12 votes
Accepted

How do we determine the "correct measure"?

Recall that any traded asset divided by a numéraire is a martingale under the measure associated to that numéraire. For the 3 interest rates you mention, the natural measure (namely the one that makes ...
11 votes
Accepted

Intuition for Stock Price Numeraire Drift

As a general principle, I would be wary of economic or financial interpretations of change of measure techniques. Changing numéraires is merely a mathematical tool to ease pricing, see for example the ...
11 votes
Accepted

Does numeraire have to be a tradable asset

This is an interesting question that I have asked myself. Below is my take. Let us consider an economy $(\Omega,\mathcal{F},P)$ equipped with a filtration $(\mathcal{F})_{t \geq 0}$ consisting on a ...
11 votes
Accepted

Numeraire correlated to the traded asset

As @ilovevolatility explains, the seminal reference for this matter is Geman, El Karoui & Rochet (1995). We assume none of the assets are dividend paying, and they are strictly positive. There are ...
8 votes
Accepted

Caplet "in arrears" pricing formula

Let $P(t, T)$ be the price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value. Consider the pricing of the caplet with payoff $(L(t_1; t_1, t_2)-K)^+$ at time $t_1$, where $0<...
  • 20.5k
8 votes

Intuition for Stock Price Numeraire Drift

The drift is the expectation of the return over an infinitesimal interval. Let $Q$ be the risk-neutral measure and $Q^S$ be measure associated with the stock price numeraire defined by \begin{align*} \...
  • 20.5k
7 votes

Using a Constant as a Numeraire

Either $r=0$ in which $B_t$ is constant and is a valid numeraire (as is any multiple of it.) or $ r \neq 0$ in which case an asset of constant value would give an arbitrage since we could take $$ B_t ...
  • 6,763
6 votes
Accepted

If there is a $T$-forward measure and a risk neutral measure, then markets are not complete?

The market is complete iff there is a unique risk-neutral measure: when every contingent claim is attainable, its unique no arbitrage price is the cost of the replicating portfolio. In the case of an ...
  • 2,400
6 votes
Accepted

Is first order stochastic dominance conserved under change of measure?

Consider a coin independently tossed 10 times. Assume under the measure $P$, $Pr(H)$ > 0.5 but not equal to 1. Let a risk neutral person be iteratively given the gamble between getting atleast $n$ ...
  • 1,662
6 votes
Accepted

Pricing an Option with payoff $\left(1-\frac{K}{S_t}\right)^{+}$

$\frac{1}{S_t}$ is log-normal If $S_t$ is a geometric Brownian motion, so is $\frac{1}{S_t}$ and indeed any power $S_t^\alpha$. Simply use Itô's Lemma and set $f(t,x)=\frac{1}{x}$, \begin{align*} \...
  • 14k
6 votes

Using a Constant as a Numeraire

A Numeraire must be a tradeable asset. If you can find a constant tradeable asset, then yes a constant can be used as a numeraire.
  • 1,329
6 votes
Accepted

Change of numeraire and reference asset

Proving the existence of a risk neutral measure is the difficult part. Once its existence is established, a simple calculation of conditional expectations allows to go from a numeraire to any other. ...
  • 3,856
6 votes

Intuition for Stock Price Numeraire Drift

I have a take on the intuition part of the question. Isn't it a simple consequence of Jensen's inequality? Thus, assuming $r=0$ for simplicity, we have in the money market measure: $E(S_T)=S_t$, ...
  • 14.3k
5 votes
Accepted

Change of Numeraire formula

We work on a probability space $(\Omega,\mathcal{N},\mathfrak{F})$ with filtration $(\mathfrak{F}_t)_{0\leq t\leq T}$ and $\mathfrak{F}_T:=\mathfrak{F}$. Let $\xi$ be a $\mathfrak{F}_T$-mesurable ...
5 votes
Accepted

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

The rolling bond $R(t)$ as defined in your question is a valid numéraire. Indeed, this bond can synthetized with the following iterative trading strategy in basic assets: At any time $T_i\in\{T_0,\...
5 votes

Why does the diffusion term remain the same when we change pricing measure?

