A Note on Forward Price and Forward Measure

Transcription

1 C Review of Quaniaive Finance and Accouning, 9: , Kluwer Academic Publishers. Manufacured in The Neherlands. A Noe on Forward Price and Forward Measure REN-RAW CHEN FOM/SOB-NB, Rugers Universiy, Levin Bldg., Rockafeller Rd., Piscaaway, NJ rchen@rci.rugers.edu; Tel.: 732) ; Fax: 732) JING-ZHI HUANG Penn Sae Universiy, Smeal College of Business, Universiy Park, PA 6802 Absrac. The forward measure is convenien in calculaing various coningen claim prices under sochasic ineres raes. We demonsrae ha cauion needs o be drawn when he forward measure is used o price coningen claims ha involve muliple cash flows. We also derive parial differen equaions for he forward price o demonsrae how forward conracs can be used for dynamic hedging and how hedges can be conduced if he payoff of a coningen claim depends on he forward price. Key words: forward measure, forward price, sochasic ineres raes JEL Classificaion: G3. Inroducion The forward measure pricing mehodology Jamshidian, 987, 989; and Geman, Karoui and Roche, 995) has been widely used in pricing securiies when ineres raes are sochasic. This echnique provides grea ease in deriving closed form soluions for various derivaive conracs wih European-syle payoffs. However, for many financial conracs whose payoffs are dependen upon forward prices, he forward measure needs o be used wih cauion. In his paper, we demonsrae, using wo popular ineres rae conracs, how roll-over expecaions under differen forward measures should be used. We also derive an alernaive parial differenial equaion PDE) o he one in Jamshidian 987) o demonsrae how he forward price can be inerpreed alernaively. This alernaive PDE is consisen wih he inerpreaion ha he forward price is an asse price, as opposed o he inerpreaion ha he forward price is an index. In doing so, we effecively ransfer an index forward price ino an forward asse price ha saisfies he Capial Asse Pricing Model, as argued in Cox, Ingersoll and Ross 98, hereafer CIR). The heoreical behavior of he forward price was firs derived and summarized by CIR. They also derived closed form soluions for forward and fuures prices under a single facor Corresponding auhor.

2 262 CHEN AND HUANG square roo process for he insananeous ineres rae. The fuures price is dependen of he uiliy funcion of he single represenaive agen in he economy. The forward price, on he oher hand, is independen of he agen s risk preference and he ineres rae assumpion. This is of course because he forward price can be derived from saic i.e., one-period) arbirage which is independen of any assumpion of he ineres rae and of he uiliy funcion of he represenaive agen. Furhermore, he saic arbirage argumen also implies ha he forward price is independen of he number of facors ha drive he economy. Recenly, he forward price is reinerpreed using he forward measure. I is shown ha he forward price is an expecaion of he erminal payoff of he underlying asse price, like he fuures price which is a risk neural expecaion, excep ha he probabiliy measure under which he expecaion is aken is differen from he risk neural measure by an adjusmen erm. This mehodology has been found very useful in calculaing a variey of forward prices. 2 We assume a single facor hroughou he paper. Wihou any loss of generaliy, our analysis is based upon he forward price of a pure-discoun bond. The paper is organized as follows. The nex secion reviews he forward measure and is use o derive ineres rae sensiive derivaives. In Secion 3, we discuss he use of forward measure. We show in wo examples how he forward measure needs o be used wih cauion. We also derive parial differenial equaions ha he forward price mus saisfy. We show how he forward price can saisfy he Capial Asse Pricing Model wih a modified discoun rae. Finally, we summarize briefly he paper in Secion A brief review of he forward measure The forward measure echnique is useful in compuing coningen claim prices under sochasic ineres raes. In his secion, we briefly review he forward measure mehodology. We limi ourselves o hose key equaions relevan o our analysis, while ineresed readers can find more deailed discussions in Rebonao 998), Musiela and Rukowski 998), or Hull 2000). Define a T -mauriy, $ face value, zero-coupon bond price as follows: P, T ) = Ê [exp ru) du ) where Ê [ ] is a condiional on ) expecaion aken under he risk neural measure. The T -mauriy forward price of an s-mauriy pure-discoun bond as follows:, T, s) = P, s) P, T ) = Ē T [PT, s 2) where < T < s and Ē T [ ] is he condiional on ) expecaion aken under he T -mauriy forward measure.

