instruction stringlengths 15 26.1k |
|---|
For an option with price C, the P&L, with respect to changes of the underlying asset price S and volatility σ, is given by
P&L=δΔS+12γ(ΔS)2+νΔσ,
where δ, γ, and ν are respectively the delta, gamma, and vega greeks.
My question is: while we are able to calculate ΔS by using the spot price of underlying S and by... |
delta-gamma-vega VaR approximation: how to calculate the delta volatility? |
Mean-Variance optimization trades off expected returns with portfolio variance. The idea is that excess variance is not desirable.
But what if you weren't averse to high variance and you wanted to maximize both expected returns and variance. Has there been any research done on this or concepts similar to this?
A... |
delta-gamma-vega approximation: how to calculate the delta volatility? |
There is no difference in real world settings.
Market practitioners usually always price a digital as a tight call spread to capture skewness. For example, setting strikes at $$± = ±1/2,$$ in the limit of $ → 0$, the payoff approaches that of a digital. The reason is that a tight call spread, by using two vanilla ... |
delta-gamma-vega VaR approximation: how to calculate the delta volatility? |
I'm implementing the solution with drift from "Dealing with the inventory risk" from Gueant, Lehalle and Tapia. I'm using the link https://arxiv.org/pdf/1105.3115.pdf as reference.
I can reproduce the standard solution with no issues, and then reproduce the surfaces in the paper. I'm trying now to implement the solu... |
Assuming that the CAPM holds, the total risk of an asset can be partitioned into systematic risk (associated with the market factor) and idiosyncratic risk. Idiosyncratic risk is asset specific. Does that imply that the error covariance matrix $\Sigma_t=\text{Var}(\varepsilon_t)$ from a cross-sectional regression
$$
... |
> Are backtests and live trading typically built upon the same abstraction or operate on the same trading/processing engine?
It could be the case, and I see many projects allow to live trade backtested strategies.
My opinion - we can achieve more if we separate the two and build more specific abstractions for eac... |
How calculate expectation and variation of stochastic integral Based on Heston model? |
$\mathbb{P}$ is the true probability measure. Measure $\mathbb{Q}$ is a measure of convenience that allows risk neutral pricing. [Stochastic discount factor $M$ takes you between the two.](https://quant.stackexchange.com/questions/38477/what-is-the-difference-between-risk-neutral-probabilities-and-stochastic-discoun/38... |
Treasury futures contract has [no carry][1], but what is its rolldown (if it exists)?
In the above answer to carry, @Helin mentioned "...bonds have expected rolldown returns that will *flow through* to futures..."
Does it mean that futures' rolldown is **exactly** the same as the rolldown of its CTD bond? Thanks... |
When fitting/calibrating a option model like heston to option data, what are some useful data handling to do?
The basic thing is to remove all options with no trade/volume, but how many maturities should one use and how far ITM/OTM?
I can't really find any scientific articles on above, but would appreciate any inputs... |
No, the CAPM does not imply the error covariance matrix is diagonal. The CAPM considers expected values, not the covariance matrix. It implies that there is a single factor that is priced, i.e. it has something to say about the expected return. However, there may be other factors that that are not priced that make the ... |
No, the CAPM does not imply the error covariance matrix is diagonal. The distinction between systematic and idiosyncratic risk under the CAPM is not as simple as indicated at the start of the question. The CAPM considers expected values, not the covariance matrix. It implies that there is a single factor that is priced... |
I need to calculate `Price-to-Earnings Ratio (PE Ratio)` of an **Equal-weighted index**.
Sum of market caps
P/E for Market-cap Weighted index = -------------------
Sum of Earnings
... |
Treasury futures contract has [no carry][1], but what is its rolldown (if it exists)?
In the above answer to carry, @Helin mentioned "...bonds have expected rolldown returns that will *flow through* to futures..."
Does it mean that futures' rolldown is **exactly** the same as the rolldown of its CTD bond in spot... |
I read in the newspaper things like,
>Interest rate swaps, which are based on market expectations about future rate decisions, are pricing in at least one Bank of Canada rate cut later this year, and additional cuts in 2024.