It has to do with the Girsanov theorem that relates the equivalent measures $\mathbb Q$ and $\mathbb P\,.$ To make intuitively clear what happens I like to give the following "baby Girsanov" ...
  • 1,543
4 votes

Intuition for Stock Price Numeraire Drift

When you try and discount everything by the stock, every price process now has extra gamma PnL naturally just due to the presence of the stochastic discount factor (due to it's quadratic variation). ...
  • 1,662
4 votes

How do we determine the "correct measure"?

I would like to add to @DaneelOlivaw answer. Your question: "How does one go about finding the right measure for a product?" Answer: One should choose any measure that will make it easy and ...
  • 5,316
4 votes
Accepted

Why can only non-dividend paying assets serve as numeraire?

Well, consider using $S_t$ as the numeraire and let the asset be the reinvested stock $S_te^{qt}$. Then this ratio equals $e^{qt}$ so can never be a martingale.
  • 14.3k
4 votes
Accepted

Bond SDE under its own forward measure

We consider a financial market with three assets: a zero-coupon bond of maturity $T_1$, a second one with maturity $T_2$ and the money market account $B_t$. Assuming the market's risk-free rate $r_t$ ...
4 votes

Does numeraire have to be a tradable asset

An obvious example is using the maturity $T$ zero coupon as numeraire, and a European option with premium paid at time $T$ hedged with maturity $T$ forward contracts. You do not need to trade the zero ...
4 votes
Accepted

Why is the numeraire in the LGM model tradeable?

The confusion is that you think that we define the numeraire as this exponential function... It is not the case. We give the numeraire properties to $N$, then we model it. Similar to any other model. ...
  • 753
4 votes
Accepted

Power Options & Forwards on Stock Squared

Consider a financial market with a filtered probability space $\left(\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb P\right)$ satisfying usual conditions equipped with a stock price process $S_t$. Suppose ...
  • 14k
4 votes

Why does the diffusion term remain the same when we change pricing measure?

Edit: the below answer contains some mistakes and needs some further work, as per the comments underneath. Intro: I think intuition is super important for this one, so my answer below focuses on the ...
  • 5,316
4 votes
Accepted

Why does the diffusion term remain the same when we change pricing measure?

Extract from my answer about what the VIX measures (more details on the notation and the conventions I am using can be found in the preceding sections from that answer): About changing the measure (...
3 votes
Accepted

On Girsanov Theorem to switch from Risk-Neutral to Stock Numeraire

(I might not be answering your question, but I feel this clarification is needed.) A random variable $X$ of $(\Omega, \mathcal{F})$ is a $\mathcal{F}$-measurable function $X : \Omega → \mathbf{R}$. So,...
  • 5,028
3 votes
Accepted

Forward starting zero-coupon bonds

$Z(t_0,t_1,t_2)$ is the $t_1$-forward price of the ZC bond with maturity $t_2$, as of $t_0$. We have: $$ Z(t_0,t_1,t_2) = E_{t_0}^{t_1}[Z(t_1,t_2)]\not= Z(t_1,t_2).$$ With a not-trivially stochastic ...
  • 5,028
3 votes

Caplet "in arrears" pricing formula

Case I Let us consider a derivative with a payoff $H(L(T_{f},T_{S},T_{E}))$ which is paid at time $T_{p}$. Note that: $T_{f}$ - LIBOR fixing date; $T_{S}$ - LIBOR start date; $T_{E}$ - LIBOR maturity ...
  • 83
3 votes
Accepted

Where does the term $\gamma$ come from when moving from measure $\mathbb Q^{N}$ to $\mathbb Q^{M}$?

From your assumption that \begin{align*} dX(t)\cdot d\frac{M(t)}{N(t)} &= \frac{M(t)}{N(t)}X(t)\gamma(t)dt,\\ dX(t) &= X(t)\big(\mu(t)dt+\sigma(t)dW(t)\big), \end{align*} under $\mathbb Q^{M}$,...
  • 20.5k
2 votes

Using a Constant as a Numeraire

Actually, all investments, retirement accounts, mutual fund accounts, utility bills, supermarket price listings are reported or stated in the Constant Numeraire, which may also be called Dollar-kept-...
  • 948

Only top scored, non community-wiki answers of a minimum length are eligible