The Limitations of NoArbitrage Arguments for Real Options Friedrich Hubalek Walter Schachermayer Working Paper No 58 October 1
October 1 SFB Adaptive Information Systems and Modelling in Economics and Management Science Vienna University of Economics and Business Administration Augasse 6, 100 Wien, Austria in cooperation with University of Vienna Vienna University of Technology http://wwwwuwienacat/am This piece of research was supported by the Austrian Science Foundation (FWF) under grant SFB010 ( Adaptive Information Systems and Modelling in Economics and Management Science )
The Limitations of NoArbitrage Arguments for Real Options F Hubalek and W Schachermayer October 5, 1 Abstract We consider an option which is contingent on an underlying that is not a traded asset This situation typically arises in the context of real options We investigate the situation when there is a surrogate traded asset whose price process is highly correlated with that of An illustration would be the cases where and model two different brands of crude oil The main result of the paper shows that in this case one cannot draw any nontrivial conclusions on the price of the option by only using no arbitrage arguments In a second step we try to isolate hedging strategies on the traded asset which minimize the variance of the hedging error We show in particular, that the naive strategy of simply replacing by fails to be optimal and we are able to quantify how far it is from being optimal 1 Introduction The success of the celebrated BlackScholes formula in the context of pricing and hedging derivative securities in financial markets is largely due to an important feature of this model: the prices obtained in the BlackScholes model do not depend on preferences but only on a noarbitrage argument This pleasant fact corresponds to a basic feature of this model: any derivative security (eg, a European call option just as well as a more complicated pathdependent option) can be perfectly replicated by an appropriate trading strategy on the underlying asset The methodology of the treatment of options in financial markets was extended to the context of real options []; the value of a real option is typically not derived from a traded asset as in the classical case, but rather from a random variable which is not traded in a liquid market Nonetheless one tries to apply similar arguments as in the case of derivatives contingent on traded assets, sometimes referring to the possibility of hedging with surrogate assets; by this we mean a traded asset whose price process is closely related (but not identical) to the price process of the underlying of the real option, which typically is not a traded asset The usual argument in favor of the use of surrogate trading strategies is that a sufficiently good surrogate asset should do just as well, or maybe almost just as well, as the underlying asset itself (if it were a traded asset) In this note we critically analyze this common belief In Section below we formalize the setting of a real option on an underlying, which is not a traded asset, but such that there is a traded assets which is close to We do this by modeling and as geometric Brownian motions with drift, correlated by a correlation coefficient which is close to one (but not equal to one!) For an intuitive illustration one might think of the following situation: An option is written on the price of some brand XYZ of crude oil We assume that there is no liquid market for this brand of crude oil (or, more realistically, for futures contracts on this brand), but there is some other brand UVW of crude oil for which a liquid futures market is available, allowing for (almost) frictionless trading The idea 1
is that the price process of these two brands should be sufficiently similar to justify the use of UVW as a surrogate asset for XYZ The main result of this note shows that for a European Call option written on the underlying we cannot conclude anything on its price by using only noarbitrage arguments: if only trading in the surrogate asset is allowed, then any number in is a possible price for this option without violating the noarbitrage principle From an economic point of view this result is, of course, absurd To fix ideas think of a European atthemoney call option on Nobody will be willing to sell the option at a price of, say, one thousandth of the price of, or vice versa to buy the option at a price of, say, ten times the price of What the theorem states is that such absurd prices are not ruled out by noarbitrage considerations (under the assumption that we are not allowed to trade in the asset ) The message of this theorem is not that applying the BlackScholes methodology to real options is not correct, but: whenever one applies the BlackScholes methodology to real options