Questions tagged [girsanov]

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Benth: Risk-neutral measure in incomplete markets

I am currently working on Benth and Benth "THE VOLATILITY OF TEMPERATURE AND PRICING OF WEATHER DERIVATIVES" and i am stuck at following paragraph at page 10, which is about risk-neutral ...
Valentin's user avatar
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Derivative pricing under $\mathbb{P}$

I recently learnt about the Girsanov-Cameron-Martin theorem, which basically says, that if $(\tilde{B}(t),t\in[0,T])$ is some Brownian motion with a (possibly stochastic) drift $\theta(t)$ defined on $...
Quasar's user avatar
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What happens trying to price derivatives starting from a non-geometric brownian motion?

To get a better understanding, I tried going through BSM-model starting from a non-geometric brownian motion. However, during the derivation I got stuck, which led me to a specific question. The set-...
Emanuele's user avatar
2 votes
0 answers
382 views

Understanding the asset pricing theory and numeraire

While reading about asset pricing theory and numeraire, I had faced some confusion. Short summary of asset pricing theory from my book We start our journey with a risky asset $S_t=\mu S_tdt+\sigma ...
emonhossain's user avatar
2 votes
1 answer
805 views

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

I am trying to price a call option on a zero coupon under the Vasicek Model using Monte Carlo method: $$C_0 = B(0,\theta) \ \mathbb{E}^{\mathbb{Q}_T}[(B(\theta,T)-K)^{+}]$$ The problem is that the ...
Feynman_kac's user avatar
2 votes
0 answers
406 views

Pricing a put-option in the Heston Model

Assume the Heston Model with dynamics under the martingale measure $Q$ given by \begin{align} dS_t &= (r-q)S_t dt + \sqrt{v_t}S_tdW_{1,t}^Q\\ dv_t &= \kappa(\theta-v_t)dt + \sigma\sqrt{v_t}dW_{...
Landscape's user avatar
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1 answer
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Market price of risk ($\lambda$) - Brigo and Mercurio

In page 52 of Interest Rate Models by Brigo and Mercurio the following is stated: Precisely, let us assume that the instantaneous spot rate evolves under the real-world measure $Q_0$ according to $dr(...
KT8's user avatar
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Where does the term $\gamma$ come from when moving from measure $\mathbb Q^{N}$ to $\mathbb Q^{M}$?

Consider two measures $\mathbb Q^{M}$ and $\mathbb Q^{N}$, as well as the two numéraires $M$ and $N$, furthermore assume that $X\frac{N}{M}$ is a $\mathbb Q^{M}$-martingale. Furthermore, the ...
user9078057's user avatar
2 votes
1 answer
158 views

Proof about discounted zero coupon bond

Hey guys I am having trouble finishing this proof: Proposition 5.1 Under the above assumptions, the process $r$ satisfies under $\mathbb{Q}$ $$ d r(t)=\left(b(t)+\sigma(t) \gamma(t)^{\top}\right) d t+\...
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10 votes
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Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?

Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete. (a) $\beta_{t}=e^{t}, S_{t}...
randorando's user avatar
1 vote
0 answers
155 views

How to use Girsanov theorem for complicated RN derivatives?

Let $W_t$ be a Brownian motion under probability measure $\mathbb{P}$. Let $X_t$ be defined as follows. $$\mathrm{d}X_t = a \mathrm{d}t + 2\sqrt{ X_t} \mathrm{d}W_t.$$ Also define: $$L_t = \exp\left(-\...
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2 answers
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Change of measure for a stochastic process to be a martingale

$\text { Give a measure change so that } X_{t}=e^{B_{t}}\left(B_{t}-t / 2\right) \text { is a martingale, } 0 \leq t \leq T$ My attempt Using Ito's lemma on $X_{t}$ we get: $-\frac{e^{B t}}{2} d t+\...
codelearner's user avatar
3 votes
1 answer
247 views

EMM for Bachelier model

The stock price is assumed to evolve as $S_{t}=S_{0}+\mu t+\sigma B_{t}$, where $S_{0}>0, \mu>0$ and the process $B_{t}$ is Brownian motion. The saving account is assumed to be $\beta_{t}=e^{r t}...
codelearner's user avatar
2 votes
0 answers
249 views

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+...
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10 votes
2 answers
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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}, ...
quantmath's user avatar
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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 ...
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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 ...
Stéphane's user avatar
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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 ...
Andrea Licata's user avatar
1 vote
1 answer
347 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. ...
d0whes's user avatar
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1 answer
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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}^{\...
Jan Stuller's user avatar
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1 answer
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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 ...
Maria Oikon's user avatar
1 vote
1 answer
140 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 ...
Taylor's user avatar
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0 answers
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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 $(W)_{t\in[0;T]}$ ...
fwd_T's user avatar
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4 votes
1 answer
454 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 ...
fwd_T's user avatar
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1 answer
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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 $0 \le t \le T_p$, ...
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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 ...
Jan Stuller's user avatar
  • 6,098
1 vote
0 answers
157 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 ...
Dreason94's user avatar
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3 votes
2 answers
297 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 ...
luca dibo's user avatar
3 votes
0 answers
107 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:...
user avatar
2 votes
1 answer
289 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_{...
user avatar
3 votes
1 answer
281 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 ...
Ile's user avatar
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4 votes
3 answers
785 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 ...
Confounded's user avatar
3 votes
0 answers
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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 ...
Kiann's user avatar
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1 answer
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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 ...
happyGiraffe's user avatar
5 votes
2 answers
2k 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 ...
dayum's user avatar
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4 votes
1 answer
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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 ...
alexbougias's user avatar
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3 votes
1 answer
575 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: $$ ...
user25844's user avatar
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1 answer
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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 $Q$ .....
Juliso's user avatar
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1 answer
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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 ...
Toby1729's user avatar
2 votes
1 answer
533 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 $...
M Smith's user avatar
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1 vote
1 answer
359 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^{-\...
Amrit Prasad's user avatar
0 votes
1 answer
393 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 + \...
Aldo Shumway's user avatar
4 votes
0 answers
537 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 ...
KACEFMA.'s user avatar
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1 answer
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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 ...
user39039's user avatar
  • 441
6 votes
1 answer
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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(-\...
K.Yan's user avatar
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4 votes
1 answer
216 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 $Q$ ...
Archetupon's user avatar
2 votes
0 answers
714 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 ...
Archetupon's user avatar
1 vote
1 answer
883 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 ...
arni's user avatar
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4 votes
1 answer
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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. ...
Politeten's user avatar
5 votes
1 answer
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 {{...
k b's user avatar
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