Questions tagged [girsanov]
The girsanov tag has no usage guidance.
44
questions
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1answer
141 views
Girsanov Theorem and Probability Measures
The Cameron-Martin-Girsanov theorem, in a simplistic way, states that:
The probability measure $\mathbb{P}$ is induced by a Wiener process $W(t)$. There exists another process $X(t)$ under the same ...
1
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0answers
85 views
How to determine exchange rate dynamics in currency derivatives
I need some guidance regarding exchange rate dynamics in currency derivatives.
Following three dynamics are defined below,
$\frac{dS(t)}{S(t)}=\alpha dt+\sigma dW(t)$ ; the stock dynamics in the ...
5
votes
1answer
287 views
Bayes Theorem with change of measure
Tomas bjork- arbitrage theory in continuous time.
Appendix B, proposition B41 says:
The proof is not clear to me.
Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the ...
0
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0answers
39 views
Swap rate in the annuity measure and Martingale Representation Theorem
As we know, swap rate evolves as a martingale in the appropriate annuity measure. Martingale representation theorem says if I can find a Brownian motion in the annuity measure and the swap rate is ...
0
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0answers
36 views
Change of numeraire/probability when asset pays dividends
So I was looking at Margrabe's formula for exchange call options in the book 'Mathematical Methods for Financial Markets' (Jeanblanc, Chesney, Yor), and I was having trouble justifying their change of ...
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0answers
39 views
Risk neutral measure in the binomial approximation of geometric Brownian motion
Suppose an asset is described by geometric Brownian motion with a drift, i.e.
$$dS_t = S_t\mu dt + S_t \sigma dW_t$$
for a Wiener process $W_t$ and $S_0=1$. By some consequence of Girsanov's theorem (...
3
votes
2answers
148 views
theoretical reason for which we can use monte carlo simulation for option pricing
The classic way to price an option is solving either analitically or numerically the associated PDE subject to the terminal and boundary conditions.
An alternative approach is to use monte carlo ...
1
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1answer
108 views
change of measure expectation
How to find expectation of this stochastic process? Also, to show that the expectation of a stochastic process expression [Xt - St] in one measure is equal to expectation of another expression (of the ...
1
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1answer
87 views
Proof standard Brownian Motion under change of measure
Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
3
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0answers
64 views
American Perpetual Put Option
I want to compute the expected payoff of a (classical) perpetual American put option in the Black-Scholes-Merton (BSM) framework with an optimal strategy of exercising the option at time $\tau=\inf\{t:...
3
votes
1answer
150 views
What is the easiest way to learn Option pricing with PDE?
I was reading about Ito's formula and Girsanov theorem, but I am still struggling to grasp how in reality these are combined to compute the price of an option. What are the main source to understand ...
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2answers
139 views
Are all changes of measures for continuous diffusion processes given by the change of drift?
In elementary discussions on change of measure for geometric Brownian motion, one often find statements like "change of measure = change of drift". Given a general continuous diffusion process of the ...
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0answers
220 views
Change of measure from physical to risk-neutral under Radon-Nikodym and Girsanov Theorem
Given a stochastic process, how do we prove and generate the change-of-measure?
I have been trying to prove the change-of-measure as under the Radon-Nikodym theorem and Girsanov Theorem, but ...
4
votes
1answer
202 views
Uniqueness of Risk-neutral measure: Probabilistic view
Suppose we are working on the Black and Scholes Framework. There are only two assets, the risk-less bank account and a stock. The discounted process is a GBM under the physical measure with drift term ...
1
vote
1answer
164 views
Why Girsanov's theorem used here?
It is written in Bjork's ArbitrageTheoryInContinuousTime that
... Assume a martingale measure Q exists. This implies (see the Girsanov theorem) that the price processes have zero drift under $Q$ .....
3
votes
1answer
233 views
Libor Market Model (LMM) under risk neutral measure
I would like to establish the equations of forward libors under risk neutral measure. Here is how I do it, and what I get :
Under the $P_{T_j} $ measure, forward Libor $L_j$ is martingale. Thus:
$$ ...