3 A NOTE ON FORWARD PRICE AND FORWARD MEASURE 263 Equaion 2) is ineresing because i shows ha he forward price is parallel o he fuures price, hey boh are expecaions of he fuure bond price. The sandard expecaion operaions show ha: [ Ê [exp ru) du PT, s) = côv exp + Ê [exp = côv [exp ru) du, PT, s) ru) du Ê [PT, s ru) du, PT, s) + P, T ), T, s) 3) where, T, s) = Ê [PT, s is a fuures price. Combining his resul wih equaion 2), we have: côv [exp T ru) du, PT, s), T, s), T, s) = 4) P, T ) I can be seen ha he difference beween he forward price and he fuures price is only he covariance beween he sochasic discoun facor and he bond price a mauriy. Since he covariance is always posiive, he forward price is always greaer han he fuures price. As a resul, he difference beween he forward expecaion and he risk neural expecaion is very inuiive. Noe ha he difference beween forward and fuures is marking o marke in he fuures marke. Marking o marke differeniaes he forward price from he fuures price only under sochasic ineres rae environmen because here exis coninuous cash flows from he fuures conrac ha need o be reinvesed a random ineres raes. Consequenly, if we incorporae he effec of he covariance beween he ineres rae and underlying asse in he forward measure, hen he forward price should iself obey an expecaion under he forward measure. Equaion 2) does no seem o be paricularly useful because he same resul of he forward price, a raio of wo bonds, can be easily obained by saic arbirage. In oher words, buying an s-mauriy bond P, s) is equivalen o buying a T -mauriy bond P, T ) and rolling over o a forward price, T, s). However, i is very useful in compuing oher coningen claim prices. For any European) coningen claim, C, T ), wih a erminal payoff of X T ), he forward measure gives he following convenien resul: C, T ) = Ê [exp = Ê [exp ru) du X T ) ru) du Ē T [X T = P, T )Ē T [X T 5)

4 264 CHEN AND HUANG As a resul, he forward measure can be used o derive a number of imporan derivaive pricing resuls. For example, i can be used o compue insananeous forward raes: d ln P, T ) f, T ) = dt = [ d P, T ) Ê = P, T ) Ê [ exp dt exp ru) du ru) du rt ) = Ē T [rt 6) and discree forward raes, f D, w, T ): f D, w, T ) =, w, T ) P, w) P, T ) = P, T ) Ê [ = Ē T [ exp Pw, T ) ] ) ru) du ] where < w < T. Pw, T ) The resul saed by equaion 7) is very useful since macroeconomiss always wonder if he forward rae is a good predicor of he fuure spo rae. Here, i is proved heoreically ha he forward rae can only predic he fuure spo rae under he forward measure. I is herefore a biased esimaor of he fuure spo rae under he normal siuaion i.e., original probabiliy space). We can also use he forward measure o re-derive he bond opion pricing resuls in CIR 985) and Jamshidian 989) and he sock opion pricing resul in Rabinovich 989). 7) 3. The use of forward measure In his secion, we presen wo examples in which he applicaions of he forward measure are no so sraighforward. For he ease of exposiion, we assume a one-facor erm srucure model while he resuls hold in a muli-facor environmen. We assume ha he insananeous ineres rae follows he following sochasic differenial equaion: dr) = ˆµr, ) d + σr, ) dŵ ) 8)

