ad over agai. Corollary B implies that the aget ca achieve a similar performace eve if the coi ca chage over time ad learig is ot possible. That is, the geeralized gradiet strategy achieves a asymptotic success rate of max{p(),1 p()} where p() deotes the fractio of heads i the first rouds. 11 This example demostrates the lik to calibratio. It implies that it may be difficult to tell whether a aget ideed kows the distributio of the coi, as the aget achieves the same performace eve whe he kows othig about the coi. This surprisig coclusio led to the 11 ote that p() eed ot eve coverge as goes to ifiity. Hece, the formal statemet of the result is that for ay choice of cois limif { X{ } max( p( ),1 p( ))} 0 almost surely, where X() deotes the success rate i the first tosses. 10
growig literature o calibratio that further ivestigates the problem of asymptotically testig whether a expert is ideed kowledgeable. 12 3. Regret-Based Bouds for Call Optios I this sectio we first describe a simple tradig model. This model is differet from the setup we described before i that it is multiplicative i ature ad ot additive. We show to adapt the origial Blackwell strategy ad costruct gradiet tradig strategies. We the show how this ca traslated to a upper boud for the value of a Europea call optio. 3.1. A Simple Fiacial Tradig Model Cosider a discrete-time -period model where time is deoted by {0,, 1, }. 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 = 1, ad assume that the asset does ot pay ay divideds. We deote by r the retur betwee 1 ad so that S (1 ) = S 1 + r. We call r = r 1,, r the price path. I additio to the risky asset we have a risk-free asset (e.g., a bod). Uless otherwise stated, we assume that the risk-free rate is zero. A dyamic tradig strategy specifies a portfolio to hold at each date. This portfolio has iitial value G 0 = 1. Each period, the curret value of the portfolio some fractio G is ivested i the assets, with w i the risky asset ad 1 w i the risk-free asset. The portfolio weight w is specified as a fuctio of the price path of the stock up to date. Sice we assume a zero iterest rate, at time + 1 its value is G+ 1 = ( wg)(1 + r+ 1) + (1 w) G = G(1 + wr+ 1) ; its fial value is each date. G. Thus, a tradig strategy determies a mappig G (r) from price paths to payoffs at I sectio 2.1 we itroduced a regret measure i a additive framework with multiple alteratives. Here we have just two alteratives: ivestig i the risk free asset or the stock. Also, because the model of returs is multiplicative, it is atural to measure regret i percetage terms, so that the aalog of the regret measure i (1) is: 12 See e.g. Foster ad Vohra (1997, 1998), Foster (1999), Dekel ad Feiberg (2006), Al-ajjar ad Weistei (2008), Wojciech ad Sadroi (2008). 11
3.2. The Gradiet Tradig Strategy max{1, S} G G = 1. (8) max{1, S } max{1, S } To adapt the costructio described i Sectio 2 to our settig, oe eeds to make a few adjustmets. First, sice returs are compouded, our setup is multiplicative rather tha additive. Secod, we seek a determiistic strategy rather tha a radomized oe; we use portfolio weights to replace radomizatio. First, we cosider two alterative bechmarks based o the two fiacial assets by lettig the payoffs i, π be the log returs of each asset: ( r ) determiistic strategy by ivestig a fractio of w Pr ( ξ 1 A 1) i the risk-free asset at time j. π,1 = l 1+, π,2 = 0. Secod, we costruct a = = i the risky asset ad 1 w Our retur at time j is give by 1+, ad our fial payoff is give by Π (1 ) 1 + wr. Sice [ 01] w, ad r > 1 we have that: 13 Hece we coclude that: 14 wr ( + wr ) w + r = E π, ξ π A 1 l 1 l(1 ), ( wr ) E, l 1+ π π = 1 = 1 ξ = Because ( π π ) = (l(1 + r )), we ca apply Corollary A to coclude that: 2 2,1,2 13 For a give [ 01] w, let f () r = 1+ wr ad g() r = (1 + r) w. ote that f(0) = g(0) = 1, f '(0) = g'(0) = w. Sice g is cocave while f is liear i r we have that f () r g() r for r > 1. 