SELF-FINANCING MARKETS AND EVENTUAL ARBITRAGE

Transcription

1 SELF-FINANCING MARKETS AND EVENTUAL ARBITRAGE J. F. CARRIÈRE Mathematical Sciences University of Alerta Edmonton, Canada Astract: In this article, we argue that the self-financing axiom with mild assumptions on the conditional expected returns yields a market with an eventual aritrage. This is accomplished y minimizing the conditional variance of a trade when the conditional expectation is a fixed constant. Examples with common processes shows that most models yield an eventual aritrage. As a final application, a cost model is applied to prices of stripped coupon and principal payments on U.S. government onds, where rates of return are estimated and lower ounds on market costs given. Key Words: Multivariate stochastic processes in discrete time, trading strategies, STRIPS. 1

2 Self-financing and aritrage 2 1 Introduction In this article, we discuss the notion of an eventual aritrage and give examples of how they can e constructed. Our main thesis is that an oservale opportunity to exercise an eventual aritrage is actually non-exploitale ecause of costs. Thus, estimates of expected gains can e interpreted as a lower ound on market costs. In the next section, we introduce the notion of a self-financing maket where costs are allowed. Next, we show that variance minimizing strategies lead to an eventual aritrage. Using price data on STRIPS, we calirate the model and construct a good predictive strategy. Using our day trader s cost model, we get an estimate on market costs. 2 Self-Financing Markets Consider a self-financing market with a finite collection of assets where the market prices at time vector are denoted y the column. Assume that is a stochastic process adapted to a history ( -field) denoted as. For our purposes, let denote the physical proaility function induced y the price process with the collection of all measurale events eing. Also let E denote the associated expectation operator. Next, let! "$#%& '( denote a trading strategy where the trading position at time *)+,- is. We assume that. 0/1 23/4 is a65, random vector that is measurale with respect to7. Let89 ;:<>= denote the value of a portfolio constructed of these assets according to a trading stategy. This is equal to 84?:@A=B C ED ' (2.1) letfg BH ) Next, denote an infusion of cash into the fund at time I that is called the dividend and let JHK) denote an outlay of cash from the fund at time that is called the cost. Definition 1. We will say that the portfolio is self-financing with cash-flow whenever the value84?:@a=

3 Self-financing and aritrage 3 has the representation D L 2 C NM+ D L 9OPFG RQ I < (2.2) IfFG S) andi 2 S), then we will simply say that the portfolio is self-financing. In this special case :< Q NM+ = D L T U). For more information aout self-financing portfolios, consult Musiela and Rutkowski (1998). Next, denote the change in prices yvw L YX1BQZ L [ 0\G \G (. We find that 89 YX1BQ]84 ^ D V_ 9OPFG RQ I ' (2.3) Moreover, we find that 84 ;:@>=` 8 # :@>=2O NM+ a c # " D V OZF Q I d (2.4) Suppose thatfg ) for all and that the cost function is equal to I feg e D(h-i (2.5) whereh-i f 0j 2j3 &k is measurale with respect to and wherej3 mlbns) foroqp]rs,t3uvwx, are the costs of holding one security. We call this the day trader s cost model. We now introduce the notion of non-exploitale markets. Definition 2. Suppose that we have a self-financing portfolio withfy z {) and I n}) for all. We will say that the market at time is non-exploitale y using a trading strategy whenever E ~ RD V_ RQ I ke7 N K).

4 Self-financing and aritrage 4 As a special case, non-exploitaility implies that using a day trader s cost model yields D fe0 e D(h-i (2.6) where the conditional expected change in prices is defined y E 0V_ e7 ƒd (2.7) 3 Self-Financing Strategies That Minimize Variance In this section, we apply a Markowitz model to portfolio optimization and find that this is sufficient for an eventual aritrage. For more information aout the Markowitz model, consult Panjer, Boyle, et al (1998). For the ensuing discussion, we assume thatfg I 2 ) and we define 2 Var 0V_ ke Nd (3.1) Now, suppose that for all then : = E 084 ;:@>='E *8 # :<>= is a martingale and for all. In this let w ˆ case, prediction is not possile. Next, denote theš -th coordinate of if w ˆŒ ). Note that for allš and set/4 Ê Ž ƒ " with Bp E 089?:<>='w 8 # :<>=2O NM+ c # then we can. Thus and so the expected value of a self-financing portfolio can e set at any level. The same cannot e said of the variance. In this paper, we will focus on strategies that minimize a variance for a fixed expectation. Var " šd V e ƒ That is, we want to choose so that is minimized suject to the constraint that E ~ D VW e where & $n ),. That is, we need to find so is minimized suject to the constraint that q K. The well-known solution is & M+ q M+ (3.2)