5 A NOTE ON FORWARD PRICE AND FORWARD MEASURE 265 where Ŵ ) is he sandard Wiener process defined in he risk-neural probabiliy space, and ˆµr, ) and ˆσr, ) are he sae- and ime-dependen drif and diffusion for he process respecively. A T -mauriy forward measure can hen be defined as: dr = [ ˆµr, ) σr, ) 2 P ] r, T ) d + σr, ) d W T 9) P, T ) in which a Girsonav ransformaion of he following is performed: dŵ ) = d W T ) + σ 2 r, ) P r, T ) d 0) P, T ) where W T ) is he Wiener process defined under he T -mauriy forward measure. 3 Under his forward measure, we can compue he forward price of an s-mauriy pure-discoun bond as shown in equaion 2). 3.. Roll-over forward measures Despie of he usefulness of his forward measure, a number of popular conracs need roll-over forward measures. For example, he Treasury bond fuures conracs raded on he Chicago Board of Trade CBOT) and he Eurodollar fuures conracs raded on he Chicago Mercanile Exchange CME) are in fac roll-over forward conracs since marking o marke akes place daily raher han coninuously. Assuming no delivery opions in hese fuures conracs, he fuure price a T one day before he delivery dae T )isinfac a forward price for one day. Hence he fuures price a T should be compued by he forward pricing formula of equaion 2). A T 2 wo days before he delivery dae), he fuures price should be he forward price of he forward price a T. The procedure repeas unil he curren ime is reached. 4 As a comparison, one can compue he forward price oday,, for he delivery a T wihou marking o marke. As shown in Chen 992b), he wo prices, a one-ime forward measure and a series of revolving forward measures, are no idenical. A similar resul can be observed in discree erm srucure models such as he Ho-Lee model 986) and he Black-Derman-Toy model 989) where he expeced value calculaed using heir binomial probabiliies is he fuures price under discree marking o marke. The same conclusion can be drawn. Anoher example is a popular conrac in credi derivaive markes forward asse swaps. Asse swaps differ from ineres rae swaps because of heir high credi risks. The credi swap raes in asse swaps can vary dramaically due o various cash flow arrangemens upon defaul. Forward asse swaps are forward conracs on asse swaps. These conracs allow invesors o lock in a specific asse swap rae. If we assume independence beween he credi risk and he ineres rae risk and he forward conrac is defaul-free, hen we can model he forward conrac as a simple forward conrac on an ineres rae swap. Since he swap

6 266 CHEN AND HUANG rae iself is a linear combinaion of forward raes, he forward swap rae is a forward price on a forward price. In boh of he above examples, we need o compue he forward price of a forward price. In he simples case where we compue he forward price of a forward price on a pure-discoun bond, we shall see ha: Ē w [ w, T, s = Ê [ w, T, s) P, w) exp [ = Ê Ê w [PT, s) Pw, T ) exp w ru) du P, w) exp w ru) du w = Ê [PT, s) Pw, T )P, w) exp T Ê [PT, s) P, T ) exp ru) du ru) du ru) du =, T, s) ) I is seen ha he forward expecaion of he forward price is no equal o he forward price which is iself an expecaion:, T, s) = Ē T [PT, s. The difference resuls from he wo forward measures, one on he LHS and one a he las line of he RHS are aken o a differen fuure poins in ime, w and T respecively. This is an ineresing conras he risk neural measure which is mauriy independen. In he discree marking o marke case menioned above, oday s fuures price is a repeaed use of equaion ). In he forward asse swap case, a linear combinaion of discree forward raes defined in equaion 7) needs o be used in conjuncion wih equaion ). In eiher case, roll-over forward measures are used Parial differenial equaions Alhough he forward measure mehodology inroduced in he previous secion is very convenien in deriving pricing resuls, i is also imporan o undersand he dynamics of he forward price. This is imporan because hrough he dynamics invesors can rade and hedge coningen claims whose payoffs are dependen upon forward prices. Forward prices are no he values of forward conracs. Raher, hey are prices o be paid o acquire underlying asses like he srike price of an opion). Hence, hey are similar o an index. CIR 98) show ha forward prices do no saisfy he Capial Asse Pricing Model. In his secion, we derive wo parial differen equaions PDE) ha he forward price has o saisfy hrough forward price dynamics. One uses he forward measure defined above. This PDE is based upon he inerpreaion ha he forward price is an index. The oher one

