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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Change of numéraire for two risky assets without bank account (Margrabe’s formula?)
I am considering two risky assets following the usual correlated GBM given by
$$\frac{\mathrm{d}S^{(i)}_t}{S^{(i)}_t}=\mu_i\mathrm{d}t+\sigma_i\mathrm{d}W^{(i)}_t,\quad i\in\{1,2\}$$
with
$$\mathrm{d}...
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Monte Carlo methods: Choosing the best measure
When pricing derivatives using Monte Carlo methods, we take outset in the risk neutral pricing formula which states that we need to calculate the expected value of the discounted cashflows. To do this,...
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Change of measure when the underlying dynamic is Ornstein-Uhlenbeck
Let the $r$ riskless rate to be constant. Let's consider the following underlying dynamic under the $\mathbf{P}$ “physical measure”
$$dS_{t}=\mu_{t}S_{t}dt+\sigma_{t}S_{t}dW_{t}^{\mathbf{P}},$$
where $...
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Effect on variance of change of measure
My current understanding:
(a) changing the probability measure of a diffusion process does not change the variance.
(b) for a general stochastic process the variance may change.
Please confirm whether ...
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RFR boostrapping using RFR OIS: Is convexity adjustment technically necessary?
For single-curve RFR bootstrapping, such as a SOFR-based discounting curve bootstrapped strictly using SOFR fixed-float OIS, I am trying to understand if convexity adjustments are technically ...
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Is homogeneity preserved under change of measure?
In a paper, Joshi proves that the call (or put) price function is homogeneous of degree 1 if the density of the terminal stock price is a function of $S_T/S_t$. In the paper I think Joshi is silently ...
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Bayes' rule for conditional expectations (Proof review)
The Baye's rule for conditional expectations states
$$ 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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Money account discounted libor rate is it a martingale under risk neutral measure?
I see that Libor $L(t,S,T)$ is a martingale under $T-$forward measure. Where we used argument that zero-coupon bonds are martingales under $T$-forward measure, as zero-coupon bond is a traded security....
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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 ...
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Why fitting $\mathbb{Q}$ vs $\mathbb{P}$ measure Heston model if both fit to market
If both models fit their closed form formulas to market prices, why should I prefer a more complex model? ($\mathbb{Q}$ version has one extra parameter $\lambda$)
Do valuation with dynamics work ...
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Calculate the amount of shares of a deposit without converting to numeraire
Let F a mutual fund with two assets A and B. Initially, F contains 1 unit of A, 1 unit of B, and there is 1 share allocated to Alice. At a later time, Bob deposits 2 units of A into F. How can I ...
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Why is the market price of risk a non-entity according to Bergomi?
I am reading Bergomi's book Stochastic Volatility Modelling. In the chapter 6 dedicated to the Heston model, page 202, he describes the traditional approach to first generation stochastic volatility ...
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University problem about Bond option [closed]
Good morning,
Next week I'll have Derivates Final test and I've a doubt about Bond Option.
If I have one ZCB, price 90 with 3 month maturity and strike 100, and I want a minimum yield of 2%, what type ...
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Does numeraire have to be a tradable asset
I thought we create replicating portfolios using underlying and the numeraire i.e. the numeraire has to be a tradable asset (assuming simple binomial model).
But I have seen some examples which ...
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How is an exchange rate process a martingale under any measure?
Suppose a process for a stock price of a US-based company traded in the USA is, under the USD money-market numeraire:
$$dS_t=S_tr_{USD}dt+S_t\sigma_SdW_1(t)$$
Using fundamental theorem of asset ...
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Change of measure for BGM (LMM) Model
I've been checking the demos for BGM (LFM) forward rate model.
Here's a short reminder to help you follow:
Now, take the following
$$\frac{dL_j(t)}{L_j(t)} = \sigma_j. dW^j(t) = \mu_{ij} dt + \...
