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I understand the concept of DV01s and when doing an interest rate future trade I need to use about a 2:1 ratio when trying to trade the 2s10s. This is explained here:
https://www.cmegroup.com/education/whitepapers/trading-the-treasury-yield-curve.html
What I don't get is what prices on the futures I should use i... |
How to choose limit prices on IRS when targeting a specific 2s10s spread? |
How to choose limit prices on IR futures when targeting a specific 2s10s spread? |
I understand the concept of DV01s and when doing an interest rate future trade I need to use about a 2:1 ratio when trying to trade the 2s10s. This is explained here:
https://www.cmegroup.com/education/whitepapers/trading-the-treasury-yield-curve.html
What I don't get is what prices on the futures I should use i... |
I'm trying to price a PIK coupon with an Embedded Option by the borrower to pay in cash. Without the Embedded Option, it is simply a zero-coupon bond paying Principal*(1 + coupon rate)^n at the end.
What I'm not sure of is how to account for the Embedded Option. What I tried to do was to imagine the borrower having ... |
How to price PIK (paid-in-kind) coupon bond with option by the borrower to pay cash? |
I'll try to convey an answer as best as possible.
**One method of screening for dividend paying stocks**: If you do not know which stocks you want to work with, you can use a stock-screener like [Tradingview stock screener][1]. Here you can filter for dividend paying stocks (from high to low) and then single out yo... |
I often see commentary saying, for example:
‘Last weeks sep/Dec SPX roll VWAP traded at FedFunds + 32bps’ and
‘Dec/Mar SPX rolls currently imply a Year End premium of FedFunds + 49bps.’
But what does the implied funding rate of the roll actually mean?
Does it mean:
with the first example, that implied f... |
This is my first post to this forum. I want to calculate security returns , so therefore, I have downloaded the PX last price from Bloomberg. My question is:
What is PX last in Bloomberg? Is this Adjusted closing price of a security? |
**Long story cut short**: I am asking why the Local Volatility function can be thought of as a function of the underlying, when in fact it appears to be a function of the strike.
Additionally, I wonder how the local vol process for the underlying, being a single unique process, achieves to generate the same option ... |
Is Local Volatility a function of the Strike or the Underlying price? |
**Long story cut short**: I am asking why the Local Volatility function can be thought of as a function of the underlying, when in fact it appears to be a function of the strike.
Additionally, I wonder how the local vol process for the underlying, being a single unique process, achieves to generate the same option ... |
I derived the pdf using the butterfly prices and the curve looks like gamma of an option at every strike. [![PDF using butterfly prices][1]][1] Is that the case or am I missing something to get the pricing of an option?
[1]: https://i.stack.imgur.com/A3pBa.png |
Does the risk neutral pdf that is derived using Litzenberger-Breeden Method correspond to gamma and it's integral correspond to delta? |
I'm going through this paper: https://arxiv.org/abs/physics/0210025
as its formulation for instantaneous mean drift seemed really interesting.
On page 14 it says that it be expressed via greeks for any european-style call:
α = - (call_vega - 0.5\*call_delta + 0.5\*call_gamma) / call_delta
However, from curr... |
I'm going through this paper: https://arxiv.org/abs/physics/0210025
as its formulation for instantaneous mean drift seemed really interesting.
On page 14 it says that it can be expressed via greeks for any european-style call:
α = - (call_vega - 0.5\*call_delta + 0.5\*call_gamma) / call_delta
However, from ... |
When pricing a vanilla option, there are at a minimum 3 yield curves to consider:
- risk free yield curve = YC
- dividend yield curve = DC (or discrete dividends for American options but not the topic here)
- borrow curve = BC
When building a process (for example BlackScholesMertonProcess) we can only pass 2 curv... |
I have fitted two competing GARCH models, one `GARCH(1,2)` model and another `EGARCH(1,1,1)` both with t-distributed errors, on the log-returns of EquinorASA (Norwegian gas equity). I will not go through the preliminary checks, but the assumptions for `GARCH(1,2)` hold.