7 A NOTE ON FORWARD PRICE AND FORWARD MEASURE 267 inerpres he forward price as an asse. We derive an implied discoun rae a which he forward price saisfies he Capial Asse Pricing Model. Applying Io s lemma o equaion 8), we can wrie he dynamics of he forward price, d, T, s), as: d = [ ] 2 σ 2 r, ) rr + ˆµr, ) r + d + r σr, ) dŵ 2) The underlying bond, P, s), can be used o hedge his innovaion in he forward price, which has he following dynamics: [ ] dp = 2 σ 2 r, )P rr + ˆµr, )P r + P d + P r σr, ) dŵ = rp d + P r σr, ) dŵ 3) The porfolio ha conains one long bond and h shor forward conracs of he following: V = P, s) h, T, s) 4) is a differen from ha of he fuures because he forward conrac does no generae d bu d )P + d, T ) over ime. Hence, he dynamics of equaion 4) is: dv = dp, s) hp + d, T ) = dp, s) hp, T ) + dp, T )) = rp, s) d + P r, s)σ dŵ hp, T ) r σr, ) dŵ [ ] h r σr, )P r, T )σ r, ) + hp, T ) 2 σ 2 r, ) rr + ˆµr, ) r + d 5) where = d, T, s). The hedge raio, h, is: h = P r, s) P, T ) r 6) This gives a PDE of he following: σ 2 [ r, ) rr + ˆµr, ) σ 2 r, ) P ] r, T ) r + = 0 7) 2 P, T ) This is a Kolmogorov backward equaion under he process for he ineres rae precisely specified by equaion 9) and hence he soluion o he forward price specified by equaion 2) is verified by he soluion o he Kolmogorov backward equaion. In addiion o he above PDE ha is derived under he forward measure, we can inerpre he forward price as an asse price and show ha he forward price should also saisfy he

8 268 CHEN AND HUANG following PDE: 5 { [ σ 2 ] r, ) Pr, T ) 2 rr + ˆµr, ) r + = σr, ) P r, s) 2 P, T ) P, s) } P r, T ) σr, )2 P, T ) = ˆr 8) This PDE shows ha he forward price should earn, in he risk neural world, an ineres rae ˆr ha is he difference beween he variance of he discoun facor and he covariance beween he discoun facor and he underlying spo. I is clear ha when he variance is equal o he covariance, he forward price is equal o he fuures price, since he fuures price saisfies he same PDE wih zero on he righ hand side. One example would be deerminisic ineres raes. In such a case, boh erms are zero and equaion 8) will become he fuures PDE. CIR poin ou ha his risk neural reurn for he forward price is negaive and herefore he forward price should always be more han he fuures price. The soluion o his PDE is a Kac funcional and idenical o he soluion o equaion 7):, T, s) = Ê [exp ˆru) du PT, s) Recognizing ha he forward price earns ˆr in he risk neural world, he maringale process for he normalized forward price can be wrien as, i.e.:, T, s) = Ê [exp = Ê [exp = Ê [exp w w ˆru) du PT, s) ) [ ˆru) du Ê w exp ˆru) du w, T, s) w ] ˆru) du PT, s) This resul is equivalen o CIR s proposiion 8 where he forward price which is no an asse price) is ranslaed ino an asse price. Like CIR, we argue ha he forward price is equivalen o he value of an asse ha earns ˆr in he risk neural world. Noe ha equaions 7) and 8) use he same boundary condiion. Tha is, a mauriy, he forward price is equal o he bond price. Therefore, he wo PDE s need o equal each oher. Eliminaing erms, we ge: P r, T ) P, T ) r = { [ ] Pr, T ) 2 P r, s) P, T ) P, s) } P r, T ) P, T ) This equaliy can be verified easily. Equaion 2) gives an alernaive derivaion for equaion 7). We can ransfer he risk neural reurn of a cerain asse ino he drif of he underlying process under which he asse earns 0 reurn so ha an expecaion can be esablished. 9) 20) 2)

