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.
103
questions
28
votes
0
answers
630
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 $Q-$ ...
- 411
19
votes
2
answers
12k
views
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
...
- 3,377
13
votes
4
answers
4k
views
Understanding $N(d_1)$ and how to use the stock itself as the numeraire?
Assume the stock price follows a geometric Brownian motion Then in Black-Scholes pricing model, $N(d_2)$ is the risk-neutral probability that the option expires in-the-money. However, it is said that $...
- 199
12
votes
2
answers
6k
views
How to use the stock as a numeraire to price a derivative with payoff of the form $(S_T f(S_T))^+$?
I have $\frac{dS_t}{S_t} = rdt + \sigma dW_t$ as usual under the money-market numéraire and I need to price options with payoffs
$$(S_T f(S_T))^+$$
How do I express the stock dynamics using the ...
- 121
11
votes
2
answers
1k
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 ...
- 692
11
votes
1
answer
862
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 ...
- 5,998
11
votes
1
answer
1k
views
How to use a change of numeraire to price this option?
I recently asked this question regarding how to price an option with payoff:
$$\text{Payoff}_T = (A_TR_T - A_T \lambda)^+ $$
Let's assume for generality that $A_t$ and $R_t$ are GMB's:
$$dA_t = \...
- 11.1k
10
votes
4
answers
2k
views
Intuition for Stock Price Numeraire Drift
I would like to ask whether there is an intuition for the drift of price processes under the Stock numeraire.
I find it intuitive that the martingale measure under the Money Market numeraire induces ...
- 5,998
9
votes
2
answers
1k
views
What is the numeraire for the real world measure $\mathbb{P}$?
We know the numeraires for the forward measure, the risk-neutral measure, etc.
What is the numeraire for the real world measure $\mathbb{P}$?
- 515
9
votes
2
answers
723
views
Change of numéraire for two risky assets without bank account (Margrabe’s formula?)
I am considering two risky assets following the usual correlated GBM given by
$$\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}...
- 609
8
votes
3
answers
2k
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-...
- 651
8
votes
2
answers
4k
views
T-Forward Price on risk-neutral measure
i have and question concerning the T-forward price definition on the Robert J.Elliot's book : Mathematics of Financial Markets. On his chapter 9, definition 9.1.3 p.249. He give the formula without ...
- 423
7
votes
3
answers
803
views
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 ...
- 83
7
votes
1
answer
5k
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 ...
- 117
6
votes
2
answers
957
views
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 ...
- 5,998
6
votes
2
answers
796
views
Option with payoff $K^2/S^2$
Given the dynamics of the risky asset ( with dividend $q$ ),
$$
\frac{dS_t}{S_t}=(\mu-q)dt + \sigma dW_t^P
$$
Consider a european option with payoff,
$$
P_0(S) =
\begin{cases}
1, & \text{...
- 514
6
votes
1
answer
2k
views
Numéraire -- couldn't understand the wiki explanation
I'm trying to understand Numéraire concept so am reading the wiki page:
I couldn't understand the last formula's 2nd equation:
$$ E_{Q}\left[\left.\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}\frac{S(T)}{N(T)}\...
- 2,211
6
votes
1
answer
440
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
$$N(t) = \frac{1}{P(0,t)} e^{H(t)X(...
- 193
5
votes
2
answers
1k
views
Libor Market Model: numeraire change
I am currently studying the Libor forward market model, and although I get the mechanics behind the main arguments, I still do not have an intuitive idea of what's exactly the objective behind ...
- 463
5
votes
1
answer
301
views
Is first order stochastic dominance conserved under change of measure?
As the title states, my question is whether first order stochastic dominance is conserved under change of measure, for instance from the $\mathbb{P}$ measure to $\mathbb{Q}$ measure and change of ...
user34971
5
votes
3
answers
170
views
Volatility of Exchange Option
I got a question and its partial solution, and have some doubts about the volatility of its geometric Brownian motion process:
Question:
How would you price an exchange call option that pays $max(S_{...
- 627
5
votes
1
answer
388
views
Drift term in rough volatility models
I'm studying rough volatility papers and was wondering, why the drift term is always missing.
See for example the paper Pricing under rough volatility by Bayer, Friz, Gatheral.
On page 2, the ...
- 268
5
votes
1
answer
603
views
On Girsanov Theorem to switch from Risk-Neutral to Stock Numeraire
Summary: long-story cut short, the question is asking for what types of functions $f(.)$, the Cameron-Martin-Girsanov theorem can be used as follows:
$$ \mathbb{E}^{\mathbb{P}^2}[f(W_t)]=\mathbb{E}^{\...
- 5,998
5
votes
1
answer
317
views
Change of measure's impact on parameter value
This is a follow-up question on Price of a prepayment-based claim.
Consider a zero-coupon bond of maturity $T$ with price $P_0$ for which the borrower can reimburse the principal $N$ at any time $\...
- 7,989
5
votes
1
answer
1k
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 ...
- 75
5
votes
1
answer
289
views
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 ...
- 1,507
5
votes
1
answer
690
views
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}{...
- 282
4
votes
4
answers
464
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 ...
- 153
4
votes
3
answers
1k
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 ...
- 245
4
votes
3
answers
1k
views
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{...
- 282
4
votes
1
answer
220
views
Pricing an "equity protection" derivative: a practical example
This is the derivative security (its underlying index is the S&P 500):
time to expiry $=4.8$Y;
payoff calculation (0): on the expiry date, give a look at S&P 500 and let its price to be $S_{T}...
- 2,111
4
votes
2
answers
2k
views
Is the money market account (MMA) numeraire and the forward measure equivalent?
Suppose we have a risk-neutral measure $\tilde{\mathbb{P}}$. The money market account is given as $M(t) = e^{\int^t_0 R(s) ds}$, while the price of the zero-coupon bond at time $t$ that matures at $T$ ...
- 245
4
votes
1
answer
239
views
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 ...
- 5,998
4
votes
1
answer
210
views
Girsanov Transform and Likelihood Process Domestic to Foreign
Working two exercises relating to $Q^d$ and $Q^f$. I'm comfortable working with transforms and likelihood processes on a risky asset between $Q$ and $Q^s$, and also on an exchange rate $X$ between $Q$ ...
- 344
4
votes
1
answer
351
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. ...
- 427
4
votes
1
answer
489
views
Equality under T-forward measure for convexity adjustment
I've been working with the convexity adjustment for interest rates that arises when changing from one measure $Q_{T_p}$ with a numéraire $N_p=P(t,T_p)$ to a measure $Q_{T_e}$ with a numéraire $N_e=P(t,...
- 282
3
votes
3
answers
3k
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 $...
user40884
3
votes
1
answer
982
views
Equivalent Martingale Measure(EMM) of Inverse of Stock Price
I met this question says how to price a vanilla call option $C(St,t,T,K) = \frac{1}{S_T}$which pays the inverse of a stock $V_{t} = \frac{1}{S_{t}}$ at maturity if the stock price follows a geometric ...
- 33
3
votes
1
answer
652
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=(...
- 5,998
3
votes
1
answer
635
views
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 ...
- 402
3
votes
1
answer
647
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 ...
- 692
3
votes
1
answer
168
views
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 ...
- 83
3
votes
1
answer
338
views
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 ...
3
votes
1
answer
2k
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 ...
- 89
3
votes
1
answer
220
views
Vanila Option self financing under Stock as numeraire
I am trying to see how the vanilla call option can be seen as self financing using the Stock as the numeraire. The case with the Bond as numeraire is quite simple and can be found in Wilmott's FAQ ...
- 402