Sometimes there will be a probability associated with the prediction, ie. markets are sa... |
Roll in futures language is typically defined in terms of the futures curve, i.e. the difference between (successive) contracts, say RXM3 minus RXH3. You can find more information on this for STIR contracts in [Jha, S. (2011), Interest Rate Markets: A Practical Approach to Fixed Income][1]. I'd say this is primarily wh... |
I'm reflecting on whether historically estimated $\beta$ is a "good" estimator of future $\beta$.
Consider the problem as follows:
- Let $r_1$, $r_2$, ...., $r_{36}$ be the last 36 months of returns for a security
- let $m_1$, $m_2$, ...., $m_{36}$ be the market returns.
You can use this data to calculate a l... |
Is beta stable over time for individual securities? |
Say I have obtained a distribution for different forward rates F_k such that:
$$
dF_k (t) = \sigma (t) * F_k (t) * dW_k(t)
$$
with
$$ dW_k(t) * dW_l(t) = \rho_{k,l} (t) dt.
$$
From this I want to simulate curves for F for k > 50. How do I go about this? It doesnt seem logical to simulate the brownian motion... |
How to simulate from instantaneously correlated Brownian motions? |
I'm reflecting on whether historically estimated $\beta$ is a "good" estimator of future $\beta$.
Consider the problem as follows:
- Let $r_1$, $r_2$, ...., $r_{36}$ be the last 36 months of returns for a security
- let $m_1$, $m_2$, ...., $m_{36}$ be the market returns.
You can use this data to calculate a l... |
This reference may help you: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4208856
You can get a good hint on what's the idea of the whole thing by just reading the first two/three sections.
|
I am trying to find a way to validate available stock market data (because I found so much disagreement between different sources) and went through a 10-K report, hoping to find any mentioning of the market price of common stock of that company (Exxon Mobil in this case). I know that the market price is somewhat irrele... |
Does a 10-K/10-Q report contain an estimate of share market price? |
On fixed rate bonds and IBOR based floating rate notes the next cashflow is known **definitively** in advance, therefore the accrued interest for a given settlement date is a trivial calculation typically following something like:
$$ accrued = cashflow * \frac{settlement - period start}{period end - period start} $$... |
Does QuantLib have functionality to value ILS Telbor swaps? |
I'm trying to fit an Heston model to market data. But market is data has some terms (<3M) with quite wide bid-ask spreads (12%-25%). Should I just use mid volatility? Is there maybe a model to pre-process market data that helps to solve this kind of problem? |
For an option with price C, the P&L, with respect to changes of the underlying asset price S and volatility σ, is given by
P&L=δΔS+12γ(ΔS)2+νΔσ,
where δ, γ, and ν are respectively the delta, gamma, and vega greeks.
My question is: while we are able to calculate ΔS by using the spot price of underlying S and by... |
In general, what does the bias-variance tradeoff look like when estimating covariance matrices with varying return frequencies (i.e. daily, weekly, monthly returns)?
From my observations I've noticed that, on average, lower frequency (e.g. monthly) returns result in higher estimates of asset variance and higher esti... |
What is the official definition of the Duncan Leading Indicator? I've looked at all links page after page on google but can't seem to find any official source of origin and therefore official definition, only third party articles which claim some definition.
And which Duncan is the original creator behind the indica... |
Consider estimating an asset pricing model such as the CAPM or a multifactor model using monthly data. [Petersen (2009)][2] section "Asset pricing application" suggests use of standard errors clustered by time, as this addresses the problem of cross-sectional correlation. (Meanwhile, serial correlation is not much of a... |
Clustered vs. GMM-based standard errors: which ones to use in asset pricing? |
How to find out if Asymmetric Laplace Distribution is having Finite/Infinite Variance? |
I am trying to find a way to validate available stock market data (because I found so much disagreement between different sources) and went through a 10-K report, hoping to find any mentioning of the market price of common stock of that company (Exxon Mobil in this case). I know that the market price is somewhat irrele... |
I was asked in an interview to get an exposure to 5Y5Y forward rate using bonds alone. Essentially it is short 5Y bond and long 10Y bond, and I needed to compute the relative weights. Regarding risk:
- The 5Y bond does not have risk on 5Y-10Y
- The 10Y bond is assumed to have equal risk on 5Y and 10Y.
