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I noticed that in the pair trading, some measure the spread on pricing and some measure the spread on return. Both methods make sense, it is intuitve to model price, but convergence of return means long stronger return and short weaker return which also leads to profit.
Has anyone compared the benefit of pair... |
I'm studying a draft of the paper “[Dealing with the Inventory Risk:
A solution to the market making problem)][1]” by Guéant et al from July 2012.
According to the paper, the closed form solution to the optimal control problem is:
$
\delta_{\infty}^{b *}(q) \simeq\frac{1}{\gamma} \ln \left(1+\frac{\gamma}{k}\... |
What are the parameters’ units in the Avellaneda and Stoikov model? |
I am a complete novice with a background in physics, currently self-studying derivatives. My primary reading resources are John Hull's book and Introduction to the Economics and Mathematics of Financial Markets by Jvaska Cvitanic.
I have a confusion regarding Marking to market.
Suppose $A$ takes a long position on... |
Why aren't the prices discounted when futures are marked to market? |
I am a complete novice with a background in physics, currently self-studying derivatives. My primary reading resources are John Hull's book and "Introduction to the Economics and Mathematics of Financial Markets" by Jvaska Cvitanic.
I have a confusion regarding marking-to-market. Suppose $A$ takes a long position o... |
Why aren't the prices discounted when futures are marked-to-market? |
When considering the value of a futures contract it often helps to take a step back to fundamentals. What is the market value of your contract? It's almost a trick question because you can observe the price directly on the exchange (ignoring any bid/ask spreads).
The day-on-day difference in market value of your con... |
How is implied volatility on future treasuries used to expect Forex volatility? |
I'm studying a draft of the paper “[Dealing with the Inventory Risk:
A solution to the market making problem][1]” by Guéant et al from July 2012.
According to the paper, the closed form solution to the optimal control problem is:
$
\delta_{\infty}^{b *}(q) \simeq\frac{1}{\gamma} \ln \left(1+\frac{\gamma}{k}\r... |
Before Google Finance changed its service, I remember I used to be able to see news or technical events on historical charts. It was beneficial for me as I was able to see the impact of the news event on the price.
AFAIK, Google Finance does not provide the feature anymore.
Is there any alternative option tha... |
Consider a standard geometric Brownian motion $V_t$ with drift $\mu<r$ and standard deviation $1$.
It holds that the discounted expectation is
$$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right] = \frac{V_t}{r-\mu}$$
Does a similar result hold for any $\mathbb{C}^2$ function? That is consider a function th... |
First of all, I realise this question might border on `meta-finance', so I'd be totally OK if it gets closed.
Having said that, the question itself:
Given a stock $S$, in the absence of default it is quite clear that the probability the stock price at time $T$ takes values in $(0,\infty)$ is $1$ for all $T$.
I... |
First of all, I realise this question might border on `meta-finance', so I'd be totally OK if it gets closed.
Having said that, the question itself:
Given a stock $S$, in the absence of default it is quite clear that the probability the stock price at time $T$ takes values in $(0,\infty)$ is $1$ for all $T$.
I... |
I am building a model to simulate people's wealth in the next years.
Say Mr X has a portfolio with an expected return of 3% (annual). From this I can simulate the return of his portfolio in the next 40 years using Geometric Brownian Motion.
I also want him to be able to add inflows/outflows (e.g. Mr X expects a sa... |
Consider a standard geometric Brownian motion $V_t$ with drift $\mu<r$ and standard deviation $1$.
It holds that the discounted expectation is
$$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right] = \frac{V_t}{r-\mu}$$
Does a similar result hold for any $\mathbb{C}^2$ function? That is consider a function th... |
Consider a standard geometric Brownian motion $V_t$ with drift $\mu<r$ and standard deviation $1$.
It holds that the discounted expectation is
$$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right] = \frac{V_t}{r-\mu}$$
provided that $r > \mu$ and diverges for $r < \mu$.
