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I have three corporate bonds with maturities 2,3 and 5 years. They pay annual fixed coupons. I know their yield to maturities. How to compute their z spreads? |
How to calculate corporate bonds Z spreads having yield to maturities and knowing that they pay annual fixed coupons? |
Can anyone help me understand as to why in CAPM's market portfolio investors will always have the assets in proportion to the market value? One of the intuitive reasonings that I have read explains that since the odds of finding over-valued and under-valued stocks in a CAPM world as random, rational investors will have... |
Why weigh assets by market values in CAPM? |
The typical representation of extrinsic value of an option is the following:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/HftqB.png
Is the gaussian the real representation of extrinsic value derived from Black and Scholes?
Thanks
|
The typical representation of extrinsic value of an option is the following:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/HftqB.png
Is the gaussian the real representation of extrinsic value derived from Black and Scholes?
Should it be a lognormal?
Thanks
|
How does truncating data deal with long right tails of data? Why is a long right tail bad? |
I was playing around with some models, and I'm able to predict a stock's current price based on the current prices of other stocks. This model is extremely accurate, although I can't see any use of this. If I wanted to know a stock's current price I would just grab it from the market, not derive it from other stock pri... |
Is there any utility to being able to predict an assets current price? |
In Canada, the CDOR index will be phased out in June 2024, and the replacement is the CORRA (Canadian Overnight Repo Rate Average), which is an OvernightIndex used for evaluating the FRN in Canada. Here is an example bond, the CUSIP is 13607HMS8, maturity date is 2025-03-04, issue date is 2021-02-25, and first settleme... |
What are some good references to study Stochastic Control with applications to Finance, like the Merton problem and other variants? Books or review papers? |
Had a few questions on MBS Dollar rolls:
1) How are dollar rolls affected by Interest rate Volatility?
2) Does the OAS of underlying pools matter to dollar rolls, and if so, how do they affect it?
3) Does the dollar roll itself potentially impart value to underlying pools? Why does that happen?
Thank You |
Suppose I have an existing FX Swap, suppose the spot leg is already settled, so only forward leg is left.
Question: What will be the P&L for this instrument - only forward leg NPV or spot leg is also marked-to-market to current spot rate?
Thanks a lot! |
The typical representation of the extrinsic value of an option is the following:
[![enter image description here][1]][1]
Is the gaussian the real representation of extrinsic value derived from Black and Scholes?
Should it be lognormal?
From an idea of quantpie I took the following graph:
[![enter image descr... |
I have studied methodologies for Moody's and S&P ratings but haven't seen any instance where the respective agencies have mentioned the reason choosing Baa3/BBB- as the dividing line between investment grade and junk issues.
Is there any research which shows that default risk shoots up at a significantly faster rate... |
Why investment grade floor is set at Baa3/BBB-? |
Write the Euler discretization of the 1-dimensional stochastic equation
$dXt = b (t, X_t) \space dt + \sigma (t, X_t) \space dW_t$
For this part I would say all right because it is a purely theoretical part that I find on any stochastic calculus book.
The problem arises on the optional part:
Set $b (t, x... |
### Spotted issues:
- The main issue with your code is that you define `S0 = 1` which in `Python` is of type **Int**. Your stock price trajectories are built upon `S0` and thus S will be of type **Int** as-well. This implies that any decimals in your estimates will be truncated. **To fix this** simply define `S0 = 1... |
### Spotted issues:
- The main issue with your code is that you define `S0 = 1` which in `Python` is of type **Int**. Your stock price trajectories are built upon `S0` and thus `S` will be of type **Int** as-well (*more specifically, Int32*). This implies that any decimals in your estimates will be truncated. **To f... |
I'm studying the way option can be priced in an incomplete market and I have found an example talking about the Cox-Ross-Rubinstein model with three path possible instead of 2, making the model incomplete. The idea was to consider a super replicating strategy and show that the initial value of this strategy is the inti... |
I was looking for both the daily adjusted close and the normal close of the S&P500. On Yahoo Finance i found data that included both of these prices, but the two were identical. These are the R-commands i've used:
library(quantmod)
getSymbols("^GSPC",src="yahoo",from="2011-01-01",to="2021-01-01")
I also ... |
Why is the adjusted close equal to the "normal" closing price of the S&P500? (Yahoo finance) |