whose underling is not a traded asset one must be very careful and one has to be aware that preferences, subjective probabilities etc have to come into the play Relying on pure noarbitrage arguments does not lead anywhere Having seen that prefect replication of an option written on the underlying is not possible by only trading in, it is natural to ask how small the hedging error can be made One way to quantify the hedging error is by considering the variance of this random variable which subsequently is to be minimized This leads to the wellknown concept of the varianceoptimal martingale measure (see [10] and [1, 13]) which is closely related to the notion of the minimal martingale measure (see [] and [3]) and gives rise to a hedging strategy using the GaltchoukKunitaWatanabe projection in an appropriate Hilbert space Following [11, Example 3] we explicitly calculate this strategy and the variance of the corresponding hedging error and analyze its asymptotic behavior as the correlation coefficient tends to one We also compare this varianceoptimal strategy with other more naive strategies These latter considerations also bear some practical relevance as they clearly show that naive strategies, such as simply replacing by, are not optimal The Main Results We consider a probability space that supports two independent standard Brownian motions and Our time horizon is so that we may write as!, as!, and the filtration "! is defined as the natural (rightcontinuous, saturated) filtration generated by and We also assume $ % which implies in particular that & is trivial Another technical assumption which we make without loss of generality is that '() Fix a constant +*,/01 and define the Brownian motion by ( 35 )6 8 : again is a standard Brownian motion whose correlation to (1) equals, which we should think of to as being close to one We now fix real numbers ;, ;, and <, and strictly positive numbers, define the asset price processes?; where the initial values and random variables and the familiar expression with ; are positive constants Using Itô s formula, it follows that the modeling the terminal values of the corresponding assets are given by GIMNH;, PO/ R BADC EFHGIJ S KADC EF GIMN ; () GIL (3) 0O/ R ()
T X two correlated Gaussian random variables The bond price process is given by T UT$0V0W The financial market is given by the traded assets! and T! and we assume as usual in the BlackScholes world that frictionless trading in continuous time is possible in these assets In contrast, we assume that the asset cannot be traded at all; but it is the underlying for a at time is given by the random variable European call option X The value of X X Y [Z6\: (5) where Z^]_ is the strike price Our goal is to investigate what can be said about the pricing and hedging of X if we only can trade in the bond and the surrogate asset The basic theme in the theory of pricing and hedging derivative securities is to determine the price X of this option at time ` and if possible to justify this price by replicating the option using an appropriate trading strategy on the available traded assets In mathematical terms this amounts to writing the random variable Xa as the sum of a constant, a stochastic integral b [c and an integral b ed, where the predictable processes c! and d!j represent the investments at time in stock and bond respectively If such a replication of an option X is possible, then the usual noarbitrage argument allows to conclude that is the unique arbitragefree price at time U for the option X As announced in the introduction in the present setting we are far away from this situation and we only obtain trivial bounds for the possible arbitragefree prices Theorem 1 Under the above assumptions, for any number f*y the price Xgh of the option at time zero is compatible with the noarbitrage principle More precisely, for every i*u, there is a probability measure j on, equivalent to, such that the discounted traded asset ( V/k W! is a j martingale and such that the pricing rule Xgmln5V k Wpomq)r s[z6 \ut (6) yields the value Xg& Hence the financial market consisting of the traded assets where the option price process X is defined by 5V k Ww k yx omq)r s[z6 \{z!, T admits an equivalent martingale measure and is therfore free of arbitrage! and vx!, t () Proof: The setting of the theorem actually is a special case of a by now wellknown topic, namely the theme of hedging under convex constraints The assertion of the theorem can be derived from [8], Exercise 56, p86 and Example 1, p5 Since the general results of [8], Chapter, require a more complicated machinery and since the subsequent direct argument also allows for some economic interpretation and understanding we give a selfcontained proof of the theorem The basic issue is to determine the set `}~ of all