Does a similar result hold for any $\... |
One of the measures to quantify credit risk is to calculate the Expected loss, which is typically quantifies as $EL = EAD \times PD \times LGD$
However, I have come across a somewhat strange calculation which goes as below.
Let say we have a exposure of $ \$1 $. Calculation time is today at $T_0$, and we are inte... |
I'm reading De Nard et al. paper ["Portfolio Selection in Large Dimensions"][1] and the authors talk about these two models, but I don't have any background on these topics from university. Can you help?
[1]: https://093116088071089126067085065087098089033069066016061000078026120065002103123097023000063039024035... |
Can someone briefly explain me what's the difference between Exact Factor Model (EFM) and Approximate Factor Model (AFM)? |
Consider a standard geometric Brownian motion $V_t$ with drift $\mu<r$ and standard deviation $1$.
It holds that the discounted expectation is
$$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right] = \frac{V_t}{r-\mu}$$
provided that $r > \mu$ and diverges for $r < \mu$.
Does a similar result hold for any $\... |
I came across the Vola Dynamics page about the W-shaped vol before an event: https://voladynamics.com/marketEquityUS_AMZN.html
I'm a bit confused by "this term does not have any butterfly arbitrage". I thought butterfly arbitrage suggests that price against strike is convex, i.e., $\partial^2 C/\partial K^2 > 0$. Bu... |
I'm reading De Nard et al. paper ["Portfolio Selection in Large Dimensions"][1] and the authors talk about these two models, but I don't have any background on these topics from university. Can you help?
[1]: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3194492 |
I am reading paper High-frequency trading in a limit order book by Marco Avellaneda and Sasha Stoikov. At the end of the paper they obtain a closed-form solution to the optimal market-maker quotes under diffusion without drift. They found that the optimal behaviour of the market-maker would be to set a bid/ask spread o... |
Let us suppose for concreteness that the 10y swap rate is 0.5% today and was 7% a year and and 6.5% a "year minus a day" ago...
>reprice the swaps for each historical scenario and calculate returns as the difference between the swaps PV from each scenario and the today's PV
This won't help you at all, really. Tak... |
Let us suppose for concreteness that the 10y swap rate is 0.5% today and was 7% a year and and 6.5% a "year minus a day" ago...
>reprice the swaps for each historical scenario and calculate returns as the difference between the swaps PV from each scenario and the today's PV
This won't help you at all, really. Tak... |
Consider a standard geometric Brownian motion $V_t$ with drift $\mu<r$ and standard deviation $1$.
It holds that the discounted expectation is
$$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right] = \frac{V_t}{r-\mu}$$
provided that $r > \mu$ and diverges for $r < \mu$.
Does a similar result hold for any $\... |
Let $X$ be any random variable with any distribution. Given that we know $\mathbb{E}[X]$, $\mathbb{E}[\max(0,X)]$, and $\mathbb{E}[\min(0,X)]$, can you write a formula for $\mathbb{E}[f(X)]$ where $f$ is any twice differentiable function?
I am guessing the solution has something to do with the arbitrage free pricing... |
Let $X$ be any random variable with any distribution. Given that we know $\mathbb{E}[X]$, $\mathbb{E}[\max(0,X)]$, and $\mathbb{E}[\min(0,X)]$, can you write a formula for $\mathbb{E}[f(X)]$ where $f$ is any twice differentiable function?
I am guessing the solution might have something to do with the arbitrage free ... |
**Re close votes**: I believe this is a fair kind of opinion-based question because it's like those ethics questions in academia se or workplace se or because it's [pedagogical][1].
---
Context: I'm actually asking this in the context of this [maths education se question][2] I have.
---
I have a pedagogical... |
I work for an oil trading company. We sell petroleum products indexed on the Brent and hedge our fixed price sales using futures to offset price fluctuations. We do not engage in speculation. I was wondering if there was any advantage to engaging in delta hedging (or delta-gamma hedging) with oil options? I could not t... |
Where can I get free options implied volatility and realized volatility data?
I used to use Quandl, but they were acquired by NASDAQ and now offer this data under a premium paywall.
For reference, I am using this for individual purposes only.