Net Issuance of Agency MBS can be thought of as driven by Existing Home Sales, New Home Sales, Cash out Refis, Amortization and Non-Agency MBS runoff. Based on this definition of net issuance, is there a reason why GNMA Net issuance decreased significantly through 2020-2021 (during the refi wave), while FNMA/FHLMC Net ... |
Why did Ginnie Mae Net issuance decrease significantly in 2020-2021? |
Net Issuance of Agency MBS can be thought of as driven by Existing Home Sales, New Home Sales, Cash out Refis, Amortization and Non-Agency MBS runoff. Based on this definition of net issuance, is there a reason why GNMA Net issuance decreased significantly through 2020-2021 (during the refi wave), while FNMA/FHLMC Net ... |
Net Issuance of Agency MBS can be thought of as driven by Existing Home Sales, New Home Sales, Cash out Refis, Amortization and Non-Agency MBS runoff. Based on this definition of net issuance, is there a reason why GNMA Net issuance decreased significantly through 2020-2021 (during the refi wave), while FNMA/FHLMC Net ... |
Net Issuance of Agency MBS can be thought of as driven by Existing Home Sales, New Home Sales, Cash out Refis, Amortization and Non-Agency MBS runoff. Based on this definition of net issuance, is there a reason why GNMA Net issuance decreased significantly through 2020-2021 (during the refi wave), while FNMA/FHLMC Net ... |
Net Issuance of Agency MBS can be thought of as driven by Existing Home Sales, New Home Sales, Cash out Refis, Amortization and Non-Agency MBS runoff. Based on this definition of net issuance, is there a reason why GNMA Net issuance decreased significantly through 2020-2021 (during the refi wave), while GSE Net issuanc... |
Why did Ginnie Mae MBS Net issuance decrease significantly in 2020-2021? |
Is it possible to construct a central bank meeting date curve, using futures prices & OIS rates, in QuantLib?
Specifically, I mean a yield curve with flat (constant) spot rates in between meeting dates, and discontinuous moves on meeting dates (or 1 day after I guess, given that typically the rate change applies the n... |
Is it possible to construct a central bank meeting date curve, using futures prices & OIS rates, in QuantLib?
Specifically, I mean a yield curve with flat (constant) forward rates in between meeting dates, and discontinuous moves on meeting dates (or 1 day after I guess, given that typically the rate change applies th... |
What are the ways one can trade mean reversion apart from using Bollinger Bands? |
In several books on asset pricing and more particularly when it concerns option pricing, I see the use of Markov process, they argue the computation is made easier with such process. Is this sufficient to use Markov Model ?
For example, when we talk about martingale property, this is the result of a fundamental theo... |
The Markov property states that the future states of a stochastic process only depend on its current state, not any previous states. In a discrete setting, this can be written as:
$$\mathbb{P}(X_{n+1}=x|X_n=x_n,X_{n-1}=x_{n-1)},...,X_0=x_0)=\mathbb{P}(X_{n+1}=x|X_n=x_n)$$
In finance, we want asset prices like sto... |
The Markov property dictates that the future states of a stochastic process only depend on its current state, not any previous states. In a discrete setting, this can be written as:
$$\mathbb{P}(X_{n+1}=x|X_n=x_n,X_{n-1}=x_{n-1)},...,X_0=x_0)=\mathbb{P}(X_{n+1}=x|X_n=x_n)$$
In finance, we want asset prices like s... |
I am not quite sure how to solve this integral to be able to do numerical calculations with it. $\lambda$ is a constant, $u$ is time, and $Z_u$ is a wiener process. Can anyone provide some direction on how to solve it please?
$f = \int_{0}^t e^{\lambda u } dZ_u$ |
Several investors track indices to gain exposure to specific asset classes. And these indices may be re-balanced on a monthly basis, based on market-cap etc which generates month-end flows from investors replicating these indices. These flows can be determined to some extent.
In Agency MBS, investors tend to track ... |
I was wondering if it's pertinent to use this interpretation of the expected utility function given by the Taylor series expansion,
$${E(U(W)}\approx{U[E(W)}]+\frac{U''[E(W)]\sigma^2_W}{2}\tag{1}$$
to delineate my optimal portfolio from the set of portafolios that lie on the efficient frontier?
So, let's say... |
Should I annualise returns for beta calculation? |
I am a young Statistics graduate.
A few days ago as an interview question, I have been asked to evaluate the predictive power of a Signal time series (supposedly output by an Artificial Intelligence system) in predicting a stock price time series and I was provided with a sample data set as follows:
|Date| Open| Hi... |
I am a young Statistics graduate.
A few days ago as an interview question, I have been asked to evaluate the predictive power of a Signal time series (supposedly output by an Artificial Intelligence system) in predicting a stock price time series and I was provided with a sample data set as follows:
|Date| Open| Hi... |
I am a young Statistics graduate.