14 ote that the use of log returs is crucial here. Cosider a two period model i which the stock price doubles i both periods with certaity. Suppose that a ivestor first chooses with equal probabilities whether to ivest his etire wealth i the stock or the bod. 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 1 with probability 0.5 ad 4 with probability 0.5 so o average 2.5. 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 2.25 with certaity. However, oce we look at log returs the radomized strategy yields o average 05l(4). = l(2) while the determiistic oe yields 2l(15). = l(225).. 12
PROPOSITIO 2. The gradiet tradig strategy implies G exp( q( r)) max{1, S } where is the quadratic variatio of the log returs, or: q 2 () r r 2. (9) = 1 qr () (l(1 + )) Cosider ow the geeralized gradiet strategy with startig poit ( x, y ). Let A ( x, y ) deote our locatio after rouds. Recall that i this case we have: + E max i A, i( x, y) π q ( r) + x + y 2 2 2 I characterizig the performace of our tradig strategy we must credit back the iitial regret of x with respect to the stock ad y with respect to the bod. Hece, we coclude that: PROPOSITIO 3. The geeralized gradiet tradig strategy implies { ( ) ( ) } 2 2 2 2 2 2 G max exp y q ( r) + x + y,exp x q ( r) + x + y S, 3.3. Price Bouds for at-the-moey Optios I the previous sectio we described a family of tradig strategies ad characterized their regret coditioal o the realized price path. We ow show how we ca use this result to provide upper bouds for the value of optios. Let Φ be a set of possible price paths for the stock. Coditioal o this set we assume that there is o arbitrage i prices. amely, for ay tradig strategy G such that G (r) > 1 for some r Φ, there exists aother price path r Φ such that G ( r ') < 1. Otherwise, ivestig i G ad shortig the bod would lead to a profit at date j give path r 1 with o possibility of a loss. 15 A Europea call optio with strike price K that matures at time has a fial payoff of max{0, S K}. Let Φ be the set of feasible stock price paths, ad let C(K Φ) be the highest value of the call optio at time 0 that is cosistet with o arbitrage. Before proceedig we should discuss the importace of imposig restrictios o the price path give by Φ. The first part of Merto s (1973) paper addresses this questio by askig what ca 15 A arbitrage opportuity is ay tradig strategy that geerates a profit i some state without the possibility of a loss. The coditio give here is sufficiet to rule out arbitrage opportuities usig the stock ad the bod. 13
be said about the value of a call optio without makig ay additioal assumptios about the price path. The aswer is that CK ( Φ) S Hece, the optio is ot more valuable tha the uderlyig asset. This is a very weak boud but caot be improved if arbitrary price paths are allowed. Our goal is to fid bouds for the optio value C(K Φ) usig the gradiet tradig strategies; i this sectio we begi by cosiderig the case of a at-the-moey optio (K=1). We begi by formalizig the lik betwee regret limitig strategies ad optio prices discussed (as a example) i the itroductio: PROPOSITIO 4. Suppose we have a dyamic tradig strategy that satisfies { } G β()max1, r S, r Φ. Let { r } * β = if r Φ β ( ) (10) The o arbitrage implies the followig upper boud for the value of a at-the-moey call optio: 0 1 C(1 Φ) * β 1 (11) Proof: Ivestig 1/β * i the tradig strategy ad borrowig $1 leads to a payoff G() r β()max{1, r S} 1 1 max{0, S 1} * * β β QED To gai some ituitio, cosider a very simple tradig strategy. Suppose we decide to use a buy ad hold strategy i which we ivest a fractio β = 12 i both assets. The future payoff of this static portfolio is { } G = 0.5 + 0.5S max 0.5, 0.5S ote that the above holds for all possible price paths. Usig the above propositio we coclude that: 1 C(1 Φ) 1 = 1 = S 0.5 0 14