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Intuition for consistent Derivative Prices under different Numeraires and Measures
This is essentially the Fundamental Theorem, however I am not asking for a thorough proof, I am more interested in the general intuition.
In words, it makes sense that whatever your unit of account (...
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Forward starting zero-coupon bonds
We trivially have that:
$$\frac{Z(t_0,t_1)}{Z(t_0,t_2)}=1+\tau L(t_0,t_1,t_2)$$
Where $L(t_0,t_1,t_2)$ is the forward Libor between $t_1$ and $t_2$, as of $t_0$.
Simply inverting this relationship ...
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question regarding relation between expectations on different measures
I am a beginner to the theory of stochastic calculus and measure change.
I have derived an equation related to expectations on different measures.
I wanted some expert opinion on whether this is true ...
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Why does the diffusion term remain the same when we change pricing measure?
Consider some Itô process $dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t}$ under the measure $\mathbb P$, where $W^{\mathbb P}$ is a $\mathbb P$-Brownian motion
In plenty of interest rate examples, I have ...
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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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If any zero coupon bond $P(T)$ can be chosen as a numéraire, then why can the rolling bond for any time discretization be chosen as numéraire
Let us consider some finite time horizon $[0,T]$, and we assume that $P(t)$, the zero coupon bond maturing in $t$ for any $t\in [0,T]$ can be chosen as a numéraire, i.e. such that the numéraire-...
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The Radon-Nikodym derivative for a sequence of dependent variables
Suppose that a probability space $(\Omega, \Sigma, \mathbb{P})$ is given. Let $W=\{W_n\}_{n\in \mathbb{N}_0}$ be a sequence of $\mathbb{P}$-i.i.d real-valued random variables on $\Omega$. Furthermore, ...
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Floating swap payoff with rate determined on current instead of previous date
I am attempting to determine the payoffs a modified swap, in which the floating payments at a time $T_k$ are made on the current date (i.e. $L(T_k,T_{k+1})\equiv L_{k+1}(T_k)$) rather than at the ...
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Are Stochastic Differential Equation diffusion terms always invariant under a change of measure?
I'm struggling with learning change of numeraire, and stochastic differential equations. I'm reading the beginning of Brigo and Mercurio's Interest Rate Models- Theory and Practice, and I'm on the ...
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Change of Numeraire technique (Cross-currency models)
Hey I have problem with understanding change of numeraire technique. For example we have
$dr^d(t)=\kappa_1(\theta_1(t)-r^d(t))dt+\sigma_1 dW_1$ (under measure $Q^1$ associated with domestic bank ...
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Pricing of LIBOR based CF settled after the LIBOR fixing by switching from risk-neutral to forward-neutral measures
When deriving the LIBOR-based swap rate formula in any interest rate model, expressions of the following types appear naturally:
Literature tells us that, switching to the – forward neutral measure, ...
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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: $$L_t = \exp\left(-\...
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Is first order stochastic dominance conserved under change of measure?
As the title states, my question is whether first order stochastic dominance is conserved under change of measure, for instance from the $\mathbb{P}$ measure to $\mathbb{Q}$ measure and change of ...
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Why does adding a negative risk premium to the short rate avoid the occurrence of inverse yield curves?
I am reading about the Vasicek One Factor short rate model and how to implement a change in measure from a risk-neutral to real-world measure, when I came across this comment:
Adding a negative risk ...
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Intuition for Stock Price Numeraire Drift
I would like to ask whether there is an intuition for the drift of price processes under the Stock numeraire.
I find it intuitive that the martingale measure under the Money Market numeraire induces ...
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Change of measure between T-forward and T*-forward contract?
I am trying to prove the need of a convexity adjustment to a forward rate by calculating the next expectation:
\begin{align*}
P(t_0, T_s)E^{T_s}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big).
\end{...