9 A NOTE ON FORWARD PRICE AND FORWARD MEASURE Summary In his paper, we reierae he convenience of he using he forward measure in pricing ineres rae sensiive coningen claims. We demonsrae in wo examples ha cauion needs o be drawn when he forward measure is used o price coningen claims ha involve muliple cash flows. For some conracs whose payoffs are dependen upon forward prices e.g., swapions), he dynamic behavior of he forward price is imporan. In view of his, we derive wo parial differenial equaions PDE), one under he forward measure and he oher under he risk neural measure. The PDE under he forward measure has he similariy o he one saisfied by he fuures price under he risk neural measure while he one under he risk measure inerpres he forward price as a raded asse. Appendix Parial differenial equaion We firs esablish he following parials: r = P r, s)p, T ) P, s)p, T ) 2 P r, T ) rr = P rr, s)p, T ) 2P r, s)p, T ) 2 P r, T ) + 2P, s)p, T ) 3 P r, T ) 2 P, s)p, T ) 2 P rr, T ) = P, s)p, T ) P, s)p, T ) 2 P, T ) A.) Applying risk neural coefficiens for he backward equaion, we have: σ 2 r, ) rr + ˆµr, ) r + 2 P rr, s)p, T ) = σ 2 2P r, s)p, T ) 2 [ P r, T ) 2 + 2P, s)p, T ) 3 P r, T ) 2 + ˆµ Pr, s)p, T ) ] P, s)p, T ) 2 P r, T ) P, s)p, T ) 2 P rr, T ) [ P, s)p, T ) ] + P, s)p, T ) 2 P, T ) { = rp, s)p, T ) rp, s)p, T ) 2 P, T ) + σ 2 Pr, s)p, T ) 2 } P r, T ) + P, s)p, T ) 3 P r, T ) 2 { = 0 + σ 2 P [ ] } r, s) P, s) P r, T ) P, s) Pr, T ) 2 + P, s) P, T ) P, T ) P, T ) P, T ) { [ ] } Pr, T ) 2 = P, T ) σ P r, s) P, s) σ P r, T ) P, T ) σ A.2)

10 270 CHEN AND HUANG Forward measure Forward measure is a probabiliy measure under which he expecaion gives rise o he forward price. The bond price has o saisfy he law of ieraive expecaions under he risk neural probabiliy space: P, s) = Ê [, T )PT, s A.3) where ), T ) = exp ru) du. A.3) can be separaed ino a produc of wo expecaions using he Radon-Nikodym derivaive: P, s) = Ê [, T Ē [PT, s = P, T )Ē [PT, s A.4) Given ha he forward price is he raio beween wo bonds, he forward measure should surely give he forward price. Now he problem is o find he Radon-Nikodym derivaive ha can help he Girsanov ransformaion. Define he derivaive as: η, T ) =, T ) P, T ) = d d ˆ A.5) Also, direcly from equaion 3), we have 0 = ln PT, T ) = ln P, T ) + rw) dw + σr, w) P rw, T ) dŵ w) Pw, T ) σr, w) P ) rw, T ) 2 dw A.6) 2 Pw, T ) As a resul, η, T ) =, T ) [ P, T ) = exp σr, w) P rw, T ) dŵ w) Pw, T ) σr, w) P ) rw, T ) 2 dw] 2 Pw, T ) A.7)