As metioed before, S 0 is the simple kow upper boud from Merto (1973). To improve upo this boud, we must cosider strategies with better performace guaratees that lead to a higher β. Cosider first the simple gradiet tradig strategy ad defie q( Φ ) = sup q( r), the highest possible realized quadratic variatio of the log returs for the price paths i Φ. Combiig PROPOSITIO 2 ad PROPOSITIO 4, we have that r Φ C(1 Φ) exp( q( Φ)) 1 (12) ow cosider geeralized gradiet tradig strategy with x=y; i this case, 2 2 ( ) β () r = exp x q () r + 2x Takig q( Φ ) ad maximizig over x we get x * ( q) = q( Φ )/ 2. Hece, we obtai the followig stroger result: PROPOSITIO 5. Based o the geeralized gradiet tradig strategy, the o arbitrage price of a at-the-moey call optio satisfies 1 ( ) C(1 Φ) exp q ( Φ) 1 (13) 2 By focusig o small q we ca quickly compare it to the Black-Scholes optio pricig formula. Usig the origial gradiet tradig strategy we have: ( ) C(1 Φ) exp q( Φ) 1 q( Φ ) for small q. (14) Usig the optimal geeralized gradiet tradig strategy we have: ( ) C(1 Φ) exp q( Φ) 2 1 q( Φ ) 2 for small q. (15) Whe cotrastig these bouds with the Black-Scholes optio pricig formula, ote that the Black-Scholes formula also assumes the stock price path is cotiuous, ad the volatility remais costat over time. Give a volatility of σ for the stock price, ad the opportuity to trade cotiuously util the optio s expiratio o date T, the Black-Scholes assumptios o the stock price path implies q(r) = σ T for all r Φ BS. We have ot made such assumptios (amely the 15
stock price ca jump ad tradig may ot be cotiuous) ad so the bouds we derive are ecessarily weaker tha that give by Black-Scholes. I particular, we have 16 C(1 Φ ) q( Φ ) 2π for small q. (16) BS BS Comparig equatios (15) ad (16), we ca see that optio price boud provided by the geeralized gradiet tradig strategy exceeds the Black-Scholes price by a amout equivalet to a icrease i the stock s volatility by a factor of π, or approximately 77%. This icrease bouds the impact o the optio price from relaxig the Black-Scholes assumptio of cotiuous price paths ad costat volatility. 3.4. Differet Strike Prices We ca geeralize the methods of Sectio 3.3 to optios with differet strike prices usig the followig result: PROPOSITIO 6. Suppose we have a dyamic tradig strategy that satisfies { } G max α( r), β( r) S, r Φ ad let { } * β = Φ β α if r ( r), ( r) / K (17) The o arbitrage implies the followig upper boud for the value of a call optio with strike price K: 1 C(K Φ) * K (18) β Proof: Ivestig 1/β * i the tradig strategy ad borrowig $K leads to a payoff G() r max{ α(), r β() r S} K K max{ K, S } max{0, } * * K = S K β β QED I PROPOSITIO 3 we demostrated that the geeralized gradiet tradig strategy with iitial regret (x, y) provides a performace guaratee of 16 To derive this result, let f ( σ) be the Black-Scholes value of a at-the-moey call optio with expiratio T whe the stock s price is 1, its volatility is σ, ad the iterest rate is zero. The f ( σ) is the correspodig vega 1 (volatility sesitivity) of the optio, ad for σ small, ( ) (0) '(0) 0 ( ) f σ f + f σ= + T σ. 2π 16
