to derive equations for asset price evolution one uses Ito lemma:

$$ d f(X_t,Y_t) = f_{x}(X)dX+f_{y}(Y)dY+f_{xy}dXdY + \frac{1}{2} f_{xx}(X) dXdX+ \frac{1}{2} f_{yy}(Y) dY dY $$

basically its the same as Taylor formula for 2 variables developed until 2nd order.

when applying Ito’s lema the following formulas are useful:

$$ dW_tdW_t=dt $$
$$ dt_tdW_t=0 $$
$$ dtdt=0 $$

Applications

it’s used for example to derive Black-Scholes equation [in partial derivatives]

to change the probability measure one uses Girsanov theorem (formula):

$$ \frac{N_a(0)}{N_a(T)} d \mathbb{P}_a=\frac{N_b(0)}{N_b(T)} d \mathbb{P}_b$$

$$N_a$$ is numeraire (price of any non dividend paying asset, usually bond or bank account) and it’s correspoding probability measure is $$P_a$$

$$N_b$$ is another numeraire with it’s probability measure $$P_b$$

Applications

expectation of short rate is forward rate under T-forward measure

– prove that $$E^T(r_T|F_t)=f_{T->T+dT}(t)$$

where $$E^T$$ is a expectation under T-forward measure (where bond $$P_{->T}$$ is numeraire
and $$f_{T->T+dT}(t)$$ is an instantaneous forward rate for time T as seen at time t

– write bond price in two forms:

$$P_{->T}(t)=E^{RN}(e^{-\int_t^Tr_sds}|F_t)$$
$$P_{->T}(t)=e^{-\int_t^Tf_s(t)ds}$$
so
$$E^{RN}(e^{-\int_t^Tr_sds}|F_t)=e^{-\int_t^Tf_s(t)ds}$$
1) differentiate both parts by T
2) change measure from Risk-neutral to T-forward measure from time t to time T (as usual rule apply girsanov formula between times when the process is stochastic ,in this case not between 0 and t for example)

Black Scholes formula

to calculate the term $$E(S_T 1_{S_T>K})$$ one uses girsanov to pass from risk neutral measure to forward measure where $$S_t$$ is numeraire

Libor In Arrears Convexity adjustment

Simple example libor in arrears convexity adjustment to change from T1 measure to T2 measure