# Questions tagged [girsanov]

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### 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 ...
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### 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 ...
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### 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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### 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 (...
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### 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 ...
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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 ...
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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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### 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 ...
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### 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 ...
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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 $$P(0,T_p)E^{Q_{T_p}}\left[S(T_0,T)\... 1answer 149 views ### Bivariate Black-Sholes Model Let us propose bivariate Black-Sholes Model. Assume, we have an arbitrage-free complete market. r_{f} is risk-free rate. Under real-world measure P: dS_{1} (t)=S_{1} (t) [\mu_{1}dt+\sigma_{1}... 1answer 4k views ### Radon-Nikodym derivative and risk natural measure I need help with my understanding of changing probability measure. Im not a mathematician so I hope for answers that are not too technical. As shown in this Wikipedia article http://en.wikipedia.org/... 1answer 2k views ### Girsanov Theorem and Quadratic Variation Girsanov theorem seems to have many different forms. I've got a problem matching the form in wiki to the one in Shreve's book, due to the difficulty of quadratic variation calculation. Below is the ... 2answers 1k views ### Uniqueness of equivalent martingale measure in Black Scholes-Model Let's consider standard Black-Scholes model with price process S_t satisfying SDE$$dS_t = S_t(bdt + \sigma dB_t), where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
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 ...
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+\...