probability measures j on, equivalent to under which the discounted price process O TNY V k W! of the traded asset is a martingale The basic insight of the seminal papers [5] and [6] was that under some regularity conditions the measures j*f } are in onetoone correspondence to the consistent, ie, arbitragefree, pricing rules via formula (6) We refer to [1] for a general version of theses issues which are dealt with in full mathematical rigor and where among other technicalities one has to pass to the concept of local martingales But in the present context these rather subtle considerations are not needed What are the equivalent martingale measures j * `}~ precisely those probability measures j on, such that the logarithmic returns HG 3 in the present situation? They are! as defined
r G G G œ x G r œ in () have drift rate < e O/ under the measure j Indeed, for an equivalent probability measure j, this latter assertion is equivalent to the j martingale property of the discounted traded asset O T Note that this requirement does not imply any restrictions on the drift of the process! under j, as the two Brownian motions and are assumed to be independent under, hence strongly orthogonal This allows us to define for arbitrary * ƒ a probability measure j, such that the drift rate of HG! is <[ O/, but the drift rate of! is with Fix h*ƒ By elementary algebra we can write We define j by + j O <6;?ADC E 5 J, where S Š Š )6 M!< hˆ [ Œ B I v )6 8 ; P 6 h Š < ; I RŽ (8) Lˆ () e& (10) is a uniformly integrable martingale on r t Girsanov s Theorem [, Chapter 35] tells us, that Š! and Š! are two independent standard Brownian motions under j In particular, the random variable w K Š U )6 Š (11) is Gaussian with mean zero and variance under j Now we let vary in ƒ : from Jœ f[z6 \ N 0V)š [Z6 \ N 0V \ ž p š [Z6 \ (1) we immediately see that y P k ª 0V \ ž p«œ š [Z6 \ U P \ ª 0V \ ž/p œ š [Z6 \ 5 (13) almost surely As the expectations, involving lognormal random variables are all finite, one easily verifies by the monotone convergence theorem that these limiting results also hold true for the corresponding expectations under j, ie, y D k ª o q V \ ž p š [Z6\ t y P \ ª o q V \ ž p«œ š [Z6\ t 5 Summing up in less formal terms what we have done so far: For arbitrary h* ƒ, we have constructed a probability measure j, equivalent to the original measure, such that under j the process of logarithmic returns HG! of the traded asset! has the correct drift, namely < [ O/, to make O T a martingale On the other hand j was fabricated in such a way that the process of logarithmic returns! on the nontraded asset has a drift coefficient equal to Speaking economically, for a given value of close to, the choice of the probability distribution j corresponds to the pricing rule applied by an agent believing that the asset will perform very well in the average; on the other hand, a given value of close to $ corresponds to the pricing rule applied by an agent believing that will perform very poorly Not too surprisingly from an economic point of view the above calculations reveal that in the former case an agent will price a (1)
call option on very high while in the latter case she will only be willing to pay a very low price for it This proves the first and the second assertion of the theorem; the third assertion now follows from the general theory as developed in [5] and [6] Remark: We have stated and proved the above theorem under the assumptions of constant coefficients B ; ; But the proof easily carries over to the case, when we assume these quantities to be optional processes We do not carry this out in detail, but only state the result in one important special case: we assume B L ; ; still to be constant, while NH! now is assumed to be an optional process taking values in r M ±/ t Defining! via? 5² )[ we are in an analogous situation as in the above theorem The conclusion generalizing the above theorem now reads as follows: either attains its values almost everywhere (with respect to 5³?, where denotes Lebesguemeasure on r t ) in ±/ µ, in which case we are in the classical situation of a complete market and a European call option on the underlying unique arbitragefree price; or attains values in / (15) can be perfectly replicated by trading in the asset and therefore has a 1 with strictly positive ³? measure, in which case again all prices in are possible arbitragefree prices for a European call option on, if only trading in is permitted The argument is the same as in the above proof with some minor technical modifications 3 Trading strategies related to minimizing the variance of the hedging error We have seen that in the setting of the previous section it is impossible to obtain a perfect replication of an option on by trading on the asset Hence we have to lower the stakes