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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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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 $Q-$ ...
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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 + \...
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American Perpetual Put Option

I want to compute the expected payoff of a (classical) perpetual American put option in the Black-Scholes-Merton (BSM) framework with an optimal strategy of exercising the option at time $\tau=\inf\{t:...
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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 ...
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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 ...
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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 ...
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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 ...
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Discrete term structure models - generalized procedure to ensure positive probabilities across multiple measures

Question: Is there a generalized procedure for building a discrete (e.g. binomial) term structure model with risk-neutral branching probabilities that ensure positive probabilities under alternative ...
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Dividend paying asset, why can't be taken as numéraire?

Why when considering numéraires, one cannot use a dividend paying asset to define a risk neutral measure? Here's where I got my question : (Shreve - Stochastic Calculus For Finance II)
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An arbitrage strategy involving forward contracts to show that LIBOR rates are martingales

I note $L_{t}^{[T_s, T_e]}$ the forward rate at time $t$ for the period $[T_s, T_e]$. Recall it is the strike making equal to $0$ the value at time $t$ of a forward contract for the period $[T_s, T_e]$...
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Expectation of expression with two currencies under forward measure

I'm trying to calculate the expected value, at time $0$, of a cashflow paid at time $T$, resetting at time $t$. The coupon is of the form: $V_0=\mathbb{E}^{T_2}\left[\frac{A_t^y(T_1,T_2)}{B_t^x(T_1,...
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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 ...
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Equivalence of formulations of Radon Nikodym derivative

Let $N$ be a numeraire associated with the probability measure $Q^N$ and $U$ be a numeraire associated with the probability measure $Q^U$, both of which are equivalent to the physical probability ...
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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 ...
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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 ...
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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, ...
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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 ...
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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(-\...
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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 ...
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How to determine exchange rate dynamics in currency derivatives

I need some guidance regarding exchange rate dynamics in currency derivatives. Following three dynamics are defined below, $\frac{dS(t)}{S(t)}=\alpha dt+\sigma dW(t)$ ; the stock dynamics in the ...
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In a multi-curve context which numéraire is used to change to the payment probability of a forward asset X paid at time T?

Should it be the coupon associated to the funding curve of the asset? Thanks.
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Hull Martingales and measures problem 27.16 7e?

Here's a question from Hull's Options Futures and Other derivatives which I'd appreciate if someone helped me to clarify. The question is from the chapter "Martingales and Measures" Suppose that the ...
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I am looking for help to derive this formula from Brigo & Mercurio

I think my use of numeraire change is incorrect and for sure my understanding is incomplete. $\frac {dQ2}{dQ1}=\frac {Pt(0,T2)P0(0,T1)}{P0(0,T2)Pt(0,T1)} = \frac {1+DeltaF_1(t)}{1+DeltaF_2(t)}$ Then ...
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Change of numeraire : quotient

Let's consider $X_1(t)$ a geometric brownian motion (with variable volatility) and $X_2(t)$ a Brownian bridge : $dX_1(t) = \mu X_1(t) dt + \sigma_1(t) X_1(t) dW(t)$ $dX_2(t) = \frac{b - X_2(t)}{T - t} ...
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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,...
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Stock Price as Numeraire, Two Stocks & One Money Market Account

We have two uncorrelated Stock price processes and the classical Money-Market (MM) account. Under the MM Numeraire, both stocks are Martingales when discounted by the MM, as usual. Question: I would ...
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Risk neutral measure & change in numeraire

There are two questions about risk neutral and change in numeraire I am not so sure if my answer is correct. Question 01: Risk neutral Let says I have 2 risky asset A and B. Each has stochastics ...
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