How to co... |
Which books on financial mathematics would you recommend for people with good background in probability, statistics and stochastic processes but without any background in financial mathematics?
If my question is too general then which books on main branches of financial mathematics would you recommend?
The only ... |
Which books on financial mathematics would you recommend for people with good background in probability, statistics and stochastic processes but without any background in financial mathematics?
The goal is the next one: to find interesting mathematical problems in financial mathematics, solve it (using methods of p... |
I am currently looking at alternatives to the Bloomberg Desktop API as I frequently reach the daily or even monthly data limit. One alternative which was proposed to me was switching from a desktop license to a Bloomberg Anywhere license since the BQNT environment which is offered under a BBA license would allow to run... |
I am interested in downloading historical short interest data from Finra for analysis but I cannot find where they keep dump files. They have the data for OTC companies but not for all of the exchanges. The only option seems to be their API, which I would have to make a ton of calls to and is still limited to just the ... |
Does FINRA have an FTD dump of historical short interest? |
Does FINRA have an FTP dump of historical short interest? |
The paper says the reservation bid/ask prices are the quotes in "the setting where no trading is allowed." I interpret this to mean "no competition." By taking the average of the bid and ask prices, you get an ideal mid-price for the market maker based on their inventory and their risk tolerance. Quoting symmetrically ... |
Which books on financial mathematics would you recommend for people with good background in probability, statistics and stochastic processes but without any background in financial mathematics?
The goal is the next one: to find interesting mathematical problems in financial mathematics, solve it (using methods of p... |
How to find interesting open math problems in quantitative finance that I could publish articles about? |
I am reading about HAR models for realised variance and they all seem to use WLS or OLS to calculate the parameters. Now I understand how that works if you just use say the 10 years of AAPL intraday history. However, if you're doing this for a univariate model on many stocks (like it seems the authors do to create stab... |
I am reading about HAR models for realised variance and they all seem to use WLS or OLS to calculate the parameters. Now I understand how that works if you just use say the 10 years of AAPL intraday history. However, if you're doing this for a univariate model on many stocks (like it seems the authors do to create stab... |
Moneyness is a good criteria for choosing the coordinates, and sure this obtains an arbitrage-free surface in the strike/moneyness dimension. It's questionable if a linear interpolation on the implied accumulated variance is necessarily globally arbitrage-free. Empirical evidence shows violation of arbitrage properties... |
I'm reading the paper by J.Gatheral and A.Schied - "**Optimal Trade Execution under
Geometric Brownian Motion in the Almgren and Chriss Framework**". On page 6 authors provide value function, which maps initial conditions with minimal cost value ($T$ - Time remaining to expire position, $X$ - value of position, $S_0$ ... |
I'm reading the paper by J.Gatheral and A.Schied (2012) - "[Optimal Trade Execution under
Geometric Brownian Motion in the Almgren and Chriss Framework](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1654151)".
On page 6, the authors provide a value function, which maps initial conditions with minimal cost va... |
How to derive this HJB equation? |
Suppose we are given a dataset with $T$ time periods and $N$ assets or portfolios. We are interested in estimating and testing the CAPM or a multifactor model. Take the CAPM:
$$
r^*_{i,t}=\alpha_i+\beta_i \mu^*_{m}+\varepsilon_{i,t} \tag{1}
$$
where $r^*_i:=(r_{i,t}-r_{f,t})$ is firm's $i$ excess return and $\mu^*_... |
I am trying to wrap my head around GMM estimation of a single factor model such as the CAPM. I started by asking [How come the cross-sectional CAPM equation produces $N$ moment conditions (not $1$)?][3] and am now following it up. Equation $(12.23)$ in Cochrane ["Asset Pricing"][1] (2005) section 12.2 (p. 241) says the... |
Reading Cochrane ["Asset Pricing"][1] (2005) section 12.2 (p. 241), I got lost in the derivation of the GMM estimator for the single-factor model. Equation $(12.23)$ says the moments are
$$
g_T(b) = \begin{bmatrix} E(R^e_t-a-\beta f_t) \\ E[(R^e_t-a-\beta f_t)f_t] \\ E(R^e-\beta \lambda) \end{bmatrix} = \begin{bmatri... |
I'm reading Natenburg's Options book, and I'm not understanding why "Buying a traditional butterfly is equivalent to selling an iron butterfly," because later in the text Natenburg says that "If we sell the June 95 / 100 / 105 iron butterfly we will take in some amount between zero and 5.00. We also hope that the under... |
I'm reading Natenburg's Options book, and I'm not understanding why "Buying a traditional butterfly is equivalent to selling an iron butterfly," because later in the text Natenburg says that "If we sell the June 95 / 100 / 105 iron butterfly we will take in some amount between zero and 5.00. We also hope that the under... |
When analysing historical movements of credit rating, sometime credit rating is put as `Non-rated` or `NR`.
