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**Astronics Corporation (ATRO) has two types of issued stock: Common stock and Corporate Class B stock.**
*The common stock (ATRO)* found on NASDAQ is the "standard" type of equity stock that gives you a 1 to 1 vote to share basis (*usually you get one vote per share owned, which is the case for ATRO*).
*The cor... |
For example with bitcoin there is both the spot market and a [cash-settled quarterly futures market][1].
As a buyer when would I want to buy quarterly futures instead of just buying on the spot market?
Spot offers the advantage of actually owning the underlying and holding a long position in perpetuity. With qua... |
What advantage does cash-settled futures have over spot? |
https://www.splithistory.com/spb/
I see a 162 for 1000 reverse split of SPB on 07/16/2018 as shown above.
But the stock open price on 07/16/2018 is 82.70, the stock close price on 07/13/2018 is 82.72.
https://finance.yahoo.com/quote/SPB/history
This does not seem to be correct as a split of 1:8 surely shoul... |
How to verify the split history of SPB? |
Let $u(x)=x^{2/3}$, $x>0$ be the utility function, $X \sim U(0, 100)$ is loss, wealth $w=\$150$.
Calculate $\mathbb{E}(u(w_r))$ if a coinsurance is $80\%$ and gross premium is $\$43$.
My attempt is:
$E(u(w_r))=\frac{1}{100}\int_0^{43}(150-43)^{2/3}dt + \int_{43}^{100} (150-0.8t)^{2/3}dt=21.3486$
But answer... |
In some documents I have seen objective function of markowitz portfolio theory is as follows.
```
minimize 1/2 * w'Σw
where w is weights
Σ is covariance matrix
```
I could not understand where does 1/2 come from? Is it there for taking half of the risk because we only care about down-side deviation?
Thank... |
Has financial derivatives of financial derivatives ever been considered in academic or practical settings? Is it useless/useful?
Say $C(S_t;K,T)$ is a European call on an underlying $S_t$ with strike $K$ and expiry $T$. Then what is the no-arbitrage price of a European call on the above call, $C^*(C(S_t;K,T);K^*,T^*... |
1. Let's use the following expression (derived in Quantuple's answer [in your link][1]), which will help us tidy up the product using [Ito's Isometry][2]
\begin{align}
\int^t_0 W^2_s ds = 2 \int^t_0 (t-s)W_s dW_s + {\frac {t^2} 2}
\end{align}
Now looking at the expectation
\begin{align}
{\mathbb E}\Bigl[ \in... |
**Astronics Corporation (ATRO) has two types of issued stock: Common stock and Corporate Class B stock.**
*The common stock (ATRO)* found on NASDAQ is the "standard" type of equity stock that gives you a 1 to 1 vote to share basis (*usually you get one vote per share owned, which is the case for ATRO*).
*The cor... |
https://finance.yahoo.com/quote/BF-A/history?period1=1517875200&period2=1518134400&interval=1d&filter=history&frequency=1d&includeAdjustedClose=true
Yahoo Finance says BF-A has a 1339:1000 split event on Feb 07, 2018. However, the split-adjusted stock has a jump from Feb 06, 2018 to Feb 07, 2018. This doesn't seem t... |
Are you sure about that formula? What is the expression on p. 90 in Shreve that implies it ?
When $X_j\sim N(0,1)$ then $\frac{t}{\sqrt{n}}X_j$ has mean zero and variance $\frac{t^2}{n}\,.$ Then
$$
\textstyle\mathbb E\Big[\exp\Big(\frac{t}{\sqrt{n}}X_j-\frac{t^2}{2n}\Big)\Big]=1.
$$
So
$$
\textstyle\mathbb ... |
Are you sure about that formula? What is the expression on p. 90 in Shreve that implies it ?