Thanks in advance for your help! |
Free data source for options IV and volatility data (US, UK, EUR Equities preferable)? |
I wanted to ask, suppose I have a portfolio of futures of gasoline and other oil products eg (ULSD, WTI) for different months. I want to compute the MonteCarlo value at risk for the positions in the portfolio. To calibrate the MonteCarlo simulations, should I use the time series of the Future expiring in the correspond... |
I wanted to ask, suppose I have a portfolio of futures of gasoline and other oil products eg ULSD (*Ultra Low Sulphur Diesel*), WTI (*West Texas Intermediate*) for different months. I want to compute the MonteCarlo value at risk for the positions in the portfolio. To calibrate the MonteCarlo simulations, should I use t... |
I have been reading about the RiskMetrics methodology. I read that RiskMetrics recommend a lambda of 0.94 for daily data and 0.97 for monthly data. I would like to convert these numbers to half-lives. I have the formula $Halflife=\frac{log(0.5)}{log(\lambda)}$.
I have 3 questions:
1. with the recommended decay fac... |
I am trying to solve this question:
Assume that CAPM is true. The risk-free rate is 3%, the expected return on the market portfolio is 10% and the standard deviation of the return on the market portfolio is 15%. Consider a portfolio P with expected return of 25% and assume that it is on the efficient frontier.
Q... |
Does a portfolio on efficient frontier also lie on CML(capital market line)? |
I'm reading this paper :
[What Does the Individual Option Volatility
Smirk Tell Us About Future Equity Returns?][1]
Yuhang Xing, Xiaoyan Zhang, and Rui Zhao∗
In section 2. A i found this equation: [![enter image description here][2]][2]
... And I don't understand how it comes up with this result, because if I com... |
How you explain that result? |
Am I correct in saying that with the leverage system in crypto perpetual futures contracts, the user does not borrow from the exchange and the exchange does not have to borrow from external sources to "fund the leverage"?
Who pays the shorts profit if the long counterparty bought without leverage?
Is each trade... |
Crypto perpetual futures contracts- How does the exchange fund the leverage? |
I am working with a vol surface that was created as a BlackVarianceSurface.
Now I would like to compute the "vol" greeks for a product and as such I need to shift that surface by a small dVol uniformally so that I can then compute:
Vega = [Price(shifted surface) - Price(base surface)] / shift
And along the same li... |
I am wondering how the industry decides the yield rate for a certain maturity when calculating the implied volatility for the SPX option.
Is it just a simple linear interpolation from the two near dividend records? Or something else?
Thanks! |
How should I decide the yield rate when calculate the spx implied volatility? |
This is my first time delving into dual curves, or multiple yield curves. A question struck me about using OIS discounting when choosing the swap rate of a new IRS.
Without multiple yield curves I thought it went something like this
$$
V_{swap} = V_{fixed}-V_{float} ~~ \text{with $N$ as par}.
$$
where at the... |
I am working on idiosyncratic volatility analysis and I would like to control for size characteristics following the method used in this paper.: https://onlinelibrary.wiley.com/doi/full/10.1111/j.1540-6261.2006.00836.x
In table 7 they implement a double sorting method which is clear to me, but I did not understand ho... |
How do you create/define concentration risk limits for credit portfolios?
I've seen a lot of recommendation for HHI for example, but how do I define what is high or low concentration?
Are there better alternatives? |
As titled states… I am trying to figure out how to solve for individual return given average weighted total return and weights of individual returns?
For example:
2% = (r1 x 0.2) + (r2 x 0.5) + (r3 x 0.3)
What is r1, r2, and r3? |
Am I able to find individual returns from total weighted average of returns? |
I am working on idiosyncratic volatility analysis and I would like to control for size characteristics following the method used in the paper "[The Cross-Section of Volatility and Expected Returns](https://onlinelibrary.wiley.com/doi/full/10.1111/j.1540-6261.2006.00836.x)" by Ang et al (2006).
In Table 7, they imple... |
I can find a bunch of phd level formulas for pricing worst of options, but haven't come across anything that translates the math formulas into a programming language.
