With this result, we can write Equation 2.87 as follows:
(2.89) where we made use of the fact that var[W(t)] t. Again, we must assume that the trajectory of W(t) is the same in Equations 2.89 and 2.87.
A slightly more complicated example that is frequently used in finance is the following:
(2.90) Applying Ito’s lemma to Y(t) logX(t) we get
(2.91) Replacing , we get
(2.92)
This can now be integrated:
(2.93) Replacing X = exp(Y), we get
(2.94) A case of importance in finance is when a and b are constant. In that case,
(2.95)
Moments of SDE Solutions
Often we are interested in computing the moments of the solutions of SDEs. One way to do this is to solve the SDE and then compute the moments. It turns out, however, that it is much easier to get ordinary differ-
X t( ) X( )0 a s( )ds 0
∫
t 1---t 0b2( )s ds∫
t W t( )+ +
=
=
dX t( ) = X t( )a t( )dt+X t( )b t( )dW t( )
= dY t( ) dX
---X 1 2--- dX
---X
2 –
=
dX ---X
dY t( ) a t( )dt b t( )dW t( ) 1
2---(a t( )dt+b t( )dW t( ))2 –
+
=
a t( ) 1 2---b t( )2dt
– +b t( )dW t( )
=
Y t( ) Y( )0 a s( ) 1 2---b s( )2
– ds
0
∫
t 0b s( )dW s( )∫
t+ +
=
X t( ) X( )0 exp a s( ) 1 2---b s( )2
– ds
0
∫
t 0b s( )dW s( )∫
t +
=
X t( ) X( )0 exp a 1 2---b2
–
t+bW t( )
=
ential equations for the moments that can then be solved either analytically or numerically.
For the process
(2.96) we can use Ito’s lemma to get an SDE for the process X(t)n:
(2.97)
We now have an ordinary differential equation for the expectation of Xn:
(2.98)
We can remark the following about this ordinary differential equation.
Depending on the nature of a(X,t) and b(X,t), we can find two different situations.
■ Equation 2.98 is part of a hierarchy, where the equations need to be solved in succession for n 1, 2,. . . In this case, it may be possible to find analytical solutions or to solve for a number of moments numerically.
■ Equation 2.98 is part of an open system of equations, where in order to find a particular moment you need to know higher order ones. In this case it is not possible to find exact analytical or numerical solutions of the exact equations for the moments. It is possible, however, to derive a sufficiently large set of equations and make assumptions about higher order moments. A practical implementation of this idea requires numerical solutions.
SDE Commonly Used in Finance
Stochastic differential equations are used primarily to model prices and interest rates. They can also be used to model other parameters, such as default intensities and volatilities.
Here we list a short sample of some of the most commonly used SDEs in finance. These models have become well established because they are analytically tractable and tend to represent the processes they are modeling fairly well. Perhaps the area where there has been the largest proliferation of models is in interest rates. Besides the first one, this short list refers to
dX t( ) = a X,( t)dt+b X,( t)dW t( )
dXn( )t nXn–1a X,( t) 1
2---n n( –1)Xn–2b2(X,t)
+ dt
=
nXn–1b X,( t)dW t( ) +
dE[Xn( )t ]
---dt = nE[Xn–1a X,( t)] 1
2---n n( –1)E[Xn–2b2(X, t)] +
=
short rates models. For practical applications in computational finance, however, LIBOR models are of greater relevance. We will discuss those in detail in Chapter 4.
There are many variations beyond the simple models mentioned here.
For a comprehensive discussion, see Duffie (1996).
Geometric Brownian Motion This is the most commonly used process in continuous stock price modeling:
(2.99) Earlier, we discussed this example in detail.
Ho and Lee Short Rate Model This is one of the earliest Gaussian models for the short rate (Ho and Lee 1986). An important consideration in using this model is that it produces negative rates.
