- Email: [email protected]
A note on an equilibrium debt option pricing model in discrete time
Journalof ELSEVIER
Journal of Banking & Finance 19 (1995) 1305-1307
BANKING & FINANCE
A note on an equilibrium debt option pricing model in discrete time Roswell E. Mathis, III * Department of Finance, College of BusinessAdministration, Florida International University, Miami, FL 33199, USA Received October 1993; final version received February 1994
Abstract In this note, the Maloney and Byme (1989) discrete time model of the term structure of interest rates for the pricing of interest rate contingent claims is corrected so that the term structures generated by the model satisfy arbitrage-free restrictions. JEL classification: G13 Keywords: Interest rate options; Valuation
1. Introduction Maloney and Byrne (1989) developed a discrete time model of the term structure of interest rates for the pricing of interest rate contingent claims. There is an error in the development of the model which results in the existence of arbitrage within the model. In this note, the model is corrected.
* Corresponding author. Phone: (1) (305)-348-2680. 0378-4266/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 3 7 8 - 4 2 6 6 ( 9 4 ) 0 0 0 7 2 - 7
R.E. Mathis, Ill~Journal of Banking & Finance 19 (1995) 1305-1307
1306
2. Illustration of the error and correction of the model
Maloney and Byme define the following:
P(i,j,k) R(i,j,k) TP
R(i,j,i + 1)
= the price in time period i and state j for a default-free zero coupon bond that pays $1 in time period k, where k = i + 1, i + 2 ..... = the continuously compounded risk-free rate at time period i and state j for a zero coupon bond maturing in period k, -- the term premium on two-period zero coupon bonds, and = uJdi-JR(O,O,1), where u and d are constants and u > 1 > d > 0.
In the model, interest rates and, thus, bond prices evolve over time according to a binomial process. To ensure that term structures generated by the model are arbitrage-free, there must be a constant, 7ri.i, at each time i and state j on the lattice so that the following equation will hold for all possible combinations of i, j, and k on the lattice:
(Tri,jP(i+ 1 , j + 1,k) + (1 - 7rij)P(i+ 1,j,k))P(i,j,i+ 1) •P(i,j,k).
(1)
The constant, ~i,j, can be interpreted as the "risk neutral" or " h e d g i n g " probability at time i and state j. Maloney and Byrne, in their Eq. (4), specify the relationship between one- and two-period term structure rates as 2 R ( i , j , i + 2) - TP = R(i,j,i + 1) [1 + 0.5(u + d ) ] .
(2)
Eq. (2) involves expectations over continuously compounded rates, whereas the arbitrage-free condition of Eq. (1) involves expectations over prices. Eq. (1) does not imply Eq. (2). To demonstrate this assertion, let the term premium equal zero. The local expectations hypothesis then describes equilibrium and, by definition, default-free zero coupon bonds of all maturities must yield the same expected return over one-period. Therefore, ~ri,j must equal the probability of an upstate at time i and state j, which Maloney and Byrne assume is equal to 0.5 for all i and j. Given the above conditions, Eqs. (1) and (2) can be used to derive expressions for P(i,j,i + 2). The arbitrage-free condition yields
P( i,j,i + 2) = P( i,j,i + 1) ( 0 . 5 P ( i + 1,j + 1,i + 2) + ( 1 -- 0 . 5 ) e ( i +
1,j,i + 2))
(3)
and Maloney and Byrne's equation yields
P(i,j,i + 2) = P(i,j,i + 1)exp{ - 0 . 5 ( u + d)R(i,j,i + 1)}. (4) Eqs. (3) and (4) only give equivalent expressions for P(i,j,i'+ 2) when u = d = 1. When this is the case, all future interest rates are known with certainty and the
R.E. Mathis, Ill~Journal of Banking & Finance 19 (1995) 1305-1307
1307
model can no longer be described as binomial. Thus, the relationship between oneand two-period term structure rates given by Maloney and Byrne is not consistent with an arbitrage-free term structure. To correct the model it is necessary to revise Maloney and Byrne's equilibrium condition in order to obtain a relationship between one- and two-period term structure rates that is consistent with the arbitrage-free condition given by Eq. (1). It is assumed that conditional on information available at time period i and state j, the expected one-period return per dollar on a two-period bond is equal to the one-period risk-free return per dollar at that time and in that state plus a constant term premium. This assumption implies the following relationship between oneand two-period term structure rates: 1
(
l+TPexp{-R(i,j,i+l)} ) 2R(i,j,i +2) = In 0.5[exp{ -dR(i,j,i + 1)} + exp{ -uR(i,j,i + 1)}] +R(i,j,i+ 1). (5) Given this relationship, the model is arbitrage-free and the portfolio duplication procedure discussed by Maloney and Byrne can be employed to derive the equilibrium rates on all other default-free zero coupon bonds.
Acknowledgements The author wishes to thank Gerald O. Bierwag and Arun J. Prakash for their many helpful comments and suggestions.
References Cox, J., J. Ingersoll and S. Ross, 1981, A re-examination of traditional hypotheses about the term structure of interest rates, Journal of Finance 36, 769-799. Cox, J., L Ingersoll and S. Ross, 1985, A theory of the term structure of interest rates, Econometrica 53, 385-408. Ho, T. and S. Lee, 1986, Term structure movements and pricing interest rate contingent claims, Journal of Finance 41, 1011-1030. Maloney, K. and M. Byrue, 1989, An equilibrium debt option pricing model in discrete time, Journal of Banking and Finance 13, 421-442.