2 2 2 α () r = exp ( y q () r + x + y ), β () r = exp ( x q ) 2 () r + x 2 + y 2 ote that i this case, ( ) the from (17), α () r = exp y x β () r. If we defie q( Φ ) = sup q( r) ad k = l K, { } l mi 0, ( ) β * = y x k + x q 2 Φ + x 2 + y 2 (19) Eq. (19) implies that to maximize β, we should choose y x = k. For if y x > k, we ca icrease β by icreasig x, ad if y x < k, we ca icrease β by icreasig y. Thus, we choose y = x + k, ad (19) reduces to r Φ l ( ) ( ) * 2 2 2 β = x q Φ + x + x+ k (20) Thus, the quality of the optio price boud provided by the geeralized gradiet strategy depeds upo the choice of x i Eq. (20). Recall that x determies the iitial regret poit, A 0 = (x, y = x + k). Because the iitial regret must be o-egative, we require x max(0, k) (21) The best boud possible usig the geeralized gradiet strategy ca thus be foud by choosig x to maximize (20) subject to (21). The optimal choice of x depeds upo the maximal quadratic variatio q 2 as follows: x ( q) = 2q + k k (22) * 1 2 2 1 2 2 Replacig x i (20) with x (q(φ)) leads to best possible performace guaratee: l β = 2 q ( Φ ) + k k (23) * 1 2 2 1 2 2 Combiig (23) ad PROPOSITIO 6, the gradiet tradig strategies imply the followig o arbitrage boud for the price of a call optio: PROPOSITIO 7. The o arbitrage price of a call optio with strike price K satisfies 2 2 ( ) 1 1 2 2 CK ( Φ) exp 2 q( Φ ) + (l K) + l( K) K (24) 17
This boud is achieved by borrowig K ad ivestig the total i the geeralized gradiet tradig strategy with iitial regret x = x (q(φ)) ad y = x + l K. Agai, we ca compare the optio price boud from the geeralized gradiet strategy with that of Black-Scholes. Figure 2 shows the geeralized gradiet boud, expressed as a Black- Scholes implied volatility, whe q = 10%. We ca iterpret the figure as showig the potetial impact o the value of the optio from droppig the Black-Scholes assumptios of a cotiuous price path with a costat volatility, where that impact is expressed i terms of the equivalet icrease i the volatility of the stock. Figure 2 Geeralized Gradiet Bouds versus Black-Scholes 4. Optimal q-based Bouds Thus far, we have demostrated that we ca obtai meaigful optio price bouds by extedig ad geeralizig Blackwell s gradiet strategy to a ivestmet tradig eviromet. These bouds deped upo the maximal quadratic variatio of the possible stock price paths, q 2 (Φ). I this sectio we take a differet approach, ad defie the set of allowable stock price paths as all paths with quadratic variatio less tha some boud: { } Φ ( q) = r q( r) q 18
We the ask, what is the best boud available for the set Φ( q )? That is, what is CK ( Φ( q ))? We demostrate how to compute this boud recursively i this sectio. It will improve o the bouds that we derived based o gradiet strategies i Sectio 3. 4.1. Optimality ad Path Idepedece The Haa ad Blackwell gradiet strategy that we examied i subsectio 3.2 is a example of a strategy that is path depedat. That is, the portfolio chose by the strategy depeds o the strategy s past performace. I cotrast, the tradig strategy i Black ad Scholes is path idepedet, i that it oly depeds o the curret stock price ad the volatility of the stock over the remaider of the optio s life. I this sectio we argue that the optimal regret miimizig strategy should also share this property of path idepedece: Defiitio: We say that a dyamic tradig strategy is path idepedet if the portfolio weights at time deped oly o the stock price at time, paths Φ. S, ad the possible future To see why a optimal strategy should be path idepedet, cosider a sceario where at time the stock