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.
81
questions
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1answer
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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 $...
5
votes
2answers
258 views
Caplet “in arrears” pricing formula
In this post here it is shown that 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 $P(t,t_2)$ ...
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0answers
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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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0answers
205 views
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}...
6
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3answers
541 views
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-...
2
votes
1answer
175 views
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}...
9
votes
3answers
539 views
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 ...
11
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1answer
480 views
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 ...
2
votes
1answer
174 views
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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0answers
375 views
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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0answers
45 views
Proof of existence of one only martingale measure
I know that:
Hypothesis 1 (Girsanov Theorem)
Let $\theta=\begin{Bmatrix}
\theta_t
\end{Bmatrix}_{t\in [0,T]}$ be a square-integrable and $\Im_t$-adapted process such that $\mathbb{E}[e^{\frac{1}{2}\...
1
vote
0answers
27 views
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 ...
1
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1answer
66 views
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 ...
1
vote
1answer
259 views
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$ ...
4
votes
1answer
159 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}^{\...
2
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0answers
163 views
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 ...
3
votes
3answers
1k views
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 $...
2
votes
1answer
339 views
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-...
6
votes
1answer
157 views
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(...
4
votes
4answers
387 views
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 ...
6
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1answer
4k views
Change of numeraire and reference asset
Learning about change of numeraire, and came across this statement:
The price of any asset divided by a reference asset (called numeraire) is a martingale (no drift) under the measure associated ...
1
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2answers
264 views
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-...
3
votes
1answer
393 views
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 ...
9
votes
2answers
669 views
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 ...
0
votes
0answers
75 views
Stock Price as Numeraire, Two Stocks & One Money Market Account
We have two uncorrelated Stock price processes and the classical Money-Market (MM) account. Under the MM Numeraire, both stocks are Martingales when discounted by the MM, as usual.
Question: I would ...
5
votes
3answers
131 views
Volatility of Exchange Option
I got a question and its partial solution, and have some doubts about the volatility of its geometric Brownian motion process:
Question:
How would you price an exchange call option that pays $max(S_{...
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0answers
33 views
Change of numeraire in exchange options with random interest rate
At time $t$, the market offers a (possibly random) bounded interest rate $r_t$ and two assets whose prices are given by
\begin{align*}
{{\rm d S^{(1)}_t \over S^{(1)}_t}} &= b^{(1)}_t d t + \...
2
votes
3answers
287 views
How can I use the Radon-Nikodym theorem to show that forward measure is indeed measure?
The following statements are taken from the Wikipedia page for forward measure.
Let
$$B(T)=\exp \left(\int _{0}^{T}r(u)\,du\right)$$
be the bank account or money market account numeraire and
$...
1
vote
1answer
138 views
Intuition for consistent Derivative Prices under different Numeraires and Measures
This is essentially the Fundamental Theorem, however I am not asking for a thorough proof, I am more interested in the general intuition.
In words, it makes sense that whatever your unit of account (...
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0answers
104 views
How to determine exchange rate dynamics in currency derivatives
I need some guidance regarding exchange rate dynamics in currency derivatives.
Following three dynamics are defined below,
$\frac{dS(t)}{S(t)}=\alpha dt+\sigma dW(t)$ ; the stock dynamics in the ...
1
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1answer
80 views
How to determine the no arbitrage price of following claim? (change of numeraire)
How do I determine the no arbitrage price for claims such as $min(S_1(T),S_2(T))$ or $max(S_1(T),S_2(T))$? We can consider a standard Black Scholes model. Hence $S_i(T)=S_i(t)e^{(r-\sigma_i^2/2)(T-t)+\...
2
votes
1answer
164 views
Asian Options-Change of Numeraire
Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model
without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$).
Show that ...
3
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0answers
83 views
Change of measure for BGM (LMM) Model
I've been checking the demos for BGM (LFM) forward rate model.
Here's a short reminder to help you follow:
Now, take the following
$$\frac{dL_j(t)}{L_j(t)} = \sigma_j. dW^j(t) = \mu_{ij} dt + \...
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votes
0answers
91 views
Risk neutral measure & change in numeraire
There are two questions about risk neutral and change in numeraire I am not so sure if my answer is correct.
Question 01: Risk neutral
Let says I have 2 risky asset A and B. Each has stochastics ...
2
votes
1answer
135 views
Proof standard Brownian Motion under change of measure
Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
3
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0answers
83 views
American Perpetual Put Option
I want to compute the expected payoff of a (classical) perpetual American put option in the Black-Scholes-Merton (BSM) framework with an optimal strategy of exercising the option at time $\tau=\inf\{t:...
1
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0answers
30 views
In a multi-curve context which numéraire is used to change to the payment probability of a forward asset X paid at time T?
Should it be the coupon associated to the funding curve of the asset?
Thanks.
2
votes
0answers
35 views
Discrete term structure models - generalized procedure to ensure positive probabilities across multiple measures
Question: Is there a generalized procedure for building a discrete (e.g. binomial) term structure model with risk-neutral branching probabilities that ensure positive probabilities under alternative ...
2
votes
1answer
211 views
Drift term in rough volatility models
I'm studying rough volatility papers and was wondering, why the drift term is always missing.
See for example the paper Pricing under rough volatility by Bayer, Friz, Gatheral.
On page 2, the ...
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3answers
719 views
The relation between exchange rate SDE and respective interest rates
The exchange rate between a domestic currency money market and a foreign currency money market can be expressed as
$$
dQ(t) = (r_d - r_f)Q(t)dt + \sigma Q(t)d\tilde{W}(t)
$$
where $r_d$ is the ...
4
votes
1answer
158 views
Pricing a call option with pay-off function max{$S_T - S_{T/2}, 0$}
Pricing a call option with payoff function $C=\max\{S_T - S_{T/2}, 0\}$, where $S_T$ is geometric brownian motion. I appreciate any help! Please close this question if this is a duplicated question. ...
5
votes
1answer
644 views
Change of numeraire between T-forward and Bank Account
I follow a course, and get to the point that one bond price discounted by another one is a martingale:
$$
\frac{P(t,T_0)}{P(t,T_1)} - \text{ is a } \mathbb{Q}^{T_1} \text{ martingale }
$$
I can not ...
2
votes
0answers
204 views
Dividend paying asset, why can't be taken as numéraire?
Why when considering numéraires, one cannot use a dividend paying asset to define a risk neutral measure?
Here's where I got my question : (Shreve - Stochastic Calculus For Finance II)
3
votes
1answer
1k views
Change of numeraire in options with currency exchange features
FV of an EUR denominated option under "COP" risk measure is given by:
$$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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2answers
612 views
What is the numeraire for the real world measure $\mathbb{P}$?
We know the numeraires for the forward measure, the risk-neutral measure, etc.
What is the numeraire for the real world measure $\mathbb{P}$?