1 The term premium in Eq. (5) is expressed as a rate that is compounded once per period.
Journal of Banking & Finance 19 (1995) 1305-1307
BANKING & FINANCE
A note on an equilibrium debt option pricing model in discrete time Roswell E. Mathis, III * Department of Finance, College of BusinessAdministration, Florida International University, Miami, FL 33199, USA Received October 1993; final version received February 1994
Abstract In this note, the Maloney and Byme (1989) discrete time model of the term structure of interest rates for the pricing of interest rate contingent claims is corrected so that the term structures generated by the model satisfy arbitrage-free restrictions. JEL classification: G13 Keywords: Interest rate options; Valuation
1. Introduction Maloney and Byrne (1989) developed a discrete time model of the term structure of interest rates for the pricing of interest rate contingent claims. There is an error in the development of the model which results in the existence of arbitrage within the model. In this note, the model is corrected.
* Corresponding author. Phone: (1) (305)-348-2680. 0378-4266/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 3 7 8 - 4 2 6 6 ( 9 4 ) 0 0 0 7 2 - 7
R.E. Mathis, Ill~Journal of Banking & Finance 19 (1995) 1305-1307
1306
2. Illustration of the error and correction of the model
Maloney and Byme define the following:
P(i,j,k) R(i,j,k) TP
R(i,j,i + 1)
= the price in time period i and state j for a default-free zero coupon bond that pays $1 in time period k, where k = i + 1, i + 2 ..... = the continuously compounded risk-free rate at time period i and state j for a zero coupon bond maturing in period k, -- the term premium on two-period zero coupon bonds, and = uJdi-JR(O,O,1), where u and d are constants and u > 1 > d > 0.
In the model, interest rates and, thus, bond prices evolve over time according to a binomial process. To ensure that term structures generated by the model are arbitrage-free, there must be a constant, 7ri.i, at each time i and state j on the lattice so that the following equation will hold for all possible combinations of i, j, and k on the lattice:
(Tri,jP(i+ 1 , j + 1,k) + (1 - 7rij)P(i+ 1,j,k))P(i,j,i+ 1) •P(i,j,k).
(1)
The constant, ~i,j, can be interpreted as the "risk neutral" or " h e d g i n g " probability at time i and state j. Maloney and Byrne, in their Eq. (4), specify the relationship between one- and two-period term structure rates as 2 R ( i , j , i + 2) - TP = R(i,j,i + 1) [1 + 0.5(u + d ) ] .
(2)
Eq. (2) involves expectations over continuously compounded rates, whereas the arbitrage-free condition of Eq. (1) involves expectations over prices. Eq. (1) does not imply Eq. (2). To demonstrate this assertion, let the term premium equal zero. The local expectations hypothesis then describes equilibrium and, by definition, default-free zero coupon bonds of all maturities must yield the same expected return over one-period. Therefore, ~ri,j must equal the probability of an upstate at time i and state j, which Maloney and Byrne assume is equal to 0.5 for all i and j. Given the above conditions, Eqs. (1) and (2) can be used to derive expressions for P(i,j,i + 2). The arbitrage-free condition yields
P( i,j,i + 2) = P( i,j,i + 1) ( 0 . 5 P ( i + 1,j + 1,i + 2) + ( 1 -- 0 . 5 ) e ( i +
1,j,i + 2))
(3)
and Maloney and Byrne's equation yields
P(i,j,i + 2) = P(i,j,i + 1)exp{ - 0 . 5 ( u + d)R(i,j,i + 1)}. (4) Eqs. (3) and (4) only give equivalent expressions for P(i,j,i'+ 2) when u = d = 1. When this is the case, all future interest rates are known with certainty and the
R.E. Mathis, Ill~Journal of Banking & Finance 19 (1995) 1305-1307
1307
model can no longer be described as binomial. Thus, the relationship between oneand two-period term structure rates given by Maloney and Byrne is not consistent with an arbitrage-free term structure. To correct the model it is necessary to revise Maloney and Byrne's equilibrium condition in order to obtain a relationship between one- and two-period term structure rates that is consistent with the arbitrage-free condition given by Eq. (1). It is assumed that conditional on information available at time period i and state j, the expected one-period return per dollar on a two-period bond is equal to the one-period risk-free return per dollar at that time and in that state plus a constant term premium. This assumption implies the following relationship between oneand two-period term structure rates: 1
(
l+TPexp{-R(i,j,i+l)} ) 2R(i,j,i +2) = In 0.5[exp{ -dR(i,j,i + 1)} + exp{ -uR(i,j,i + 1)}] +R(i,j,i+ 1). (5) Given this relationship, the model is arbitrage-free and the portfolio duplication procedure discussed by Maloney and Byrne can be employed to derive the equilibrium rates on all other default-free zero coupon bonds.
Acknowledgements The author wishes to thank Gerald O. Bierwag and Arun J. Prakash for their many helpful comments and suggestions.
References Cox, J., J. Ingersoll and S. Ross, 1981, A re-examination of traditional hypotheses about the term structure of interest rates, Journal of Finance 36, 769-799. Cox, J., L Ingersoll and S. Ross, 1985, A theory of the term structure of interest rates, Econometrica 53, 385-408. Ho, T. and S. Lee, 1986, Term structure movements and pricing interest rate contingent claims, Journal of Finance 41, 1011-1030. Maloney, K. and M. Byrue, 1989, An equilibrium debt option pricing model in discrete time, Journal of Banking and Finance 13, 421-442.
1 The term premium in Eq. (5) is expressed as a rate that is compounded once per period.