# 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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### 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,...
88 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 ...
35 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....
1 vote
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### 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 ...
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
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### 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 ...
113 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 ...
431 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 ...
126 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 ...
165 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 ...
762 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 ...
1 vote
223 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
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### 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
64 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 ...
168 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
254 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
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