instruction stringlengths 15 26.1k |
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Lets talk about the Monte Carlo simulation of forward rates in Euler discretization scheme under the $T_N$-forward measure, a so called terminal measure. Suppose that we have a number of time steps parameter in the Monte Carlo simulation which calculates the time step as
$$dt = \frac{T_{N-1}}{\mbox{num of steps}}$$
... |
I am reading a news article regarding how OIS swap rates behave when we receive or pay OIS swap rates.
The article states that when we receive OIS swap rates it makes swap rates fall. When we pay swap rates, the swap rates rise.
What would be a logical explanation for this? |
How OIS swap rates behave when we receive or pay OIS swap rates? |
This might sound like a trivial question but would appreciate the answer.
How would you calculate the return of the portfolio consisting of only equity and credit instruments? For example, consider only two assets S&P 500 and CDX IG and assume that they have equal weighting at 50%.
In order to get daily portfolio ret... |
I'm trying to apply the Kelly Criterion to poker. Poker players have been stuck using bankroll management techniques for decades, and I want to change that.
My goal is to graph the log growth of playing poker with respect to the size of your bankroll, given some edge or return distribution.
For most poker formats... |
I'm trying to apply the Kelly Criterion to poker. Poker players have been stuck using outdated bankroll management techniques for decades, and I want to change that.
My goal is to graph the log growth of playing poker with respect to the size of your bankroll, given some edge or return distribution.
For most poke... |
I'm trying to apply the Kelly Criterion to poker. Poker players have been stuck using outdated bankroll management techniques for decades, and I want to change that.
My goal is to graph the log growth of playing poker with respect to the size of your bankroll, given some edge or return distribution.
For most poke... |
I'm trying to apply the Kelly Criterion to poker. Poker players have been stuck using outdated bankroll management techniques for decades, and I want to change that.
My goal is to graph the log growth of playing poker with respect to the size of your bankroll, given some edge or return distribution.
For most poke... |
I have seen the question [here](https://quant.stackexchange.com/questions/68515/how-to-derive-forward-price-on-stock-with-continuous-dividend?noredirect=1&lq=1) and have gone through the answer, but I still don't fully understand why the approach below, based on no-arbitrage, yields a different answer.
To summarize... |
I have seen the question [here](https://quant.stackexchange.com/questions/68515/how-to-derive-forward-price-on-stock-with-continuous-dividend?noredirect=1&lq=1) and have gone through the answer, but I still don't fully understand why the approach below, based on no-arbitrage, yields a different answer.
To summarize... |
I have seen the question [here](https://quant.stackexchange.com/questions/68515/how-to-derive-forward-price-on-stock-with-continuous-dividend?noredirect=1&lq=1) and have gone through the answer, but I still don't fully understand why the approach below, based on no-arbitrage, yields a different answer.
To summarize... |
Well, just my opinion- I would say look at the weather through the window instead of trying to model meteological events ;) most interest rates are being fixed once a day on a Act/ 365 dcc . Why would you want to interpolate your IR curve in shorter dt than daily ? ; I gues it is more important to have your tenors ma... |
I was conviced to read Bergomis book on stochasic volatility to learn how options are traded in practice. He basically writes that the probabilisitc side is rather useless and that one only uses the PDE as an accouting tool, which makes sense to me given the difficulty to model stocks. Anyway, I am just on the 1st chap... |
I was conviced to read Bergomis book on stochasic volatility to learn how options are traded in practice. He basically writes that the probabilisitc side is rather useless and that one only uses the PDE as an accouting tool, which makes sense to me given the difficulty to model stocks. Anyway, I am just on the 1st chap... |
I was conviced to read Bergomis book on stochasic volatility to learn how options are traded in practice. He basically writes that the probabilisitc side is rather useless and that one only uses the PDE as an accouting tool, which makes sense to me given the difficulty to model stocks. Anyway, I am just on the 1st chap... |
I have seen the question [here](https://quant.stackexchange.com/questions/68515/how-to-derive-forward-price-on-stock-with-continuous-dividend?noredirect=1&lq=1) and have gone through the answer, but I still don't fully understand why the approach below, based on no-arbitrage, yields a different answer.
