We can now eliminate the additional dimension, z, by replacing U(r,S, z,t) ezV(r, S, t) in Equation 3.95. This gives the two-dimensional pricing equation
(3.95)
with the end condition
(3.96)
An Application of Jump Processes: Credit Derivatives
The purpose of this section is to illustrate issues in the derivation of pricing equations when jumps are involved. The reader interested in gaining a broader understanding of credit derivatives may consult some of the numer- ous articles in the literature (e.g., Duffie and Singleton, 1999; Schoenbucher, 1997).
Credit derivatives are instruments sensitive to changes in either the credit quality of the issuer of an underlying security, or to so-called
“credit events” affecting the issuer of the underlying security. Among the credit events that affect the quality of the issuer are defaults and changes in credit rating. In principle, a corporate bond can be viewed as a credit derivative much the same way a nondefaultable bond can be viewed as an interest rate derivative. The market has agreed to call credit deriva- tives instruments specifically designed to manage credit risk, either by mitigating the risk or by gaining exposure to credit risk.
The two main approaches for pricing credit derivatives are structural methods, which attempt to establish relationships between the capital structure of the issuer of securities underlying the credit derivative, and reduced methods, where pricing is done by postulating models for the sto- chastic processes involved, without particular regard for capital structure considerations.
We will concentrate on the applications to reduced models. We will illustrate the derivation of the pricing equations by considering a derivative that depends on a credit event such as default or credit change. We will assume that the credit event can be characterized as a pure jump process. If the credit event is a default, the value of the jump process at time t is an integer number that indicates the number of defaults that have occurred up to time t.
The process for the value of the derivative will be a function of the sto- chastic process that represents the credit event, as well as a function of any other relevant diffusion processes, such as the short rate.
In Chapter 2 we discussed the application of Ito’s lemma to the case of processes with jumps. Consider the combined jump-diffusion process x (for additional details, please refer to Chapter 2):
(3.97)
If we now set 0 and 0, we obtain a pure jump process with jumps at time t of magnitude z, distributed according to (x(t), z)
dx t( ) = +h x t( ( ),t)
∫
z(x t( ),z)dzdt+dW t( )+dJx= =
(3.98) If in addition, we also assume that (x(s), z) (z 1), where (.) is the Dirac delta function, we obtain a jump process that can jump by one at time t. We will assume that this is a useful representation of the process of credit events we are interested in. Notice that at this stage we are not precisely specifying what the credit event is. The event intensity, h, can depend on both the event count, x(t), and time. If the credit event were a default and we restricted our attention to the first event of default, the intensity would be only a function of time. The process for the credit event becomes
(3.99) Additionally, we may assume that the intensity of occurrence of the credit event follows a diffusion process
(3.100) where h and h can be functions of time and the other continuous processes in the problem. To illustrate the derivation of the pricing equation for credit derivatives, let’s also assume that the short rate is described by a diffusion process
(3.101) where the drift and the volatility can be functions of time and the other continuous processes in the problem. Using Ito’s lemma with jumps, the value of the derivative, V(t), is governed by the following jump-diffusion process:
(3.102) where the last term represents the random shocks undergone by the deriva- tive’s price as a result of the jumps in default process x. Carrying out the integral in Equation 3.102, we get
dx t( ) = h x t( ( ), t)
∫
z(x t( ),z)dzdt+dJx( )t= –
dx t( ) = hdt+dJx
dh t( ) = hdt+hdWh
dr = rdt+rdWr
dV t( ) ∂V
---∂t r∂V
---∂r h∂V
---∂h r2 ---2 ∂2V
∂r2
--- rhr,h∂2V
∂r∂h--- h2 ---2 ∂2V
∂h2 ---
+ + + + +
dt
=
h x t( ( ), t)
∫
[V x t( ( )+z,r,t)–V x t( ( ),r,t)](z–1)dz
dt
+
∂V∂r
---rdWr ∂V
---∂hhdWh dJV
+ + +
(3.103)
Assume now that the market fully diversifies the shocks due to default.
This means that the risk neutral expectation of dV(t) must be equal to rV(t). If we assume that the processes for h,r, and x are in the risk neutral measure, this expectation gives us the partial differential equation govern- ing the derivative price. Taking the expectation of Equation 3.103,
(3.104)
where we used the fact that Wr,Wh, and JV are martingales.
For clarity of the discussion, assume now that the credit event charac- terized by x(t) is the default process. Notice that x does not contribute to the dimensionality of the problem because it does not appear on the left- hand side of the pricing equation. It is convenient to rewrite the pricing equation as follows:
(3.105)
where V is the known change in value of the derivative if default occurs.