(2.100) Vasicek Interest Rate Model This is a very common process for the short rate (Vasicek 1977). Like the Ho-Lee model, this model can produce nega- tive rates. The innovation consists of the introduction of mean reversion in the drift:
(2.101) Hull and White Short Rate Model This model has been used extensively in fixed income pricing (Hull and White 1990). It is also known as the extended or generalized Vasicek model:
(2.102) Cox-Ingersoll-Ross Short Rate Model This is the simplest model that pre- vents rates from being negative and also allows for analytical valuation of simple interest rate products (Cox 1985):
(2.103) The Markov Property of Solutions of SDE
A process X(t) is said to satisfy the Markov property, or to be Markovian, if the random properties of the process at time s t, conditional on information at time t, only depend on the value of the process at time t. This means that the process does not have a memory of events before the observation time that will influence its stochastic properties beyond the observation time. In
dS t( ) = S t( )dt+S t( )dW t( )
dr t( ) = a t( )dt+dW t( )
dr t( ) = [a–br t( )]dt+dW t( )
dr t( ) = [a t( )–br t( )]dt+dW t( )
dr t( ) = [a t( )–b t( )r t( )]dt+ r t( )dW t( )
≥
other words, the behavior of the process beyond the observation time does not depend on the trajectory that the process followed up to the observation time.
Subject to technical conditions, the solution of the SDE,
(2.104) is Markovian. Intuitively, the reason for this is that if the solution is known at a given time, t0, all the properties at a later time, t1, are uniquely deter- mined by the value of the process at t0,X(t0). This is the case because X(t0) is the initial condition of the solution that determines the behavior of the process at t1≥t0.
There are several ways to express the Markovian property of a stochas- tic process. In our case, we are interested in writing this property in terms of an expectation, because our immediate use for the Markovian property is the derivation of the Feynman-Kac theorem.
We use the following notation: E(t,x) means the expectation given t and x. Compare this with Et, which means expectation conditional on informa- tion at time t.
For a Markovian process, the expectation of a function of the stochas- tic process f(X) satisfies
(2.105) This expression says that the process does not remember what hap- pened before the observation time, t. In other words, the only relevant part of the information set at time t is the value of the process, X(t).
The Feynman-Kac Theorem
The Feynman-Kac theorem states that we can find the (time-dependent) expectation of a function of a Markovian stochastic process by solving a partial differential equation, subject to appropriate boundary and end conditions.
Since, as we will see in the next chapter, we price derivatives by eval- uating expectations of (properly discounted) cash flows, in many cases the Feynman-Kac theorem allows us to derive a PDE for the derivative’s price.
The Feynman-Kac theorem states that given an SDE,
(2.106) the expectation of a function of X(T), for 0≤t≥T, given by
(2.107) dX t( ) = a X,( t)dt+b X,( t)dW t( )
Et[f X s( ( ))] = E(t,X t( ))[f X s( ( ))],s≥t
dX t( ) = a X,( t)dt+b X,( t)dW t( )
g t,( x) = E(t,X t( )=x)[f X T( ( ))]
satisfies the partial differential equation
(2.108) subject to the end condition (EC):
(2.109) Notice that g(t,x) is not a random variable. If we replace x with X(t) in g(t,x), we have a stochastic process. We will show that this stochastic process is a mar- tingale and then use this fact to show that g(t,x) must satisfy the PDE above:
(2.110) Replacing the definition of g in Et[g(s,X(s))] we get
(2.111) Since we are assuming that the X(t) is Markovian, we get
(2.112) Replacing in Equation 2.111, we get
(2.113) From the properties of expectations, we get
(2.114) (2.115) Invoking the Markovian property again, the result is
(2.116) Replacing in Equation 2.115, we get
(2.117) Finally, the result is
(2.118) This shows that g(t,X(t)) is a martingale. Since g is a function of time and X(t), we can use Ito’s lemma to get the SDE that governs g(t):
∂g∂t
--- a x,( t)∂g
∂x--- 1
2---b2(x,t)∂2g
∂x2 --- +
+ = 0
g t( =T,x) = f x( )
Et[g s,( X s( ))] = g t,( X t( )),s≥t
Et[g s,( X s( ))] = Et{E(s,X s( ))[f X T( ( ))]}
E(s,X s( ))[f X T( ( ))] = Es[f X T( ( ))]
Et[g s,( X s( ))] = Et{Es[f X T( ( ))]}
Et{Es[f X T( ( ))]} = Et[f X T( ( ))] Et[g s,( X s( ))] = Et[f X T( ( ))]
Et[f X T( ( ))] = E(t,X t( ))[f X T( ( ))]
Et[g s,( X s( ))] = E(t,X t( ))[f X T( ( ))]
Et[g s,( X s( ))] = g t X t( , ( ))
(2.119)
Since g is a martingale, the drift of dg(t) must be zero. This gives us the PDE satisfied by g(t,x).