# Questions tagged [numeraire]

Numeraire is a unit of account in which all other assets in a given model are denominated. Most importantly, one can borrow and lend at the Numeraire rate.

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### Is there a relationship between Risk Neutral Pricing framework and Nash Equilibria?

Based on the Fundamental Theorem of Asset Pricing, the risk neutral price of a contingent claim on an asset in a liquid, arbitrage free market can be determined by switching to an equivalent $Q-$ ...
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### Bayes' rule for conditional expectations (Proof review)

The Baye's rule for conditional expectations states $$E^Q[X|\mathcal{F}]E^P[f|\mathcal{F}]=E^P[Xf|\mathcal{F}]$$ With $f=dQ/dP$ - thus being the Radon-Nikodyn derivative and $X$ being ...
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### Change of numeraire between T-forward and Bank Account

I follow a course, and get to the point that one bond price discounted by another one is a martingale: $$\frac{P(t,T_0)}{P(t,T_1)} - \text{ is a } \mathbb{Q}^{T_1} \text{ martingale }$$ I can not ...
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### Change of numéraire for non-Normal distributions

I'm looking for a resource, a book or an article, that describes the framework of change of numéraire in a broader context than just Brownian motions or Normal distributions. I'm only really ...
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### Martingale measure result application for interest rates under T-forward measure?

I've got a question about the way the equivalent martingale measure result is used for pricing derivatives. Hull states the result as the next equality: \begin{align*} f_o = g_0 E^{g}\big(\frac{f_T}{...
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### Using a Constant as a Numeraire

Please provide steps to justify the below. 1) Can we use a constant as a numeraire? Related Question: Scaling Stock Price and Strike etc. by a Constant The rest of standard Geometric Brownian ...
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### The relation between exchange rate SDE and respective interest rates

The exchange rate between a domestic currency money market and a foreign currency money market can be expressed as $$dQ(t) = (r_d - r_f)Q(t)dt + \sigma Q(t)d\tilde{W}(t)$$ where $r_d$ is the ...
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### Change of measure between T-forward and T*-forward contract?

I am trying to prove the need of a convexity adjustment to a forward rate by calculating the next expectation: \begin{align*} P(t_0, T_s)E^{T_s}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big). \end{...
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### Deriving Black Scholes PDE under stock as a numeraire

There are many ways to derive the Black Scholes PDE. The Martingale way would be to demand the option price is driftless according to particular measures. Below I derive the correct PDE using the bank ...
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### Bond SDE under its own forward measure

I am trying to write the SDE for a forward bond, $dP(t,T_1,T_2)$, under the $T_1$-Forward measure, $Q_{T_1}$. I can easily do this by: Writing the equation of $dP(t,T_1)$ and $dP(t,T_2)$ under the ...
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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 ...
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### Change of numeraire to the forward measure in the Vasicek model

I am working through the Brigo/Mercurio book on Interest Rate Models (Second Edition) and I am having some trouble with the change of numeraire in chapter 3.2.1, page 59 to be exact, formula 3.9. It ...
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 ...