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## Hot answers tagged girsanov

12 votes

### Heston stochastic volatility, Girsanov theorem

Consider the Heston (1993) model under the real world measure ($\mathbb{P}$) \begin{align*} \mathrm{d}S_t&=\mu^\mathbb{P} S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}^\mathbb{P}, \\ \mathrm{d}v_t&...
• 16k
10 votes
Accepted

• 14.7k
7 votes
Accepted

### Problem with pricing a call option using the Monte Carlo Vasicek model

To make sure that I understand the problem: you are trying to price a call option expiring at time 0.5, which will exercise into a unit notional zero-coupon bond with a maturity of 1.0 at a strike (...
• 845
6 votes
Accepted

• 2,220
3 votes

### Bayes Theorem with change of measure

Nothing new in this answer - I have just consolidated what others have said in the answer and the comments, and put the explanation next to each step. I have to move the original equations so as to ...
3 votes

### Girsanov Theorem application to Geometric Brownian Motion

Denote $B_t=e^{rt}$ the discount factor. Requiring $S_t/B_t$ to be a martingale it would mean the equation $S_0/B_0=E[S_t/B_t]$ hold. Therefore we can calculate the price of an option by discounting ...
• 522
3 votes

### Girsanov Transform and Likelihood Process Domestic to Foreign

Let $(V_t)_{t \geq 0}$ denote a self-financing wealth process in foreign currency units. In the absence of arbitrage, the former process should emerge as a martingale when expressed in the foreign ...
• 14.7k
3 votes

### Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?

I assume all three models are stated under the money-market measure: then there is no arbitrage if the discounted pay-off is a martingale under the money-market Numeraire. Therefore to show no ...
• 6,118
3 votes
Accepted

### Where does the term $\gamma$ come from when moving from measure $\mathbb Q^{N}$ to $\mathbb Q^{M}$?

From your assumption that \begin{align*} dX(t)\cdot d\frac{M(t)}{N(t)} &= \frac{M(t)}{N(t)}X(t)\gamma(t)dt,\\ dX(t) &= X(t)\big(\mu(t)dt+\sigma(t)dW(t)\big), \end{align*} under $\mathbb Q^{M}$,...
• 21.1k
2 votes

### Understanding Girsanov's theorem in Bjork's book

The process $L$ is strictly positive. This implies that $Q$ is equivalent to $P$ and not merely absolutely continuous.
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2 votes

### Girsanov theorem and default rates in bond credit rating

Suppose I give you objective probabilities $\mathbb{P}(S_T \geq K)$ of an equity finishing above a certain level $K$ at a future time $T$ (or in your case a survival probability in the form of default ...
• 14.7k
2 votes
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### Libor Market Model (LMM) under risk neutral measure

We assume that, under the $T_j$-forward probability measure $P_{T_j}$, \begin{align*} \frac{dP(t, T_j)}{P(t, T_j)} = \mu_P(t, T_j) dt + \sigma_P(t, T_j) dW_t^{T_j}, \end{align*} where $\mu_P(t, T_j)$ ...
• 21.1k
2 votes
Accepted

### Change of numeraire in options with currency exchange features

Notations $S_T$ and $K$ are expressed in EUR; $D^{CCY}(t,T) = \frac{\beta^{CCY}_t}{\beta^{CCY}_T}$ where $\beta^{CCY}$ is the money market account in currency $CCY$). In other words, it is the (...
• 2,220
2 votes

### Are all changes of measures for continuous diffusion processes given by the change of drift?

I have read that for diffusion processes, indeed the volatility must be preserved under a change of measure. This old question appears to be relevant : Version of Girsanov theorem with changing ...
• 17.1k
2 votes

To give some background, I give you some pieces from Geman, El Karoui and Rochet (1995) which use a change of numéraire. A fundamental observation is the change of measure formula $\mathbb{E}^\mu[X]=\... • 16k 2 votes ### Change of measure for a stochastic process to be a martingale (Now that I saw Gordon's solution, I can finish my attempt; I had noticed$dB_t +1/2dt$immediately from product rule for$V_t$, zero quadratic covariation between$t/2$and$e^{B_t}\$, but hours later ...
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