To summarize... |
I have seen the question [here](https://quant.stackexchange.com/questions/68515/how-to-derive-forward-price-on-stock-with-continuous-dividend?noredirect=1&lq=1) and have gone through the answer, but I still don't fully understand why the approach below, based on no-arbitrage, yields a different answer.
To summarize... |
I have seen the question [here](https://quant.stackexchange.com/questions/68515/how-to-derive-forward-price-on-stock-with-continuous-dividend?noredirect=1&lq=1) and have gone through the answer, but I still don't fully understand why the approach below, based on no-arbitrage, yields a different answer.
To summarize... |
I have seen the question [here](https://quant.stackexchange.com/questions/68515/how-to-derive-forward-price-on-stock-with-continuous-dividend?noredirect=1&lq=1) and have gone through the answer, but I still don't fully understand why the approach below, based on no-arbitrage, yields a different answer.
To summarize... |
I was conviced to read Bergomis book on stochasic volatility to learn how options are traded in practice. He basically writes that the probabilisitc side is rather useless and that one only uses the PDE as an accouting tool, which makes sense to me given the difficulty to model stocks. Anyway, I am just on the 1st chap... |
I have seen the question [here](https://quant.stackexchange.com/questions/68515/how-to-derive-forward-price-on-stock-with-continuous-dividend?noredirect=1&lq=1) and have gone through the answer, but I still don't fully understand why the approach below, based on no-arbitrage, yields a different answer.
To summarize... |
I am working with implied vol surfaces for money market options and trying to understand the dynamics of the CME's options on one-month SOFR futures. I want to establish whether trading ceases before commencement of the period over which the settlement rate is measured because, if not, the vol surface will need radical... |
I have the following code:
```
cashflows = pd.DataFrame({
'Nominal': cf.nominal(),
'AccrualStartDate': cf.accrualStartDate().ISO(),
'AccrualEndDate': cf.accrualEndDate().ISO(),
'AccrualPeriod': cf.accrualPeriod(),
'Price': cf.price(disco_yts),... |
This is not an answer but more a comment to the given answer, that I am happy to delete afterwards.
Did you compute this value? It explodes for spot below strike, is roughly 1 for ATMF and declines to 0 if spot is above strike. In the gif below, I set interest $r$ to 3% and no dividends $q$. The x-axes shows differ... |
I have a time series $Q={\{q_t\}}$ of known standard deviation $\sigma$, and its EMA of parameter $\alpha$ : $\{EMA_t(\alpha)\}$.
**My question is** : I'm looking for a formula that would give the standard deviation $\sigma_{\alpha}$ of the time series made by the difference between $Q$ and its EMA for any $\alpha$... |
Standard deviation of the difference between a time series and its EMA? |
Does the local volatility flattens (existing not forward) the skew faster than what we observed in the implied volatility surface?
The process is:
1. Get market implied volatilities
2. Fit a IV model (i.e. SVI)
3. Using IVs and Dupire equation obtain LV, price options and imply IV from those options.
What wou... |
Does the local volatility flattens the (existing not forward) skew faster than what we observed in the implied volatility surface?
The process is:
1. Get market implied volatilities
2. Fit a IV model (i.e. SVI)
3. Using IVs and Dupire equation obtain LV, price options and imply IV from those options.
What wou... |
BSM gives the following formula for option gamma
$$
\Gamma = \frac{e^{-qT-\frac{d_1^2}{2}}}{S\sigma\sqrt{2\pi T}}
$$
where
$$
d_1=\frac{\ln\frac{S}{K}+(r-q+\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}
$$
Then, to calculate its limit as $S\rightarrow 0^+$, the key is to calculate
$$
\lim_{S\rightarrow 0^+}\frac{e^{-... |
BSM gives the following formula for option gamma
$$
\Gamma = \frac{e^{-qT-\frac{d_1^2}{2}}}{S\sigma\sqrt{2\pi T}}
$$
where
$$
d_1=\frac{\ln\frac{S}{K}+(r-q+\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}
$$
Then, to calculate its limit as $S\rightarrow 0^+$, the key is to calculate
$$
\lim_{S\rightarrow 0^+}\frac{e^{-... |
Lets say I have an alpha generating model that forecasts expected returns for SP500 stocks. I formulate a portfolio with 100 stocks having the highest expected return. What is the simplest way of reducing tracking error relative to the S&P500? |
So I was wondering, how I can convert for example a 20 pips charge is(Spot: 1.0250 with pips 1.0270) on EURUSD into EURCNH Pips with (Spot EURCNH at 7.3005). Is there a general formula and short-cut?
Thinking from Trader perspective and someone wants to buy
Thank you very much in advance! |
PIP Value conversion - How can I convert my Pips? general formula |
Assuming no dividends, the put-call parity equation says:
$c + Ke^{-rT} = p + S$
where $c$ is the price of the European call, $p$ is the price of the European put, $S$ is the current stock price, $K$ is the option strike price, $r$ is the risk-free rate, $T$ is the time to expiry.
In *You Can Be a Stock Market... |
Please tell me the difference between CS01 and RST 1% (Relative spreads tightening by 1%) and how these two are used to monitor the credit flow traded product's exposures. Why would you use the spread tightening to monitor the risk? |
is there a way to quickly calculate the PV01 of for example for a swap EUR fix floating 10 years?
And is it possible to calculate the Notional, with a given PV01 and the years of the trades or duration?
Thank you very much in advance!
|
How to quickly calculate PV01? Or quickly calculate notional back given PV01 and duration? |
So this is very confusing. If you read online they say that:
> "SPY is an exchange-traded fund (ETF) that tracks the Standard &
> Poor's 500"
Now the word "track" is super confusing and sound passive. I may understand one of these:
1. If it track those 500 companies, that means they affect SPY, but SPY doe... |
How SPY ETF really works? |
While reading about asset pricing theory and numeraire, I had faced some confusion.
## Short summary of asset pricing theory from my book
1. We start our journey with a risky asset $S_t=\mu S_tdt+\sigma S_t dW_t$.
2. Define a risk-free stochastic process, $M_t=r_t M_t dt$ (as numeraire).
3. Form a relative pr... |
While reading about asset pricing theory and numeraire, I had faced some confusion.
## Short summary of asset pricing theory from my book
1. We start our journey with a risky asset $S_t=\mu S_tdt+\sigma S_t dW_t$.
2. Define a risk-free stochastic process, $M_t=r_t M_t dt$ (as numeraire).
3. Form a relative pr... |
If you have only two assets, with the same volatility and return, regardless the correlation, they should split 50%/50% no matter the methodology used (may deviate due to numerical procedure).
Call the first asset A and the second B, your algorithm gives you 70%/30% weights. As the assets are (for this algorithm) in... |
Edit: Can complete hold even if $d < u \le 1+R$ or $1+R \le d < u$ ?
---
In Tomas Björk's [Arbitrage Theory in Continuous Time][1] (or [here][2]), $\exists$ this proposition
> ![enter image description here][3]
It seems that to show that the model is complete, we must show that the claims are reachable, i.e... |
do you use the LGD as a risk management tool when trading credit cash bonds? Or it's more for trading the loan product. |
Consider the (liquid) options on a very liquid underlying, say the SPX option. Do the calls and puts all have the same strikes? If that is not generally true, what about the strikes close to the forward? |
Do calls and puts have the same strikes? |
Do you use the LGD as a risk management tool when trading credit cash bonds? Or it's more for trading the loan product. |
It will typically be on the net asset value (NAV) of the fund, which has a precise definition, and in your hypothetical case, is *around* the $10M value.
The long answer is: it depends on how it's legally drafted.