A few days ago as an interview question, I have been asked to evaluate the predictive power of a Signal time series (supposedly output by an Artificial Intelligence system) in predicting a stock price time series and I was provided with a sample data set as follows:
|Date| Open| Hi... |
You might find more traction at https://stats.stackexchange.com/, since most ML and data kind of problems are handled there.
I am no ML expert but let me brainstorm for you
> FIRST Question: how could one evaluate the predictive power of Signal given only the data above?
This looks like a regression type task, p... |
I'm studying G2++ Model with Brigo(2007)'s book.
The model constructed like this
Short rate $r(t) = x(t) + y(t) + φ(t), r(0) = r_0$.
And $dx(t) = -ax(t)dt + σdW_1(t), x(0) = 0$, $dy(t) = -by(t)dt + ηdW_2(t), y(0) = 0$, $dW_1(t)*dW_2(t) = ρdt$.
When we calculate the variance, There is something that I cannot der... |
Investing in cryptocurrencies is a wild ride. There is obviously a lot of speculation involved but my question is another one: what are good models to evaluate the fundamental value of cryptocurrencies? Does it help to price them as you would price traditional currencies or are they fundamentally different? If so, why ... |
How to fundamentally value cryptocurrencies? |
Suppose we have a universe of $n$ stocks, and for each time period $t$ we have $n$ predictions for their future returns. Now we can calculate the information coefficient for our predictions in two ways:
1. Time series $IC_{ts}$: for each stock, calculate the Pearson correlation of its prediction and realized return... |
Time series strategy versus cross section strategy? |
What is the rule (assuming there is one) specifying which day count convention should prevail in a cross-currency swap?
For example, where EUR follows ACT/360 and GBP follows ACT/365, which of the two conventions would apply to the calculation of a EURGBP forward price?
Bloomberg DES function refuses to show a D... |
### Spotted issues:
- **The main issue:** You define `S0 = 1` which in `Python` is of type *Int*. Your stock price trajectories are built upon `S0` and thus `S` will be of type *Int* as-well (*more specifically, Int32*). This implies that any decimals in your estimates will be truncated. **To fix this** simply defin... |
> Investing in cryptocurrencies is a wild ride. There is obviously a lot of speculation involved but my question is another one: what are good models to evaluate the fundamental value of cryptocurrencies?
One of the best models to value anything is (modified) discounted cash flow. Note that there can be other model... |
> Investing in cryptocurrencies is a wild ride. There is obviously a lot of speculation involved but my question is another one: what are good models to evaluate the fundamental value of cryptocurrencies?
One of the best models to value anything is (modified) discounted cash flow. Note that there can be other model... |
> Investing in cryptocurrencies is a wild ride. There is obviously a lot of speculation involved but my question is another one: what are good models to evaluate the fundamental value of cryptocurrencies?
One of the best models to value anything is (modified) discounted cash flow. Note that there can be other model... |
> Investing in cryptocurrencies is a wild ride. There is obviously a lot of speculation involved but my question is another one: what are good models to evaluate the fundamental value of cryptocurrencies?
One of the best models to value anything is (modified) discounted cash flow. Note that there can be other model... |
You might find more traction at https://stats.stackexchange.com/, since most ML and data kind of problems are handled there.
I am no ML expert but let me brainstorm for you, since these kind of task is slightly more an art than a science.
> FIRST Question: how could one evaluate the predictive power of Signal given... |
You might find more traction at https://stats.stackexchange.com/, since most ML and data kind of problems are handled there.
I am no ML expert but let me brainstorm for you, since this kind of task is slightly more an art than a science. You need be creative than just following a det of rules.
> FIRST Question: ho... |
What is the rule (assuming there is one) specifying which day count convention should prevail in a cross-currency swap?
For example, where EUR follows ACT/360 and GBP follows ACT/365, which of the two conventions would apply to the calculation of a EURGBP forward price?
Bloomberg DES function refuses to show a D... |
there is one question I got wrong that I am confused about and that is the following :
If you believe that yield will increase in the future which of the following should you choose?
the two possibilities are.
A )REPO
B )REVERSE REPO
I answered REPO because if the yield goes up this means that price goes down and... |
there is one question I got wrong that I am confused about and that is the following :
If you believe that yield will increase in the future which of the following should you choose?
the two possibilities are.
A )REPO
B )REVERSE REPO
I answered REPO because if the yield goes up this means that price goes down... |
there is one question I got wrong that I am confused about and that is the following :
If you believe that yield will increase in the future which of the following should you choose?
the two possibilities are.