Are there any known references to textbooks with actual code examples to check out?
Been searching for code that outputs fair value and respectiv... |
So assuming I am entering a trade to long A stock at 10 and short B stock at 20. And assume we want the position to be dollar neutral, so we are going to long 2 shares of A and short 1 share of B. Expected return for A is 0.3% and A's daily vol is 2%; expected return for B is 0.2% and B's daily vol is 2% too. Correlati... |
Suppose the consumer Solves $\max -e^{-\gamma W}$ where $W=X^T D -X^Tp R_f$ where $X$ is the vector invested in a risky asset and $D\sim N(E[D],\Sigma^2_D)$ and $R=\sim N(E[R],\Sigma^2_R)$. Then ${ X=(\gamma \Sigma_R)^{-1}(E[R-R_f])}$. Is this formula correct?
My reasoning is as follows:
$e^{-\gamma W}=e^{-\gamma ... |
Is this equation correct for portfolio optimization for CARA normal with N risky and one riskless asset? |
I'm looking for an open source that provides intraday data (minute by minute). I wasn't able to find one so far. Any suggestions?
Thank you. |
The most intuitive explanation I have found is to consider two different options for a bank in need of funds:
1. Emit a bond, let's say a zero-coupon bond with nominal 1 which expires in one year, or
2. Create a synthetic bond using an uncollateralized derivative where we receive some up-front premium and pay 1 in ... |
The most intuitive explanation I have found is to consider two different options for a bank in need of funds, they could either:
1. Emit a bond, let's say a zero-coupon bond with nominal 1 which expires in one year, or
2. Create a synthetic bond using an uncollateralized derivative where we receive some up-front pr... |
The most intuitive explanation I have found is to consider two different options for a bank in need of funds, they could either:
1. Emit a bond, let's say a zero-coupon bond with nominal 1 which expires in one year, or
2. Create a synthetic bond using an uncollateralized derivative where we receive some up-front pr... |
How does Bloomberg calculate the 2 year swap rate using the current eurodollar futures prices? |
How do you create/define concentration risk limits for credit portfolios?
I've seen a lot of recommendation for HHI for example, but how do I define what is high or low concentration?
The idea of the analysis is not regulatory but for my own control of the portfolio positions and credit risk distribution. The por... |
From my understanding, a typical test of the CAPM after having estimated it via time-series regressions reads as follows:
$R_{i,t+1} = \alpha + \gamma \beta_{i} + \epsilon_{i,t+1}$,
where $\beta_i$ is the CAPM-beta of stock $i$. Researchers say that the CAPM-beta is expected to predict the next period returns, i.... |
From my understanding, a typical test of the CAPM after having estimated it via time-series regressions reads as follows:
$R_{i,t+1} = \alpha + \gamma \beta_{i} + \epsilon_{i,t+1}$,
where $\beta_i$ is the CAPM-beta of stock $i$. Researchers say that the CAPM-beta is expected to predict the next period returns, i.... |
How do you create/define concentration risk limits for credit portfolios?
I've seen a lot of recommendation for HHI for example, but how do I define what is high or low concentration?
The idea of the analysis is not regulatory but for my own control of the portfolio positions and credit risk distribution. The por... |
How do you create/define concentration risk limits for credit portfolios?
I've seen a lot of recommendation for HHI for example, but how do I define what is high or low concentration?
The idea of the analysis is not regulatory but for my own control of the portfolio positions and credit risk distribution. The por... |
From my understanding, a typical test of the CAPM after having estimated it via time-series regressions from period $t-k$ to $t-1$ reads as follows:
$R_{i,t+1} = \alpha + \gamma \beta_{i} + \epsilon_{i,t+1}$,
where $\beta_i$ is the CAPM-beta of stock $i$. Researchers say that the CAPM-beta is expected to predict ... |
Typically, one estimates the CAPM beta of stock $i$ of a regression of stock excess returns $R_t$ on the market excess return $MRP$:
$R_{t} = \alpha + \beta MRP_t + \epsilon_t$,
where $t$ denotes a period (i.e. month hereafter).