I have used Kevin Shepard's well-known [arch]... |
I have fitted two competing GARCH models, one `GARCH(1,2)` model and another `EGARCH(1,1,1)` both with t-distributed errors, on the log-returns of EquinorASA (Norwegian gas equity). I will not go through the preliminary checks, but the assumptions for `GARCH(1,2)` hold.
I have used Kevin Shepard's well-known [arch]... |
A real life principal protected note pays exactly the index return: (end_price/start_price-1)*50%
ie.. with a 50% participation, no commission. It does have a max return of 30% over 3y total.
Is it true that a call option plus a strip bond always has the exact same return as the index (for the total principal and s... |
How does a principal protected note pay exactly the return on the market? |
A real life principal protected note pays exactly the index return:
total_principle*(end_price/start_price-1)*50%
ie.. with a 50% participation, no commission - it does have a max return of 30% over 3y total.
Is it true that a call option plus a strip bond always has the exact same return as the index (for the t... |
A real life principal protected note pays exactly the index return:
total_principle*(end_price/start_price-1)*50%
ie.. with a 50% participation, no commission - it does have a max return of 30% over 3y total.
Is it true that a call option plus a strip bond always has the exact same return as the index (for the t... |
What's the difference between the credit curve and the CDS curve? Can I read the CDS curve from the Bloomberg terminal? for both single name and index?
Also, can someone please explain the difference between the CDX IG and HY? I recall one is quoted in spreads, and another is quoted in price. |
How does a principal protected note pay exactly the return on the market, on the whole principal? |
How does a principal protected note pay the return on the market, on the whole principal? |
Assume that you observe $T$ returns, then choose some $T_1<T$ and do the following:
1. Estimate your model on $r_1,\dots,r_{T_1}$ and use your estimated parameters to predict $\hat{\sigma}_{T_1+1}^2=E(\epsilon_{T_1+1}^2\vert \mathcal F_{T_1})$.
2. Estimate your model on $r_1,\dots,r_{T_1+1}$ and use your estimate... |
I have been told there is a multiplicative CLT. It says that - no matter the shape of returns distributions - if you *multiply* consecutive iid RVs (centered at 1.1, for instance), then a lognormal is the limiting distribution for prices/wealth. The only criteria I know of is that, for this to work, both the mean and... |
Does anyone know of a proof of the multiplicative central limit theorem? |
A GARCH model is a model not only for the conditional variance but for the entire conditional distribution of a random variable of interest. To assess the model's **statistical adequacy** out of sample, you would use time series cross validation via rolling or expanding windows as described in Lars's answer. You would ... |
A GARCH model is a model not only for the conditional variance but for the entire conditional distribution of a random variable of interest. To assess the model's **statistical adequacy** out of sample, you would use time series cross validation via rolling or expanding windows as described in Lars's answer. You would ... |
When talking to traders I hear this sentence a lot
I am a buyer/seller of X
where X = {vol, gamma, vega}
Is X basically all the same -- they are just saying -- I think implied volatility is cheap or expensive?
Or is it something fancier, where they basically isolated the Greeks and are trading them one by o... |
I believe the answer is no, as minimum value of call option is S - PV(K), which can never be below S-K.
The reason for the question is this paragraph in Natenberg, pg 109:
> Is it ever possible for an option to have a positive theta such that if nothing
changes, the option will be worth more tomorrow than it is ... |
Can european call option on stock have positive theta? (assume positive interest rate) |
I'm going through the paper "[Pricing European Options in Realistic Markets](https://arxiv.org/abs/physics/0210025)" by Schaden (2002) as its formulation for instantaneous mean drift seemed really interesting.
On page 14, the author says that it can be expressed via greeks for any European-style call:
α = - (call... |
Regarding *swaps*, the current preferred fixings for IRS in various currencies are given below. As with all OTC instruments, you're free to use whatever you like when you agree a deal, though most banks will stick to particular fixings.
Ccy Dom Int Alt Int
AUD BBSW BBSW LIBOR
CAD ... |
A real life principal protected note pays exactly the index return with 50% participation and a max 3y return of 30%:
total_principle*(end_price/start_price-1)*50%
These are all of the features, including fineprint.