price is give by S ad the value of our portfolio is w G. Let G, deote the payoff at time of a strategy that starts with a wealth of $1 at ad follows a dyamic portfolio strategy w. The the regret miimizig strategy must solve w w GG, G max mi, w r G Φ w r max 1, max 1, max mi Thus, the optimal strategy is idepedet of = { S } Φ { S } G. A immediate implicatio is that the Haa-Blackwell gradiet tradig strategy is suboptimal. The claim exteds to origial additive setup; also there the Haa ad Blackwell does ot miimize regret over fiite horizo. This is clearly ot a criticism of their result as whe the horizo is ifiite the average regret of their strategy is optimal. Oe way of makig the gradiet strategy to be path idepedet ca be based o our geeralized gradiet strategy. Recall from Sectio 2.2 that we ca iterpret the geeralized gradiet strategy as though we started the game with a arbitrary iitial relative performace vector A 0. The i 19
Sectio 33.3 we computed the optimal geeralized gradiet tradig strategy, ad showed that the optimal startig positio depeded oly o the remaiig quadratic variatio of the price paths, ad the ratio of the curret stock price to the optio strike price. That is, it is optimal at each poit i time to forget the actual past performace of the strategy, ad choose the curret portfolio i a path idepedet maer. I the remaider of this sectio, we implemet this idea by usig backward iductio to solve for the optimal regret miimizig tradig strategy. 4.2. The Super-Replicatig Portfolio We solve for the optimal price boud by computig the lowest cost of a super-replicatig portfolio, which is a portfolio of the stock ad the bod whose value exceeds the value of the call optio after ay move of the stock price. Let V(S, q 2 ) be the cost of the super-replicatig portfolio, give a curret stock price of S, a strike price of 1, ad a future retur path with a quadratic variatio of at most q 2. Obviously, at the boudary q 2 = 0, we have V(S,0) = max{0, S 1}. (25) For q 2 > 0, the fuctio V is a upper boud for the value of the optio if i additio to (25), V satisfies 2 V( S, q ) = mi Δ S+ B Δ, B r r 2 2 ( ) s.. t Δ Se + B V Se, q r for all r such that r q 2 2 (26) That is, V is the miimum cost of a portfolio, which holds Δ shares of the stock ad B uits of the bod, such that give ay feasible retur for the stock, the ew value of the portfolio will be sufficiet to purchase the optio give the ew stock price ad the remaiig quadratic variatio of the returs. The optimal boud is the smallest fuctio V that satisfies (25) ad (26). To see that V is welldefied, ote that V(S, q 2 ) = S for q 2 > 0 is a solutio, ad that if V 1 ad V 2 are solutios, so is mi(v 1, V 2 ). Fially V is bouded below by the itrisic value of the optio. 20
4.3. A Lower Boud While we caot solve for isight ito the form of we derive a aalytic lower boud for true boud. * V aalytically, we ca compute it umerically. But to gai further * V, as well as the behavior of the worst case price paths, i this sectio * V that turs out to be a very good approximatio of the The limit o the quadratic variatio of the stock price path bouds the size of ay jumps i the stock price. Specifically, the costrait r 2 q 2 implies that the percetage jump i the stock price must be withi the rage [ q, q ], where + q q = + e 1 ad 1 q q = e (27) I this sectio we argue that a lower boud for V is equal to the fuctio ˆ V defied as follows: qq + q + q 1/ q for S 1 2 + ˆ(, ) qq + 1/ q+ + 1 for 1 V S q q + q + S S S S (28) To motivate the fuctio ˆ V, suppose S = 1 ad = 1. The, the highest value for the optio is obtaied if the stock price moves