A )REPO
B )REVERSE REPO
I answered REPO because if the yield goes up this means that price goes down... |
I know dr of risk-neutrual measure is
[![enter image description here][1]][1]
There is a price of a pure-discount bond can be derived by computing the expectation, I get:
[![enter image description here][2]][2]
where A and B are:
[![enter image description here][3]][3]
why dr becomes to:
[![enter ... |
I'm currently developping forex robot using Python API for Metatrader 5. Among different things, this robot places trades given by external signals (with price, SL and TP).
I use euro funded account, trade XAUUSD, and have a 1:100 leverage ratio.
In order to implement 1% strategy (meaning never risk more than 1% of... |
**I'm studying G2++ Model in Brigo(2007)'s book.**
The model constructed as follows,
$$
r(t) = x(t) + y(t) + φ(t), \quad r(0) = r_0\\
$$
with the dynamics of $dx(t)$ and $dy(t)$ described by:
\begin{align}
dx(t) &= -ax(t)dt + σdW_1(t), \quad x(0) = 0,\\
dy(t) &= -by(t)dt + ηdW_2(t),\quad y(0) = 0,\\
\end{... |
In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$
$$
S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})(T - t) + \sigma \sqrt{T - t} N(0,1)} \text{,}
$$ M. Joshi in *The concepts and practice of mathematical finance* starts from the stochastic process for a deli... |
Can anyone explain the difference between THE US Federal reserve interest on reserve balances (IRRBIOER) and effective fed funds rate (EFFR)?
IRRBIOER is currently at 0.9% and EFFR 0.83%. There seems to be a spread and I am unsure why:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/... |
Difference between US FED interest on reserve and EFFR? |
I am trying to understand the paper "Option Pricing: A Simplified Approach" by Cox-Ross-Rubinstein (available online [here][1]).
To my frustration, I already don't understand the paper starting from just page 3. It is said there:
> Consider forming the following levered hedge:
> (1) Write three calls at $C$ ea... |
I'm new to the world of option market, but after having studied CRR model I'm wondering if European call and put option are very useful since a talk with my professor that piqued ma curiosity. In the following I focus on call option. I mean, from what I have understood there are 2 uses of call option : 1) limit the amo... |
Are European call and put option useful ? [Cox-Ross-Rubinstein model] |
**What are some good techniques for proxying time series?**
My purpose is for risk management / modelling and I would like proxy missing series. An example could be the Facebook stock prior to its listing?
But what are some good methods when if I need to take volatility scaling, correlation, etc. into account?
... |
Since integrated variance can be estimated via the sum of squared returns (which connects closly to the realized volatility, can I also asusme that "the integrated volatility is an estimator of the realized volatility?" Thank you. |
In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$
$$
S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})t + \sigma \sqrt{t} N(0,1)} \text{,}
$$ M. Joshi in *The concepts and practice of mathematical finance* starts from the stochastic process for a delivery contr... |
In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$
$$
S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})t + \sigma \sqrt{t} N(0,1)} \text{,}
$$ M. Joshi in *The concepts and practice of mathematical finance* starts from the stochastic process for a delivery contr... |
In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$
$$
S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})t + \sigma \sqrt{t} N(0,1)} \text{,}
$$ M. Joshi in *The concepts and practice of mathematical finance* starts from the stochastic process for a delivery contr... |
**What are some good techniques for proxying time series?**
My purpose is for risk management / modelling and I would like proxy to missing series.
Given that I also have to account for volatility, correlation, etc., what methods can I then apply?
I am currently working on some simulated data generated from Ge... |
In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$
$$
S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})t + \sigma \sqrt{t} N(0,1)} \text{,}
$$ M. Joshi in *The concepts and practice of mathematical finance* starts from the stochastic process for a delivery contr... |
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 or if I am making a mistake.
Let $X(t) = \frac{USD}{EUR}(t)$ be a FX spot. Let $B^{USD}(t) = e^{\int^t_0 r_{USD}(u)... |
In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$
$$
S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})t + \sigma \sqrt{t} N(0,1)} \text{,}
$$ M. Joshi in *The concepts and practice of mathematical finance* starts from the stochastic process for a delivery contr... |
i am doing research on a so-called carbon beta where I made a portfolio that takes a short position in 'green' stocks and a long position in 'brown' stocks, from a period of 2005 until 2020. So from the brown minus green portfolio I generated one beta per company in my sample (750) through 750 time series regressions.... |
How do you cross-sectionally standardize a variable? |
I understand the put-call parity and am trying to derive the results based on a CME article in their education section and also the Wikipedia explanation in the Black Scholes model where Put-call parity is derived for European call and put.
