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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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 $Q-$ ...
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}...
0 votes
1 answer
128 views

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,...
1 vote
1 answer
109 views

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 $...
1 vote
1 answer
440 views

Effect on variance of change of measure

My current understanding: (a) changing the probability measure of a diffusion process does not change the variance. (b) for a general stochastic process the variance may change. Please confirm whether ...
2 votes
1 answer
2k views

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 ...
2 votes
0 answers
98 views

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 ...
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 ...
0 votes
0 answers
38 views

Money account discounted libor rate is it a martingale under risk neutral measure?

I see that Libor $L(t,S,T)$ is a martingale under $T-$forward measure. Where we used argument that zero-coupon bonds are martingales under $T$-forward measure, as zero-coupon bond is a traded security....
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 ...
1 vote
0 answers
104 views

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 ...
0 votes
0 answers
41 views

Calculate the amount of shares of a deposit without converting to numeraire

Let F a mutual fund with two assets A and B. Initially, F contains 1 unit of A, 1 unit of B, and there is 1 share allocated to Alice. At a later time, Bob deposits 2 units of A into F. How can I ...
1 vote
0 answers
104 views

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 ...
0 votes
1 answer
118 views

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 ...
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 ...
2 votes
1 answer
460 views

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 ...
3 votes
0 answers
204 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 + \...
1 vote
1 answer
258 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 (...
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 ...
2 votes
1 answer
138 views

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 ...
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 ...
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 ...
1 vote
1 answer
257 views

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-...
1 vote
0 answers
30 views

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, ...
1 vote
1 answer
65 views

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 ...
2 votes
0 answers
173 views

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 ...
1 vote
0 answers
289 views

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 ...
1 vote
2 answers
182 views

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, ...
1 vote
0 answers
144 views

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(-\...
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 ...
2 votes
0 answers
82 views

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 ...
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 ...
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{...
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 ...
0 votes
2 answers
554 views

Show a process is Martingale

$$Z(t)=(\frac{S(t)}{H})^p$$where $S$ has a standard Black-scholes Dynamics for a stock, $H$ is a postive constant and $p =1 - \frac{2r}{\sigma^2}$. How can I show that $Z(t)/Z(0)$ is a postive Q-...
1 vote
1 answer
215 views

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 $...
2 votes
0 answers
79 views

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 ...
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-...
2 votes
1 answer
244 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}...
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 ...
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=(...
1 vote
0 answers
44 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 vote
1 answer
73 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
1 answer
930 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 $S_t^2$ ...
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}^{\...
2 votes
0 answers
196 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
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 $...
2 votes
1 answer
2k 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-...