and look for trading strategies on the asset such that the outcome is close to the random variable IZ6 \ The concept of close will be interpreted in terms of the variance of the hedging error, which we shall minimize For the sake of simplicity and in order not to overload the presentation by too many constants, (we are afraid, the reader had already a sufficient dose in the above proof), we shall assume throughout this section that Z, <Š ;N ;N, and N We observe that the assumptions ', <i%, and 6 6' are essentially just normalizing assumptions and do not really restrict the generality (as regards the assumption <¹ one may always take the bond as numeraire to reduce to this case) On the other hand, the assumptions ;º ;ŠY indeed reduce the generality of presentation as this implies that and are already martingales with respect to the original measure The more realistic case ;» and/or ;?» requires more involved arguments and will be treated elsewhere The classical BlackScholes theory applied to the financial market yields the representation where! NHADC¼EJ DÁŠ O/ [Z6\6 à À J[ZMÁº! and T S Áº! F½ / (16) (1) 5
ÍÐ Ð É Ç É É ˆ o É É O with! Ä Å O Z6uÆ, O/ (18) As discussed above, this corresponds to the setting where is a traded asset, in which case perfect replication of the option ¹Z \ by a trading strategy in the asset is possible But our question is: what is the wisest thing to do, if we only can trade in the asset? To formalize this question, we call a predictable process! an admissible trading strategy for the asset if the stochastic integral b ÃÉ É8! is an Ê bounded martingale As in view of our assumptions on the constants ; and we have we also may write the above considered stochastic integral as ÃÉ É!É (1) From basic facts of stochastic integration we infer that a predictable process is an admissible trading strategy iff o+r b tpë We now can formalize our problem of minimizing the variance of the hedging error i ÅKÏ Var f[z6 \ Ì ÃÍ (1) ˆ) ¹ with minimization over all Í*ºƒ, and all admissible trading strategies The term Í b denotes the result of a trading strategy starting with an initial investment Í at time and subsequently holding units of the asset at time ½ (As we assumed for simplicity, that T here is no term corresponding to trading on the bond T ) Our aim is to determine the pair 1Ð ÍF K Ð such that b minimizes 1 à Proposition 1 The optimal solution to the optimization problem (1) is given by the following pair 1Ð ÍF B Ð : 1 Íh Ð DÁŠ F[ZMÁŠ, ie, ÍÐ is just the BlackScholes price of the option s[z6 \ 5 Ð Ñ1Ò Ñ1Ò š, ie, the optimal strategy for trading on simply equals the fold of the optimal strategy for trading on (if this were permitted) times the fraction 3 The variance of the hedging error defined in (1) is given by )6 Var r Proof: Setting Obviously o+r [Z6 \pt we have [Z6 \ ÍÐ ÍÐ s[z6 \ut ÓÐ Í and Var r [Z6\ t oõô The hedging error for starting with capital Ð ÊSN [Z6 \ ÓÐ Í % ˆ Ö ÃÉ )6 Í and trading according to the strategy Ð Ð 6 6? O (0) () (3) is ()
o o Ç and so o r Ê t U and Var r Ê t Y)6 To prove that Ð result of any admissible trading on Let N)6 Var r,z \ t is in fact optimal, it suffices to show, that the hedging error Ê is orthogonal to the d?í with arbitrary Ís*ƒ and arbitrary admissible Then o+r dpês t {[!Í?Í ˆ_Ø (5) (6) () since and are orthogonal martingales Remark: The variance of the hedging error as a function of the correlation is a parabola; it is Fh times the constant Var r 6Z6 \pt As we can write { Y) D!SÙ {, thus the variance of the hedging error tends linearly to zero in )6 It is not difficult to compute explicitly an expression for the constant Var r iz6\ t o r Z6 \ t o+r s[z6 \ut, namely! Var r à [Z6 \pt V Áº 1J ZMÁº Z Áº r PÁŠ F[ZMÁŠ t (8) with and!, as above & Å O Z6 Comparison with a naive strategy We recall once more that, if trading in perfectly, [Z6\6 ºÚ O/ were allowed, we could replicate the option _ À () ºZ6 \ with the BlackScholes price and the corresponding deltahedging strategy Therefore we could try somewhat naively to simply replace by the traded asset We will call this strategy the imitation strategy What is the result of this naive strategy: it simply replicates the option on, ie, s[z6 \ / The hedging error Ê is given by the random variable Ê Y [Z6 \ ( f[z6 \ (30) (31) (3) Again for (mainly notational) simplicity we only discuss the case ;s M/ ;? M/ <± m/ &/Z /U