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Caplet "in arrears" pricing formula
The forward Libor rate $L(t,t_1,t_2)$, with $0 \leq t \leq t_1$, must be a martingale under the T-forward measure associated with the zero coupon bond $P(t,t_2)$ that matures at time $t_2$.
Pricing a ...
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Show a process is Martingale
$$Z(t)=(\frac{S(t)}{H})^p$$where $S$ has a standard Black-scholes Dynamics for a stock, $H$ is a postive constant and $p =1 - \frac{2r}{\sigma^2}$. How can I show that $Z(t)/Z(0)$ is a postive Q-...
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If there is a $T$-forward measure and a risk neutral measure, then markets are not complete?
I am trying to understand the connection between market completeness and risk neutral measures.
A market is complete if and only if the equivalent martingale measure is unique.
But if I change to the $...
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Stock price under Bond numeraire
The Radon-Nikodym derivative going from the bank-acount Numeraire $N(t)$ to the bond numeraire $P(t,T)$ is:
$$\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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How do we determine the "correct measure"?
Frequently I come across the statement that the "correct measure" for a product is this-or-that measure. For example,
Eurodollar Futures or Stock returns - Risk neutral measure
Libor forward rate - T-...
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Arithmetic Asian Option
Assume the risk-free bond Bt and the stock St follow the dynamics of the Black & Scholes model
without dividends (with interest rate r, stock drift $μ$ and volatility $σ$).
Let $A_T:=\frac{1}{T}...
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Numeraire correlated to the traded asset
The Fundamental Theorem of Asset Pricing states that:
\begin{align*}
\frac{X_0}{N_0} &= \mathbb{E}^N{ \left[ \frac{X(t)}{N(t)}|\mathcal{F}_0 \right] }
\end{align*}
The usual conditions apply (both ...
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Pricing an Option with payoff $\left(1-\frac{K}{S_t}\right)^{+}$
Let $S_t=S_0 \exp\left\{rt+0.5\sigma^2t+\sigma W_t\right\}$ be the usual GBM model for a Stock price under the money-market numeraire.
Suppose we want to price an option with payoff at maturity: $C_T=(...
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Change of numeraire between t1-forward mesure and t2-forward mesure
Let denote $\mathbb{Q}_{t_1}$ the $t_1$-forward mesure associated to zero coupon bond $B(.,t_1)$.
Let denote $\mathbb{Q}_{t_2}$ the $t_2$-forward mesure associated to zero coupon bond $B(.,t_2)$.
I am ...
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We have a two LIBOR contracts, how to compare their values by change of change of numeraire
We have two LIBOR contracts:
contract 1 pays $L\left(T_{1},\:T_{2}\right)-K$ at time $T_{1}$
contract 2 pays $L\left(T_{1},\:T_{2}\right)-K$ at time $T_{2}$.
Now, $F_{1}$ is the par strike such that ...
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Power Options & Forwards on Stock Squared
Short story: the process for Stock price squared is not a martingale when discounted by the money-market numeraire under the risk-neutral measure. How can we then compute derivative prices on $S_t^2$ ...
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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}^{\...
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Value of an option to exchange an asset for another
I'm working out the examples in the paper "Changes of Numeraire, Changes of Probability Measure and Option Pricing", corollary 3.
An option of exchanging asset 2 against asset 1 at time T, its time-0 ...
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How to prove martingality of forward rate under T-forward measure
Let $P(t,T)=\mathbb{E}_{Q_{R}}[e^{\int^{T}_{t}r(u)du}|\mathcal{F}_{t}]$ be the price of a 1-euro zero-coupon bond with maturity $T$ and $r(u)$ the interest rate process. Consider the the forward rate $...
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Change of Numeraire formula
The general change of Numeraire formula gives the following Radon-Nikodym derivative:
$$ \frac{dN_2}{dN_1}(t)|\mathcal{F}_{t_0}=\frac{N_1(t_0)N_2(t)}{N_1(t)N_2(t_0)} $$
I am able to derive this Radon-...
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