5 V D Self-financing and aritrage 5 whenever, the inverse M+ exists and M+ nœ)!,. As a special case, if then Ž. In general, more constraints can e put on the strategy ut we will not e investigating the general prolem. Now, consider the variance Var ~ ED V e ƒe Let us calculate the variance of89. This is equal to (ž M+ (3.3) NM+ Var g89 ŸR a a NM+ c # ˆc # Cov " D V & ˆ+D V_ˆmd (3.4) SŠ Cov " D V & ˆ(D V ˆ E ) If then ecause Dk p is non-stochastic, y construction. thatp Š Cov " Note that no independent-increment assumption is made. In the case we get & D V E E Var ~ D V e Y. Thus we get NM+ Var 089 ŸR a c # ž E -: M+ = M+v (3.5) In the constant variance model, we T have ž E : Var 084 Nq že NM+ c # (ž for all and so M+. Also, if ž = M+' ž n )+ (3.6) f n[) for all then Var 084 Ÿ$ ª «: =. Next, assume that8 # *) and consider the standardized value, defined y 89 RQ E 089 N Var g89 Ÿ (3.7)

6 Self-financing and aritrage 6 Using Cheyshev s Inequality, we find that g89 J K)(E * Q E 089 Ÿ Var 089 Ÿ;± S e ²esH E g89 Ÿ Var 084 Ÿ± Var 084 Ÿ : E g89 NN=ž NM+ c # (ž ž NM+ c # ž QE³ )+ as ³ E 089 ŸE «: =, whenever (which is true when Var g89 ŸE µ: žm = is ounded for all ) and for someģn ). Thus, we have shown that there exists an eventual aritrage, defined as follows. Definition 3. We will say that a strategy admits an eventual aritrage whenever8 # *) and ¹ºƒ» Y¼ g89 n )E ^,t (3.8) It is instructive to state the conditions on the model that lead to this result. First, the market must e self-financing. Second, the inverse M+ must exist and M+ q n*) Var 084 Ÿ2 œ µ: žm =. Third, for someģn ) which means that the variance is growing slowly relative to the expectation of84. that ½: =B * ½:d)¾=+ 3ÀvÁErv: yq ž žã= OWÄf: = x Example 1. Suppose whereä : = is a Weiner process and q fjr( µ:då =ÇÆ)W fål x has w q E 0\G e7 N` µ:. In this case, the geometric Weiner proces =: È QÉ, = /4 7 j 0 ½: =&:dè QÉ, =<M+ Ê ) Ë$ L Var \y e Nµ 0 ½: and, as long as. Moreover =<ž È ž X9Ì&Í Var 0/1 D \G e N j ž È Ì&Í Var 089 Ÿ j ž È Ì&Í. Thus and. Moreover, under the optimal strategy we find that89 is a sum of independent and identically distriuted random variales and so the Central Limit Theorem implies that converges in distriution to a standard normal variate and 84 admits an eventual aritrage. Example 2. Using a day trader s cost model withi $nc) andfg š.) for all, we find that84 will

7 Self-financing and aritrage 7 yield an eventual aritrage whenever the trading strategy is exploitale. That is, E ~ D V EQ*eg e D(h $e( Nn )v{î 4 The Yield Data In this section, we present the yield data that we used to test our hypothesis. One of the est sources of data for yield rates are the yield rates on U.S. Treasury Strips, as reported y Bear, Stearns & Co. via Software Technologies Inc. and pulished in the Wall Street Journal. Strips are pure discount onds that arise from stripping the principal and coupons from government onds and trading these strips separately. This data was used previously Carrière (2001). Let us descrie the data in detail. In this article we used yield rates from all the trading days in the years of 1993 to 1997, inclusive. In all, we had 1250 trading days with 250 trading days per year. For each trading day, the data consisted of the id and asked yield rates for stripped coupon interest, stripped Treasury Bond principal, and stripped Treasury Note principal at various maturities. For our purposes, the yield of the ond at a fixed maturity was the average of the id and asked yields for all coupon and principal strips with that maturity. In all, we had 100 onds with distinct maturities. Specifically, these onds have maturity dates at every three months during the 25 year period of l ow,t3u, utï-) Ð9Ñ Ò 1998 to 2022, inclusive. Let for, denote the trading times and let for Ó,-3uvw,%)-) denote the maturity dates. Thus, for each time and maturity we oserved the yield rates,ôw: ld²ð+ñ&=. The corresponding prices are calculated as follows: ½: l ²Ð+Ñ&=Õ 3ÀvÁÖrsQz: Ð+ÑqQ lÿ=+ô: l ²Ð+Ñ&= x (4.1) To get an idea of the type of data that we have, we present Figure 1 where µ: l@²ðk=, \GlYK µ: l X1&²Ðk=>QS µ: l@²ðk= and ½: l ²Ð#<# =, \GlƒØ#<# Ù ½: l X1òÐ2#<# =>QS ½: l@?ð2#<# = are plotted versus l.

8 Self-financing and aritrage 8 Figure 1: The first column shows the prices and their differences versus time for a strip that matures in Feruary, The second column shows the prices and their differences versus time for a strip that matures in Feruary, 2022.