Repeated for different stocks $i$, one gets a cross section of betas $\beta_i$. Ac... |
Typically, one estimates the CAPM beta of stock $i$ of a time series regression of stock excess returns $R_t$ on the market excess return $MRP$:
$R_{t} = \alpha + \beta MRP_t + \epsilon_t$,
where $t$ denotes a period (i.e. month hereafter). The sample for this regression spans from $t-k$ up to $T$.
Repeated fo... |
Typically, one estimates the CAPM beta of stock $i$ of a time series regression of stock excess returns $R_t$ on the market excess return $MRP$:
$R_{t} = \alpha + \beta MRP_t + \epsilon_t$,
where $t$ denotes a period (i.e. month hereafter). The sample for this regression spans from $T-k$ up to $T$.
Repeated fo... |
I've been wondering how do institutional investors actually perform portfolio optimization, i.e., do the traditional methods work when combined together with superior data collections and quality, and state-of-art optimization methods? Simple example would be CAPM + additions + GARCH models. You can go very complex or ... |
Is CAPM + additions actually used in industry-level portfolio optimization? |
I am currently researching the article by Julien Guyon & Jordan Lekeufack: Volatility Is (Mostly) Path-Dependent.
Their model is quite impressive in both its simplicity, as well as its efficacy. However, I have one fundamental question in doing research about the joint calibration of the SPX and VIX.
The question... |
The Holy Grail of Volatility Modelling: The SPX & VIX - Why? |
I am currently researching the article by Julien Guyon & Jordan Lekeufack: [Volatility Is (Mostly) Path-Dependent][1].
Their model is quite impressive in both its simplicity, as well as its efficacy. However, I have one fundamental question in doing research about the joint calibration of the SPX and VIX.
The que... |
I bootstrapped a swap curve and then, having the zero rates, I used Nelson-Siegel to estimate a terminal rate at $t+1$. However, because I am using a cubic spline, the new long term rate changes the spline between the zeros at $t-1$ and $t$.
This has caused my final swap to mis-price, when originally it was prici... |
Today given the strengthening of USD, many Central banks are trying to lift their domestic currencies by selling their USD holding.
My question is how a central bank earns and save USD in their kitty? One possibilities I can see is by selling their Gold holdings. Is this one only way they can earn USD?
Any insig... |
Today given the strengthening of USD, many Central banks are trying to lift their domestic currencies by selling their USD holding.
My question is how a central bank earns and save USD in their kitty? One possibilities I can see is by selling their Gold holdings. Is this one only way they can earn USD?
Also, whe... |
I am currently researching a pre-print article by Julien Guyon & Jordan Lekeufack (2022): [Volatility Is (Mostly) Path-Dependent][1].
Their model is quite impressive in both its simplicity, as well as its efficacy. However, I have one fundamental question in doing research about the joint calibration of the SPX and ... |
The Gordy Formula used for measuring Credit Risk as proposed in Basel Rules is based on the asymptotic single risk factor model. It is derived from a Merton Model. The Merton Model only knows to stati, i.e. performing or defaulted.
Is it therefore fair to say that the formula calculates default risk, but not the ris... |
Does Gordy Formula measure default risk & downgrade risk? |
From the pricing theory, processes need to be martingales when divided by the numeraire asset.
A classical example is a stock option:
Consider a money market $B$ being the numeraire asset. When we price a stock option with a payoff $h(S(T))$, then the money-market discounted stock price process $S/B$ has to be a ma... |
Does every process need to be a martingale under martingale measure? |
From the pricing theory, processes need to be martingales when divided by the numeraire asset.
A classical example is a stock option:
Consider a money market $B$ being the numeraire asset. When we price a stock option with a payoff $h(S(T))$, then the money-market discounted stock price process $S/B$ has to be a ma... |
How does bloomberg calculate the discount rate from EUR estr curve? |
I was asked this interview questions for an analyst level structuring role and it has been bothering me since I can't figure it out:
_Assuming the price of an equity is 100, what is the minimum price to quote a 150 strike European call option to guarantee you make no losses?_
- You have not information about Gre... |
What is the minimum price of an option, given no information about Greeks? |
How do you create/define concentration risk limits for credit portfolios?
