2. A Basic Model of Bond Markets 19
2.3 Defining the No-Arbitrage Condition
In an arbitrage portfolio, the arbitrage profit might not be obtained at time zero. It might be obtained at some future time and given the structure of the market it may not be possible to find an arbitrage portfolio for which the profit is obtained at time zero. Our modification of the optimization problem (2.2) should take this into account.
In order to construct the portfolio completely free of risk we have no other alternative but to assume that a dollar in period one can cover at most a dollar liability in period two. If we allow for more than a dollar liability at time two, some risk will be involved in the position. If even an
“infinitesimal” risk is involved in the portfolio, it is no longer an arbitrage
portfolio. Thus, as in the above example, we can take the excess cash obtained at time j and use it to cover a shortage, if such exists, at time
j+1.
This is done by introducing the additional variablesZ0,Z1, andZ2, etc.
which are constrained to be nonnegative. The variable Z0 represents the cash we move from period zero to period one. Thus it can be used to cover any shortage that might arise in period one. In the same manner,Z1is the amount we might want to transfer from period one to period two. Hence we subtractZjfrom the cash flow obtained at timejand add it to the cash flow obtained at time j+1. TheZs are constrained to be positive, since negative values mean we allow cash to be moved backward through time. (In other words, negative values ofZ mean borrowing at the current period against excess cash which will be obtained in the next time period.) However, as of time zero, if we would like to have a portfolio free of risk, we have no way of knowing for sure what a dollar obtained in some future period 1<j might be worth at time j−1.
We bring this final example to motivate our definition of the NA con- dition. Consider a market with 2 bonds, the price of each is $95. Bond B1 has a coupon rate of 10% and matures in one year,B2is a zero coupon which matures in two years and its face value is $110. That is, the market is represented by what we call a payoff matrixA
A=
110 0 0 110
and a price vectorP,P= 95
95
. The proceeds from the sale are−B1ã95−B2ã95.
The cash flow in period 1 isB1ã110+B2ã0.
The cash flow in period 2 isB1ã0+B2ã110.
Clearly there is an arbitrage opportunity in this market, can you the reader stipulate it? However, our structure would not identify it. Let us verify below that indeed the output of this structure isB1=0 andB2=0.
> simplex[maximize](-95*B1-95*B2,\
> {110*B1+0*B2>=0, 0*B1+110*B2>=0});
{B1=0,B2=0}
Let us introduce the additional variables Z0 and Z1 which are con- strained to be nonnegative. The variable Z0 represents the cash we move
from period zero to period one. Thus it can be used to cover any shortage that might arise in period one. In the same manner Z1 is the amount we might want to transfer from period one to period two. Hence we subtract it from the cash flow obtained at time one and add it to the cash flow obtained at time two. We also define variables for the cash flow in each periodC0, C1andC2(these can be avoided but are added for clarity). Hence we now have
C0=−B1ã95−B2ã95−Z0 C1=110ãB1+Z0−Z1
C2=110ãB2+Z1.
In an arbitrage portfolio,C0,C1 andC2are all nonnegative. Arbitrage will be realized if either; at least one of the Cs is strictly positive, i.e., C0+C1+C2>0, or ifC0=C1=C2=0 and at least one of theZs is strictly positive. That is, arbitrage is realized if and only ifC0+C1+C2+Z0+Z1>
0. Hence to identify an arbitrage portfolio we can solve:
Max C0+C1+C2+Z0+Z1 subject to
C0=−B1ã95−B2ã95−Z0 C1=110ãB1+0ãB2+Z0−Z1
C2=0ãB1+110ãB2+Z1 Z0≥0,Z1≥0,C0≥0,C1≥0,C2≥0
If the optimal solution of this problem is Z0=Z1=C0=C1=C2= B1=B2=0, no arbitrage portfolio exists and the no arbitrage condition is satisfied. If this is not the case, the problem has no finite solution, as the arbitrage profit will be unfounded. As before, an arbitrage portfolio can be identified in these cases by imposing constraints on short sales.
Here is the solution of this optimization.
> simplex[maximize](C0+C1+C2+Z0+Z1,{C0=-B1*95-B2*95-Z0,\
> C1=B1*110+B2*0+Z0-Z1,C2=B1*0+B2*110+Z1,\
> Z0>=0,Z1>=0, C0>=0,C1>=0,C2>=0});
As we see, no output is generated as the problem is unbounded. We thus proceed to impose a constraint on bond 2 short position. (An exercise
at the end of the Chapter will ask you to explain why imposing a constraint on the short position of bond 2 will suffice but posting it on bond 1 will not generate an output showing an arbitrage portfolio.)
