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 ...
- 7,301
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 ...
- 7,301
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 ...
- 7,301
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 ...
- 7,301
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 ...
- 7,301
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,\...
- 7,301
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$ ...
- 7,301
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 ...
- 5,552
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
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