How would you achieve this?
Could a call option plus a strip bond always achieve the exact same ... |
You can compare the losses against each model and determine the "best" model to be the one with the smallest losses. In many cases for larger studies, the results might be ambiguous where one or more models are favored by different loss functions. **Therefore, we want to know if we can construct statistical tests that ... |
Thought asking around on a problem I'm currently facing.
I have a hypothetical multi-asset portfolio of equities and bonds, on which I'm trying to measure it's liquidity risk in stressed periods. I've read Roncalli's four-piece paper on Liquidity Stress Testing in the Asset Management Industry, and in it they suggest... |
What is selling intensity, loss intensity, and how can I calibrate them? |
Thought asking around on a problem I'm currently facing.
I have a hypothetical multi-asset portfolio of equities and bonds, on which I'm trying to measure it's liquidity risk in stressed periods. I've read Roncalli's four-piece paper on Liquidity Stress Testing in the Asset Management Industry ([link][1]), and in it ... |
@nbbo2 and @Quantuple already answered the question in their comments but if in doubt, I always think computer coding is very helpful because you can simply try it out and run a lot of calculations in one go. Since option prices and Greeks (assuming Black Scholes) are [well defined](https://en.wikipedia.org/wiki/Greeks... |
@nbbo2 and @Quantuple already answered the question in their comments but if in doubt, I always think computer coding is very helpful because you can simply try it out and run a lot of calculations in one go. Since option prices and Greeks (assuming Black Scholes) are [well defined](https://en.wikipedia.org/wiki/Greeks... |
Not sure where to ask this, but I had this question that asked to do a differential analysis (pretty sure that's the english term, the class is in French) where a company currently sells 60k units a year, with a revenue of 2M\$, variable cost of 1M\$ and fixed cost of 500k\$. They sell the old equipment for 180k\$ and ... |
Does differential analysis include the cost of initial investment? |
consider the following problem i am trying to find the monthly covariance matrix given daily data. i have the following codeimport datetime
```
import pandas as pd
import yfinance as yf
import numpy as np
tickers = ['AAPL', 'AMZN', 'XOM']
start_date1 = datetime.date(2010, 1, 2)
end_date1 = datetime.date(2019, ... |
consider the following problem i am trying to find the monthly covariance matrix given daily data. i have the following codeimport datetime
```
import pandas as pd
import yfinance as yf
import numpy as np
tickers = ['AAPL', 'AMZN', 'XOM']
start_date1 = datetime.date(2010, 1, 2)
end_date1 = datetime.date(2019, ... |
@nbbo2 and @Quantuple already answered the question in their comments but if in doubt, I always think computer coding is very helpful because you can simply try it out and run a lot of calculations in one go. Since option prices and Greeks (assuming Black Scholes) are [well defined](https://en.wikipedia.org/wiki/Greeks... |
**Question**\
Let $\mathbf{C}$ be a $n\times n$ symmetric matrix with diagonal elements all equal to 1, and the non-diagonal ones all equal to $\rho$ such that $-1\leq\rho\leq1$ that is $\mathbf{C}$ is a correlation matrix. Which range of values is admissible for $\rho$?
**Solution**\
Let $X_1,\dots,X_n$ be a sequ... |
I have an implementation of an LMM+ model (a shifted Libor Market Model with rebonato volatility function) and am seeing recently that the calibrated parameters are becoming more unstable over time; I have been observing this mostly since the war in Ukraine started. In terms of the implied volatility surface, we have s... |
@nbbo2 and @Quantuple already answered the question in their comments but if in doubt, I always think computer coding is very helpful because you can simply try it out and run a lot of calculations in one go. Since option prices and Greeks (assuming Black Scholes) are [well defined](https://en.wikipedia.org/wiki/Greeks... |
@nbbo2 and @Quantuple already answered the question in their comments but if in doubt, I always think computer coding is very helpful because you can simply try it out and run a lot of calculations in one go. Since option prices and Greeks (assuming Black Scholes) are [well defined](https://en.wikipedia.org/wiki/Greeks... |
I'm new to Quantitative Finance. I've data which I need to fit a CIR model and estimate its parameters.