When $X_j\sim N(0,1)$ then $\frac{t}{\sqrt{n}}X_j$ has mean zero and variance $\frac{t^2}{n}\,.$ Then
$$
\textstyle\mathbb E\Big[\exp\Big(\frac{t}{\sqrt{n}}X_j-\frac{t^2}{2n}\Big)\Big]=1.
$$
So
$$
\textstyle\mathbb ... |
# Empirical analysis of realized measures
The paper [Does anything beat 5-minute RV? A comparison of realized measures across multiple asset classes](https://www.sciencedirect.com/science/article/pii/S0304407615000329) [[LPS2015]](https://www.sciencedirect.com/science/article/pii/S0304407615000329) does a very good ... |
I am trying to understand the following. If I have $100, how do I determine how much to allocate to the long and short of a pairs spread?
You might say "use the hedge ratio you calculate, as that will tell you the number of shares of stock B to long/short for every share of stock A". The hedge ratio, though, is basi... |
What is the proper capital split/allocation between the long and short in a pairs trading? |
What is the proper capital split/allocation between the long and short in a pairs trade? |
https://www.marketvolume.com/technicalanalysis/wildersvolatility.asp
I see this page describes wilder's moving average. But the first step `ATR = SMA(TR)` is not clear. How many periods should be used to compute this simple moving average? Should it be N?
```
...
Step #2: Apply the moving average to the defin... |
**On p. 21 (bottom)** of the original book [Wilder, J. W. (1978). *New concepts in technical trading systems.* Trend Research.](https://books.mec.biz/tmp/books/218XOTBWY3FEW2CT3EVR.PDF) the author writes the following about the Volatility Index Indicator:
> In order for the range to be a meaningful tool as a measur... |
### Yes, the SMA should be computed with $N$ periods in your situation:
In the original book of **[Wilder, J. W. (1978). *New concepts in technical trading systems.*](https://books.mec.biz/tmp/books/218XOTBWY3FEW2CT3EVR.PDF)** the author writes the following about the Volatility Index Indicator (**see p. 21 (bottom)... |
I am new to pairs trading, and I have come up with an idea of how to allocate capital between the long and short leg of a pairs trade. I feel that there is a problem with it, and I want to figure out what I'm missing.
Let's say I have stock A and B, and I have a trading signal that it's time to enter a long/short pa... |
What would be the problem with this pairs trading allocation scheme? |
candlestick plot does not have information on whether the high price or the low price is achieved first. Is there a commonly accepted way to represent this info on a candlestick plot? |
candlestick plot with the order of high and low achieve time represented? |
In making a choice among financial strategies, each of which has some estimated return distribution, some strategies will clearly be better than others. But many times, the choice is a question of risk tolerance. That said, there's still a problem: the risk/reward tradeoff is often too coarse -- because different distr... |
What problem(s) are you trying to solve here? If you're trying to convince yourself that free data on the Internet is not fully trustworthy, then you have already succeded hundredfold. If you're trying to verify the details of Brown-Forman's 2018 split (more precisely, stock dividend), then start with their fy 2018 10K... |
One way is to use a smile for the FX rate and another for compo options (the equity denominated in the other currency). FX smiles are liquid, so you have freedom to choose the compo smile. Then use this model, [Repricing the Cross Smile][1], to value quantos, and calibrate the compo smile to hit them. You can value qua... |
One way is to use a smile for the FX rate and another for compo options (the equity denominated in the other currency). FX smiles are liquid, so you only have freedom to choose the compo smile. Then use this model, [Repricing the Cross Smile][1], to value quantos, and calibrate the compo smile to hit them. You can valu... |
### Yes, the SMA should be computed with $N$ periods in your situation:
In the original book of **[Wilder, J. W. (1978). *New concepts in technical trading systems.*](https://books.mec.biz/tmp/books/218XOTBWY3FEW2CT3EVR.PDF)** the author writes the following about the Volatility Index Indicator (**see p. 21 (bottom)... |
I am using a couple of resources ([here][1] and [here][2]) to calculate the mean reversion half-life of a time series. This method of calculating it is also presented in Ernest Chan's *Algorithmic Trading* on page 47.
The method, as I understand it, is to use the formula: halflife = -log(2)/G
Where G is the coeff... |
Why is my mean-reversion half-life completely wrong? |
The following website gives the specifications of the CME Term SOFR reference rates: [CME Term SOFR](https://www.cmegroup.com/education/articles-and-reports/cme-term-sofr-reference-rates.html).
