When applying the Girsanov theorem, we assume that a technical con- dition called the Novikov condition holds:
(2.139) It is also possible to derive a multidimensional version of the Girsanov theorem, in which case W and are vector processes. For a detailed deriva- tion of the Girsanov theorem, refer to Oksendal (1995).
So, as long as no jumps occur, Ito’s lemma will apply. Ito’s lemma applies without any changes in between jumps. However, we want to find a way to express Ito’s lemma such that it applies across jumps as well, not only in between jumps.
The Poisson Jump Model
The Poisson model for jumps is commonly used in finance to model unan- ticipated changes such as stock price jumps or the occurrence of default. The Poisson model states that the probability of the occurrence of one jump in the interval t is ht plus higher order terms, where h is called the jump intensity.
Notice that the word intensity here refers to the probability of occurrence, not to the magnitude of the jump. In general, the jump intensity may itself be a stochastic process. For now we will assume that jump intensity is a deter- ministic function of time.
The probability of one jump over the infinitesimal interval dt is equal to h(t)dt. The probability that no jump has occurred in the interval (0,t) is called the survival probability and is denoted by ps(t). The change in proba- bility that no jump has occurred in the interval (0,t) is given by
(2.141) Integrating this equation gives us an expression for the survival probability:
(2.142) The probability that at least one jump has occurred in the interval 0,t is simply 1 –ps(t).
Defining a Pure Jump Process
Assume that the process S(t) undergoes pure jumps. We will define the change of the process as a result of jumps by djS(t), where the subscript indicates that in general this is not an infinitesimal quantity:
(2.143) Here S(t–) is the value of the process as it approaches t from the left. The value of djS(t) is zero until a jump happens. The jump intensity, or the proba- bility that the jump happens in the interval t, t dt, may depend on S. If it does, it will be a function of the process immediately before the jump, S(t–).
If the jump happens, its magnitude, z, is drawn from a distribution (S(t–),z). The expected magnitude of the pure jump is then
(2.144) dps( )t = –ps( )h tt ( )dt
ps( )t exp h s( )ds 0
∫
t–
=
djS t( ) (S t( +t)–S t( )_ ) tlim→0
=
+
E[djS t( )] h S t( ( )– )dt z(S t( )– ,z)dz
∫
z=
We can now arrange Equation 2.143 as follows:
(2.145) The process is called a compensated process and is a martingale. We can now write the pure jump process as
(2.146) where
(2.147) Defining a Jump-Diffusion Process
To define a jump-diffusion process, just add the jump component, djS(t), to a drift-diffusion process.
If we add the jump component to the drift-diffusion process
(2.148) we get the jump-diffusion process
(2.149)
Ito’s Lemma in the Presence of Jumps
Since Ito’s lemma is valid between jumps, the application of Ito’s lemma to a function g(S(t),t) of the process given by Equation 2.149 gives
(2.150)
If the jump in S(t) happens, g(t) jumps by an amount g. This amount is drawn from a distribution g(), which may depend on other variables such as g(t–), the jump magnitude of S(t), and S(t–). As before, we can write (2.151) Replacing in the equation for dg(t), we get
djS t( ) = E[djS t( )]+ djS t( )–E[djS t( )] dJs( )t
dJs( )t
djS t( ) h S t( ( )_ ) z(S t( )_ ,z)dz+dJS( )t
∫
z=
dJS( )t = S t( )–S t( )_ – h S t( ( )_ )
∫
zz(S t( )_,z)dz dtdS t( ) = ( )dtt +( )dW tt ( )
dS t( ) ( )t h S t( ( )_ ) z(S t( )_ ,z)dz
∫
z+ dt+( )t dW t( )+dJS( )t
=
dg t( ) = Ito’s lemma applied to the drift-diffusion part of S t( ) g t( )–g t( )_
Jump in g due to jump in S +
g t( )–g t( )_ h t( )dt gg( ). dg+dJg( )t g
∫
=
(2.152)
In general, we may not have a way to specify the jump magnitude dis- tribution of g as a function of the jump distribution of S. We will see exam- ples of this in the next chapter when we discuss defaultable instruments.
dg t( ) ∂g
∂t
--- ∂g∂---S 1 2---2∂g
∂S2
--- h t( ) gg( ). dg
∫
g + + +
= dt
∂g
∂S
---dW t( ) dJg( )t
+ +
3
41
Pricing in Continuous Time
y formulating the pricing problem in continuous time we can bring to bear the analytical tools of stochastic calculus and partial differential equations. The basic formulation of the continuous time pricing problem consists of expressing the current value of a derivative security as an expec- tation of properly discounted future cash flows.
If B(t) is the value of a traded asset that does not pay dividends, the basic relationship for computing the value of a European derivative secu- rity, V(t), is of the form
(3.1) where the expectation is taken in a measure determined, or induced, by asset B(t).
Asset B(t) is called a normalizing asset or numeraire. This expression tells us that the value of a derivative, in units of B, is a martingale if the sto- chastic processes involved in V(t) are expressed in a measure associated with B(t). Equation 3.1 determines the measure under which other deriva- tives can be priced. For example, if (t) is the price of another European derivative,
(3.2) where the expectation is taken in the same measure as in Equation 3.1.
Once we have formulated this expectation, we have at least three alterna- tives to compute its value. We can compute the expectation analytically (this is easy to do in the case of simple Black and Scholes European options), we can value the expectation through simulation, or we can express the expectation in terms of a partial differential equation. We can then solve the partial differential equation either analytically or numerically. With some
B
V t( ) B t( )
--- EtB V T( )
B T( ) ---
=
V˜
V˜( )t B t( )
--- EtB V˜( )T
B T( ) ---
=
significant changes, a similar picture carries over to the case of derivatives with early exercise features, such as Bermudan or American options. We will discuss European derivatives first and derivatives with early exercise later on.
The most commonly used normalizing asset in continuous time is the money market account, (t), defined as the value of a continuously rein- vested unit of account.
(3.3) The measure under which the value of a derivative normalized with the money market account is a martingale is called the risk neutral measure.
Risk neutral pricing, where the money market account is the normaliz- ing asset, is the standard framework for pricing in continuous time. We will develop this framework in sufficient detail. Similar frameworks can also be set up for pricing with other normalizing assets.
We will start by pricing a derivative on a single asset in the risk neutral measure. We will then extend the same logic for the case where the normal- izing asset is something other than the money market account. After that, we will consider the multidimensional case of risk neutral pricing.