**As per Black Scholes equation and put call parity**
**Url : https://en.wi... |
I am reading about the LMM with lognormal-mixture dynamics. Consider the following dynamics for the forward rate $F_{i}(t)$ fixing at $T_{i-1}$ and paying at $T_i$:
\begin{align}
dF_{i}(t) = (F_i (t) +\gamma) \sigma_i^I dt \ ,
\end{align}
where $\gamma$ is a constant shift and $\sigma_i^I$ is a random variable for ... |
Why is the LMM with mixture dynamics (Brigo & Mercurio) inconsistent for the pricing of exotics? |
I'm trying to justify the lower bound for the price of a contingent claim (a European one) which is not marketable in an arbitrage free market. I would like to have your advice on my way to do it:
First, I start with the set $\{V_0(\theta) : V_T(\theta)\leq H\; and\;\theta\; admissible\}$ where $V_0(\theta)$ is the ... |
I am reading about the LMM with lognormal-mixture dynamics. Consider the following dynamics for the forward rate $F_{i}(t)$ fixing at $T_{i-1}$ and paying at $T_i$:
\begin{align}
dF_{i}(t) = (F_i (t) +\gamma) \sigma_i^I dt \ ,
\end{align}
where $\gamma$ is a constant shift and $\sigma_i^I$ is a random variable for ... |
Can I assume that "the long run volatility forecast of a GARCH(1,1) is higher in periods of high volatility than in periods of low volatility? |
Can someone help if I am thinking correctly?
If R(t,i) is the i(th) log-return (for i = 1.....M) of day t (for t = 1.....T). Can I assume that the daily realized volatility (denoted RV(t)) is a consistent estimate of the true daily volatility
denoted QV(t)] in the sense that RV(t)--> QV(t) when T--> ∞ ? |
Is there an equivalent BS volatility formula for the Heston model, something like Hagan's formula for the SABR model? Of course, such a formula will be an approximation as in Hagan's formula.
Under the Heston model, we can price European options with the inverse Fourier transform (or FFT) quite precisely. So it's p... |
Equivalent BS volatility formula under the Heston model? |
**To keep it brief:** *the realized variance estimator, $RV_t$, is only a consistent estimator of Quadratic Variation (QV) under absence of microstructure noise.*
Following the paper of **[Barndorff‐Nielsen, O. E., & Shephard, N. (2002)](https://www.jstor.org/stable/3088799?seq=1)** they show how the realized varia... |
Can someone help if I am thinking correctly?
If $R(t,i)$ is the i'th log-return for $i = 1\ldots,M$ of day $t$ for $t = 1\ldots,T$.
Can I assume that the daily realized volatility (denoted $RV(t)$) is a consistent estimator of the true daily volatility
denoted $QV(t)$] in the sense that $RV(t)\rightarrow QV(t)... |
**To keep it brief:** *the realized variance estimator, $RV_t$, is only a consistent estimator of Quadratic Variation (QV) under absence of microstructure noise.*
Following the paper of **[Barndorff‐Nielsen, O. E., & Shephard, N. (2002)](https://www.jstor.org/stable/3088799?seq=1)** they show how the realized varia... |
I try to understand the equal risk contribution (ERC) portfolio as described in [On the Properties of Equally-Weighted Risk Contributions Portfolios](https://www.researchgate.net/publication/45397778_On_the_Properties_of_Equally-Weighted_Risk_Contributions_Portfolios?enrichId=rgreq-e57c2f5c85f2f372662ff322fd9faf92-XXX&... |
Answering the question in a way coming from the motivation at the back: There is not one universal approximation, but different ones depending on which asymptotics you want to look into. Just to state a few:
- For the extreme strike you have the results by Friz, Gerhold, Gulisashvili and myself, https://arxiv.org/ab... |
I found this method to represent time series to improve performance of some ML models, any thoughts about this method? some applications or use cases in financial markets? |
Application of Gramian Angular Field to financial series? |
What is the rule (assuming there is one) specifying which day count convention should prevail in a cross-currency swap?
For example, where EUR follows ACT/360 and GBP follows ACT/365, which of the two conventions would apply to the calculation of a EURGBP forward price?
Bloomberg DES function refuses to show a D... |
Can one use the GARCH model to estimate the realized volatility, [such as done in this paper][1], rather than forecast volatility?
[1]: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=306002 |
You can use GARCH to estimate volatility. If you have high-frequency data, you can use realized volatility to estimate volatility. You would normally not use GARCH to estimate realized volatility (why estimate an estimate?), though I suppose it is technically possible to view your fitted volatilities from a GARCH model... |
Can one estimate rather than forecast volatility using the GARCH model? |
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