Point 1 in the link above specifies that the tenors that are currently supported are 1m, 3m, 6m, and 12m. Point 2 specifies... |
I'm trying to get a better grasp on the suitability of Geometric Brownian Motion as a model for asset prices, and I've stumbled across a hurdle which I can't seem to get over. Assuming GBM, the asset price is given by
$ S_t = S_0 \exp\{ (\mu - \frac{1}{2}\sigma^2)t + \sigma W_t \} $,
which implies that daily lo... |
This question pertains to a [whitepaper](https://cdn.cboe.com/resources/futures/vixwhite.pdf) published by the [CBOE](https://www.cboe.com/) that explains how the [VIX index](https://markets.cboe.com/tradable_products/vix/) is calculated.
Near the bottom of page 5 of the whitepaper, it explains that two risk-free in... |
I recently came across an equation in a paper.
In short, suppose that $I(t)$ denotes a longevity index at time $t$. An informative indicator that is useful in the absence of any information about the market prices, would be the risk premium per annum $\delta$ derived by the following equation
\begin{equation}
\su... |
I have replicated the complete CBOE VIX calculation approach in my working paper [No Model No Cry?][1]. Code and documentation are available in the accompanying `R`-Package [R.MFIV][2], where I also explain the complete VIX-procedure.
> What inputs would be needed?
The CBOE uses CMT rates which you can obtain fro... |
I have replicated the complete CBOE VIX calculation approach in my working paper [No Model No Cry?][1]. Code and documentation are available in the accompanying `R`-Package [R.MFIV][2], where I also explain the complete VIX-procedure.
> What inputs would be needed?
The CBOE uses CMT rates which you can obtain fro... |
I recently came across an equation in a paper.
In short, suppose that $I(t)$ denotes a longevity index at time $t$. An informative indicator that is useful in the absence of any information about the market prices, would be the risk premium per annum $\delta$ derived by the following equation
\begin{equation}
\su... |
[Longevity risk](https://en.wikipedia.org/wiki/Longevity_risk) is the risk that pensioners (or policy holders) live longer than expected, which can lead to a considerable stress to (future) cash flows of a company.
The formula you present seems to imply that the risk-neutral expected value of the longevity index i... |
Conventionally, when 10Y T yield is up, investors run away from growth stock, and vice versa, as it affect the risk free rate, and caused changes to their DCF model.
I am here trying to breakdown the logic in it.
First, if 10Y T yield is up, is because it's bond price is down;
Because investors are selling bon... |
Is High Treasury yield a bullish signal to stock market? |
Let's say I would like to obtain an incremental discounting rate (as in IFRS-16) for a company $X$ with a rating score of $Y$ and belonging to a sector $Z$. Usually, any data provider such as Bloomberg and Reuters only displays the yields or CDS spreads for some of the sectors and only for liquid ratings: That is, even... |
Someone told me that trading systems are very time-dependent and they must use network protocols that ensure speed.
Which network protocols do they use to ensure fast transmissions?
Thank you |
Which network protocols do trading systems use for speed? |
I am investigating ways to calculate the mean reversion half life of a mean reverting series. I am encountering things like the Ornstein – Uhlenbeck Process and various types of regression to estimate this value (the half-life).
What I want to understand is this: If I have a series, why can't I just deduce the half-... |
Why do I need fancy methods to calculate half-life of mean reversion? |
I am using univariate spline to interpolate betweeen a number of dots I have (in blue - it is a yield curve). The code is the following:
a = originalVals['Ttm'].to_numpy()
b = originalVals['YTM'].to_numpy()
spline = UnivariateSpline(a, b, s=5)
for i in range(0,100):
spline.set_smoothing... |
I think your notion of half life is interesting but, technically, it's not the definition of what half life is. The AR(1) is the best way to see it but
keep in mind that the principle is the same regardless of what model is being considered.
