16 votes

What does instantaneous forward mean?

1. Observable instruments, spot rates, and forward rates First remember that something observable means that you can observe/find the rate in the market by looking at traded rate instruments or ...
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12 votes
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What does instantaneous forward mean?

Given a forward rate, for example: $ F(t, T, T+\delta)$ The instantaneous forward rate $f(t,T)$ fixed in $t$ is the limit when $\delta \rightarrow 0$ of your forward rate. If the relation between ...
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9 votes
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Calculating instantaneous forward rate from zero-coupon yield curve

Your overall approach is correct. However to my knowledge it is formally more appealing to work with a parameterized and smoothed yield curve. Basically one assumes that the yield curve can be ...
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  • 3,337
8 votes
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Why are multiple custom curves (swap) built for one desk?

Chapter 1: Goldilocks is ousted by the bears Once upon a time, the banks used a fixing called LIBOR as a measure of the risk-free interest rate. Then the big hairy crisis came along and ate all our ...
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  • 3,589
6 votes
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FX Forward rate agreement valuation in quantlib

You are not giving the constructor a discountCurve. The constructor is: ...
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  • 5,285
5 votes
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U.S. Rate Hike Prediction

The CME' Fed Fund Futures are what you are looking for. http://www.cmegroup.com/trading/interest-rates/stir/30-day-federal-fund.html On settlement day they settle at the average overnight rate set ...
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  • 739
5 votes
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why $f(t,u) \neq E_t^Q [r(u)]$ when $r$ is random?

Your equations are flawed. Also there is no expectation if the process $\{r_s\}$ is deterministic. The correct reasoning is, assuming $\{r_s\}$ is stochastic: $$ f(t,u)=-\frac{d}{du}\ln P(t,u)=-\...
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5 votes

How to prove martingality of forward rate under T-forward measure

By definition, $$Fo(t,T)=E^T[S_T|F_t]$$ Note that expectation is taken under $T$-forward measure. Now, evaluating at $s<T$: $$E^T[Fo(t,T)|F_s] = E^T[E^T[S_T|F_t]|F_s] = E^T[S_T|F_s] = Fo(s,T)$$ (...
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5 votes

Instantaneous forward rate within the HJM framework

This is known as the classical Leibniz rule. The link sends to Wikipedia, where you can find a proof. It allows to differentiate under the integral sign. A general statement of the formula is: $$\text{...
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5 votes
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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

The rolling bond $R(t)$ as defined in your question is a valid numéraire. Indeed, this bond can synthetized with the following iterative trading strategy in basic assets: At any time $T_i\in\{T_0,\...
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4 votes
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Practical implementation of Libor Market Model

For a swap, we have a sequence of re-setting and payment dates. The # of forward rates corresponding to the # of payment dates. For example, let us assume that we have $n$ payment dates $t_1, \ldots, ...
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  • 20.4k
4 votes
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Formula for the forward rates?

The price of the zero-coupon bond is the discount factor for this maturity. In the world of exponential compounding formulas are of the form $\exp(\sum \cdots)$. With a replication argument if we want ...
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  • 13.2k
4 votes

forward vs spot simply-compounded spot interest rate

The flaw is $L(T,S)$ is a future spot rate that is determined at time $T>t$ and unknown at present. It is correct that $$F(t,T,S)=\frac{1}{S-T}\left[\frac{P(t,T)}{P(t,S)}-1\right] \iff P(t,S)(S-...
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  • 3,250
4 votes

Mid-curve swaption

You can only infer forward vol by pairing a mid-curve option with a spot option. It's easier to go through an example (I'll use 5y x 5y vol since I have the sketch below handy...) One decomposition of ...
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  • 10.9k
4 votes
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continuously compound forward rate formula

In the simple case, you have as per first equation on your last slide: $\frac{P(t,T_0)}{P(t,T)}=1+\delta F(t,T_0, T)$ The continuous time equivalent, assuming constant piecewise rate, as per your ...
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4 votes
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Approximation of Forward Rates in discrete time

You may show it as follows: \begin{align*} f_{t,T}&= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{T-t}}-1\\ &=e^{\frac{1}{T-t} \left[\ln (1+r_T)^T - \ln (1+r_t)^t \right]} -1\\ &\...
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  • 20.4k
4 votes

How to calculate one-year forward one-year rate?