$ dX_{t+1} = a(b-X_{t})dt + \sigma \sqrt{X_t}dW_{t} $
While I can fit and obtain parameparameterates using Euler-maurayana discretization and then linear regression closed form solution, I'd like to explore ... |
I'm new to Quantitative Finance. I've data which I need to fit a CIR model and estimate its parameters.
$ dX_{t+1} = a(b-X_{t})dt + \sigma \sqrt{X_t}dW_{t} $
While I can fit and obtain parameparameterates using Euler-maurayana discretization and then linear regression closed form solution, I'd like to explore ... |
I am currently reading Market Liquidity by Foucault, Pagano and Röell. In chapter one they describe the limit order book markets and dealer markets.
I am confused about two rules that so called "broker-dealers" have to follow. These are the Quote Rule and the Limit Order Display Rule.
Reading the SEC on the rules... |
A relevant working paper is [Forecasting Market Volatility: The Role of Earnings Announcements][1]
[1]: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3219384 |
In his paper, "The Statistics of Sharpe Ratio", Andrew Lo writes
> "hence, the ratio will increase as the square root of q, making a
> longer horizon investment seem more attractive. This interpretation is
> highly misleading and should not be taken at face value. Indeed, the
> Sharpe ratio is not a complete sum... |
Can someone please explain to me how most banks calculate their stress VaR. is there a regulatory-defined time series, such as the 2008 to 2009 period, to apply to the current position? My understanding of the VAR calculation is a rolling window of 251 days. So, how would some credit spreads/ratings in 2008 affect the ... |
I hope this is an okay place to ask this:
Case: Assume you find a corporate bond you want to invest in. You then invest in it below par several times over the years, and you also sell bits of your holdings above par. It then matures, and you get your remaining nominal amount back at par. You therefore have uneven ca... |
Calculate combined return on corp. bond traded multiple times? |
Given data that arrives at a daily frequency, I aggregated it to a weekly frequency, and estimated an OLS regression on it. Given that there are roughly 5 trading days per week, I can construct 5 different OLS models using 5 different starting points. For example - one model uses returns from Monday-Monday, the next Tu... |
I'm reading Grinold & Kahn (2000) for the proof of the Fundamental Law of Active Management.
I can't understand formula (6A.20) on page 168, which says:
> Finally, by assuming that all the signals have equal value,
> $$\zeta_b^2=\rho^2=IC^2 \qquad \text{(6A.20)}$$
According to (6A.18) at the end of page 167,... |
In the book, section 4.8 class weights, Marcos suggests applying class weight, which I agree because sometimes you have more bullish price action than bearish price action e.g. 52% of the time is bullish.
But he doesn't talk about splitting the dataset. My intuition is that if the training set has 2 years of data, a... |
Should we split data into several periods before calculating class weight? (Advances in Financial Machine Learning) |
i'm facing a new and interesting task: We are calculating a time series of (hypothetical) behavioral portfolios, for which i need a few parameters to calculate the portfolio's weights in each asset. I'm using an observed portfolio as starting point, from which i need to extract the implied utility parameters (in the ca... |
A better calibration method available? |
i'm facing a new and interesting task: We are calculating a time series of (hypothetical) behavioral portfolios, for which i need a few parameters to calculate the portfolio's weights in each asset. I'm using an observed portfolio as starting point, from which i need to extract the implied utility parameters (in the ca... |
Let us consider a two day option:
- 1 day having a baseline vol of 16%
- 1 day having an event for which we want to find the implied move, higher than 16%
Is it possible that the price of the straddle in % of spot is lower than the implied move we would have found.
For example let us imagine the the straddl... |
Can the break-even of a straddle be lower than the implied move? |
**Question**\
Let $\mathbf{C}$ be a $n\times n$ symmetric matrix with diagonal elements all equal to 1, and the non-diagonal ones all equal to $\rho$ such that $-1\leq\rho\leq1$ that is $\mathbf{C}$ is a correlation matrix. Let its associated variance matrix be $\mathbf{V}=I$ where $I$ is the unit matrix. **Which rang... |
**Question**\
Let $\mathbf{C}$ be a $n\times n$ covariance matrix such that all diagonal elements are equal to 1, and the non-diagonal ones to $\rho$ with $-1\leq\rho\leq1$. Which range of values is admissible for $\rho$?