And your question broadly asks "how funds incur fee[s] onto its investors", so I should note that th... |
How do funds assess fees to investors? |
This question may seem basic but it is not. It concerns about the details of the market structure. So please do not rush to close it. Consider the (liquid) options on a very liquid underlying, say the SPX option. Do the calls and puts all have the same strikes? If that is not generally true, what about the strikes clos... |
I am going through the 2nd edition of Paul Wilmott on Quantitative Finance, and came across the following,
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/A2Y7T.png
Shouldn't the last equality be
$$
F(t;T) = y(t;T) + \color{red}{(T-t)}\frac{\partial y}{\partial T}
$$
or maybe... |
Typo in Wilmott's Forward rate formula? |
When you run PCA on some financial assets, let’s say stocks, do you calculate covariance on levels, standardized levels, returns or standardized returns?
I’ve seen several papers and posts that suggest to calculate covariance on the series that has been standardized (removed mean and divided by standard deviation) w... |
Is the a public list of all DAX constituents over time? I was thinking of a table like:
symbol, start_date, end_date
It would be nice if I didn't have to resort to webscraping to get at least the list of current stocks. |
This is a rather simple question, so this is maybe not the right place, but...
I have done quite a bit of reading on predicting asset returns, i.e. determining return expectations. I have now started reading some fundamental literatur on DCF valuation. For example, one might perform company valuation by doing as fol... |
I'm trying to get the basics of bonds by going from yield to price (and vice-versa hopefully).
What I want to do is from publicly available source go from the treasury bond yield to the price. So for example using MarketWatch I see:
[![2Y Treasury Data][1]][1]
So I assume the remaining coupons are paid 4 and pai... |
I'm trying to get the basics of bonds by going from yield to price (and vice-versa hopefully).
What I want to do is from publicly available source go from the treasury bond yield to the price. So for example using MarketWatch I see:
[![2Y Treasury Data][1]][1]
So I assume the remaining coupons are paid 4 and p... |
```
import QuantLib as ql
startdate = ql.Date(31,1,2020)
maturityDate = ql.Date(31,1,2021)
calendar = ql.NullCalendar()
tenor = ql.Period(ql.Monthly)
calendar.addHoliday(ql.Date(31,3,2020))
aschedule =ql.Schedule(startdate,maturityDate,tenor,calendar,ql.Following,ql.Unadjusted,ql.DateGeneration.Forward,T... |
Is End of Month (Eom) Rule overrides convention rule in Quantlib Schedule? |
I have implemented a Levenberg-Marquardt(LM) based method to calibrate the Heston model against market data by minimizing a weighted $L^2$-norm of differences of market vs model prices. Pretty standard stuff.
The calibration works pretty well and fits the market data as good as could be expected.
As a reference, th... |
Useful methods to avoid degenerate calibration? (Heston model in my case) |
For an US investor, buying an European index futures contract doesn't generate EURUSD exposure to the portfolio, since this trade is equivalent to borrowing EUR and buying the underlying.
Can I follow that same reasoning for commodity futures priced in EUR, for example carbon trading? Does trading it entail any FX e... |
**This post is Quant Stack Exchange's master list of data sources.**
Please append your links to other data sources to the list below.
## Economic Data ##
See [What are the most useful sources of economics data?][1] on Cross Validated SE.
**World**
- <https://macrovar.com/macrovar-database/> includes free... |
We all know that the traditional BS equation is:
$$\frac{\partial \mathrm V}{ \partial \mathrm t } + \frac{1}{2}\sigma^{2} \mathrm S^{2} \frac{\partial^{2} \mathrm V}{\partial \mathrm S^2}
+ \mathrm r \mathrm S \frac{\partial \mathrm V}{\partial \mathrm S}\ -
\mathrm r \mathrm V = 0$$
Suppose we want to wri... |
Is the a public list of all DAX constituents over time? I was thinking of a table like:
symbol, start_date, end_date
It would be nice if I didn't have to resort to webscraping to get at least the list of current stocks.
