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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Verifying if a Function is a Radon-Nikodym Derivative for changing the numeraire
I am analyzing the following function within a financial mathematics framework:
$$
f(t) = \dfrac{B(S; S) \cdot m(t)}{B(t; S) \cdot m(S)}
$$
where:
$$
B(t; S) := \mathbb{E}_{t}^{\mathbb{P}} \left[\exp\...
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Complete market price and incomplete market price specification
We know that if a liquid market of an asset exists, then the standard derivative pricing theorem implies an equivalent martingale measure exists, not necessarily unique, under which the discounted ...
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Functional Form of Radon-Nikonym Derivative
Suppose I have empirical values of dQ/dP in the context of risk-neutral and real-world probabilities of asset returns. Would it be possible to fit a functional form of the derivative? I'm particularly ...
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Incompatibility between LFM and LSM
In an excellent textbook by Brigo & Mercurio (2006), there is an equation I coundn't derive. It's (6.40) (p.245, shown below) about the forward-rate dynamics under the forward-swap measure.
I'd ...
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Radon–Nikodym derivative of $\alpha$-measure
I'm struggling a bit to understand how the Radon–Nikodym derivative is computed in pag.3 of this paper written by Mark Joshi titled "THE USE OF POWER NUMERAIRES IN OPTION PRICING". Given a ...
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I am looking for help to derive this formula from Brigo & Mercurio
I think my use of numeraire change is incorrect and for sure my understanding is incomplete.
$\frac {dQ2}{dQ1}=\frac {Pt(0,T2)P0(0,T1)}{P0(0,T2)Pt(0,T1)} = \frac {1+DeltaF_1(t)}{1+DeltaF_2(t)}$
Then ...
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Intuition of drift in Libor market model
I studied the classical Libor market model, where the dynamics of rate $F_k$ from time $T_{k-1}$ to $T_k$ are given by
$$
dF_k(t)/F_k(t) = \sigma_k(t) dZ_k(t)
$$
under the forward measures $Q^k$ (...
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numéraires change
I am currently reading the Brigo and Mercurio's book about interest rate models.
I am stuck at understanding the numéraire change proof for the 2.3.1 proposition.
Could someone be kind enough to ...
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Equivalence of formulations of Radon Nikodym derivative
Let $N$ be a numeraire associated with the probability measure $Q^N$ and $U$ be a numeraire associated with the probability measure $Q^U$, both of which are equivalent to the physical probability ...
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Computing Derivative Security with Change of Numeraire
Under Black-Scholes, price a contract worth $S_T^{2}log(S_T)$ at expiration.
This is a question from Joshi's Quant Book (an extension question).
Ok, so I solved this with 3 different methods to make ...
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Change of numeraire : quotient
Let's consider $X_1(t)$ a geometric brownian motion (with variable volatility) and $X_2(t)$ a Brownian bridge :
$dX_1(t) = \mu X_1(t) dt + \sigma_1(t) X_1(t) dW(t)$
$dX_2(t) = \frac{b - X_2(t)}{T - t} ...
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Change of measure when the underlying dynamic is Ornstein-Uhlenbeck
Let the $r$ riskless rate to be constant. Let's consider the following underlying dynamic under the $\mathbf{P}$ “physical measure”
$$dS_{t}=\mu_{t}S_{t}dt+\sigma_{t}S_{t}dW_{t}^{\mathbf{P}},$$
where $...
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Monte Carlo methods: Choosing the best measure
When pricing derivatives using Monte Carlo methods, we take outset in the risk neutral pricing formula which states that we need to calculate the expected value of the discounted cashflows. To do this,...
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Is homogeneity preserved under change of measure?
In a paper, Joshi proves that the call (or put) price function is homogeneous of degree 1 if the density of the terminal stock price is a function of $S_T/S_t$. In the paper I think Joshi is silently ...
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Why fitting $\mathbb{Q}$ vs $\mathbb{P}$ measure Heston model if both fit to market
If both models fit their closed form formulas to market prices, why should I prefer a more complex model? ($\mathbb{Q}$ version has one extra parameter $\lambda$)
Do valuation with dynamics work ...
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Why is the market price of risk a non-entity according to Bergomi?
I am reading Bergomi's book Stochastic Volatility Modelling. In the chapter 6 dedicated to the Heston model, page 202, he describes the traditional approach to first generation stochastic volatility ...
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University problem about Bond option [closed]
Good morning,
Next week I'll have Derivates Final test and I've a doubt about Bond Option.
If I have one ZCB, price 90 with 3 month maturity and strike 100, and I want a minimum yield of 2%, what type ...
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How is an exchange rate process a martingale under any measure?
Suppose a process for a stock price of a US-based company traded in the USA is, under the USD money-market numeraire:
$$dS_t=S_tr_{USD}dt+S_t\sigma_SdW_1(t)$$
Using fundamental theorem of asset ...
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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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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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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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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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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 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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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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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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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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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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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
$$\mathrm{d}...
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Pricing an Option with payoff $\left(1-\frac{K}{S_t}\right)^{+}$
Let $S_t=S_0 \exp\left\{rt+0.5\sigma^2t+\sigma W_t\right\}$ be the usual GBM model for a Stock price under the money-market numeraire.
Suppose we want to price an option with payoff at maturity: $C_T=(...
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Change of numeraire between t1-forward mesure and t2-forward mesure
Let denote $\mathbb{Q}_{t_1}$ the $t_1$-forward mesure associated to zero coupon bond $B(.,t_1)$.
Let denote $\mathbb{Q}_{t_2}$ the $t_2$-forward mesure associated to zero coupon bond $B(.,t_2)$.
I am ...
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We have a two LIBOR contracts, how to compare their values by change of change of numeraire
We have two LIBOR contracts:
contract 1 pays $L\left(T_{1},\:T_{2}\right)-K$ at time $T_{1}$
contract 2 pays $L\left(T_{1},\:T_{2}\right)-K$ at time $T_{2}$.
Now, $F_{1}$ is the par strike such that ...
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Power Options & Forwards on Stock Squared
Short story: the process for Stock price squared is not a martingale when discounted by the money-market numeraire under the risk-neutral measure. How can we then compute derivative prices on $S_t^2$ ...
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On Girsanov Theorem to switch from Risk-Neutral to Stock Numeraire
Summary: long-story cut short, the question is asking for what types of functions $f(.)$, the Cameron-Martin-Girsanov theorem can be used as follows:
$$ \mathbb{E}^{\mathbb{P}^2}[f(W_t)]=\mathbb{E}^{\...
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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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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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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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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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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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Why is the numeraire in the LGM model tradeable?
I'm trying to understand the LGM model, which Hagan defines as follows. The state variable $X$ evolves according to
$$dX(t) = \alpha(t) dW^N(t)$$
wrt the numeraire
$$N(t) = \frac{1}{P(0,t)} e^{H(t)X(...
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Change of measure for BGM (LMM) Model
I've been checking the demos for BGM (LFM) forward rate model.
Here's a short reminder to help you follow:
Now, take the following
$$\frac{dL_j(t)}{L_j(t)} = \sigma_j. dW^j(t) = \mu_{ij} dt + \...
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Why can only non-dividend paying assets serve as numeraire?
In Kerry Back, A Course in Derivative Securities, Sect. 1.4 (page 29), the author stated the FTAP in the following form (in boldface):
If there are no arbitrage opportunities,
then for each (non-...
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