**Solution**\
Let $X_1,\dots,X_n$ be a sequence of independent random variables with unit var... |
Let us consider a two day option:
- 1 day having a baseline vol of 16%
- 1 day having an event for which we want to find the implied move, higher than 16%
Is it possible that the price of the straddle in % of spot is lower than the implied move we would have found.
For example let us imagine the the straddl... |
I am reading some corporate documentation, including the notion of market risk factor proxy. Could you provide some examples?
What I find obvious is the observable ON interest rate as a proxy for instantaneous rate, or the same one for implied volatility. Hope there is a diverse series of examples. |
The paper you are looking for is [Large tick assets: implicit spread and optimal tick size](https://arxiv.org/abs/1207.6325) by Dayri and Mathieu Rosenbaum 2013.
they explain the role of the tick in the relation between market makers and investors. In short their main measure (coming from another paper on "uncertainty... |
The equation for $\frac{dC}{dt}$ in a European call option is:
$\frac{SN'(d_1)\sigma}{2\sqrt{T-t}}-rKe^{-r(T-t)}N(d_2)$. This is calculated by taking the derivative of a European call option with respect to 't'. However, S is treated as a constant here! Isn't S a function of t as well? $S = e^{(\mu-\frac{\sigma^2}{2})... |
Why is 'S' treated as a constant when calculating the greek Theta (dC/dt) for a European call option? |
**Question**\
Let $\mathbf{C}$ be a $n\times n$ covariance matrix such that all diagonal elements are equal to 1, and the non-diagonal ones to $\rho$ with $-1\leq\rho\leq1$. Which range of values is admissible for $\rho$?
**Solution 1**\
Let $X_1,\dots,X_n$ be a sequence of independent random variables with unit v... |
The CME describes how to put on a 2s10s trade in this screenshot:
https://i.stack.imgur.com/2yPzW.jpg
Looking at current 2 and 10 year futures the CTD is roughly a 2 year and 7 year respectively.
Am I right in saying that using the current 2 and 10 year futures to put on a 2s10s trade, you're actually not ca... |
How does one put on a 2s10s trade using 2 and 10 year treasury futures contracts when the CTDs are not 2 and 10 year bonds? |
I am reading some corporate documentation, including the notion of market risk factor proxy. Could you provide some examples of risk factors and their proxies?
What I find obvious is the observable ON interest rate as a proxy for instantaneous rate, or the same one for implied volatility. Hope there is a diverse ser... |
In a nutshell, if S is dependent on 't', why treat it as a constant when calculating the partial derivative $\frac{dC}{dt}$?
The equation for $\frac{dC}{dt}$ in a European call option is:
$\frac{SN'(d_1)\sigma}{2\sqrt{T-t}}-rKe^{-r(T-t)}N(d_2)$. This is calculated by taking the derivative of a European call option ... |
Since $S = e^{(\mu-\frac{\sigma^2}{2})t+\sigma W_t}$, why treat it as a constant when calculating the greek Theta (dC/dt) for a European call option? |
Let me heed @Bob's suggestion and turn my comment into a full answer:
Like other disciplines, finance uses lots of shortcuts to achieve brevity and convenience. That can be awfully confusing for students. For example, we write $\text{d}S_t$ instead of integrals, just because it's shorter. Formally, $\text{d}S_t$ has... |
Suppose a process for a stock price of a US-based company traded in the USA is, under the USD money-market numeraire:
$$dS_t=S_tr_{USD}dt+S_t\sigma_SdW_1(t)$$
Using fundamental theorem of asset pricing, we must have that:
$$\frac{S_0}{1}=\mathbb{E}^Q_{USD}\left[\frac{S_t}{B_{USD}(t)}\right]=\mathbb{E}^Q_{USD}\... |
How is an exchange rate process a martingale under any measure? |
Suppose a process for a stock price of a US-based company traded in the USA is, under the USD money-market numeraire:
$$dS_t=S_tr_{USD}dt+S_t\sigma_SdW_1(t)$$
Using fundamental theorem of asset pricing, we must have that:
$$\frac{S_0}{B_{USD}(t_0)}=\frac{S_0}{1}\stackrel{?}{=}\mathbb{E}^Q_{USD}\left[\frac{S_t}... |
Suppose a process for a stock price of a US-based company traded in the USA is, under the USD money-market numeraire:
$$dS_t=S_tr_{USD}dt+S_t\sigma_SdW_1(t)$$
Using fundamental theorem of asset pricing, we must have that:
$$\frac{S_0}{B_{USD}(t_0)}=\frac{S_0}{1}\stackrel{?}{=}\mathbb{E}^Q_{USD}\left[\frac{S_t}... |
I am working with options data, and I am using Breeden-Litzenberger formula to derive the risk-neutral terminal stock price PDF.
