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This is more of an academic question. The results are SO close, I think they are ACTUALLY THE SAME FORMULAS. So someone published a paper with a "new" method to adjust Kirk's formula to make it more accurate, and the reality is, I think it's just a rearranged version of another paper's formula. Or that's what it app... |
This is more of an academic question. The results are SO close, I think they are ACTUALLY THE SAME FORMULAS. So someone published a paper with a "new" method to adjust Kirk's formula to make it more accurate, and the reality is, I think it's just a rearranged version of another paper's formula. Or that's what it app... |
This is more of an academic question. The results are SO close, I think they are ACTUALLY THE SAME FORMULAS. So someone published a paper with a "new" method to adjust Kirk's formula to make it more accurate, and the reality is, I think it's just a rearranged version of another paper's formula. Or that's what it app... |
This is more of an academic question. The results are SO close, I think they are ACTUALLY THE SAME FORMULAS. So someone published a paper with a "new" method to adjust Kirk's formula to make it more accurate, and the reality is, I think it's just a rearranged version of another paper's formula. Or that's what it app... |
I'm reading Shreve Stochastic Calculus II , theorem 5.4.9 (Second fundamental theorem of asset pricing) ,
This is the part that confuses me :
"suppose there is only one risk-neutral mesure. This means first of all that the filtration for the model is generated by the d dimensional Brownian m... |
What type of debt instruments can be used to estimate short-rate model parameters?
How can I find the history of short-rate?
My guesses are overnight LIBOR/SOFR or extrapolation through the yield curve fitted on T-bills.
According to my understanding yield curve may also be used. In the case of discrete-time, it... |
What instruments can be used to calibrate short-rate models? |
I have owned a penny stock for a while now that typically trades between \$0.0002 and \$0.0003. The thing is, though, that it most frequently trades BETWEEN those two price points and doesn't actually hit them very often. The platform I use to trade (along with any others I've so far investigated) only allow trading ou... |
This is more of an academic question. The results are SO close, I think they are ACTUALLY THE SAME FORMULAS. So someone published a paper with a "new" method to adjust Kirk's formula to make it more accurate, and the reality is, I think it's just a rearranged version of another paper's formula. Or that's what it app... |
This is more of an academic question. The results are SO close, I think they are ACTUALLY THE SAME FORMULAS. So someone published a paper with a "new" method to adjust Kirk's formula to make it more accurate, and the reality is, I think it's just a rearranged version of another paper's formula. Or that's what it app... |
Since options contracts are created by open interest in the contract, it is conceivable that the notional of the total options contracts can exceed the value of the underlying. If that happens, does the underlying become the derivative of the options?
This is more than just a theoretical question. I have started t... |
When does the underlying become the derivative? |
Since options contracts are created by open interest in the contract, it is conceivable that the notional of the total options contracts can exceed the value of the underlying. If that happens, does the underlying become the derivative of the options?
This is more than just a theoretical question. I have started t... |
the autocorrelation is the correlation of a process $X$ and its lagged version, hence you have to consider it from a probabilistic viewpoint.
Use the $\sigma_\ell$ notation for the operator that shifts a process:
$${\cal A}(X;\ell):=\frac{\mathbb{E}\big((X - \mathbb{E}X) \cdot (\sigma_\ell\circ X - \mathbb{E}\sigma_... |
https://school.stockcharts.com/doku.php?id=technical_indicators:moving_averages
It says "First, calculate the simple moving average for the initial EMA value. An exponential moving average (EMA) has to start somewhere, so a simple moving average is used as the previous period's EMA in the first calculation."
Why ... |
There seems to be some newer papers on this, though maybe not a consensus on the approach?
https://link.springer.com/chapter/10.1007/978-3-030-89477-1_99
https://www.sciencedirect.com/science/article/abs/pii/S1062940821001169 |
This market-making question comes from a prop trading final round which I failed.
I was told to make a market on the number of prime numbers between 1-100.
