Suppose that there are multiple martingale measures $Q1$ and $Q2$ that attain the minimal variance. Then the convex combination $Q* := \frac{1}{2}Q1 + \frac{1}{2}Q2$ is also a martingale measure. Due... Read More

$$ Zt = f(St) := \left( \frac{St}{H} \right)^p $$ $$ dZt = \partialx f(St) dSt + \frac{1}{2} \partial^2{xx} f(St) d\langle S \ranglet = p\frac{St^{p-1}}{H^p} dSt + \frac{1}{2} p(p-1) \frac{St^{p-2}}{... Read More

In the equilibrium models you can assume that there exists so called Alpha, i.e. an opportunity that can be exploited. Most of the buy side models (i.e. asset allocation, portfolio construction) are... Read More

Ad. 1. You are right: $Y{t}=e^{-rt}S{t}$ $dY{t}=d(e^{-rt}S{t})=-re^{-rt}S{t}dt+e^{-rt}dS{t}=(\mu-r)e^{-rt}S{t}dt+\sigma e^{-rt}S{t}dW{t}=(\mu-r)Y{t}dt+\sigma Y{t}dW{t}$ Now we have: $\hat{W}{t}=\frac... Read More

By definition $Q^{Ts}$ is risk neutral for the numeraire $P(t,Ts)$, and $Q^{Te}$ is risk neutral for the numeraire $P(t,Te)$, hence $$ \left(\frac{dQ^{Ts}}{dQ^{Te}}\right)t = \frac{P(t,Ts)}{P(t,Te)}... Read More

Try $$\mathbb{E}\frac{1}{T} \int0^T Vt dt = \frac{1}{T} \int0^T \mathbb{E} Vt dt$$ and use $$\frac{1}{dt}\mathbb{E} Vt = \kappa\theta - \kappa \mathbb{E}Vt + (\lambda0 + \lambda1\mathbb{E} Vt)\muV, $... Read More

From what I remember, there is no real relation between Markov and Martingale, and my intuition was confirmed by this post. Basically, it says that you can say neither of the following: If A is Marko... Read More

A martingale must have constant expectation, such that adding a deterministic finite variation process $(b-r)dt$ would break the martingale property (except for when its a constant, which it is not b... Read More

The proofs seem good to me. An alternative answer to the first question is, with $s $$\begin{align} E\left(BtWt|\mathscr{F}s\right)&= E\left((Bt-Bs+Bs)(Wt-Ws+Ws)|\mathscr{F}s\right) \\ &= E\left((Bt-... Read More

Based on Ito's isometry, \begin{align} E_t (r^2_{t+1}) &= E_t \bigg(\int_t^{t+1} \sigma_s dW_s \int_t^{t+1} \sigma_s dW_s\bigg)\\ &= E_t \bigg(\int_t^{t+1} \sigma_{\tau}^2 \,d\tau\bigg) \\ &= E_t\big... Read More