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.

Filter by
Sorted by
Tagged with
23
votes
0answers
399 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 $Q-$ ...
13
votes
1answer
10k views

Baye's 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 ...
12
votes
4answers
3k views

Understanding $N(d_1)$ and how to use the stock itself as the numeraire?

Assume the stock price follows a geometric Brownian motion Then in Black-Scholes pricing model, $N(d_2)$ is the risk-neutral probability that the option expires in-the-money. However, it is said that $...
11
votes
2answers
4k views

How to use the stock as a numeraire to price a derivative with payoff of the form $(S_T f(S_T))^+$?

I have $\frac{dS_t}{S_t} = rdt + \sigma dW_t$ as usual under the money-market numéraire and I need to price options with payoffs $$(S_T f(S_T))^+$$ How do I express the stock dynamics using the ...
11
votes
1answer
512 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 ...
11
votes
1answer
1k views

How to use a change of numeraire to price this option?

I recently asked this question regarding how to price an option with payoff: $$\text{Payoff}_T = (A_TR_T - A_T \lambda)^+ $$ Let's assume for generality that $A_t$ and $R_t$ are GMB's: $$dA_t = \...
10
votes
2answers
694 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 ...
9
votes
3answers
622 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 ...
9
votes
0answers
227 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}...
8
votes
2answers
651 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 $\mathbb{P}$?
8
votes
2answers
3k views

T-Forward Price on risk-neutral measure

i have and question concerning the T-forward price definition on the Robert J.Elliot's book : Mathematics of Financial Markets. On his chapter 9, definition 9.1.3 p.249. He give the formula without ...
7
votes
2answers
345 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 ...
7
votes
1answer
4k views

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 ...
6
votes
3answers
654 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-...
6
votes
2answers
552 views

Option with payoff $K^2/S^2$

Given the dynamics of the risky asset ( with dividend $q$ ), $$ \frac{dS_t}{S_t}=(\mu-q)dt + \sigma dW_t^P $$ Consider a european option with payoff, $$ P_0(S) = \begin{cases} 1, & \text{...
6
votes
1answer
2k views

Numéraire — couldn't understand the wiki explanation

I'm trying to understand Numéraire concept so am reading the wiki page: I couldn't understand the last formula's 2nd equation: $$ E_{Q}\left[\left.\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}\frac{S(T)}{N(T)}\...
6
votes
1answer
166 views

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(...
5
votes
2answers
1k views

Libor Market Model: numeraire change

I am currently studying the Libor forward market model, and although I get the mechanics behind the main arguments, I still do not have an intuitive idea of what's exactly the objective behind ...
5
votes
3answers
135 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_{...
5
votes
1answer
141 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 ...
5
votes
1answer
239 views

Change of measure's impact on parameter value

This is a follow-up question on Price of a prepayment-based claim. Consider a zero-coupon bond of maturity $T$ with price $P_0$ for which the borrower can reimburse the principal $N$ at any time $\...
5
votes
1answer
682 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 ...
5
votes
1answer
228 views

Change of numéraire for non-Normal distributions

I'm looking for a resource, a book or an article, that describes the framework of change of numéraire in a broader context than just Brownian motions or Normal distributions. I'm only really ...
5
votes
1answer
520 views

Martingale measure result application for interest rates under T-forward measure?

I've got a question about the way the equivalent martingale measure result is used for pricing derivatives. Hull states the result as the next equality: \begin{align*} f_o = g_0 E^{g}\big(\frac{f_T}{...
4
votes
4answers
395 views

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 ...
4
votes
3answers
972 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{...
4
votes
1answer
205 views

Pricing an “equity protection” derivative: a practical example

This is the derivative security (its underlying index is the S&P 500): time to expiry $=4.8$Y; payoff calculation (0): on the expiry date, give a look at S&P 500 and let its price to be $S_{T}...
4
votes
2answers
1k views

Is the money market account (MMA) numeraire and the forward measure equivalent?

Suppose we have a risk-neutral measure $\tilde{\mathbb{P}}$. The money market account is given as $M(t) = e^{\int^t_0 R(s) ds}$, while the price of the zero-coupon bond at time $t$ that matures at $T$ ...
4
votes
1answer
205 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}^{\...
4
votes
1answer
174 views

Girsanov Transform and Likelihood Process Domestic to Foreign

Working two exercises relating to $Q^d$ and $Q^f$. I'm comfortable working with transforms and likelihood processes on a risky asset between $Q$ and $Q^s$, and also on an exchange rate $X$ between $Q$ ...
4
votes
1answer
171 views

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. ...
4
votes
1answer
344 views

Equality under T-forward measure for convexity adjustment

I've been working with the convexity adjustment for interest rates that arises when changing from one measure $Q_{T_p}$ with a numéraire $N_p=P(t,T_p)$ to a measure $Q_{T_e}$ with a numéraire $N_e=P(t,...
3
votes
3answers
1k 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 $...
3
votes
1answer
704 views

Equivalent Martingale Measure(EMM) of Inverse of Stock Price

I met this question says how to price a vanilla call option $C(St,t,T,K) = \frac{1}{S_T}$which pays the inverse of a stock $V_{t} = \frac{1}{S_{t}}$ at maturity if the stock price follows a geometric ...
3
votes
1answer
456 views

Deriving Black Scholes PDE under stock as a numeraire

There are many ways to derive the Black Scholes PDE. The Martingale way would be to demand the option price is driftless according to particular measures. Below I derive the correct PDE using the bank ...
3
votes
1answer
440 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 ...
3
votes
1answer
250 views

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 ...
3
votes
3answers
753 views

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 ...
3
votes
1answer
1k views

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 ...
3
votes
1answer
184 views

Vanila Option self financing under Stock as numeraire

I am trying to see how the vanilla call option can be seen as self financing using the Stock as the numeraire. The case with the Bond as numeraire is quite simple and can be found in Wilmott's FAQ ...
3
votes
0answers
89 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 + \...
3
votes
0answers
88 views

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:...
3
votes
1answer
305 views

Convexity adjustment when payment if after interest natural term?

I've been working with a convexity adjustment for an interest rate payoff and the next question came to me: The usual problem that gives rise to the convexity adjustment I'm referring to is as ...
2
votes
1answer
422 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-...
2
votes
1answer
171 views

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 ...