I was confident the number was around 20-30. (It's 25). So I made my market at 20-25. (Interviewer only allowed a spread of 5). They proceeded to buy my sell ... |
the autocorrelation is the correlation of a process $X$ and its lagged version, hence you have to consider it from a probabilistic viewpoint.
Use the $\sigma_\ell$ notation for the operator that shifts a process:
$${\cal A}(X;\ell):=\frac{\mathbb{E}\big((X - \mathbb{E}X) \cdot (\sigma_\ell\circ X - \mathbb{E}\sigma_... |
That is a very funny interview question, and I will keep my answer related to true quant finance questions about market making.
1. if you think about [Kyle's model][1], that is very basic but contains the essence of market making when the MM is confronted to informed and noise traders:
- as a MM, you try to gues... |
Optimizing collateral is a hot topic in the financial industry. I came across the term *cheapest collateral*. What does it actually mean in the context of collateral optimization, please ? |
What is the definition of "cheapest collateral"? |
**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... |
Is there a research showing a way to use SABR LMM with new RFRs such as SOFR, i.e. pricing exotic path-dependent RFR derivatives with volatility smile and skew?
I'm aware that
- [Looking Forward to Backward-Looking Rates: A Modeling Framework for Term Rates Replacing LIBOR][1] and [Looking Forward to Backward-Lo... |
There seems to be some newer papers on this, though maybe not a consensus yet on the approach?
An Optimal Investment Portfolio Constructed with Fractal Analysis and Long Memory Models, by Robert Garafutdinov (Chapter in conference proceedings book, 2021)
https://link.springer.com/chapter/10.1007/978-3-030-89... |
Going through Duffie and Kan (1996) I came up with a question.
After Eq. $(3.4)$ the claim the following:
> By $(2.4)$, we also know that $\mathcal{D} F (X_t, t) - R(X_t) F(X_t, t) = 0$. Since $F$ is strictly positive valued, from $(3.4)$ we have [...].
and then they equate what they claim together with equat... |
From my understanding, there are (most generally speaking) two approaches for bootstrapping the yield curve (with an exact method). We can either interpolate between the market quotes (interbank deposits, futures, FRAs, swaps, etc.) and then infer the discount factors, this would not require a minimization technique as... |
Let me post an answer to my own question in case someone would need it...
*> Should I first obtain a caplet volatility surface, i.e. do what is known as a spot volatility stripping from given flat volatilities, and then calibrate SABR on caplets since these are the "real" options?*
This is the right approach. Fir... |
In the Vanna-Volga (VV) paper by Castagna and Mercurio they state that, once you build up a curve of prices by interpolating-extrapolating on $K$, you can recover the same exact curve by redefining the curve, now using three prices result of using three new different strikes and the VV method:
> We now state two imp... |
I'm doing a study at Rutgers on the TUT spread. The TUT spread is composed of 2 2-year treasuries and 1 10-year treasury per spread. I was trying to estimate the historical volatility of the spread, and only have historical volatility of each product individually.
Would the spread volatility just be the sum of the ... |
Is there a modified Bachelier's futures spread option model with adjustments for skew and kurtosis? |
Going through [Duffie and Kan (1996)][1] I came up with a question.
After Eq. $(3.4)$ the claim the following:
> By $(2.4)$, we also know that $\mathcal{D} F (X_t, t) - R(X_t) F(X_t, t) = 0$. Since $F$ is strictly positive valued, from $(3.4)$ we have [...].
and then they equate what they claim together with ... |
Going through [Duffie and Kan (1996)][1] I came up with a question.
After Eq. $(3.4)$ the claim the following:
> By $(2.4)$, we also know that $\mathcal{D} F (X_t, t) - R(X_t) F(X_t, t) = 0$. Since $F$ is strictly positive valued, from $(3.4)$ we have [...].
and then they equate what they claim together with ... |
I can't wrap my head around how to determine the interest rates to calculate the forward rates of any currency. At this point, I don't even know if this data is actually available to do the calculation myself.
