# Questions tagged [girsanov]

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### 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 ...
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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) $\beta_{t}=e^{t}, S_{t}... • 103 1 vote 0 answers 122 views ### How to use Girsanov theorem for complicated RN derivatives? Let$W_t$be a Brownian motion under probability measure$\mathbb{P}$. Let$X_tbe 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(-\... • 589 3 votes 2 answers 186 views ### 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+\... • 373 3 votes 1 answer 136 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}... • 373 2 votes 0 answers 120 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+... • 303 8 votes 2 answers 1k 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}, ... • 131 2 votes 0 answers 66 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)}$$Suppose I now want to price an ... • 5,306 1 vote 1 answer 320 views ### What is the link between the SDF in the Black-Scholes-Merton model and the exponential process in Girsanov's theorem? Question I have been toying around to get some understanding of what the stochastic discount factor look likes in Black-Scholes-Merton and how it relates to the exponential process in Girsanov's ... • 2,336 1 vote 0 answers 46 views ### Change of measure to get a determined drift let's say I have a real stochastic process dX_t=dt+\frac{1}{B_t}dB_t on [0,T], with B_t Brownian in \mathbb{P} (not centered in 0) in [0,\tau] with \tau some adequate stopping time that ... 1 vote 1 answer 223 views ### Objective probability of default from CDS spread I have the risk neutral probability of default extrapolated from the market data of the CDS spreads. How can I empirically estimate the market risk price of the objective probability of default (i.e. ... • 47 5 votes 1 answer 465 views ### On Girsanov Theorem to switch from Risk-Neutral to Stock Numeraire Summary: long-story cut short, the question is asking for what types of functions f(.), the Cameron-Martin-Girsanov theorem can be used as follows:$$ \mathbb{E}^{\mathbb{P}^2}[f(W_t)]=\mathbb{E}^{\... • 5,306 1 vote 1 answer 240 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$units of stock share ... 1 vote 1 answer 118 views ### true or false: the risk-neutral measure is useless in this situation Example 2 of this Wiki article on the risk-measure describes how a stock price$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 ... • 534 2 votes 0 answers 114 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$(W)_{t\in[0;T]}$... • 540 4 votes 1 answer 314 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 ... • 540 1 vote 1 answer 343 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$0 \le t \le T_p$, ... • 589 0 votes 1 answer 332 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 ... • 5,306 1 vote 0 answers 122 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 ... • 311 3 votes 2 answers 233 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 ... • 101 3 votes 0 answers 98 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:... 195 views

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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 $Q$ .....
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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 ...
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### 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 + \...
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### 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 $Q$ ...
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### 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 ...
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### 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 ...
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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. ...
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### 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 {{...
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