4.5 Appendix to Chapter 4
4.5.1 Derivation of the Bond Pricing Equation
The crucial assumption in our model is that the value CBdef of the default-risky bond is a fraction P of the default-free bond value, CBnodef, and that the illiquid bond’s value CBilliq
is a fraction L of the perfectly liquid bond’s value CBliq:
CBdef = P ·CBnodef, CBilliq = L·CBliq
Naturally, the size of the factors P and Lwill depend on the period during which the bond is subject to default risk or illiquidity and to the extent of the default risk and illiquidity. A simple interpretation for P is developed by Duffie and Singleton (1997) who view P(t, t+ ∆t) = Et
h
exp−Rt+∆t
t λ(s)ds
i
as the conditional survival probability between t and t+ ∆t.
Attaching an interpretation to L is somewhat more difficult. For our purposes, it will suffice to assume that selling an illiquid bond involves random searching costs (1−L) proportional to the bond value. This yields the relation CBilliq = L · CBliq. We
assume that L has a similar exponential-affine representation as P, i.e. L(t, t+ ∆t) =
Et
h
exp−Rt+∆t
t γ(s)ds
i
. The liquidity intensity γ can then be interpreted as the continuous-time rate formulation of the searching costs which arise for each infinitesimally small time interval until the maturity of the bond.
We now show our argument in a three-period model with independent interest rate, default risk, and liquidity factors for notational simplicity. However, this model can easily be extended to dependent risk factors by replacing the expectations operator for D, P, and L with the joint expectations operator. Our goal is then to price a coupon-bearing, default-risky, illiquid bond at time 0 with a fixed coupon c paid at times t = (1,2,3), notional F, and maturity in 3. For ease of exposition, we also assume a recovery rate of 0. This bond will pay the coupon c and the face value F at time 3 if no default has occurred prior to 3. Therefore, the dirty price of the bond at time 3, CB(t= 3) is equal to the payment F +c since there is no default or liquidity risk as well as no time delay until the payment is made.
If we go back one time step to 2, the value of the bond is equal to the coupon c plus the value of the claim on the payment F +c at 3. The (dirty) price of a perfectly liquid,
default-free bond with identical payment structure at 2 is given by CBliqnodef(t= 2,dirty) = c+CBnodefliq (t = 2,clean)
= c+E2
CBliqnodef(t= 3,dirty) = c+D2,3·CBliqnodef(t= 3,dirty)
= c+D2,3·(F +c),
where D2,3 is the default-free interest rate discount factor that applies between 2 and 3.
If the claim is subject to default risk between 2 and 3, then the price is equal to CBdefliq(t= 2,dirty) = c+CBliqdef(t= 2,clean)
= c+P2,3·CBliqnodef(t= 2,clean)
= c+P2,3·E2
CBliqnodef(t= 3,dirty) = c+P2,3·D2,3·CBliqnodef(t= 3,dirty)
= c+P2,3·D2,3·(F +c)
= c + P2,3·D2,3 ·c
+P2,3 ·D2,3·F.
If, in addition, the claim is also subject to liquidity risk, that is searching, trading, or transaction costs are incurred if the claim on the payment at time 3 is sold prior to time 3, the price is given by
CBilliqdef(t = 2,dirty) = c+CBilliqdef(t= 2,clean) = c+L2,3·CBliqdef(t= 2,clean)
= c+L2,3·P2,3·CBliqnodef(t= 2,clean)
= c+L2,3·P2,3·D2,3·CBliqnodef(t = 3,dirty)
= c+L2,3·P2,3·D2,3·(F +c)
= c + L2,3·P2,3·D2,3·c
+L2,3·P2,3·D2,3·F.
to default risk and illiquidity both between 2 and 3 but not between 1 and 2 equals CBilliqdef,t,t=2=2(t= 1,dirty) = c+CBilliqdef,t,t=2=2(t= 1,clean)
= c+E1
h
CBilliqdef,t,t=2=2(t= 2,dirty)
i
= c+D1,2·CBilliqdef(t = 2,dirty)
= c+D1,2·[c+L2,3·P2,3·D2,3·(F +c)]
= c + D1,2·c
+L2,3·P2,3·D1,2·D2,3·c
+L2,3·P2,3·D1,2·D2,3·F.
If above claim is also subject to default risk between 1 and 2, the value equals CBdefilliq,t,t=1=2(t = 1,dirty) = c+CBilliqdef,t,t=1=2(t = 1,clean)
= c+P1,2·CBilliqdef,t,t=2=2(t= 1,clean)
= c+P1,2·D1,2·CBilliqdef(t= 2,dirty)
= c+P1,2·D1,2·[c+L2,3·P2,3 ·D2,3·(F +c)]
= c + P1,2·D1,2·c
+L2,3·P1,2·P2,3·D1,2·D2,3·c
+L2,3·P1,2·P2,3·D1,2·D2,3·F.
Adding liquidity risk between 1 and 2 gives the value of the default-risky, illiquid bond as CBilliqdef,t,t=1=1(t= 1) = c+L1,2·CBilliqdef,t,t=1=2(t= 1,clean)
= c+L1,2·P1,2·CBilliqdef,t,t=2=2(t= 1,clean)
= c+L1,2·P1,2·D1,2·CB def,t=2 illiq,t=2(t= 2,dirty) = c + L1,2·P1,2·D1,2·c +L1,2·L2,3·P1,2·P2,3·D1,2·D2,3·c +L1,2·L2,3·P1,2·P2,3·D1,2·D2,3·F.
The choice of the form of the default-free discount factor, the survival probability and the liquidity discount factor yields thatD1,2·D2,3 =D1,3,P1,2·P2,3 =P1,3, andL1,2·L2,3 =L1,3.
Therefore, we can write above equation as CBilliqdef,t,t=1=1(t= 1) = c· 3 X i=1 L1,i·P1,i·D1,i+F ·L1,3·P1,3·D1,3, where L1,1 =P1,1 =D1,1 = 1.
Adding a non-zero recovery rate and allowing for pricing at any point-in-time t yields the bond pricing equation.