From Investing.com ([link][1]) I wish to determine the 1Y and 1M forward rate for starters. I figured out t... |
I have a couple of comments to add to your question. First, are you taking into account the time component in your formula? It looks like you're computing the forward rate as
$$ S_T = \dfrac{1 + r_d}{1 + r_f} S_0 ,$$
where what it should read is
$$ S_T = \dfrac{1 + r_d T}{1 + r_f T} S_0 .$$
Moreover, note that ... |
Suppose I am willing to buy a contract which I believe has a 15% chance to settle to $100 and 0 otherwise. The EV of this contract is therefore 15. How much should I buy this for?
I would answer at most 15 (in order to ensure I make money on this contract). But going beyond this, how can the price on (0, 15) be dete... |
How should the spread be determined after calculation of expected value? |
Below is a problem I did. I am hoping somebody can confirm I did it correctly, or tell me where I went wrong.
Problem:
ABC Corp. has just paid a dividend of \$3 per share. You—an experienced analyst—feel
quite sure that the growth rate of the company’s dividends over the next 10 years will be
15\% per year. A... |
Below is a problem I did. I am hoping somebody can confirm I did it correctly, or tell me where I went wrong.
Problem:
ABC Corp. has just paid a dividend of \$3 per share. You—an experienced analyst—feel
quite sure that the growth rate of the company’s dividends over the next 10 years will be
15\% per year. A... |
I'm trying to learn to bootstrap and am taking some bonds from the Treasury curve:
https://docs.google.com/spreadsheets/d/1vA7s4ZfFzGfTji_d9cLUid5rqyaugRrI0etCW_3Jb6w/edit?usp=sharing
For some reason the zero rate I'm ending up with for the 1.5 year bond (cell B8) is less than the YTM of that bond (cell F4). Thi... |
Would it make sense to calculate the Sharpe Ratio with the semi-standard dev. So as to standardize/compare asset returns to their downside risk? |
I'm trying to learn to bootstrap and am taking some bonds from the Treasury curve:
https://docs.google.com/spreadsheets/d/1vA7s4ZfFzGfTji_d9cLUid5rqyaugRrI0etCW_3Jb6w/edit?usp=sharing
For some reason the zero rate I'm ending up with for the 1.5 year bond (cell B19) is less than the YTM of that bond (cell F4). Th... |
I'm trying to come up with a measure for the volatility risk premium (VRP) for a strategy I want to implement, but I'm not entirely sure how to proceed. My situation is as follows.
1) The underlying is a futures contract that expires yearly (December).
2) There are four option contracts on this futures contract e... |
I am working on a pricer for an Autocallable product in Python. Coupons are accumulated at each observation and paid once the AC barrier is reached. At maturity, short down-and-in put.
I am trying to price it with local volatility model, but have some trouble to implement it. I have several questions :
- I guess I wo... |
I was reading some papers on delta-hedged option returns and came across an intriguing paper that I found quite interesting.
However, I was a bit confused on the authors' methodology of computing option returns for their strategies. For some context, the paper is looking at the returns of a gamma selling option stra... |
I was reading some papers on delta-hedged option returns and came across an intriguing paper that I found quite interesting.
However, I was a bit confused on the authors' methodology of computing option returns for their strategies. For some context, the paper is looking at the returns of a gamma selling option stra... |
I need to test strike (butterfly) arbitrage on a **discrete** implied volatility grid.
I know that the traditional procedure for continuous case is (for a given maturity T):
1) See the Dupire formula in terms of implied volatility (and not in terms of option prices). You can find it [here][1].