We will next discuss two simple examples of application of the pricing equation.
Defaultable Bonds
In the event of default of the issuer of a bond, the bond drops in value to a level called the recovery value of the bond. The way the recovery value of the bond is characterized can have a significant influence on the pricing of the bond. Let’s assume that upon default, the holder of the bond receives a given fraction, R(t), of the contemporaneous market value of the bond. In that case, the right-hand side of Equation 3.105 can be written as
(3.106) dV t( ) ∂V
∂t
--- r∂V
---∂r h∂V
---∂h r2 ---2 ∂2V
∂r2
--- rhr,h∂2V
∂r∂h--- h2 ---2 ∂2V
∂h2 ---
+ + + + +
dt
=
h V x t[ ( ( )+1, r, t)–V x t( ( ),r,t)]dt +
∂V
---∂rrdWr ∂V
---∂hhdWh dJV
+ + +
∂V
---∂t r∂V
---∂r h∂V
∂h
--- r2
---2 ∂2V
∂r2
--- rhr,h∂2V
∂r∂h--- h2 ---2 ∂2V
∂h2 ---
+ + + + +
V x t( ( ))r–h V x t[ ( ( )+1)–V x t( ( ))]
( )
=
∂V∂t
--- r∂V
---∂r h∂V
---∂h r2 ---2 ∂2V
∂r2
--- rhr,h∂2V
∂r∂h--- h2 ---2 ∂2V
∂h2 ---
+ + + + +
Vr–hV
=
Vr–hV = Vr–h RV V( – ) = Vr+hLV = V r( +Lh)
where L is the loss fraction in the event of default. Of course, R(t) can be a stochastic process. This assumption about recovery is known as recovery of market value (Duffie and Singleton, 1999). The pricing equation is now
(3.107)
In addition, we need an end condition at maturity and boundary condi- tions for r 0, r ,h 0,h . If we view the bond as a contract that terminates upon default, the end condition will be equal to a known payment equal to the notional amount. Although the boundary conditions will be dis- cussed in greater detail in Chapter 7, here we can make an observation that brings up the significance of assuming the recovery to be a given fraction of market value as opposed to a given amount. If we ask ourselves what the value of the bond is as h , it appears intuitively clear that if the recovery is equal to a given amount to be paid in case of default, the value of the bond should be precisely this amount. The reason for this is that if the intensity of default is infinitely large, the bond will default immediately and the holder will receive the known amount immediately. If, on the other hand, recovery is given in terms of market value, the pricing equation will contain a source term of the form LVh, which cannot be balanced by the finite terms on the left-hand side if V 0. The only way for the equation to make sense as h (assuming V remains smooth) is for V 0 as h→ .
The suitable boundary condition for h 0 is the solution to the PDE describing a default-free bond.
Full Protection Credit Put
A credit put compensates the holder for the loss of value of defaultable bonds as a result of default, credit degradation, or both. Here we formulate a simple example as follows. Consider a derivative that pays the holder the following amount if default occurs at time t, where 0 t T,
(3.108) where R(t) is the recovery rate, B(t,TB) is a risk-free bond maturing at time TB, Bd(t,TD) is a defaultable bond maturing at time TD, and K is a constant. We will assume that the occurrence of a default is enough to trigger the payment.
In such a case, the contract terminates. If no payments triggered by default have occurred until maturity, at maturity the derivative pays the following:
(3.109)
∂V∂t
--- r∂V---∂r h∂V---∂h r2 ---2 ∂2V
∂r2
--- rhr,h∂---∂r∂2Vh h2 ---2 ∂2V
∂h2 ---
+ + + + +
V r( +Lh)
=
= = = =
→
→ ≠ →
=
< <
KB t,( TB)–R t( )Bd(t,TD)
max [0, KB T,( TB)–Bd(T,TD) ]
This is a full-protection credit put because it protects both against defaults and drop of value as a result of increased yields.
The right-hand side of Equation 3.105 in this case is
(3.110) The pricing equation of the credit put can be written as
(3.111)
with the end condition
(3.112) and suitable boundary conditions.
Notice that in order to solve Equation 3.111 we need to know B(t,TB) and Bd(t,TD) at every point in the solution space. This means that in addi- tion to this PDE, we need to solve the two additional ones representing the values of the riskless and defaultable bonds. For the riskless bond we have
(3.113) and for the defaultable bond,
(3.114)
In addition, we need suitable boundary conditions. The issue of boundary conditions will be discussed in detail in Chapter 7. The pricing problem is then the solution of a system of the three partial differential Equations 3.111, 3.113, and 3.114.