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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54 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 $...
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Caplet “in arrears” pricing formula

In this post here it is shown that 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 $P(t,t_2)$ ...
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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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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}...
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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-...
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1answer
175 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}...
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539 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 ...
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480 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 ...
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1answer
174 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=(...
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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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Proof of existence of one only martingale measure

I know that: Hypothesis 1 (Girsanov Theorem) Let $\theta=\begin{Bmatrix} \theta_t \end{Bmatrix}_{t\in [0,T]}$ be a square-integrable and $\Im_t$-adapted process such that $\mathbb{E}[e^{\frac{1}{2}\...
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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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66 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 ...
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1answer
259 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 $S_t^2$ ...
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159 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}^{\...
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163 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 ...
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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 $...
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339 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-...
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Why is the numeraire in the LGM model tradeable?

I'm trying to understand the LGM model, which Hagan defines as follows. The state variable $X$ evolves according to $$dX(t) = \alpha(t) dW^N(t)$$ wrt the numeraire $$N(t) = \frac{1}{P(0,t)} e^{H(t)X(...
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Using a Constant as a Numeraire

Please provide steps to justify the below. 1) Can we use a constant as a numeraire? Related Question: Scaling Stock Price and Strike etc. by a Constant The rest of standard Geometric Brownian ...
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Change of numeraire and reference asset

Learning about change of numeraire, and came across this statement: The price of any asset divided by a reference asset (called numeraire) is a martingale (no drift) under the measure associated ...
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Why can only non-dividend paying assets serve as numeraire?

In Kerry Back, A Course in Derivative Securities, Sect. 1.4 (page 29), the author stated the FTAP in the following form (in boldface): If there are no arbitrage opportunities, then for each (non-...
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393 views

Bond SDE under its own forward measure

I am trying to write the SDE for a forward bond, $dP(t,T_1,T_2)$, under the $T_1$-Forward measure, $Q_{T_1}$. I can easily do this by: Writing the equation of $dP(t,T_1)$ and $dP(t,T_2)$ under the ...
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669 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 ...
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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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131 views

Volatility of Exchange Option

I got a question and its partial solution, and have some doubts about the volatility of its geometric Brownian motion process: Question: How would you price an exchange call option that pays $max(S_{...
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Change of numeraire in exchange options with random interest rate

At time $t$, the market offers a (possibly random) bounded interest rate $r_t$ and two assets whose prices are given by \begin{align*} {{\rm d S^{(1)}_t \over S^{(1)}_t}} &= b^{(1)}_t d t + \...
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287 views

How can I use the Radon-Nikodym theorem to show that forward measure is indeed measure?

The following statements are taken from the Wikipedia page for forward measure. Let $$B(T)=\exp \left(\int _{0}^{T}r(u)\,du\right)$$ be the bank account or money market account numeraire and $...
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1answer
138 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 (...
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104 views

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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1answer
80 views

How to determine the no arbitrage price of following claim? (change of numeraire)

How do I determine the no arbitrage price for claims such as $min(S_1(T),S_2(T))$ or $max(S_1(T),S_2(T))$? We can consider a standard Black Scholes model. Hence $S_i(T)=S_i(t)e^{(r-\sigma_i^2/2)(T-t)+\...
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1answer
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Asian Options-Change of Numeraire

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Show that ...
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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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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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Proof standard Brownian Motion under change of measure

Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
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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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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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Drift term in rough volatility models

I'm studying rough volatility papers and was wondering, why the drift term is always missing. See for example the paper Pricing under rough volatility by Bayer, Friz, Gatheral. On page 2, the ...
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The relation between exchange rate SDE and respective interest rates

The exchange rate between a domestic currency money market and a foreign currency money market can be expressed as $$ dQ(t) = (r_d - r_f)Q(t)dt + \sigma Q(t)d\tilde{W}(t) $$ where $r_d$ is the ...
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Pricing a call option with pay-off function max{$S_T - S_{T/2}, 0$}

Pricing a call option with payoff function $C=\max\{S_T - S_{T/2}, 0\}$, where $S_T$ is geometric brownian motion. I appreciate any help! Please close this question if this is a duplicated question. ...
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644 views

Change of numeraire between T-forward and Bank Account

I follow a course, and get to the point that one bond price discounted by another one is a martingale: $$ \frac{P(t,T_0)}{P(t,T_1)} - \text{ is a } \mathbb{Q}^{T_1} \text{ martingale } $$ I can not ...
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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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Change of numeraire in options with currency exchange features

FV of an EUR denominated option under "COP" risk measure is given by: $$V_t^{COP} = D^{COP} \mathbb{E}_t^{COP} \left[X_T(S_T -K)^+\right]$$ where $X_T$ is the exchange rate COP/EUR. Pricing the ...
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612 views

What is the numeraire for the real world measure $\mathbb{P}$?

We know the numeraires for the forward measure, the risk-neutral measure, etc. What is the numeraire for the real world measure $\mathbb{P}$?