11 A NOTE ON FORWARD PRICE AND FORWARD MEASURE 27 This implies he Girsanov ransformaion of he following: Ŵ ) = W ) + σr, w) P rw, T ) Pw, T ) dw A.8) and his complees he proof. A more general version of he above resul is o le he erminal dae be an arbirary choice. Then, he Radon-Nikodym derivaive is: η, u) =, u)pu, T ) P, T ) =, T )Pu, T ) u, T )P, T ) =, T ) P, T ) Pu, T ) u, T ) A.9) From he derivaion above, we know ha: his implies ln, T ) ln P, T ) = ln u, T ) ln Pu, T ) = The wo ogeher gives: σr, w)p r w, T )/Pw, T ) dŵ u u 2 [σr, w)p rw, T )/Pw, T 2 dw A.0) σ 2 r, w)p r w, T )/Pw, T ) dŵ 2 [σ 2 r, w)p r w, T )/Pw, T 2 dw A.) [ u η, u) = exp σr, w)p r w, T )/Pw, T ) dŵ u ] 2 [σr, w)p rw, T )/Pw, T 2 dw A.2) Acknowledgmen We would like o hank an anonymous referee for valuable suggesions. All errors are ours. Noes. In heir proposiion, CIR show ha he fuures price saisfies he CAPM bu no he forward price. 2. See, for example, Jamshidian 987, 989), Longsaff 990), Chen 992a, b), Chen and Sco 992), Geman, Karoui and Roche 995) and Schroder 999). Deailed discussions of he forward measure can be found in Musiela and Rukowski 998).

12 272 CHEN AND HUANG 3. Noe ha σ 2 r, )P r, T )/P, T ) = E[dr)dP/P is he insananeous covariance beween he discoun facor and he underlying ineres rae. This corresponds o equaion 4). 4. Exac deails can be found in Chen 992b) for he Vasicek erm srucure model. 5. The derivaion is given in an appendix. 6. This is checked wih he Vasicek model. References Black, F., E. Derman and W. Toy, A One-Facor Model of Ineres Raes and Is Applicaion o Treasury Bond Opions. Financial Analyss Journal 46, 33 39, 990). Chen, R. and L. Sco, Pricing Ineres Rae Opions in a Two Facor CIR Model of he Term Srucure. Review of Financial Sudies 54), , 992). Chen, R., Exac Soluions for Fuures and European Fuures Opions on Pure Discoun Bonds. Journal of Financial and Quaniaive Analysis 27), March, 97 07, 992a). Chen, R., A New Look a Ineres Rae Fuures Conracs. Journal of Fuures Markes, Ocober, , 992b). Cheng, S., On he Feasibiliy of Arbirage-Based Opion Pricing when Sochasic Bond Price Processes are Involved. Journal of Economic Theory 53, 85 98, 99). Cox, J., J. Ingersoll and S. Ross, A Theory of he Term Srucure of Ineres Raes. Economerica 532), , 985). Cox, J., J. Ingersoll and S. Ross, The Relaion Beween Forward Prices and Fuures Prices. Journal of Financial Economics , 98). Geman, H., N. El Karoui and J. Roche, Changes of Numeraire, Changes of Probabiliy Measure and Opion Pricing. Journal of Applied Probabiliy 32, , 995). Ho, T. and S. Lee, Term Srucure Movemens and Pricing Ineres Rae Coningen Claims. Journal of Finance, 0 029, 986). Hull, Opions, Fuures, Opions, and Oher Derivaives, 4h ediion, Prenice Hall, Jamshidian, F., Pricing of Coningen Claims in he One-Facor Term Srucure Model. Working paper, Merrill Lynch Capial Markes, 987. Jamshidian, F., An Exac Bond Opion Formula. Journal of Finance, , 989). Longsaff, F., The Valuaion of Opions on Yields. Journal of Financial Economics 26), 97 22, 990). Musiela, M. and M. Rukowski, Maringale Mehods in Financial Modeling, Springer-Verlag, 998. Rabinovich, R., Pricing Sock and Bond Opions when he Defaul-Free Rae is Sochasic. The Journal of Financial and Quaniaive Analysis 244), , 989). Rebonao, R., Ineres-Rae Opion Models: Undersanding, Analyzing and Using Models for Exoic Ineres-Rae Opions, Wiley Series in Financial Engineering, 2nd ediion, 998. Schroder, M., Changes of Numeraire for Pricing Fuures, Forwards, and Opions. Review of Financial Sudies 2, 43 63, 999). Vasicek, O., An Equilibrium Characerizaion of The Term Srucure. Journal of Financial Economics, 77 88, 977).