2) Check if the deno... |
I'm not totally confident on how to merge these two monthly CRSP data sets. As I write this, it comes from two databases: `crsp.mse` and `crsp.msf` (the documentation on [their site](https://wrds-www.wharton.upenn.edu/pages/get-data/center-research-security-prices-crsp/annual-update/stock-security-files/monthly-stock-f... |
I'm reading Shreve Stochastic Calculus II, theorem 5.4.9 (Second fundamental theorem of asset pricing),
This is the part that confuses me :
> *suppose there is only one risk-neutral measure. This means first of all that the filtration for the model is generated by the $d$-dimensional Brownian... |
According to the first bullet point in [this][1] explanation, "In the stock market, open interest is the number of buy orders submitted before the market opens. When the open interest is high, people are ready to add shares to their positions or initiate new positions, which means that the stock is likely to go up in p... |
I have set up a mean variance optimization problem,
$$min:{W}^{\prime}{\Sigma_{\varepsilon}{W}}$$
$$s.t:{W}^{\prime}{\alpha}=R_B\;,\;\;W^{\prime}l={1},\;\;W'\beta=0,\;\;W'Z=\beta_p$$
where, $W$ is an ($n+1$)vector and the last value ($W_{n+1}=\sum_{i=1}^{n}w_i\beta_i$ ),
$\;\Sigma_{\varepsilon}$ is the covari... |
You are mixing up your time fractions. you used 0.5 and 1 for 6M and 1Y respectively, but calculated the 18M rate using actual year fractions (~1.545) instead of 1.5. Hence the lower value.
You should instead calculate the PV of the coupons using the actual year fractions for 6M and 1Y (~.49 and ~.93 resp.) and then... |
You are mixing up your time fractions. you used 0.5 and 1 for 6M and 1Y respectively, but calculated the 18M rate using actual year fractions (~1.545) instead of 1.5. Hence the lower value.
You should instead calculate the PV of the coupons using the actual year fractions for 6M and 1Y (~.49 and ~.93 resp.) and then... |
I'm interested in modeling NFT Floor Price. Specifically, I'm trying to answer the question:
>**Given current bid-ask info on an NFT collection, what is the probability distribution of the lowest ask on any NFT in the collection at some future time t?**
So far, I've tried:
1. Searching for existing literatu... |
Recently I study some interest rate models.
When I moved on to forward rate models, I see this documents
https://en.wikipedia.org/wiki/Heath-Jarrow-Morton-_framework
It said "***HJM-type models capture the full dynamics of the entire forward rate curve, while the short-rate models only capture the dynamics of ... |
Random thought I had around what would be an ideal analytical basket formula. If the formula gave prices of each asset instead of a single basket price, you could price any number of exotic payoffs. Which would in theory make it as useful as MC.
Could you emulate the same with closed form solutions that are alread... |
Has anyone ever derived an analytical basket option which gives asset prices individually? |
Random thought I had around what would be an ideal analytical basket formula. If the formula gave terminal prices of each asset instead of a single basket price, you could price any number of exotic payoffs. Which would in theory make it as useful as MC.
Could you emulate the same with closed form solutions that a... |
Random thought I had around what would be an ideal analytical basket formula. If the formula gave terminal prices of each asset instead of a single basket price, you could price any number of exotic payoffs. Which would in theory make it as useful as MC.
Could you emulate the same with closed form solutions that a... |
Random thought I had around what would be an ideal analytical basket formula. If the formula gave terminal prices of each asset instead of a single basket price, you could price any number of exotic payoffs. Which would in theory make it (nearly) as useful as MC.
Could you emulate the same with closed-form solutio... |
I am looking into BRC's, and I keep reading about their relatively high coupon rates which are pre-determined by the issuer. However, I can't seem to find any good resources on HOW they pre-determine the coupon rate.