Edit:
Maybe it wasn't clear to the quick reader and editor: I was asking for the symbo... |
What you often obeserve in implied volatiltiy are higher levels of implied volatility for upcoming events like earnings or presentation of pharma data. For a human being which collects manual the information it is possbile to interpret this "spikes" or higher levels in implied volatility. Higher implied volatility in a... |
Identify upcoming stock price gaps in Implied Volatiltiy (quant / standardized approach)? |
This is a rather simple question, so this is maybe not the right place, but...
I have done quite a bit of reading on predicting asset returns, i.e. determining return expectations. I have now started reading some fundamental literature on DCF valuation. For example, one might perform company valuation by doing as fo... |
When you run PCA on some financial assets, let’s say stocks, do you calculate covariance on levels, standardized levels, returns or standardized returns?
I’ve seen several papers and posts that suggest to calculate covariance on the series that has been standardized (removed mean and divided by standard deviation) w... |
PCA on levels or returns, and standardized or not? |
>Black-Scholes volatility is average of local volatilities.
It is from:
https://bookdown.org/maxime_debellefroid/MyBook/all-about-volatility.html
First what's the meaning of `the average of all the paths between spot and the maturity and strike of the option`? Here does `path` mean the underlying path satisfying... |
I have read in a book from Emanuel Derman that Goldman Sachs manufactured a derivative in the early 90's that consisted of buying cheap puts on the Nikkei index (and paid in Yen) and combining them which a Yen-USD protection and resold them to clients. This construction was called the "Kingdom of Denmark" puts.
To h... |
>Black-Scholes volatility is average of local volatilities.
It is from:
https://bookdown.org/maxime_debellefroid/MyBook/all-about-volatility.html
First what's the meaning of `the average of all the paths between spot and the maturity and strike of the option`? Here does `path` mean the underlying path satisfying... |
>Black-Scholes volatility is average of local volatilities.
It is from:
https://bookdown.org/maxime_debellefroid/MyBook/all-about-volatility.html
First what's the meaning of `the average of all the paths between spot and the maturity and strike of the option`? Here does `path` mean the underlying path satisfying... |
What I saw in the references is
>Heston model can matches market option prices perfectly and SABR cannot.
Is it correct? But for my understanding, a model matches market option prices perfectly only when its equivalent local vol model should strictly follow Dupire equation (Dupire local vol model). |
Does Heston and SABR match market vol smile perfectly (arbitrage free)? |
I confused about the correct formula to compute monthly realized variance from daily data? What is the first sigma in the picture: sum or average? I mean after subtracting each observation from monthly mean and then squaring each raw , should I just take the sume for each month or take average?
[![formula of month... |
I am confused about the correct formula to compute monthly realized variance from daily data. What is the first sigma in the picture: sum or average? I mean, after subtracting each observation from monthly mean and then squaring each difference, should I just take the sum for each month or take the average?
[![formu... |
What are the sources one can search for or view / download research articles and other publications on quantitative finance in addition to the Internet search engines? |
The question mixes up various concepts:
* Exact fit of model to market prices. Neither Heston nor SABR can claim to fit exactly market prices. There are two reasons: both have a limited number of parameters (5 for Heston, 4 for SABR), and their corresponding models further restrict the shapes attainable with those p... |
I am reading Advanced Equity Derivates by Sebastien Bossu and trying to do the exercises.
In chapter 1 we have the following question :
Consider an exotic option expiring in one, two, or three years on an underlying
asset S with the following payoff mechanism:
* If after one year S1 > S0 the option pays off 1 + ... |
Good morning,
this is my first question on this forum, I'm writing from Milan (Italy) and I have a question about a University Problem.
The problem is about entering in a Long Range Forward (buy a call option and sell a put option). Now, the goal is to calculate the Strike of the option and get the volatility smile w... |
Is it possible to have only one volatility surface for american options (that fits both calls and puts)? |
We are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $\mathbb P$-Brownian motion.