maximally, to (1 + q + ) or (1 q - ). The risk-eutral probability 17 that the stock price rises is the q ( q + q ), so that the value of the optio is + ˆ 2 [ q ( q + q )][(1 + q ) 1] = q q ( q + q ) = V(1, q ). Thus, this must be a lower boud for V + + + + whe S = 1. ext suppose S < 1. Suppose the stock price drifts up slowly at some rate ˆr dt util the stock price equals 1, but with some chace stock price drops q - ad stays there. If there is o drop, the stock price will reach 1 at time t such that Se rt ˆ =1. Because the risk-eutral hazard rate that the stock price will drop is ( rq ˆ ) dt, 18 the risk-eutral probability that there is o drop before time t is ( rq ˆ ) t 1 q e = S. Give the value of the optio whe S = 1 calculated earlier, this leads to the 17 That is, the probability that makes the expected retur of the stock equal to the risk-free rate of zero. 18 If the arrival rate of the drop is λdt, the for the expected retur to be zero, λ q - dt = ˆr dt. 21
lower boud for V whe S < 1 show i (28). 19 A symmetric calculatio (assumig the stock price drifts dow to 1 or jumps up maximally) leads to the lower boud for V whe S > 1. 20 Thus, we have show by costructio the followig result: 2 2 PROPOSITIO 8. If V satisfies (26), the V( S, q ) Vˆ ( S, q ). For a at-the-moey call optio, this lower boud implies that exp( q) + exp( q) 2 C(1 Φ) q( Φ) 2 exp( q) exp( q) for small q. (29) Comparig (29) with (15) ad (16), we see that there is room for substatial improvemet beyod the geeralized gradiet strategy, but that (ot surprisigly) there is still a sigificat icrease i the potetial value of the optio from relaxig the Black-Scholes assumptios. 4.4. umerical Computatio of V We ca compute the optimal boud V via backward iductio o the umber of remaiig price movemets. The limitig case as becomes large is equivalet to the solutio whe there is o limit to the umber of price movemets. Specifically, let V(S,q 2,) be the value with periods remaiig, ad ote that V S q 2 (,,0) max{0, S 1} = (30) The we ca compute the boud for > 0 iductively as follows: 2 V( S, q, ) mi Δ S+ B Δ, B r r 2 2 ( ) s.. t Δ Se + B V Se, q r, 1 for all r such that r q 2 2 It is easy to show by iductio that for all, V is weakly icreasig i ad V V * (31). Thus, V coverges mootoically to * V. ote also that to facilitate computatio, we ca use the result of sectio 4.3 ad start the umerical calculatio with 19 ote that if the stock price drifts smoothly, the remaiig quadratic variatio is uchaged at q 2, as a smooth cotiuous drift has o quadratic variatio. 20 ote that we ca compute the value of the call optio as the value of a put optio plus the stock plus $1 of debt. 22
ˆ 2 2 V( S, q,0) = V( S, q ) (32) i place of (30). We fid umerically that while the optimal boud does ideed exceed ˆ V, the differece is rather egligible. For example, for q = 50%, the optimal boud V * exceeds ˆ V by leas tha 0.03%, ad the gap disappears as q approaches 0. Thus ˆ V provides a very useful approximatio to the optimal boud. 4.5. Compariso with Black-Scholes Agai, we ca compare the optimal q-based boud with the optio price that would be obtaied if the Black-Scholes assumptios were satisfied, as well as the geeralized gradiet strategy of Sectio 3. Figure 3 shows the compariso i terms of the correspodig Black-Scholes implied volatility. Figure 3 Optio Price Bouds versus Black-Scholes I Figure 4, we show the optimal hedgig strategy for the optimal q-based boud. Specifically, give the boud V, we ca compute the umber of shares of the stock Δ to hold at each poit i time i order to super-replicate the optio at the lowest possible cost. Ideed, from Eq. (26), we have Δ = V 1(S,q 2 ), show i Figure 4 for the case q = 0.10. As a compariso, we also show the 23