It seems to be strongly related to the valuation of the down-and-in put option on the underlying as... |
How does the issuer of a Barrier Reverse Convertible determine the coupon? |
Given that $S(T)$ is the value of an asset in a foreign currency, $X(T)$ is the spot domestic/foreign rate, $P_t$ is the value of a Portfolio in the the domestic currency (invested in $S$) and $z^{f}$ and $z^{d}$ are Discount Factors (foreign and domestic),
could somebody explain me what $P_t$ is, given that $ \frac{d... |
I'm looking for historical end of day price data for non US-stocks. Preferably as bulk download for analysis, but an API would do also. Specific markets of intrest:
- China
- Japan
- Major European markets: UK, France, Germany
- India
- Canada
- Korea
- Australia
- Any others would be great
Does a ... |
Random thought I had around what would be an ideal analytical basket formula. If the formula gave terminal prices of each asset instead of a single basket price, you could price any number of exotic payoffs. Which would in theory make it (nearly) as useful as MC.
Could you emulate the same with closed-form solutio... |
Has anyone ever derived an analytical basket option which gives terminal asset prices individually, by asset? |
How does one go about properly backtesting and visualising their strategy using their trading signals and historical prices where the trading signals are 1 for long, 0 for no position -1 for short?
Obviously the end result should look something like this, this visual originates from a simple moving average strategy ... |
I'm looking for historical end of day price data for non US-stocks. Preferably as bulk download for analysis, but an API would do also. Specific markets of interest:
- China
- Japan
- Major European markets: UK, France, Germany
- India
- Canada
- Korea
- Australia
- Any others would be great
I'm us... |
From Colin Bennet's trading volatility (pg 117), he says:
"A forward on a volatility future is short vol of vol. This means it is possible to back out the implied vol of vol from the price of this volatility future. This implied vol of vol can be used to price options on variance or even options on volatility future... |
I'm reading the stochastic local volatility model literature, e.g., the Heston Stochastic Local Volatility model (https://ir.cwi.nl/pub/22747/22747D.pdf); but I'm a bit unsure about its calibration and pricing.
It seems to me that the calibration and pricing of an LSV model consist of the following steps:
1. Use ... |
Great question. Let me try to provide some insights and thoughts regarding the points and questions you raised. It may not be a full answer but hopefully it will help connecting the contents in the paper/book with some trading intuition:
1) From a theoretical perspective, I don't see any mistake in your thinking reg... |
I have closing prices for multiple equities in NumPy arrays (or a pandas timeseries DataFrame).
I like to calculate the % change numbers for the closing prices in the DataFrame the way it is shown in mainstream charting platforms like the one below:
[![enter image description here][1]][1]
Is there any common ... |
Can NumPy calculate the % change the way it is shown in multiple instrument charts? |
According to the following thread:
https://quant.stackexchange.com/questions/12886/how-can-i-calculate-the-strike-price-or-implied-volatility-from-a-given-delta
To back out some strike given some Delta, you simply use realized vol (plus a few other inputs) and within a few steps, you can convert any Delta to a sp... |
According to the following thread:
https://quant.stackexchange.com/questions/12886/how-can-i-calculate-the-strike-price-or-implied-volatility-from-a-given-delta
To back out some strike given some Delta, you simply use realized vol (plus a few other inputs) and within a few steps, you can convert any Delta to a sp... |
Theoretically, the answer to the question is yes, a correlation matrix for potential pairs trades can be computed in $O\left((n^2t)^{(\omega+\epsilon)/3}\right)$ time, for any $\epsilon > 0$, where $\omega < 2.38$ is the so-called <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.1070"><b>exponent of... |
Is there any research on applying state-space or dynamic linear models to forecasting equity risk premia on a security-by-security basis with a medium term horizon (say 3 month to 12 months horizon)?
Markov Models are a special case of state-space models where the states are discrete. There's an avalanche of resear... |
If I have a forecasted volatility of the log returns of say, 0.03, this is obviously transformed relative to the log I took of the returns. It strikes me that I should raise `e` to the power of the volatility I forecasted to in order to get back something "normal" looking. When I do this I get `~1.03`, which seems real... |
Amidst bankruptcy situation in Sri-lanka, India is said to have come to rescue Sri-lanka, and entered into a currency swap deal as referenced here - https://www.cbsl.gov.lk/en/news/cbsl-enters-into-bilateral-currency-swap-agreement-with-rbi.
