Some calculations with exponential martingales

Wojciech Szatzschneider

School of Actuarial Sciences

Universidad Anáhuac México Norte

Mexico

Introduction

Girsanov theorem in Practice.

Under P : physical world

if

And Risk Neutral world Q exists.

Under Q law

Always

a martingale

Problem

When is true martingale and not only local martingale?

Novikov:

Local martingale is true martingale if and

only if is of class (DL), if for every a>0, the family of random variables XT, (T all stopping times < a), is uniformly integrable.

martingale

Set

Kazamaki

submartingale

martingale

Novikov for small intervals:

for some

then martingale

… Simply note

choose

and use “Tower property”

Basic fact

If under Q , is true martingale > 0 and if under P, > 0 a. e. then Q ~ P is a martingale.

If possible, we will take advantage of

Change of measure

Let Yt be a process such that:

and rt = Yt.

Let Xt be a BESQ process

And consider the continuous exponential local martingale

Where M(s)=Xs-s. It results that

The local martingale is bounded (Xs ≥ 0 and < 0). Therefore, Zt is a martingale, and the change of drift via the Girsanov’s theorem is justified.

At time zero (to simplify the notation), we have

We look for Fu(s) = F(s) such that F2(s)+F’(s)=2+2 for s[0, u), F(u)=.

In this part, the most suitable approach to the Riccati equation, that defines F, is via the corresponding Sturm-Liouville equation. WritingF(s)=’(s)/(s), s[0, u), (0) =1, we get

where ¯ is the left-hand side derivative.

The exponential martingale that correspondsto F will be called Z,

(Z corresponds to F in the same way as to ).

The solution of the equation ’’(s)/’(s)=c2 in [0,u) is clearly Aecs+Be-cs with conditions A+B=1 (because of (0)=1) and

We obtain after elementary computations,here X(0)=1, u=1.

and reproduce easily Cox, Ingersoll & Ross formula.

The same method can be applied in the case of 2-2 >0 with c2=2-2.

Case 2

In this case we have

Lemma 1

Lemma 2If

then, as before we have

Define

Now

Proofs

i. Here and as before

Therefore

, and

ii. Now

and the bounded solution exists in [0,1] (after elementary calculations) for and using (1) we obtain the result.

Linear Risk Premia

We will clarify what can be done and what can not in one dimensional financial market driven by Brownian motion, and asset prices that in the RW (under the law P) follow a Geometric Brownian Motion:

Set (discounted prices)

where

and is the spot IR in the RW.

Now,

The RNW is defined as the probability law Q (Q ~ P), t < T that under Q

being W* another Brownian Motion.

It can be shown that if r(t) is CIR (in real world and driven by the same BM), then such Q does not exist.

An easy argument is based on explosion until T = 1 of the process defined by:

But what we really want is CIR in the RNW. We prove the following:

Theorem, If under P

Then for any T>0, there exists Q ~ P, for the process considered until time T such that under Q the interest rates follow:

Longstaff Model

In 1989, Longstaff proposed the so-called double square root model defined in Risk Neutral World by:

where

In 1992 Beaglehole & Tenney showed that Longstaff’s wrong formula for Bond Prices in his model gives the correct bond prices in the case of:

Longstaff uses Feynman-Kac approach and obtains the formula for bond prices of the form:

This calculations repeat all over the world in several textbooks

For some functions m,n,p and x=rt . However to apply Feynman-Kac representation, P(t,T) should be of class C2 with respect to x.

Some relaxation of this assumption is imaginable, but there is no possibility to make adjustments that could work for P(t,T). The problem is of course at zero.

We will show how to calculate:

and

iff in (0,t)

This matching procedure does not work in the original Longstaff model, it means for calculations of:

An application of Girsanov theorem leads to

CIR and intensity based approach

Our goal is to calculate

for r(s) and (s) dependent CIR processes.

Assume that one can observe correlations between r(s) (default free rate), and (s) (the intensity of default). We will use very special dependence structure between r(s) and (s) that can approximately generate the correlation one, and this structure will produce explicit formulas.

Our modelling is as follows:

Set:

with independent r1 and r2, 1 and 2 .

Also r1 is independent of 1 .

Now set 2(t) = r2(t) for some

consequently

With the use of Pythagoras theorem define:

with the restriction that:

Using a well known formula for the variance of the CIR we can write:

This method can be extended easily to multifactor CIR

Model with saturation

We want to prove that

is true martingale and not only local martingale.

satisfies

The law of this process we will call P. The best way to prove that H(t) is true martingale is through some equivalences of laws of processes on , .

Note that V (t) is Pearl-Verhulst model.

Let under the law Q, dV(t)=V(t)dt+V(t)dW(t). The law Q is equivalent to the law Q1. Under Q1, V(t)=eW(t)-½t.

Now, under Q1

ZT is clearly true martingale and by Girsanov theorem changes the law Q1 into such that under ,

Moreover , under measure.

Therefore on , and we have proved that ZT is P martingale. The last equivalence is obvious.