delta from the Black-Scholes tradig strategy give a volatility of q, as well as the implied volatility of the optio if it were tradig at the price implied by the boud. Figure 4 Delta Hedgig Strategy for the Optimal q-based Boud versus Black-Scholes Fially, we compute the cost of hedgig a call optio usig the tradig strategy from the optimal q-based boud. By defiitio, the cost of hedgig the optio is at most V (S,q 2 ). But because the tradig strategy oly super-replicates, rather tha replicates, the optio, the cost will be less tha this amout for almost all price paths. I Figure 5 we compute the rage of hedgig cost for a at the moey optio (S = 1) with a iitial quadratic variatio of q = 0.50. We also show the same rage whe we restrict the price paths to be cotiuous. As a bechmark we compare this with the rage for the Black-Scholes strategy. ote that with a cotiuous price path ad a fixed quadratic variatio, the cost usig Black-Scholes is idetically equal to the Black-Scholes optio value for all such price paths. But oce we allow for jumps, the worst case uder Black- Scholes is ideed worse tha that uder the optimal strategy. 24
Figure 5 Hedgig Cost usig the Optimal q-based Boud versus Black-Scholes 5. Coclusio We have examied the relatio betwee cocepts of approachability ad regret miimizatio i games ad o arbitrage price bouds i fiacial markets. We demostrate that the simple strategies of Haa ad Blackwell ca be used to derive dyamic tradig strategies that imply robust upper bouds for the value of a call optio uder differet assumptios. We the showed that these strategies are path depedet ad thus suboptimal. We the solved for the optimal upper boud for the optio price, which simultaeously implies the regret miimizig strategy i the dyamic tradig game. The optio price bouds that we derive are for stock price paths whose quadratic variatio is bouded, as i the Black-Scholes framework, but where we have dropped the assumptio of cotiuity ad cotiuous tradig. Oe strikig result of our aalysis is that the optimal quadratic variatio based bouds that we derive are close to the optio prices derived i a Black-Scholes framework. For at-the-moey call optios, our bouds lead to Black-Scholes implied volatilities that are roughly 25% higher tha what oe would calculate based o the quadratic variatio of the price path. I other words, relaxig the Black-Scholes assumptio that price paths are cotiuous leads to relatively small chage i the price of the optio. Moreover, 25
the implied volatilities predicted by our model are much closer to those foud empirically. This result suggests that our bouds may be relevat for optio pricig i practice. 6. Refereces Al-ajjar, abil, ad Joatha Lewis Weistei. 2008. "Comparative Testig of Experts," Ecoometrica., forthcomig Bergma D. ad K. Schlag (2005) "Robust Moopoly Pricig: The Case of Regret," workig paper, Yale Uiversity. Bertsimas D., L. Koga ad A.W. Lo (2001) Hedgig derivative securities ad icomplete markets: a ε-arbitrage approach, Operatio Research 49(3) 372-397. Bolle.P ad R. Whaley (2004) Does et buyig Pressure Affect the Shape of Implied Volatility Fuctios?, Joural of Fiace 59(2) 711-754 Berardo A. E. ad O. Ledoit (2000) Gai, loss ad asset pricig, Joural of Political Ecoomy, 108:144 172. Black F. ad M. Scholes (1973) The pricig of optios ad corporate liabilities, Joural of Political Ecoomy, 81:637 654. Blackwell D. (1956), A aalog of the MiiMax theorem for vector payoffs, Pacific Joural of Mathematics, 6:1 8. Borodi A. ad