Is there any industry acceptance definition of `Non-rated`? Or is it just non availability of credit rating the cause of which may include Data issue?
Appreciate any insight on this subject. |
The CAPM is a single-period model that says
$$
\mathbb{E}(R^*_i)=\beta\mathbb{E}(R^*_m)
$$
where $R^*_i:=R_i-r_f$ is an asset's excess return, $R^*_i:=R_m-r_f$ is the market's excess return and $\beta:=\frac{\text{Cov}(R^*_i,R^*_m)}{\text{Var}(R^*_m)}=\frac{\text{Cov}(R_i,R_m)}{\text{Var}(R_m)}$. The latter equal... |
Definition and estimation of $\beta$: raw or excess returns? |
Suppose we are interested in testing the CAPM using the [GRS test][2]. Consider $N$ assets observed for $T$ time periods. Using the notation of Cochrane ["Asset Pricing"][1] (2005), the GRS test amounts to running $N$ time series regressions of the form
$$
R^{ei}_t=\alpha_i+\beta_i f_t+\varepsilon^i_t \tag{12.1}
$$
... |
Testing the CAPM: does GRS account for errors in variables (measurement error)? |
Suppose we are interested in testing the CAPM using the [GRS test][2]. Consider $N$ assets observed for $T$ time periods. Using the notation of Cochrane ["Asset Pricing"][1] (2005), the GRS test amounts to running $N$ time series regressions of the form
$$
R^{ei}_t=\alpha_i+\beta_i f_t+\varepsilon^i_t \tag{12.1}
$$
... |
[Fabozzi, Ma and Oliphant (2008)][1] have this very nice comparison between the return on the overall stock market and the return on sin stocks across multiple countries. They have done this for the period 1970-2007. Does anyone know where to find an updated estimate of this graph?
[![enter image description here]... |
The realized variance under classical Black Scholes where the stock price process follows a GBM is given as
$$V_T = \frac1T\int_0^T\sigma_s^2ds\qquad (1)$$
however, the texts I have been reading do not give a derivation of this fact. Further, it is stated, that in the case of the merton jump diffusion model, term $(... |
How is variance derived in BS? |
I'm trying to calibrate Hull White model in VietNam market to value IRS, CSS products which are not publically traded.
dr(t)=(θ(t)−αr(t))dt+σ(t)dW(t)
I only have historical VNIBOR curves data, we do not have trading interest derivatives yet.So basically we have R_i(0,1/250), R_i(0,1/4), R_i(0,1/2), ... f... |
How to calibrate short-rate model (Hull-White) using historical domestic IBOR curve without other derivative price? |
I have a confusion about how to calculate RWA for a exposure with many types of CRM (says, collaterals and guarantee) in IRB approach.