After applying the formula, here is a scatter plot of strike price vs Breeden-Litzenberger pdf:[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/7fQRJ.png
At t... |
I am trying to understand if the conventional spread column in Markit's CDS database simply represents the CDS spread, measured in bps, or should I make some adjustments (in case I would like to make some comparisons between CDS's with different coupns, e.g., 1% or 5%)?
Thank you! |
In which units the conventional CDS spreads in Markit's data are measured? |
they are different books for quants addressing market microstructure
- [The Financial Mathematics of Market Liquidity](https://www.routledge.com/The-Financial-Mathematics-of-Market-Liquidity-From-Optimal-Execution-to/Gueant/p/book/9781498725477) 2016, by Guéant
- [Algorithmic and High-Frequency Trading](https://www... |
That's true that in market making papers, like [Dealing with the Inventory Risk. A solution to the market making problem](https://arxiv.org/abs/1105.3115) 2012 by Guéant, L and Fernandez-Tapia, the volatility is often taken as constant.
What does it means? indeed you have two nature of uncertainties on the table whe... |
they are different books for quants addressing market microstructure
- [The Financial Mathematics of Market Liquidity](https://www.routledge.com/The-Financial-Mathematics-of-Market-Liquidity-From-Optimal-Execution-to/Gueant/p/book/9781498725477) 2016, by Guéant
- [Algorithmic and High-Frequency Trading](https://www... |
They are a lot of branches of financial maths:
- **Modern Portfolio Theory** is one of them: how to express portfolio management using mathematical formulas (and how does it leads to different solutions)? Think about Markowitz portfolios of course, but also about Black and Litterman ones too, or about Merton's inter... |
I've been checking the demos for BGM (LFM) forward rate model.
Here's a short reminder to help you follow:
[![BGM Definition][1]][1]
Now, take the following
$$\frac{dL_j(t)}{L_j(t)} = \sigma_j. dW^j(t) = \mu_{ij} dt + \sigma_j. dW^i(t) $$
if we consider the same brownians as in the definition of the ... |
I am working with options data, and I am using Breeden-Litzenberger formula to derive the risk-neutral terminal stock price PDF.
After applying the formula, here is a scatter plot of strike price vs Breeden-Litzenberger pdf:[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/7fQRJ.png
At t... |
From a statistical viewpoint, it is not standard to be in a situation to
1. have a "*noisy CDF*",
2. sample points from it,
3. deduce another CDF that would be "less noisy".
You can repeat point (2) and (3) to "bootstrap the CDF", but what would it means?
First, you have to know that **bootstrap is not magi... |
Taking into account an old post [maths.stackexchange][1], I recall the following:
On the one hand, we know that the Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process.
An Ornstein–Uhlenbeck process, $s_t$, satisfies the following stochastic differ... |
Can I extend the private information model of Kyle in in a continuous analogue, e.g. the Ornstein–Uhlenbeck process? |
Taking into account an old post of [maths.stackexchange][1], I recall the following:
On the one hand, we know that the Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process.
An Ornstein–Uhlenbeck process, $s_t$, satisfies the following stochastic dif... |
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