R. El-Yaiv. (1998) Olie Computatio ad Competitive Aalysis, Cambridge Uiversity Press. Bose, S., E. Ozdorze ad A. Pape (2004) Optimal Auctios with Ambiguity, workig paper, Uiversity of Texas, Austi ad Uiversity of Michiga. Cochrae J.H. ad J. Saa-Requejo (2000) Beyod arbitrage: Good-deal asset price bouds i icomplete markets, Joural of Political Ecoomy, 108:79 119. Cover T. (1991), Uiversal portfolios, Mathematical Fiace, 1:1-29. Cover T. (1996), Behavior of sequetial predictors of biary sequeces, Trasactios of the Fourth Prague Coferece o Iformatio Theory.. 26
Cover T. ad E. Ordetlich.(1996) Uiversal portfolios with side iformatio, IEEE Trasactios o Iformatio Theory, 42: 348--368. Cover T. ad E. Ordetlich.(1998) The cost of achievig the best portfolio i hidsight, Mathematics of Operatios Research, 960--982, Darrell Duffie D. (2001). Dyamic Asset Pricig Theory, Priceto Uiversity Press, 2001. Dekel, E. ad Y., Feiberg 2006, o-bayesia Testig of a Stochastic Predictio Review of Ecoomic Studies, 73, 4, pp. 893-906. Eraker B. (2004) Do stock prices ad volatility jump? recocilig evidece from spot ad optio prices, Joural of Fiace, 59:[ 1367-1403. Eraker B., M. Johaes, ad. G. Polso. (2003) The impact of jumps i returs ad volatility, Joural of Fiace, 53:1269-1300. Dea Foster (1999), A proof of Calibratio via Blackwell's Approchability Theorem," Games ad Ecoomic Behavior, 73-79. Foster D., ad R. Vohra. (1993) A radomized rule for selectig forecasts, Operatios Research, 41:704-709. Foster D., ad R. Vohra. (1997) Regret i the o-lie decisio problem, Games ad Ecoomic Behavior, 21:40-55. Foster D., ad R. Vohra. (1998) Asymptotic calibratio, Biometrika, 85:379-390. Foster D., D. Levie ad R. Vohra. (1999) Itroductio to the Special issue,, Games ad Ecoomics Behavior, 29:1-6. Fudeberg D., ad D. Levie (1998) Theory of Learig i Games, Cambridge, MA: MIT Press Gilboa I. ad D. Schmeidler (1989) Maxmi expected utility with o-uique prior, Joural of Mathematical Ecoomics, 18:141-153. Grudy B. ad Z. Wieer (1999) The aalysis of Deltas, State Prices ad Var: A ew Approach, Workig paper 27
Haa J. (1957) Approximatio to bayes risk i repeated plays, I M. Dresher, A. Tucker, ad P. Wolfe, editors, Cotributios to the Theory of Games, 3: 97--139. Priceto Uiversity Press, 1957. Hart S. (2005) Adaptive Heuristics, Ecoometrica 73:5:1401-1431 Hart S., ad A. Mas-Colell (2000) A simple adaptive procedure leadig to correlated equilibrium, Ecoometrica, 68:1127-1150. Hayashi T. (2005) Regret aversio ad opportuity depedece, workig paper Uiversity of Texas, Austi. Merto, R. C. (1973) Theory of ratioal optio pricig, Bell Joural of Ecoomics ad Maagemet Sciece, 4 (1):141-183. Milor J. (1954) Games Agaist ature, I Decisio Processes ed R.M. Thrall, C.H. Coombs & R.L. Davis. ew York Riley. Lo A.W. (1987) Semi-parametric upper bouds for optio prices ad expected payoffs, Joural of Fiacial Ecoomics 19:373-387. Myklad P. (2000) Coservative delta hedgig, The Aals of Applied Probability, 664 683. Pa J. (2002) The jump-risk premia implicit i optios: Evidece from a itegrated time-series study, Joural of Fiacial Ecoomics, 63: 3-50. Shreve, S.,. El Karoui, ad M. Jeablac-Picque (1998) Robustess of the Black ad Scholes formula, Mathematical Fiace, 8:93-126. Sio M. (1958), O geeral mimax theorems, Pacific Joural of Mathematics, 8, 171-176. Shafer G. ad V. Vovk (2001) Probability ad Fiace: It's Oly a Game! Wiley Sleator D. ad R. E. Tarja (1985) Amortized efficiecy of list update ad pagig rules, Commuicatios of the ACM, 28:202-208. Wojciech O. ad A. Sadroi (2008) Maipulability of Future-Idepedet Tests, Ecoometrica, forthcomig. 28