- In BCBS128, point 206, "In the case where a bank has multiple CRM techniques covering a single exposure (e.g. a bank has both collateral and guarantee partially covering an exposure... |
I'm trying to calibrate Hull White model in VietNam market to value IRS, CSS products which are not publicly traded.
dr(t)=(θ(t)−αr(t))dt+σ(t)dW(t)
I only have historical VNIBOR curves data, we do not have trading interest derivatives yet.So basically we have R_i(0,1/250), R_i(0,1/4), R_i(0,1/2), ...... |
As far as I understand, perpetual contracts are like regular futures except they can be held open indefinitely. However, I don't know about any other brokers out there, but in Binance, when you open a perpetual contract, you do so by interacting with an order book, meaning, for every position you open there is an oppos... |
I am pricing CDS calibrating the default probabilities intensities using CDS spreads from Markit.
Those spreads are given with a recovery.
I have two questions regarding the recovery:
1. Hull wrote that, provided we use the same recovery for calibrating default probabilities and pricing, the recovery rate has l... |
Short answer: for equity asset pricing, working with returns in excess of the risk free rate (i.e. $r - r_f$) tends to make more sense.
- By subtracting off the risk free rate, practically what you're doing is setting the dynamics of the risk free rate aside as a separate problem.
- e.g. it makes returns from h... |
Lets say I want to run series of regressions for zero cost portfolio Y that goes long on stocks based on high variable x and short stocks with low variable of x. How do I run the regression for example for CAPM? is it:
Y - rf = a + b(market-rf) + e
or
Y = a + b(market-rf) + e
Is it the first one only if we... |
CAPM model formula for zero cost portfolio? |
I am using historical 30yr US treasury rates for a project. When I downloaded the rates from Bloomberg by queuing the history of the USGG30YR index, I found the numbers different from what US Department of Treasury has.
A few examples:
[![USGG30YR on Bloomberg][1]][1]
https://home.treasury.gov/resource-center... |
**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... |
I have a number of different annualized realized volatility estimates (for the same point in time) that I'd like to combine. Is a simple average over these appropriate? Or should I do this in the variance space? $\sqrt{(\sum_{i} \sigma_i^2)/ n}$
Or would a geometric mean be more appropriate?
These volatility esti... |
I'm familiar with the volatility surface for equity options with the smile/skew dynamic and flattening with increased maturity, and the explanation/intuition behind its shape. However, today I came across the volatility surface for FX options, and I noticed that, unlike equity options, volatility is increasing with mat... |
Suppose we are given a dataset with $T$ time periods and $N$ assets or portfolios. We are interested in estimating and testing the CAPM or a multifactor model. Take the CAPM:
$$
r^*_{i,t}=\alpha_i+\beta_i \mu^*_{m}+\varepsilon_{i,t} \tag{1}
$$
where $r^*_i:=(r_{i,t}-r_{f,t})$ is firm's $i$ excess return and $\mu^*_... |
Suppose we are given a dataset with $T$ time periods and $N$ assets or portfolios. We are interested in estimating and testing an augmented CAPM or a multifactor model with an additional factor: the asset's idiosyncratic volatility. In the case of the augmented CAPM, the model is
$$
r^*_{i,t}=\alpha_i+\beta_i \mu^*_{... |
Here is my guess:
If we were to estimate a simple CAPM by GMM, we could use the equation $(12.23)$ from Cochrane ["Asset Pricing"][1] (2005) section 12.2 (p. 241). In his notation, the moments are
$$
g_T(b) = \begin{bmatrix} E(R^e_t-a-\beta f_t) \\ E[(R^e_t-a-\beta f_t)f_t] \\ E(R^e-\beta \lambda) \end{bmatrix} = ... |
Suppose we are given a dataset with $T$ time periods and $N$ assets or portfolios. We are interested in estimating and testing an augmented CAPM or a multifactor model with an additional factor: the asset's idiosyncratic volatility. In the case of the augmented CAPM, the model is
$$
\mu^*_{i}=\beta_i \mu^*_{m}+\gamma... |
Suppose we are given a dataset with $T$ time periods and $N$ assets or portfolios. We are interested in estimating and testing the CAPM or a multifactor model. Take the CAPM:
$$
r^*_{i,t}=\alpha_i+\beta_i \mu^*_{m}+\varepsilon_{i,t} \tag{1}
$$
where $r^*_i:=(r_{i,t}-r_{f,t})$ is firm's $i$ excess return and $\mu^*_... |
I've seen the formulae for Call valuation with the Heston model for non-dividend paying Stocks.
