martingale


stochastic calculus - unique equivalent martingale measure in incomplete markets

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


itos lemma - Show that Z(t)/Z(0) is a positive mean-1 martingale

$$ Zt = f(St) := \left( \frac{St}{H} \right)^p $$ $$ 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


emh - Assumptions based on non-martingale?

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


brownian motion - Calculating the stochastic integral of $\exp(-rt)S_t$

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


martingale - Change of measure between T-forward and T*-forward contract?

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


stochastic volatility - Realized variance in SVJJ (Heston with jumps) model

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


equities - Is the stock price process a martingale or a Markov process?

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


stochastic calculus - Uniqueness of equivalent martingale measure in Black Scholes-Model

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


Three proofs regarding brownian motions and martingales

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


volatility - On an application of Ito's lemma

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