Û Ú å Ø Ú Var r Ê t æ V1ÁŠ Ü ÁŠ F with coefficient ærv0áº Ü Áº F Áº è é Ú á å Ø Ø Then Ê YV š _1 \ (V _1 \ (33) with denoting a bivariate normal random variable with mean O/, variance, and correlation To state the following result we need the bivariate standard normal distribution function, see [, Appendix 11B] Proposition The variance of the imitation strategy is Var r Ê t Û [Û where H! K (3) (35)!J Þß Here Á denotes the univariate standard normal distribution function and Þ the bivariate standard normal distribution function Asymptotically, as, we have with coefficient V0Áº Ü áuâ äã Ú å Var r Ê t Ù V0ÁºDÜ (36) )[ (3) The technical proof of this result is given in the appendix Let us compare this with the optimal solution of the previous section: The minimal variance of the hedging error is which asymptotically yields Var r Ê t çævv0áºdü Jå Áº Û UV1ÁŠDÜ L ÁŠ H auvpý1þßh à Þ' P Áº ÁŠ Pè 0è )6 (38) )6! (3) This shows that the variance of the hedging error of the imitation strategy tends to zero as with the same order as the minimal variance, but the leading coefficient is almost twice as high Hence, roughly speaking, the price of being naive is that one ends up with a variance of the hedging error which is twice as high as it can be by acting a little wiser References [1] F Delbaen and W Schachermayer, A General Version of the Fundamental Theorem of Asset Pricing, Math Ann 300, 63 50, 1 [] A K Dixit and R S Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, NJ, 1 [3] H Föllmer and M Schweizer, Hedging of contingent claims under incomplete information, in: Applied Stochastic Analysis, M H A Davis and R J Elliott, eds Gordon and Breach, London, New York, 381, 11 [] H Föllmer and D Sondermann, Hedging of nonredundant contingent claims, in: Contributions to Mathematical Economics in Honor of Gérad Debreu, W Hildenbrand and A MasColell, eds, North Holland, Amsterdam, 05 3, 186 8
3 optimal imitation 5 15 1 05 0 1 08 06 0 0 0 0 0 06 08 1 Figure 1: Comparison of variance of hedging error for optimal and imitation strategy as function of the correlation coefficient ê [5] J M Harrison and DM Kreps, Martingales and Arbitrage in Multiperiod Securities Markets, J Econom Theory 0, 381 08, 1 [6] M Harrison and S Pliska, Martingales and Stochastic Integrals in the Theory of Continuous Trading, Stochastic Process Appl 11, 15 60, 181 [] J C Hull, Options, Futures, and Other Derivatives, Prentice Hall, 18 [8] I Karatzas, Lectures on the Mathematics of Finance, American Mathematical Soc, Providence, RI, 1 [] I Kartzas and S Shreve, Brownian Motion and Stochastic Calculus, Springer, Berlin, Heidelberg, New York, 188 [10] M Schäl, On quadratic cost criteria for option hedging, Math Oper Res 1, 11 131, 1 [11] M Schweizer, Option hedging for semimartingales, Stoch Proc Appl 3, 33 363, 11 [1] M Schweizer, Varianceoptimal hedging in discrete time, Math Oper Res 0, 1 3, 15 [13] M Schweizer, Approximation pricing and the varianceoptimal martingale measure, Ann Appl Prob, 0636, 16 A Appendix We start with the following elementary formulas: Var r Ê t Var r V _1\ t Cov r V š _1\g1V _1\ t (0) Var r V _1 \Ft o r V 1 \ t o+r V _1 \Ft (1) Cov r V š _1 \ 1V (1 \Ft o+r V š _1 \ V (1 \Ft o+r V _1 \Ft ()
Þ Á ð ø ú ø ñ þ Á x ý ø ø V ñ ñ ñ ñ r Let ë5 ]( ]( ¼µ Completing the squares in the exponent (or more sophisticated) by using the bivariate Esscher transform, we obtain where Þ and o+r V š _1 \ V UVPÝ~Þ'H is the standard bivariate normal distribution function, ñ Þ'HÍ ñ a Í!S /ò Ší/î k ª )6 ¼ Ší1ï k ª,ð ADC¼E ñ ñ ñ 8ñ ñ ñ & ñ (5) the standard bivariate normal density We are interested in from below Since ñ ñ behaves rather irregularly as ñ ñ 01 it is not clear that or how a simple ð trivariate Taylor expansion can be applied, so we need the following detour Let Þ'HÍF lnóþßhíp ÍF, HÍF ô!gln5õ ÞßHÍp ôk O õ Í and Þ HÍF {lnuþ HÍF Íp Lemma 1 Using the above notation we have the following asymptotic relations: ÞßH Þß _1 \Ft o+r V š \ aì ë t o r J Þß à :ì ë t Þ' ë t (3) P () à!s Þß Ü Ü L Þ SUÞ'!J[Þ (6) ö Ü Ü D [ ) () _ö D&6 )6! Ü (8) Proof: A careful look at the bivariate Taylor expansion of ÞßHÍp ôk with respect to Í and ô shows that the remainder term is uniformly ö! This is not true for Þ, but there appears! k, which reduces the quadratic error term to ö )6 Ü Next we look at ÞßHÍp as By conditioning we obtain in particular Since Áfû û ûhíbsnhí _1 uniformly in ñ as with ÞßHÍ Í!S í î k ª Á ø Í 6 ¼ñ {6!ù[ú Ší ͹6 ¼ñ Þ'HÍF Íp : ñ8 k ª )[ ù ú HÍB is bounded we get from Taylor s Theorem for fixed Ís*ƒ ͹[ 8ñ UÁ {[ ù Á û û ø ͹,ñ Á û ø {6 ù ͹,ñ ñ {6 ö ) ù Plugging this expression into (50) yields ÞßHÍp 3 ýpþ8híf a ýp HÍp! í k ª ý HÍp! Á w þ 10 HÍp!Pü 6 ö {6 Í+[ñ )6 ù ú ñ Í [ñ )6 ù ñ ü )[ () (50) (51) ñ Ü )[ Ü (5) (53) )[ Ü (5) ñ8 (55)
Þ úç ý ú Í úç ö ú Ç ú ò ú ò ú ø ÿ ÿ ÿ With a simple substitution and partial integration we rewrite ýp HÍp!a ÁºHÍB /ò )6 ª )[ ù ÁºHÍ+ (56) Watson s Lemma tells us now The integrals ý and ý ÁºHÍBJ Combining these results yields Also HÍ /ò {6 ¼ û HÍ å )6 ö )[ Ü and ý can be computed explicitly, and after painful elementary calculations we see ý ÞßHÍp a ÁŠHÍ J HÍp : HÍB HÍB 6 HÍ Í Ç HÍB å ö {6 )6 ) (5) )6! (58) ö _ö (5) )! Ü (60) {6! Using this expansion in () we obtain (3) (61) 11