How should I modify it to use it in an FX pair (which has two risk free rates: local currancy rate $r$ and foreign currency rate $q$)?
Formulas for Stock (only $r$) i'd like with foreign currency rate $q$:
$$
C(\th... |
I am looking for an API to download the list of securities that will need to be in the 13F filings from asset managers. Does the SEC have such an API or a non-PDF file that can help with getting the data electronically?
Thanks! |
Consider a set of alternative forecasts. The literature on forecast combinations* shows that an **average forecast often tends to beat the estimated (ex ante) best forecast** from the set. This is because of two reasons. First, different forecasts make different mistakes, and averaging across forecasts tends to make th... |
I'd like to do a montecarlo simulation of a $\Delta$ hedged strategy (long OTM call) to see how the PnL distributes on cases like:
- $\sigma_{bought} < \sigma_{realized}$
- $\sigma_{bought} > \sigma_{realized}$
- $\sigma_{bought} = \sigma_{realized}$.
For this, I calculate the purchased option price with $\s... |
The information I have found about delta hedging frequency and (gamma) PnL on this site and numerous others all reiterate the same thing: that the frequency at which you delta-hedge only has an effect on the smoothness and variance of your PnL.
> Nivel Egres: From the perspective of gamma pnl, the only thing that ma... |
Delta-hedging frequency directly affects PnL, and not just PnL smoothness and variance? |
I've read even in recent papers all the anguish over whether to reflect or truncate negative variances whilst simulating the Heston process and the subsequent ramifications of what doing that entails, etc.
It seems the solution is so simple it might have escaped them, but why not just reject the sample if it results... |
$$
\newcommand{\cbkt}[1]{\left\{{#1}\right\}}
\newcommand{\rbkt}[1]{\left({#1}\right)}
\newcommand{\sqbkt}[1]{\left[{#1}\right]}
$$
Shreve volume I, defines an American derivative security as follows:
---
**Definition**. An American derivative security is any contract that can be exercised at times $0,1,\l... |
$$
\newcommand{\cbkt}[1]{\left\{{#1}\right\}}
\newcommand{\rbkt}[1]{\left({#1}\right)}
\newcommand{\sqbkt}[1]{\left[{#1}\right]}
$$
Shreve volume I, defines an American derivative security as follows:
---
**Definition**. An American derivative security is any contract that can be exercised at times $0,1,\l... |
$$
\newcommand{\cbkt}[1]{\left\{{#1}\right\}}
\newcommand{\rbkt}[1]{\left({#1}\right)}
\newcommand{\sqbkt}[1]{\left[{#1}\right]}
$$
Shreve volume I, defines an American derivative security as follows:
---
**Definition**. An American derivative security is any contract that can be exercised at times $0,1,\l... |
Has there been any historical evaluation of the quality of credit ratings provided by agencies such as Fitch, S&P, and Moody's? Are there any academic resources available on this topic? I have checked Google Scholar, but could not find much. I'm curious to learn more about this topic, including the methodologies that c... |
I've read even in recent papers, and the [authoritative book by Jim Gatheral][1], all the anguish over whether to reflect or truncate negative variances whilst simulating the Heston process and the subsequent ramifications of what doing that entails, etc.
It seems the solution is so simple it might have escaped them... |
I've read even in recent papers, and on page 21 of the [authoritative book by Jim Gatheral][1], all the debate over whether to reflect or truncate negative variances whilst simulating the Heston process and the subsequent ramifications of what doing that entails, etc.
It seems the solution is so simple it might have... |
I've read even in recent papers, and on page 21 of the [authoritative book by Jim Gatheral][1], all the debate over whether to reflect or truncate negative variances whilst simulating the Heston process and the subsequent ramifications of what doing that entails, etc.
[![enter image description here][2]][2]
It seems ... |
If you look at just a single example, it may seem like the frequency of hedging directly effects the EV/Avg(Pnl), like in the situation you described where hedging every minute proved to be more profitable. But you need to think about the question in a bigger picture sense. How would hedging frequency affect the result... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.