I've seen a lot of recommendation for HHI for example, but how do I define what is high or low concentration?
The idea of the analysis is not regulatory but for my own control of the portfolio positions and credit risk distribution. The por... |
I was reading the Methodology that S&P uses in order to construct VIX index [VIX S&P Methodology][1] If I understood it right, the VIX Short-Term Futures Index works as follows:
<img src="https://latex.codecogs.com/svg.image?IndexER_{t}&space;=&space;IndexER_{t-1}&space;*&space;\frac{CRW_{\text{VIX&space;First&space... |
For one call date,
The call probability is just the probability that the swap rate for the remaining life of the swap is below the strike rate. This is easily obtainable in a normal vol model, it is :
N((Strike−ForwardRate)/NormVol∗Sqrt(T))
where T is time from now until call date, where N is the cumulative Norma... |
If my one step ahead forecasts from GARCH(1,1)-X are:
\begin{equation}
\hat{h}_{t+1} = \hat{\alpha}_0 + \hat{\alpha}_1 \hat{u}^2_t + \hat{\beta}_1 \hat{h}_t + \hat{\psi} X_t
\end{equation}
Where $\hat{\alpha}_0,\, \hat{\alpha}_1,\,\hat{u}^2_t,\, \hat{\beta},\, \hat{h}_t$ and $ \hat{\psi}$ denote the GARCH(1,1... |
I am constructing a classic pairs trading strategy in which I use a linear estimator to model the spread of two assets opening a long-short market neutral position during times of divergence. I am struggling to understand how to estimate the appropriate hedge ratio via regression. Namely, I am confused regarding the ro... |
I am constructing a classic pairs trading strategy in which I use a linear estimator to model the spread of two assets opening a long-short market neutral position during times of divergence. I am struggling to understand how to estimate the appropriate hedge ratio via regression. Namely, I am confused regarding the ro... |
I am constructing a classic pairs trading strategy in which I use a linear estimator to model the spread of two assets opening a long-short market neutral position during times of divergence. I am struggling to understand how to estimate the appropriate hedge ratio via regression. Namely, I am confused regarding the ro... |
I am constructing a classic pairs trading strategy in which I use a linear estimator to model the spread of two assets opening a long-short market neutral position during times of divergence. I am struggling to understand how to estimate the appropriate hedge ratio via regression. Namely, I am confused regarding the ro... |
Item 31.12 from [FRTB's documentation][1] states the following regarding risk factor eligibility for the expected shortfall model:
> A bank must determine which risk factors within its **trading desks that
> have received approval to use the internal models approach** as set out
> in [MAR32] are eligible to be inc... |
Item 31.12 from [FRTB's documentation][1] states the following regarding risk factor eligibility for the expected shortfall model:
> A bank must determine which risk factors within its **trading desks that
> have received approval to use the internal models approach** as set out
> in [MAR32] are eligible to be inc... |
Hi James: I find your question and results interesting and I can't answer or say anything definitive but I experienced similar issues back in the day when I tried to use cointegration for pairs. Just a couple of things as far as some intuition and comments. I was hoping someone else would say something because I doubt ... |
Hi James: I find your question and results interesting and I can't answer or say anything definitive but I experienced similar issues back in the day when I tried to use cointegration for pairs. Just a couple of things as far as some intuition and comments. I was hoping someone else would say something because I doubt ... |
Item 31.12 from [FRTB's documentation][1] states the following regarding risk factor eligibility for the expected shortfall model:
> A bank must determine which risk factors within its **trading desks that
> have received approval to use the internal models approach** as set out
> in [MAR32] are eligible to be inc... |
How do you fit a GARCH model to the returns of a stock given the dates of past earnings announcements? Volatility will tend to higher than a GARCH model would predict on the announcement day. |
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