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.
simply consist of the decision to exercise or not exercise at any given time between the inception of the contract and its maturity date. In general, however, the exercise policy can consist of complex rules stipulated in the contract. The market has evolved specialized names for specific types of exercise policies. Here, however, we call an American derivative any deriva- tive whose cash flows can be affected by the holder in a nontrivial manner (the trivial manner to affect the cash flows is to sell the contract).
Relationship between European and American Derivatives
For simplicity of exposition, we will consider the case of a single underlying process, S(t). We denote the set of exercise strategies by c(.), whose argu- ments may include items such as the underlying process, properties of the past history of the underlying processes, time, and so on. In the simple case of an American put, c(S,t) is a binary variable representing the exercise or don’t exercise decisions.
If markets are complete, there is a straightforward conceptual relation- ship between European and American derivatives. If the market is com- plete, the cash flows generated by any derivative security can be replicated through a dynamic trading strategy with other securities. Fixing an exercise strategy means that the holder of the security will not be able to influence the cash flows. This means that for every exercise strategy of an American security there is a corresponding European security. Consider a derivative whose price, V, depends on an exercise strategy, c. Consider now a particu- lar exercise strategy , such that
(3.115) The exercise strategy is called an optimal exercise strategy. It is easy to see that in order to prevent arbitrage, the value of the American security must equal V( ). Assume that the American security’s price is less than V( ). In this case, we sell the portfolio that replicates V( ) and we purchase the Amer- ican derivative. We now follow an optimal exercise strategy, thereby match- ing the cash flows between the American security and V( ), and we keep the initial risk-free profit. If, on the other hand, the price of the American security were greater than V( ), we can buy the portfolio that replicates V( ) and sell the American derivative. The holder of the American security will follow an optimal exercise strategy, thus causing the cash flows between the American security and V( ) to be matched. Again, we keep the initial risk-free profit.
Notice that because we assume that markets are complete, there is no prob- lem in replicating V( ).
cˆ
V c( )ˆ max c V c( )
= cˆ
cˆ cˆ
cˆ
cˆ
cˆ cˆ
cˆ cˆ
We see that the valuation of American options is no different than the valuation of European options, provided we know the optimal exercise strategy . Of course, we don’t know ahead of time what the optimal exer- cise strategy is. In practice, the optimal exercise strategy is found simulta- neously with the price.
In derivatives pricing, optimal exercise strategies are associated with the concept of free boundaries. For illustration, consider the case of the simple exercise–don’t exercise version of exercise strategy. The free boundary separates the region in (S,t) where it is optimal to exercise the option from the region where it is optimal to hold the option. The free boundary is also referred to as the exercise boundary. The argument in the previous paragraph indicates that the price of the American security will have the same description as a European security on the side of the free boundary where it is optimal to hold the security, and it will be equal to the exercise value on the side where it is optimal to exercise.
Since the European pricing problem is described by a partial differential equation, the presence of the exercise boundary poses the question of suitable conditions at those boundaries. In the approaches that we fol- low in this book, it is not necessary to be concerned about the details of what happens at the exercise boundaries. The reason is that in the least squares Monte Carlo method and in the linear complementarity imple- mentation of finite differences, our methods of choice for dealing with early exercise, the exercise boundary is resolved as part of the solution.
In order to better understand the behavior of the solution, however, the next two sections discuss the conditions that must be satisfied at the exercise boundaries. It is possible to pose the problem such that the exer- cise boundary is also a boundary of the computational domain. This is easy to do if the problem has one space variable. The resolution of the boundary as an integral part of the solution, however, is a much more practical approach, especially in several dimensions (this is true of simu- lation and finite differences).
For the purpose of clarifying the conditions at the exercise boundaries, the next two sections address the American pricing problem in the context of dynamic optimization.
American Options as Dynamic Optimization Problems
This discussion is not intended as a real solution strategy, but as a device to derive the continuity conditions at the exercise boundaries. For the purpose of illustration, consider an American derivative with an exercise–don’t exercise strategy that applies in the interval 0 t T. Here, at every time t the decision must be made whether to exercise the option or to continue to hold it. Notice that direct optimization of the option value over the space
cˆ
≤ ≤
of exercise strategies will in general lead to a problem with a very large number of dimensions, since the number of possible exercise strategies is very large. We can, in principle, parameterize the problem by a suitable description of the free boundaries and solve an optimization problem over a reduced number of dimensions. Although this is often done in practice (Ingersoll, 1998), it requires a priori knowledge of the features and location of the free boundaries.
In the case when the optimal exercise strategy depends only on the cur- rent value of the underlying processes and time, there is a powerful alternative for determining the optimal strategy and thereby the option value, known as the Bellman principle of dynamic programming (Dixit and Pindyck, 1994).