> simplex[maximize](C0+C1+C2+Z0+Z1,\
> {C0=-B1*95-B2*95-Z0, C1=B1*110+B2*0+Z0-Z1, C2=B1*0+B2*110+Z1,\
> Z0>=0,Z1>=0, C0>=0,C1>=0,C2>=0,B2>=-1});
{B1=1, B2=−1,C0=0, C1=0, C2=0,Z0=0,Z1=110}
We are finally ready to have the general formulation of the No- Arbitrage condition.
Definition: (The No-Arbitrage Condition)
Let aijbe the payoff (coupon payment or face value) from bond i in period j, where i=1, . . . ,N and j=0, . . . ,K. The No Arbitrage condition is satisfied if the optimal value of the optimization problem below is zero.
Maxx1,...,xn,z0,...,zk
"
K
∑
j=0
Cj+
K−1
∑
j=0
Zj
#
such that
N i=1∑
(−xiPi)−z0=C0
N
∑
i=1
xiai j+zj−1−zj=Cj,j=1, . . . ,K−1
N
∑
i=1
xiai j+zk−1=Ck
0≤zj,j=0, . . . ,K−1,0≤Cj,j=0, . . . ,K (2.3) We conclude this subsection by introducing the procedureNarbitBthat determines if the no arbitrage condition in a market is satisfied. The input to the procedure is the prices of the bonds and their payoff. The presumption of this procedure is that the time intervals are equal. That is, the time in- terval between the current time (on which the price of the bonds are given) and the first coupon payment is equal to the time between the first and sec- ond coupon payment etc. The prices of the bonds are represented by an
array with its elements being the prices of the bonds. Hence if there are two outstanding bonds the price of the first being 95 and the second being 95 this will be entered as [95,95]. The cash flow from each bond is entered in the same manner. If the first bond’s cash flow is $110 followed by $0 it will be entered as [110,0] and if the second bond’s cash flow is $0 followed by $110 it will be entered as [0,110]. The above example can represent a case where the first bond matures in a year, the second bond in two years and the prices are observed a year before the first bond matures. Both the second bond and the first bond are zero coupon bonds. The cash flow from the bonds will be entered as the first parameter to the procedure in the form [[110,0], [0,110]] and the understanding is that the first price is the price of the first bond and the second is the price of the second bond. To find if the NA is satisfied in this market you run:
> NarbitB([[110,0],[0,110]],[95,95]);
The no-arbitrage condition is not satisfied An arbitrage portfolio is:
Buy, 1
110,of Bond, 1 Short, 1
110, of Bond, 2 The cost of this portfolio is zero
This portfolio produces income of,1, at time,1 This portfolio produces income of, −1, at time,2
The procedure identifies an arbitrage portfolio if it exists, and reports its composition. If an arbitrage portfolio does not exist the procedure states this, and produces further output that will be discussed shortly. The reader may already anticipate the meaning of this output given the discussions above. Note that the arbitrage amount is not known at the time the positions are taken. The arbitrage is actually generated from investing the $1 cash inflow obtained at time 1, (a year after the initiation of the portfolio) for a year and then using it to pay the cash outflow of $1 at time 2 (a year after the $1 was invested). At time 2, the $1 invested at time 1 for a year will grow to 1+r, but the r is not known at time zero. Hence the arbitrage amount is therobtained at time 2, but the value ofr, the one-year spot rate as of time 1, will be known only at time 1.
Let us see what happens if we change the prices of the bonds so that the price of the first bond is 95 and the second is 94:
> NarbitB([[110,0],[0,110]],[95,94]);
The no-arbitrage condition is satisfied.
The discount factor for time, 1,is given by, 19
The interest rate spanning the time interval,[0,1],is given by,22 0.1579
The discount factor for time, 2,is given by, 47
The interest rate spanning the time interval,[0,2],is given by,55 0.1702
The function Vdis ([c1,c2,..]), values the cashflow [c1,c2,..]
Let us examine another example of a market with 3 bonds and 3 time periods as stipulated below
> simplex[maximize](-94.5*B1-97*B2-89*B3,
{105*B1+10*B2+8*B3>=0, 110*B2+8*B3>=0,108*B3>=0});
{B1=0,B2=0,B3=0} and then run it again using theNarbitBprocedure.
> NarbitB([[105,0,0],[10,110,0],[8,8,108]], [94.5,97,89]);
The no-arbitrage condition is satisfied.
The discount factor for time,1, is given by,0.9000000000 The interest rate spanning the time interval,[0,1],is given by, 0.111
The discount factor for time,2, is given by,0.8000000000 The interest rate spanning the time interval,[0,2],is given by, 0.250
The discount factor for time,3, is given by,0.6981481481 The interest rate spanning the time interval,[0,3],is given by, 0.432
The function Vdis ([c1,c2,..]), values the cashflow [c1,c2,..]