Let $dX_t = \theta_tdt +dW_t$ be an Ito process where $(\theta_t)_{t \in [0,T]}$ is $\mathscr{F}... |
There is also a related question on the Economics site:
https://economics.stackexchange.com/questions/4679/what-are-some-good-repositories-for-economic-data
Answers from there: The [American Economic Association has a list of resources for economists, including a page for data][1]. There you find links to many inst... |
By [Bayes' rule for conditional expectation][1] (or [here][2]),
$$E_{\mathbb Q}[X_t | \mathscr F_u] E[L_T| \mathscr F_u] = E[X_tL_T| \mathscr F_u]$$
$$ \to E_{\mathbb Q}[X_t | \mathscr F_u] L_u = E[X_tL_T| \mathscr F_u]$$
$$\to E_{\mathbb Q}[X_t | \mathscr F_u] = \frac{1}{L_u} E[ \frac{X_tL_t}{1} | \mathscr F_... |
I am coming from a machine learning/time series forecasting background and are currently studying Asset Pricing.
I have a good understanding of what Markowitz Mean-Variance Optimization (MVO) does, and have read that CAPM results directly from MVO and it's hypotheses e.g. the Efficient Market Hypothesis (EMH).
S... |
I am coming from a machine learning/time series forecasting background and are currently studying Asset Pricing.
I have a good understanding of what Markowitz Mean-Variance Optimization (MVO) does, and have read that CAPM results directly from MVO and it's hypotheses e.g. the Efficient Market Hypothesis (EMH).
S... |
In the well-known article by Mr. Piterbarg "Funding Beyond Discounting". he demonstrates that the price of a derivative product in a multi-curve universe:
[![enter image description here][1]][1]
Who also expresses it but without demonstrating it as:
[![enter image description here][2]][2]
[1]: http... |
FVA demonstration? |
I am trying to use Quantlib to price FloatingRateBond, Thanks to some other posts on the stack I managed to do it for Ibor indexes. I tried to use the SOFR, but I didn't succeed, I couldn't get around the SOFR being fixed at the end of the period as opposed to Ibor, as well as the compounding of the fixing. Is there a ... |
Does Quantlib FloatingRateBond support SOFR index? |
I was trying to understand why the Black and Scholes PDE for the value of an option, $V (F , t)$, with the forward price, $F$, as underlying is
$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2F^2\frac{\partial ^2V}{\partial F^2} - rV = 0$$
Why does the delta term vanish? |
To get a better understanding, I tried going through BSM-model starting from a non-geometric brownian motion. However, during the derivation I got stuck, which led me to a specific question.
The set-up:
$dS_t = \mu dS_t + \sigma dW_t^P$
We want $e^{-rt}S_t $ to be a martingale. Applying Ito, Girsanov and ... |
What happens if I try pricing derivatives starting from a non-geometric brownian motion? |
What happens trying to price derivatives starting from a non-geometric brownian motion? |
I am starting to explore this area. My ultimate aim is to build a 3 month LIBOR forward curve.
1. I wish to know what exactly 'Dual Curve Bootstrapping' is (If someone could explain it in clear words).
2. What are the 2 curves I will be using for coming up with 3Month Libor forward curve
3. Can this process be... |
To get a better understanding, I tried going through BSM-model starting from a non-geometric brownian motion. However, during the derivation I got stuck, which led me to a specific question.
The set-up:
$dS_t = \mu dt + \sigma dW_t^P$
We want $e^{-rt}S_t $ to be a martingale. Applying Ito, Girsanov and en... |
Good morning,
Next week I'll have Derivates Final test and I've a doubt about Bond Option.
If I have one ZCB, price 90 with 3 month maturity and strike 100, and I want a minimum yield of 2%, what type of option I have to buy?
Is it correct think that if I've a ZCB and I want to make a floor for yield I've to bu... |
Can someone enlighten me regarding the market convention for quoting an overnight fx swap where one leg is USD and today is a USD-holiday (but not a holiday in the other currency)? An example is attached below from bbg where today = 24/11 (=thanksgiving holiday in the US). It seems there is a quote for an o/n fx swap f... |
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