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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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 $...
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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 ...
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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
...
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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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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 $\...
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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:
$$dA_t = \...
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T-Forward Price on risk-neutral measure
i have and question concerning the T-forward price definition on the Robert J.Elliot's book : Mathematics of Financial Markets. On his chapter 9, definition 9.1.3 p.249. He give the formula without ...
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Numéraire -- couldn't understand the wiki explanation
I'm trying to understand Numéraire concept so am reading the wiki page:
I couldn't understand the last formula's 2nd equation:
$$ E_{Q}\left[\left.\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}\frac{S(T)}{N(T)}\...
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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,
$$
P_0(S) =
\begin{cases}
1, & \text{...
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Caplet "in arrears" pricing formula
The forward Libor rate $L(t,t_1,t_2)$, with $0 \leq t \leq t_1$, must be a martingale under the T-forward measure associated with the zero coupon bond $P(t,t_2)$ that matures at time $t_2$.
Pricing a ...
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Change of measure between T-forward and T*-forward contract?
I am trying to prove the need of a convexity adjustment to a forward rate by calculating the next expectation:
\begin{align*}
P(t_0, T_s)E^{T_s}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big).
\end{...
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Using a Constant as a Numeraire
Please provide steps to justify the below.
1) Can we use a constant as a numeraire?
Related Question: Scaling Stock Price and Strike etc. by a Constant
The rest of standard Geometric Brownian ...
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The relation between exchange rate SDE and respective interest rates
The exchange rate between a domestic currency money market and a foreign currency money market can be expressed as
$$
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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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$ ...
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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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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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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 ...
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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 ...
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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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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)} $$
I am able to derive this Radon-...
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question regarding relation between expectations on different measures
I am a beginner to the theory of stochastic calculus and measure change.
I have derived an equation related to expectations on different measures.
I wanted some expert opinion on whether this is true ...
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RFR boostrapping using RFR OIS: Is convexity adjustment technically necessary?
For single-curve RFR bootstrapping, such as a SOFR-based discounting curve bootstrapped strictly using SOFR fixed-float OIS, I am trying to understand if convexity adjustments are technically ...
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Asian Options-Change of Numeraire
Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model
without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$).
Show that ...
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How can I use the Radon-Nikodym theorem to show that forward measure is indeed measure?
The following statements are taken from the Wikipedia page for forward measure.
Let
$$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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