You, the reader, may wish to explore this procedure further, changing the prices or the payment structure, before reading on. For example, let us examine the effect of making bond 2 mature at time 3, instead of at time 2, while leaving its price unchanged.
> NarbitB([[105,0,0],[10,10,110],[8,8,108]], [94.5,97,89]);
The no-arbitrage condition is not satisfied An arbitrage portfolio is:
Buy, 0.02910762160, of Bond,1 Short, 1.0,of Bond,2 Buy, 1.058981233, of Bond, 3
Buying this portfolio produces income of,2.10−8at time, 0 This portfolio produces income of,1.528150132,at time,1 This portfolio produces income of, −1.528150136, at time,2
This portfolio produces income of,4.3699732, at time,3
Here is another example: Note that in this market there are two bonds and three time-periods. A market where certain cash flows cannot be gen- erated is referred to as an incomplete market. A market where the number of time periods is larger than the number of bonds must be an incomplete market. We will investigate this concept shortly.
> NarbitB([[3,103,0],[11,11,111]],[100,122]);
This is an incomplete market.
The no-arbitrage condition is satisfied.
The set of discount factors is,
d1=−103
3 d2+100
3 , d3=1100
333 d2−734 333, 97
103<d2,d2<50 53
The reader is encouraged to experiment with other combinations of bond prices and bond payoffs to arrive at an appreciation for the features of the debt market.
The NarbitB procedure is also capable of handling a situation when the period between the current time and the next coupon payment is not the same as the time between coupon payments. Assume that the above example reflects a case where the bonds pay annual payments and there is a year between the current time and the next coupon payment. If we now change the assumption that time between the current time and the next coupon payment is only half a year we need to use this as a basic unit of time. Hence the cash flow from the bonds will also be reported based on the time of unit of half a year. Consequently the cash flow from the first bond will be represented as [0,3,0,103,0,0] and from the second bond as [0,11,0,11,0,111], and if the prices of the bonds are now 90 and 130 we
have to run:
> NarbitB([[0,3,0,103,0,0],[0,11,0,11,0,111]],[90,130]);
This is an incomplete market.
The no-arbitrage condition is not satisfied An arbitrage portfolio is:
Buy, 119
87 ,of Bond, 1 Short,1, of Bond, 2 Buying this portfolio produces income of, 200
29 , at time,0 This portfolio produces income of,0, at time,1 This portfolio produces income of,−200
29 ,at time, 2 This portfolio produces income of,0, at time,3 This portfolio produces income of, 11300
87 ,at time, 4 This portfolio produces income of,0, at time,5 This portfolio produces income of, −111, at time,6
When a bond is purchased between coupon payments, the next coupon is divided between the old owner and the new owner in a prorated fashion.
Assume coupon payments are every six months and a bond was purchased two months before a coupon payment. Thus, the first coupon will be di- vided such that the old owner will receive 4
6 of it and the new owner will receive 2
6 of it. Hence, the new owner, upon purchasing the bond, pays 4 of a coupon to the old owner. 6
Quoted bond prices (in North America) do not include the payment of the fraction of the first coupon to be paid to the old owner. That is, the quoted price of a coupon bond does not include accrued interest, but only the present value of the future cash flow. A bond price that does not include accrued interest is referred to as a clean price, while if the accrued interest is included, it is referred to as a dirty price. The latter, in the main, is quoted in European bond markets. Hence, Dirty Price = Clean Price +Accrued Interest. A price of a bond that is quoted immediately after a coupon payment is clean and dirty, as there is no accrued interest to include.
Clean prices change for economic reasons only, while dirty prices change due to the passage of time as well, because the prorated portion changes.
Thus when bond prices are quoted at a time that is not immediately after a coupon payment, clean prices should be used to infer the interest rates that are implicit in bond prices.
Note that in the above representation clean prices should be used. As well, the market becomes an incomplete market as there are more time periods than bonds. If we change the prices of the bonds so that the no arbitrage condition is satisfied, e.g., to [90,110] and re-run NarbitB, we have the following output:
> NarbitB([[0,3,0,103,0,0],[0,11,0,11,0,111]],[90,110]);
This is an incomplete market.
The no-arbitrage condition is satisfied.
The set of discount factors is,
d2=−103
3 d4+30, d6=1100
333 d4−220 111, 87
103<d4,d1<1, d3<−103
3 d4+30, d4< 45
53,d4<d3,d5<d4, −103
3 d4+30<d1, 1100
333 d4−220 111<d5
The results regarding the set of discount factors are not so applicable since in an incomplete market there is a set of such factors. While the concept of incomplete market is an important one, in the context of the bond markets it is suppressed due to some features of the bond market. This will become apparent when we deal with the estimation of the discount factors or equivalently of the term structure.