My question is, how exactly this type of deals help the donar country? With... |
Resampling is a popular method for portfolio optimization. We repeatedly draw samples from a distribution, compute the optimal mean-variance portfolio and finally average over all allocations.
However, from a mathematical point of view, I do not understand why we would gain anything from this procedure. Have we not ... |
What's the point of resampling? |
The books "The Elements of Statistical Learning" by Trevor Hastie, and "Advances in Financial Machine Learning" by Lopez De Prado are highly recommended books for ML. They both deal with machine learning algorithms, and the statistics involved in ML algorithms. De Prado's book deals with ML for finance while Trevor Has... |
[This](https://www.sec.gov/files/ib_interestraterisk.pdf) SEC document claims that increasing the ocupon on a bond *decreases* the interest rate risk (bottom of page 3):
[![enter image description here][1]][1]
And the Finra SIE exam states the same also.
I cannot understand the logic behind this statement, it ... |
How are leveraged futures paid in the case of large order-book spread? |
What's the optimal way to size a limit order? |
Resampling is a popular method for portfolio optimization. We repeatedly draw samples from a distribution, compute the optimal mean-variance portfolio and finally average over all allocations.
However, from a mathematical point of view, I do not understand why we would gain anything from this procedure. Have we not ... |
The short answer is “no.” The long answer is, “it depends on what you intend to do with the ranking.”
First, let us ignore asset classes because you are assuming something that is a priori impossible. We never know the actual population parameters. If we did, we would not need the field of statistics. Nonetheles... |
The short answer is “no.” The long answer is, “it depends on what you intend to do with the ranking.”
First, let us ignore asset classes because you are assuming something that is a priori impossible. We never know the actual population parameters. If we did, we would not need the field of statistics. Nonetheles... |
I'm looking for a (free) paper I read which showed that even a "perfect" market timing strategy wasn't very good compared to buy-and-hold. There were some restrictions to the timing, something like: you buy at the lowest point between all time highs and sell at each all time high...and they concluded that it still wasn... |
Paper on returns from perfect market timing? |
I'd like to optimize a bond portfolio with different bond classes (government bonds, corporates, ...) and different ratings as well as maturities. Is this even possible to optimize such a portfolio? And how should I do that in practice? Suggestions or recommendations for research or similar would be very helpful. Thank... |
How can I optimize a Bond Portfolio in Practice? |
How to break down yield to maturity to different components? |
Fidelity now offers direct indexing for individuals for 40 bps per year, as well as zero-fee total market index funds. Which one should I dollar cost average (I.e., invest $x every n-th month) in? Does the problem boil down to an average correlation / dispersion parameter?
Ignore the tax benefits of direct indexing.... |
Which is more valuable: a basket of optimal stopping problems or an optimal stopping problem on a basket? |
I think I understand. You are trying to calculate the IRR of the a-cash flows and the b-cash flows individually ? But there are multiple solutions: you can partition the PV into PV(a) and PV(b) and solve for IRRs of a and b separately with only the constraint that PV= PV(a)+PV(b). |
I know what PB ratio is and I am looking forward to calculate the sector PB ratio.
Here's the sector PB ratio of State Bank of India is **2.23**. [Link](https://www.tickertape.in/stocks/state-bank-of-india-SBI?checklist=basic&chartScope=1d)
![Screenshot (70)](https://user-images.githubusercontent.com/103634312/1654... |
Does it matter that Bachelier IV differs from BS IV for a given price?
In one sense, yes, obviously: which is the hurdle realised volatility?
In another sense, no, it’s just an accounting convention.
Which is correct? |
In one sense, it’s just an accounting convention, so it doesn't matter. In another sense, the implied volatility can be interpreted as the minimum realised volatility which implies that your option price was ≤ fair value (realized via dynamic hedging/gamma scalping, see https://quant.stackexchange.com/questions/39619/g... |
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