3
votes
1answer
773 views
Change of numeraire in options with currency exchange features
FV of an EUR denominated option under "COP" risk measure is given by:
$$V_t^{COP} = D^{COP} \mathbb{E}_t^{COP} \left[X_T(S_T -K)^+\right]$$ where $X_T$ is the exchange rate COP/EUR.
Pricing the ...
1
vote
1answer
258 views
Question about the Cameron-Martin-Girsanov (CMG) theorem
Within my lecture notes, the following definition of the CMG theorem is given:
Under the probability measure $\mathbb{\tilde{P}}$ with density $\gamma_T = \exp(cW_T - \frac{c^2}{2}T)$, the process $...
1
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1answer
130 views
Girsanov's Theorem for Multiple Risky Assets
Girsanov's theorem provides the measure transformation from probability measure P to Q such that-
$dW_t^Q=dW_t^P+\lambda dt\implies \xi_tW_t^Q$ is a martingale under the P measure where $\xi_t=e^{-\...
0
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1answer
228 views
Equivalent Martingale Measure result Hull?
I've been reading Hull's chapter about Martingales and measures where he states that if you have the dynamics of two securities as follows:
\begin{align}
\frac{df}{f} = (r + \lambda \sigma_f) dt + \...
4
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1answer
1k views
Girsanov Theorem application to Geometric Brownian Motion
I recently read this from a book on mathematical finance
The important example for finance the (unique) EMM for the geometric
Brownian. Let $S_{t}$ be the price of an asset, $${{d{S_t}} \over
{{...
4
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0answers
275 views
Girsanov theorem and stopping time
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, equipped with a filtration $(\mathcal{F})_{0 \leq t \leq T}$ which is a natural filtration of a standard Brownian motion $(W_{t})_{0 \leq ...
0
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1answer
597 views
Change of Numeraire to price European swaptions
In the pricing of a European swaption, it is common to use the annuity factor $A(t)$ as the Numeraire. I was trying to write down the pricing formula via the bank account as numeraire to see if they ...
5
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1answer
1k views
Girsanov Theorem, Radon-Nikodym Derivative backward
Given a filtered probablity space $(\Omega,\mathcal{F},{\mathcal{F}}_t,\mathbb{P})$ and a standard Brownian motion $W_t$.
Normally, in Girsanov Theorem, we use the exponential martingale $Z_t=\exp(-\...
6
votes
1answer
722 views
SDE simulation: P or Q?
Let's take a GBM under $P$:
$dS=\mu dt+\sigma dW_{t}^{P}$
and then under $Q$
$dS=r dt+\sigma dW_{t}^{Q}$, where $dW_{t}^{Q} = dW_{t}^{P} + (\mu - r)/\sigma dt $
Now, let's say that I have ...
4
votes
1answer
151 views
Girsanov Transform and Likelihood Process Domestic to Foreign
Working two exercises relating to $Q^d$ and $Q^f$. I'm comfortable working with transforms and likelihood processes on a risky asset between $Q$ and $Q^s$, and also on an exchange rate $X$ between $Q$ ...
2
votes
0answers
232 views
Normalized Gains Process is a Q-Martingale - Proof and Intuition
I'm trying to work the proof that the normalized gains process, $G^z_t = \frac{S_t}{B_t}+\int^t_0\frac{1}{B_s}dD_s$ is a Q-martingale under Q (the risk-neutral measure). I'll show what I've worked ...
0
votes
1answer
372 views
Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire [duplicate]
Let $S$ be a GBM with dynamics $dS_t/S_t=rdt+\sigma dW_t$. We want to compute the following expected value:
\begin{align*}
\mathbb{E}(S_T\log(S_T)).
\end{align*}
Using a change of measure we can write
...
4
votes
1answer
184 views
Understanding Girsanov's theorem in Bjork's book
In Bjork's arbitrage theory in continuous time, he writes
S, essentially we define $Q$ using $h_t$, and then pick $h_t$ so $Q$ is a martingale measure.
But, $Q$ needs to be an equivalent measure. ...
2
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0answers
186 views
How to understand the integral in the Girsanov theorem?