Let $\{r_t\}_{t>0}$ be the spotrates and $f_{t,T}$ be the forward rate from time $t$ to $T$ for $t<T$. Then the general formula to compute $f_{t,T}$ is $$ (1+r_T)^T=(1+r_t)^t(1+f_{t,T})^{T-t} $$...
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  • 1,585
4 votes
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How to prove martingality of forward rate under T-forward measure

For the instantaneous forward, please see the last page of this note: T-Forward Measure by Fabrice Douglas Rouah (http://www.frouah.com/finance%20notes/The%20T-Forward%20Measure.pdf). For the simple ...
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4 votes

How to prove martingality of forward rate under T-forward measure

The answer by @Prabhnoor Duggal is correct. Here, I would like to further expand to make it more streamlined (see also Section 2.5 of the book Interest Rate Models - Theory and Practice). Let $Q$ and $...
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  • 20.4k
4 votes
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Half of the bid-ask spread as transaction cost

The idea of assuming that the transaction cost is one half of the bid-offer spread comes from several assumptions: the positions are marked-to-market at mid; you can actually execute at bid or ask (...
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4 votes

Which measure is used to price a swap?

I want to propose a different answer here. I think mathematical expectation (under any measure) is not used in valuing an interest swap. Years ago I used to explain swaps to beginners by speaking in ...
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  • 9,347
4 votes
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Pricing a Forward Rate Agreement using QuantLib Python

For a 3x6 FRA, you probably want to write something like: ...
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3 votes

FX Forward pricing with correlation between FX and Zero-Cupon

We consider the expectation \begin{align*} E^{Q_d^{t_f}} \Big(P_d(t_f, T) X_{t_f} \mid \mathcal{F}_t \Big), \end{align*} where $Q_d^{t_f}$ is the $t_f$-forward measure, and $P_d(t_f, T)$ is the ...
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  • 20.4k
3 votes
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forward space vs zero space in finance jargon

In interest rate land you can look at the yield curve in 3 ways: par space (a chart of the par swap rates of different maturities) , zero space (the zero coupon swap rates) and forward space (usually ...
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  • 13.7k
3 votes
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Why can a swap option be regarded as a type of Bond option?

Consider a payer swaption with maturity $T_0$ and strike $K$. Here the strike $K$ is the fixed rate paid on the fixed leg of the underlying fixed-for-floating swap with reset dates $T_0, \ldots, T_{n-...
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  • 20.4k
3 votes

Dual discounted forward curve

Which currency are you looking at ? Say that your 1y swap would have yearly fixed payments vs 3M floating payments. Your 1.5y swap would probably have: a fixed payment 6m after effective date and ...
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3 votes

Why is the forward rate used for the underlying in Black's model?

As a trader I used Black model (amongst others) to value swaptions, where the forward swap rate is the key observable underlying rate. Any market where the forward is the traded instrument would ...
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  • 13.7k
3 votes

Why is the spot price not used as the forward price when a forward contract is created?

it's easiest to see in terms of replication. The pay-off of a forward contract is $$ S_T - K. $$ We can replicate this precisely and statically by buying one unit of stock, $S_0,$ and $Ke^{-rT}$ ...
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  • 6,743
3 votes
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meaning of discount term in FRA value

A very good and up-to-date question. Whether you use the LIBOR-rate or any other rate for discounting depends on what you decide to be the fundamental rates in the market. Before the crisis LIBOR-...
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  • 3,337
3 votes

Derive instantaneous forward rate

You can start with $$P(t,T)=exp({-\int_t^T f_t(u).du})$$ then take derivative wrt to T $$R_F(0,T)=f_0(T)=-\frac{\partial} {\partial T}{ ln(P(0,T))} $$
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