Questions tagged [girsanov]
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55
questions
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Girsanov Theorem example
My reference is here : https://arxiv.org/pdf/1504.05309.pdf
My question is related to the example 2.6.1 : page 21-22; 2.6 Girsanov Theorem
It said in equation (2.8) $Z_t = exp(-\frac{1}{2}∫^t_0θ_s^2ds+...
3
votes
2answers
257 views
Heston stochastic volatility, Girsanov theorem
How can we apply Girsanov's theorem to a stochastic volatility model?
In Heston's model the dynamics are given by
\begin{align*}
dS_t &= \mu S_t dt + \sqrt{v_t}S_t d\widehat{W}^\mathbb{P}_{1,t}, ...
2
votes
0answers
47 views
Stock price under Bond numeraire
The Radon-Nikodym derivative going from the bank-acount Numeraire $N(t)$ to the bond numeraire $P(t,T)$ is:
$$\frac{dP}{dN}(T|\mathcal{F}_t)=\frac{1}{N(T)P(t,T)}$$
Suppose I now want to price an ...
0
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0answers
68 views
Apply Girsanov theorem to derive a formula for price [duplicate]
Let $[S_t]_{t\ge 0}$ be a geometric Brownian Motion with drift $r$ and volatility $\sigma$.
The dynamics of $S$ are:
$$dS_t=rS_tdt+\sigma S_tdW_t$$
Instead of completing the square technique, how to ...
1
vote
1answer
96 views
What is the link between the SDF in the Black-Scholes-Merton model and the exponential process in Girsanov's theorem?
Question
I have been toying around to get some understanding of what the stochastic discount factor look likes in Black-Scholes-Merton and how it relates to the exponential process in Girsanov's ...
1
vote
0answers
40 views
Change of measure to get a determined drift
let's say I have a real stochastic process $dX_t=dt+\frac{1}{B_t}dB_t$ on $[0,T]$, with $B_t$ Brownian in $\mathbb{P}$ (not centered in 0) in $[0,\tau]$ with $\tau$ some adequate stopping time that ...
1
vote
1answer
135 views
Objective probability of default from CDS spread
I have the risk neutral probability of default extrapolated from the market data of the CDS spreads. How can I empirically estimate the market risk price of the objective probability of default (i.e. ...
0
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0answers
47 views
Proof of existence of one only martingale measure
I know that:
Hypothesis 1 (Girsanov Theorem)
Let $\theta=\begin{Bmatrix}
\theta_t
\end{Bmatrix}_{t\in [0,T]}$ be a square-integrable and $\Im_t$-adapted process such that $\mathbb{E}[e^{\frac{1}{2}\...
4
votes
1answer
191 views
On Girsanov Theorem to switch from Risk-Neutral to Stock Numeraire
Summary: long-story cut short, the question is asking for what types of functions $f(.)$, the Cameron-Martin-Girsanov theorem can be used as follows:
$$ \mathbb{E}^{\mathbb{P}^2}[f(W_t)]=\mathbb{E}^{\...
1
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1answer
139 views
Black Scholes model without using Girsanov's theorem? It might happen?
We can calculate the stock price by the equation: $\frac{dS_t}{dt} = \mu dt + \sigma dB_t$,where $B_t$ is a Brownian motion.
First i create a portfolio that consists of $\Phi$ units of stock share ...
1
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1answer
95 views
true or false: the risk-neutral measure is useless in this situation
Example 2 of this Wiki article on the risk-measure describes how a stock price $S_t$ that is modeled with Geometric Brownian motion with drift $\mu$
$$
dS_t = \mu S_t dt + \sigma S_t dW_t
$$
can be ...
2
votes
0answers
89 views
Last step step in Girsanov's theorem proof
I consider the version of Girsanov's theorem presented in this question.
Let us take the particular case that $\mathbb{F}$ is the filtration generated by standard Brownian motion $(W)_{t\in[0;T]}$ ...
4
votes
1answer
246 views
Are all change of measure operations between equivalent probability measures Doléans-Dade exponentials?
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be ...
0
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1answer
150 views
How do we derive the Radon-Nikodym derivative for T-forward measures?
Let $Q^{T_e}$ denote the $T_e$-forward measure and let $Q^{T_p}$ denote the $T_p$-forward measure.
I have seen the following Radon-Nikodym derivative being used in derivations.
For $0 \le t \le T_p$,
...
0
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1answer
241 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
106 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 ...
3
votes
2answers
178 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 ...
3
votes
0answers
88 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:...
2
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1answer
150 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
votes
1answer
237 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 ...
2
votes
2answers
319 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 ...
3
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0answers
553 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 ...
1
vote
1answer
134 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 ...
5
votes
2answers
658 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 ...
4
votes
1answer
363 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 ...
3
votes
1answer
343 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:
$$ ...
1
vote
1answer
192 views
Why Girsanov's theorem used here?
It is written in Bjork's ArbitrageTheoryInContinuousTime that
... Assume a martingale measure Q exists. This implies (see the Girsanov theorem) that the price processes have zero drift under $Q$ .....
3
votes
1answer
1k views
Change of numeraire in options with currency exchange features
FV of an EUR denominated option under "COP" risk measure is given by:
$$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 ...
2
votes
1answer
374 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
vote
1answer
186 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
votes
1answer
278 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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0answers
358 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
1k 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 ...
6
votes
1answer
2k 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(-\...
4
votes
1answer
172 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
373 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 ...
1
vote
1answer
521 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
218 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. ...
4
votes
1answer
2k 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
{{...
2
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0answers
208 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
223 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
830 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 \in ...
1
vote
2answers
264 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
98 views
On the construction of a Brownian motion from a Gaussian process
Let $X$ a Gaussian process defined by
$$
X_t=\int_{0}^{t}\left(\frac{1}{\sigma}\left(r_s-\frac{\sigma^2}{2}\right)-\rho\sigma_P(s,T)\right)\mathrm{d}s+\sqrt{1-\rho^2}Z_2(t)+\rho Z_1(t);\;\;t\in[0,T]
$...
7
votes
1answer
1k views
How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?
Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
9
votes
2answers
2k views
Ito, Stochastic Exponential and Girsanov
This is a two-part question relating to the change of measure density used in Girsanov and secondly to the Stochastic Exponential.
Whilst reading notes relating to Girsanov it is stated that the ...
11
votes
2answers
4k views
Radon-Nikodym derivative and risk natural measure
I need help with my understanding of changing probability measure. Im not a mathematician so I hope for answers that are not too technical.
As shown in this Wikipedia article http://en.wikipedia.org/...
1
vote
1answer
174 views
Bivariate Black-Sholes Model
Let us propose bivariate Black-Sholes Model.
Assume, we have an arbitrage-free complete market.
$r_{f}$ is risk-free rate.
Under real-world measure $P$:
$dS_{1} (t)=S_{1} (t) [\mu_{1}dt+\sigma_{1}...
2
votes
2answers
347 views
Why do we need $dS_t=r S_tdt+\sigma S_tdW_t^Q$?
Suppose $S_t$ is the stock price and follows the dynamics $$dS_t=\mu S_tdt+\sigma S_tdW_t$$. According to Girsanov, we can apply change of measure and obtain $dS_t=r S_tdt+\sigma S_tdW_t^Q$, this ...
5
votes
2answers
2k views
Uniqueness of equivalent martingale measure in Black Scholes-Model
Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