Let $W^P$ be a $d$-dimesional $P$-wiener procss. Define $L_t =
> e^{\int_0^t \phi_s^T dW_s^P - \frac{1}{2} \int_0^t \| \phi_s\|^2
> ds}$.Assuming $E^PL_T = 1$, then the measure given by $dQ = ...
2
votes
1answer
203 views
Girsanov theorem and default rates in bond credit rating
Default rates are kind of probabilities, right?
Is it possible to use the Girsanov theorem in that context? For example if we have a table of real world probabilities, could we use the Girsanov ...
9
votes
1answer
657 views
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
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2answers
194 views
Simulate drifted geometric brownian motion under new measure
I have a very fundamental question regarding simulation of DRIFTED geometric brownian motion.
We have the standard Blackos Scholes model:
$dS(t)=r S(t)dt+\sigma S(t) dW^{\mathbb{P}}(t)$, where $W^{\...
1
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0answers
94 views
On the construction of a Brownian motion from a Gaussian process
Let $X$ a Gaussian process defined by
$$
X_t=\int_{0}^{t}\left(\frac{1}{\sigma}\left(r_s-\frac{\sigma^2}{2}\right)-\rho\sigma_P(s,T)\right)\mathrm{d}s+\sqrt{1-\rho^2}Z_2(t)+\rho Z_1(t);\;\;t\in[0,T]
$...
6
votes
1answer
821 views
How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?
Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
2
votes
2answers
329 views
Why do we need $dS_t=r S_tdt+\sigma S_tdW_t^Q$?
Suppose $S_t$ is the stock price and follows the dynamics $$dS_t=\mu S_tdt+\sigma S_tdW_t$$. According to Girsanov, we can apply change of measure and obtain $dS_t=r S_tdt+\sigma S_tdW_t^Q$, this ...
8
votes
2answers
2k views
Ito, Stochastic Exponential and Girsanov
This is a two-part question relating to the change of measure density used in Girsanov and secondly to the Stochastic Exponential.
Whilst reading notes relating to Girsanov it is stated that the ...
3
votes
1answer
344 views
Girsanov theorem in CMS convexity derivation
I am going through the derivation of CMS convexity from the notes of
Lesniewski
There is a transformation from $T_p$ forward measure to annuity measure $Q$ as
$$
P(0,T_p)E^{Q_{T_p}}\left[S(T_0,T)\...
1
vote
1answer
153 views
Bivariate Black-Sholes Model
Let us propose bivariate Black-Sholes Model.
Assume, we have an arbitrage-free complete market.
$r_{f}$ is risk-free rate.
Under real-world measure $P$:
$dS_{1} (t)=S_{1} (t) [\mu_{1}dt+\sigma_{1}...
8
votes
1answer
4k views
Radon-Nikodym derivative and risk natural measure
I need help with my understanding of changing probability measure. Im not a mathematician so I hope for answers that are not too technical.
As shown in this Wikipedia article http://en.wikipedia.org/...
12
votes
1answer
2k views
Girsanov Theorem and Quadratic Variation
Girsanov theorem seems to have many different forms. I've got a problem matching the form in wiki to the one in Shreve's book, due to the difficulty of quadratic variation calculation.
Below is the ...
5
votes
2answers
1k views
Uniqueness of equivalent martingale measure in Black Scholes-Model
Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
8
votes
1answer
1k views
How to compute the Radon-Nikodym derivative?
Suppose $B(t)$ is a standard Brownian motion, and $B_{1}(t)$ is given by
$dB_{1}(t)=\mu dt+dB(t)$. Suppose $P$ is the Wiener measure induced by $B(t)$ on the $C[0,\infty)$, and $P_{1}$ is the Law ...
8
votes
1answer
553 views
Use of Girsanov's theorem in bond pricing
Assume that we want to calculate the time $t=0$ price of a bond:
$B(0,T) = E_P[\exp(-\int_0^T r_s ds)]$,
where $r$ is the interest rate following the SDE
$dr_t=k(\theta-r_t)dt+\sigma dB_t=b(r_t)dt+\...
