Quad_variation.pdf

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Stochastic Modelling in Finance - It o Lemma and Quadraticvariation

In the background reading the focus is placed on a small class of processes as thingsare easier here:

Denition 0.1 (2.3.3). A stochastic process (X t )t [0,T ] is a Ito process if it is of the form

X t = X 0 + t

0s ds +

t

0s dW s

where (t )t [0,T ] and (t )t [0,T ] are adapted stochastic processes which

T

0|s | ds < and

T

0|s |2 ds < P a.s.

The rst condition implies that the ordinary (Riemann-Stieltjes) integral is well-dened while the second ensures that the stochastic integral is well-dened.

Theorem 0.2 (It o formula) . Let (X t )t 0 be a continuous process and f : [0, T ] R Rbe a C 1,2 function. Then for t [0, T ]

f (t, X t ) = f (0, X 0) + t

0f t (s, X s ) ds +

t

0f x (s, X s ) dX s +

12

t

0f xx (s, X s ) d[X, X ]s

where f t (t, x ) = t f (t, x ), f t (t, x ) = x f (t, x ) and f t (t, x ) =

2

x 2 f (t, x ).

The nal term contains the quadratic variation term d[X, X ]s . Before giving somemore facts about quadratic variation for those who did not take Stochastic calculus, letssee how the previous theorem works when (X t )t 0 is a Ito process.

Theorem 0.3 (2.3.5). Let (X t )t 0 be a It o process and f : [0, T ] R R be a C 1,2

function. Then for t [0, T ]

f (t, X t ) = f (0, X 0) + t

0f t (s, X s ) ds +

t

0f x (s, X s ) s ds

+

t

0f x (s, X s )s dW s +

1

2 t

0f xx (s, X s )2s ds

where f t (t, x ) = t f (t, x ), f x (t, x ) = x f (t, x ) and f xx (t, x ) =

2

x 2 f (t, x ).

So in this case d[X, X ]t = 2t dt . The next result is quite useful as well

Proposition 0.4 (Integration-by-parts) . Suppose that (X t )t 0 and (Y t )t 0 are continuous processes then

X t Y t = X 0Y 0 + t

0X s dY s +

t

0Y s dX s + [Y, X ]s .

where [X, Y ]s is the quadratic covariation between X and Y .

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Proof. You can prove this very simply by applying the Ito formula to (X t + Y t )2, X 2t andY 2t and subtracting the latter two quantities from the rst.

The extra term in the integration by parts formula is due to the presence of stochastic

integrals which do not quite behave like ordinary (Riemann-Stieltjes) integrals.In fact the previous result is often used to dene the quadratic variation of any process

X (this differs from the mesh based denition supplied by Goran in Stochastic calculusbut the denitions are equivalent - see Theorem 23 in Chapter II of Protter StochasticIntegration and Differential Equations, 2nd Ed. Springer (2005)).

Denition 0.5. The Quadratic variation of a process X is dened as

[X, X ]t := X 2t X 20 2 t

0X s dX s

Similarly the quadratic covariation between two continuous processes X, Y is dened as

[X, Y ]t := X t Y t X 0Y 0 t

0Y s dX s

t

0X s dY s

The next result collects up a few properties of quadratic variation so that they dontget lost:

Proposition 0.6 (Properties of Quadratic variation) . Take X, Y to be adapted processes the quadratic variation (when it is well-dened) has the following properties

(i) Symmetric [X, Y ]t = [Y, X ]t for all t 0

(ii) Polarisation identity

[X, Y ]t = 12

([X + Y, X + Y ]t [X, X ]t [Y, Y ]t ) t 0

(iii) Linear form [X + Y,X ]t = [X, X ]t + [X, Y ]t

(iv) Suppose that X is of nite variation (contains no It o integral) and Y is continuous

and of innite variation then [X, Y ]t = 0 for all t 0.Example 0.7 (Abstract example of integration-by-parts) . Suppose that (X 1t )t 0 and (X 2t )t 0 are It o processes with respect to the same Brownian motion so that

X 1t = X 10 + t

01s ds +

t

01s dW s

X 2t = X 20 + t

02s ds +

t

02s dW s

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in this case [X 1, X 2]t = t

0 1s 1s ds and hence

X 1t X 2t = X 0Y 0 +

t

0X 1s (1s ds + 1s dW s )

+ t0 X 2s (2s ds + 2s dW s ) + t0 1s 2s ds.where [X, Y ]s is the quadratic covariation between X and Y .

A straightforward example

Example 0.8. Find an expression for tW t where W t is a Brownian motion. Suppose that X 1t = t and X 2t = W t are It o processes with 1t = 1, 1t = 0, 2t = 0, 2t = 1 for all t 0.Thus using the previous example:

tW t = t0 s dW s + t0 W s ds.because X 1 is nite variation so [X 1, X 2]t = 0 for all t 0.

Proposition 0.9 (Martingale generated by quadratic variation) . Suppose that (X t )t 0and (Y t ) t 0 are continuous local martingales with well-dened quadratic variations [X, X ]and [Y, Y ] then the process (Z t )t 0 dened using

Z t = X t Y t [X, Y ]t

is a martingale.

In particular, if we take two Brownian motions ( W t )t 0 and (B t )t 0 such that for eacht 0 the random variable COV( X t , Y t ) = E [W t B t ] = t then for > 0 the process(W t B t ) t 0 is a submartingale (and for < 0 the process (W t B t )t 0 is a supermartingale)so according to the previous proposition

Z t := W t B t t

denes a process (Z t ) t 0 which a martingale and [W, B ]t = t . We use this property asthe denition of correlated Brownian motions .

Denition 0.10. Two Brownian motions (W t )t 0 and (B t ) t 0 are correlated with corre-lation coefficient [ 1, 1] if

[B, W ]t = t t 0.

In particular, when COV( X t , Y t ) = 0 for all t 0 the process (W t B t ) t 0 is a martingaleand [W, B ]t = 0 for all t 0. This is the intuitive reason why quadratic variation of behaves similarly to the variance of the process.

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