The Bellman principle leads to a recursive argument that states the optimal strategy in terms of two components. In the case of an American option with an exercise–don’t exercise strategy, the Bellman principle in continuous time can be phrased as follows. At a given time, the optimal strategy corresponds to the maximum of either the exercise value or the value associated with selecting an optimal strategy an instant later. This idea can be expressed in what is known as the Bellman equation of dynamic programming. For simplicity, consider only one underlying price process, S(t), and an exercise value, F(S(t)), that depends only on S(t).
Notice that we assume the exercise value itself does not depend on the exer- cise strategy. The Bellman equation is
(3.116) where PVt stands for present value at time t.
Notice that the recursive structure of Equation 3.116 allows us to solve for both the optimal strategy and the value of the security if we know end conditions and work backward in time. The commonly used backward induction techniques implemented through trees for American option pric- ing are particular implementations of solutions of the Bellman equation.
We will use the Bellman equation to discuss the boundary conditions at exercise boundaries.
Conditions at Exercise Boundaries
The purpose of this section is to better understand the way the solution behaves near exercise boundaries. In practice, we don’t need to concern ourselves with the properties of the exercise boundary, since the exercise boundary is automatically captured by simulation or by the linear comple- mentarity formulation. There are two conditions that must be satisfied at exercise boundaries. The first condition is that the exercise value and the continuation value of the option must be the same at the exercise boundary.
V S t( ( ),t) = max {F S t( ( )), PVt[V S t( ( )+dS t( ), t+dt)] }
This is a way of characterizing the exercise boundary as a region of indif- ference between exercising and not exercising the option.
The second condition is that the gradient of the option value with respect to the underlying variables must be continuous at the exercise boundary. This condition is known as smooth pasting and it states that the gradient of the exercise value of the option must be equal to the gradient of the continuation value.
To prove the condition of smooth pasting, consider the implications of Equation 3.116 when we are at the exercise boundary. If we are at the exer- cise boundary, we must have V(S,t) F(S). Assume that an upward move- ment in S would place us in the exercise region, while a downward movement would place us in the continuation region (this only requires continuity of V(S,t) at the exercise boundary). If there is an upward movement in S, the option payoff will be
(3.117) If there is a downward movement, the option value will be
(3.118) Assume now that the probability of an upward movement is equal to p.
The option value can be written as
(3.119) where PV(.) denotes present value. Since at the exercise boundary we are indifferent between exercising and waiting, V(S,t) must equal the discounted expectation of V(S dS,t dt). Replacing this in Equation 3.119, we get
(3.120) Expanding in Taylor series and replacing F V, we get
(3.121) If the underlying process is a diffusion process of the form dS = dt +dW, then infinitesimal changes in S are proportional to . In addition, the probability of upward or downward movements also deviates from one proportionally to . This means that the last equation can be written as follows:
=
F S( +dS)
V S( +dS, t+dt)+Et+dtdV
V S,( t) = PV[F S( +dS)p+(V S( +dS, t+dt)+Et+dtdV)(1–p)]
+ +
PV[V S( +dS, t+dt)–F S( +dS)p V S( +dS, t+dt)(1–p)–Et+dtdV
( )(1–p) ]= 0
–
= PV ∂F
∂S
---dS ∂V
∂S ---dS
– ∂V
---dt∂t
–
p+(Et+dtdV)(1–p) = 0
dt dt
(3.122) Neglecting higher order terms, we get the following relationship that must be satisfied at the exercise boundary:
(3.123) This condition is known as smooth pasting.
Linear Complementarity Formulation of American Option Pricing
The value at time t of an option that can be exercised at times is given by the following expectation:
∫ (3.124)
where are stopping times conditional on information at time t, and F(.) is the payoff if the option is exercised at time equal to the stopping time (Lamberton and Lapeyre, 1996).
It can be shown that Equation 3.124 will hold if the following system of partial differential inequalities is satisfied (Lamberton and Lapeyre, 1996):
(3.125)
Intuitively, this system can be understood in the following manner. The first inequality expresses the fact that at all times the value of the option cannot fall below its intrinsic value. The second inequality reflects the fact that if the value of the option grows more slowly than that of a riskless bond, the option is exercised. The third condition enforces the fact that if the value of the option is above its intrinsic value, the price of the option is described by the same partial differential equation that describes the corre- sponding European option. The last condition indicates that the option value at maturity equals its intrinsic value. In Chapter 7 we will describe in some detail a standard method for solving Equation 3.125.
PV ∂F
---∂Sᏻ( dt) ∂V
---∂Sᏻ( dt)
– ∂V
---dt∂t
–
p+(Et+dtdV)ᏻ dt = 0
∂V∂S
--- ∂F
∂S---
=
V r S t( , , ) sup
Er,S,t e–
∫
tr( )dF s( ( ) )=
V≥F
∂V∂t
--- rS∂V
∂S
--- 1
2---2S2
+ + ∂2V
∂S2 ---≤rV
∂V
---∂t rS∂V
---∂S 1
2---2S2∂2V
∂S2 ---–rV
+ +
V–F
( ) = 0
V S,( T) = F S( )