During the whole semester, topic 3 was probably the easiest. Because I have known the material of Ito integral and stochastic differential equation(SDE) in advance of the lecture. But it was still good as a complement to my knowledge.

Everyone should know what probability space, filtration, stochastic processes, adapted processes, martingale and Brownian motion are, cause these concepts are the very basics of mathematical finance. One should also know that Ito integrals are martingales and if the integrand is non-random then Ito integral and Stratonovich integral are the same thing, which means that the product rule for Riemann integral applies to Ito integral as well. Why do we choose Ito integral instead of Stratonovich integral given that Stratonovich integral preserves the product rule? The reason is when we construct Ito integral, we choose the left end points for the integrand so we don't look into the future, and this is essential to financial application.

One of the most important theory we need to use is Ito's lemma. It basically expands the concept of derivative we already know in calculus. We know the Taylor expansion:

\(df(x)=f'(x)dx\)

and now for Ito integral we have

\(dg(t,W(t))=g_tdt+g_xdW(t)+\frac{1}{2}g_{xx}dt\)

so essentially we are using the fact that \(dW(t)dW(t)=dt\).

In order to compare different securities and remove the price appreciation due to the time value of money effect, we introduce the concept of numeraire and prices divided by numeraire are called dicounted prices. For any numeraire there exists a corresponding martingale measure which is equivalent to the probability measure that discounted prices are martingales under such measure. Actually this is not precisely correct cause martingale measure does not always exist, so this sentence I am saying is more with respect to change of numeraire. That is if you change the numeraire then the martingale measure changes and vice versa.

The first fundamental theory of asset pricing says if there exists a martingale measure then there is no arbitrage in the market. The converse is not true and you need to add another condition called no free lunch with vanishing risk to ensure that martingale measure exists given there isn't any arbitrage opportunity. One thing new I learnt from this lecture is that if martingale measure exists for one numeraire then for another numeraire we could also find martingale measure. The most commonly used numeraire are banking account and zero coupon bond.

Change of measure is another important technique with respect to mathematical finance. You first need to know what is Radon-Nikodym derivative from real analysis. Then you should know the Girsanov's Theorem:

If W is a Brownian motion under P, then for

\(dP^*=\rho dP\), \(\rho=\exp{-\int^t_0\theta(s)dW(s)-\frac{1}{2}\int^t_0\theta^2(s)ds}\)

\(dW^*(t)=dW(t)+\theta(t)dt\)** is a Brownian motion under \(P^*\).

Martingale measure according to banking account is called risk neutral measure which is named after the fact that the market price of risk is zero in this case. Under risk neutral measure, all the prices discounted by momey market account are martingales and in case of random interest rate model we often change the numeraire to forward measure because of simplicity of calculation(this is not the only reason!). Forward measure is the EMM with respect to zero coupon bond and one of the professors said to us that if someone knows what forward measure is then he/she could skip the course of Mathematical Finance I. Notably, forward price is a martingale under forward measure.

In fixed income modeling, one rule I didn't know is that conditional expectation of short rate is equal to the forward rate: \(E^T_t[r(T)]=f(t,T)\).