9 Ý l Ý l Self-financing and aritrage 9 Note that the price of our onds approaches one when we approach maturity. Also note that the volatility approaches zero when approaching maturity. All the graphs and calculations in this article were done with Gauss, a matrix programming language. 5 Caliration of the Model In this section, we present a model of and that is calirated with half the data, while the other half is used to construct a variance-minimizing and self-financing strategy. We will report on the portfolio value 89, as it evolves over our prediction period. For eachú,tuve,)t) we assume that \Gl *Û VWlƒM+ O]ÜE: l@?ð =; %Ý (5.1) whereüe: ²ÐT= Ý is a function (possily stochastic) having the property,üe: => C), and where E l e ßN1 *) is fixed and non-stochastic. We also require that, and Var e7 ßNR Ó, Ý, and Cov l lmˆ<e7 ßNE Sà ˆpá Q6,t,d pþ Some examples ofüe: ²ÐT= are Üw: ²ÐT=B.:ŸÐ Q ='ât Üw: ²ÐT=B.:'Q ¹ƒã ½: ²ÐT='N=;ä Üw: ²ÐT=B. µ: ²ÐT='Yå-: Ð Q =<â¾ Üw: ²ÐT=B.:? 0 ½: ²ÐT=<å :;Q ¹ƒã µ: ²ÐT='N=;ä (5.2)

10 Ûè ê Self-financing and aritrage 10 where nf),æánf) andç né). This is a classical regression prolem where a generalized leastsquares solution is optimal. We denote the estimates as follows:èû,è,è à ˆ. The estimators are: [ 0é éúm+ é ê èž ë, ê IQ]éÓ é éúm+ é ê àè ˆ9 ê IQ]éÓ 0é ë ékm+ é ê ˆ (5.3) è è4ˆ whereë is the numer of oservations and íì ÜE: \î &²Ð To define the design matrixé, we first define ñ Êì \ # Ø Üw: # ²Ð Thus = w \ ï ÜE: ï ²Ð = \ ïõm+;ø Üw: ïõm+&?ð =%ð =%ð (5.4) (5.5) Let A. gû &2Û1. In this model, é. ñ &2 ñ òt (5.6) E gv_ ²e NE AV NM+ (5.7)

11 è è è è Self-financing and aritrage 11 and 2 Var 0V e ƒe Cr Cov 0\G Therefore, Note that k\y ˆ'e7 ƒx q èav_ NM+ è 2 ^r Üw: ²Ð =+ÜE:?ÐwŶ= è è M+ è è M+ è è4ˆ àè ˆ1 ë, ê ؈c ;ØóóóƒØ where Cov \G è4ˆ àè ˆ x ؈c ;ØóóóNØ \G ˆ'e E SÜE:?Ð =;ÜE: ²ÐEŶ=; 4ˆà (5.8) IQ]éÓ 0é ésm+ é ê ˆ (5.9) ˆ. Prediction è Using the estimates, which are calirated with the data at time ô,-3uv ë, we predict according to the optimal rule for the nextë oservations and oserve the value of our self-financing portfolio with a starting value of81ï] *). Thus, žï 8 žï] a c ï è D V (5.10) The result is plotted in Figure 2, where we find a succesful result, That is the fund value is increasing linearly since thatüw: ²ÐT=B ÓQ µ: ²ÐT= ¹ƒã ½: ²ÐT= for all. In this demonstration, we assumed. žïc ï è D V Finally. an estimate of the average return per unit traded is sinceeè žï c ï eè e D¾õ *uv"ötu 5,%) M (5.11) edõ is the total numer of trades at time. With 250 trading days in a year, we can calculate the nominal annual rate of return when trading with a variance-minimizing strategy. The rate is

12 Self-financing and aritrage 12 Figure 2: Pot of the Value from a Self-Financing Portfolio.

13 Self-financing and aritrage 13,%)t) 5 utï-) 5 uv~ö-u Â,%) Óg)tøtùtú, which is miniscule. Thus, it seems very unlikely that this market is exploitale. Thus, a lower ound on the cost of uying or selling a million dollars in onds isûtu ~ötu. This ound can e improved y developping etter predictive models. References Carrière J. (2001) A Gaussian Process of Yield Rates Calirated with Strips. North American Actuarial Journal, Vol. 5, No. 3. pp Musiela, M. and Rutkowski, M. (1998): Martingale Methods in Financial Modelling. New York, N.Y.: Springer-Verlag. Panjer, H. (Editor), Boyle, P., Cox, S., Dufresne, D., Gerer, H., Mueller, H., Pedersen, H., Pliska, S., Sherris, M., Shiu, E., Tan, K.S. (1998): Financial Economics: With Applications to Investments, Insurance and Pensions. Schaumurg, Ill.: The Actuarial Foundation. Last revised: August 20, 2002