Suppose one has:
$y_t = \phi y_{t-1} + \epsilon_t $
Also, assume tha... |
I think your notion of half life is interesting but, technically, it's not the definition of what half life is. The AR(1) is the best way to see it but
keep in mind that the principle is the same regardless of what model is being considered.
Suppose one has:
$y_t = \phi y_{t-1} + \epsilon_t $
Also, assume tha... |
Someone told me that trading systems are very time-dependent and they must use network protocols that ensure speed.
Which network protocols do they use to ensure fast transmissions?
What’s an example of such a system?
Thank you |
I know a bond pays an annuity cashflow of coupon payments and then at maturity it pays the face value. But, at maturity, does it pay an additional coupon payment on top of the face value or are we supposed to think about this payment of its face value as its final coupon payment? |
Does a bond pay a coupon at maturity? |
I have been trying to implement GARCH(1,1) model for pricing call options. Suppose I have calibrated Garch(1,1) model for modelling the conditional volatility using the historical data of an equity through MLE. Now I want to price some call options for that particular equity for that I had tried to use the following ri... |
I'm trying to get a better grasp on the suitability of Geometric Brownian Motion as a model for asset prices, and I've stumbled across a hurdle which I can't seem to get over. Assuming GBM, the asset price is given by
$ S_t = S_0 \exp\{ (\mu - \frac{1}{2}\sigma^2)t + \sigma W_t \} $,
which implies that daily lo... |
I think your notion of half life is interesting but, technically, it's not the definition of what half life is. The AR(1) is the best way to see it but
keep in mind that the principle is the same regardless of what model is being considered.
Suppose one has:
$y_t = \phi y_{t-1} + \epsilon_t $
Also, assume tha... |
I have some technical questions about what are the best settings in Bloomberg to calculate the interest rate risk of a swap.
When Bloomberg calculates the DV01, it simply bumps the par swap curve by +/-10bps, then calculates the difference in NPVs, and divides it by 20. When Bloomberg does this, it does not touch t... |
Assume we have an SDE
$$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$
where $\sigma>0$ and $W_t$ is a Wiener process. Is there a transformation $y(X_t)$ that will make the dynamics of the transformed process $Y_t=y(X_t)$ have constant volatility? |
All you can do is fit the values of the parameters of a model you have already assumed.
> Now my problem is: how would you go about verifying this with empirical data?
As Kermittfrog mentioned, after fitting a particular model, you can use test to check whether your initial assumptions make sense or not.
> I... |
When pricing FX forwards, what type of curves are used? OIS, LIBOR, SOFR, etc? |
What type of curve is used for pricing fx forwards? |
I understood that given two listed assets, the one with the lower effective spread is more liquid, and if one has effective spread lower than the quoted one, it means there has been a price improvement. Which other conclusions could be figured by comparing one asset’s effective spread with another?
Thanksssss |
I understood that given two listed assets, the one with the lower effective spread is more liquid, and if one has effective spread lower than the quoted one, it means there has been a price improvement. Which other conclusions could be figured by comparing one asset’s effective spread with another?
|
I'm pretty new in structured products area and I have some basics questions regarding autocall :
- Why the autocall has an automatic redemption feature ? I mean an Investor could be interested in earnings coupons during more time if they are attractive. What is the purpose of this calling mechanism ? same question fo... |
https://help.tc2000.com/m/69445/l/754439-historical-volatility-ratio
"Historical volatility is calculated by taking the standard deviation of the natural log of the ratio of consecutive closing prices over time."
https://www.thebalance.com/how-average-true-range-atr-can-improve-trading-4154923
ATR is defined a... |
Relation between historical volatility and ATR? |
So I have a simple enough assignment question. I was asked to find the EWMA for daily stock returns of a given firm. When I was answering the Garch problem, I had multiple variables- alpha, beta, unconditional variance and I used the log likelihood as well.
I ran the initial EWMA using a lambda of 0.94, but my profe... |
I was wondering how Kelly criterion can be used for portfolio optimisation in the case one would like to optimise the portfolio for minimum variance.
I understand how the Kelly criterion can be used to decide the allocation for a certain stock, but what if I also would like to make sure I am diversified (= I want to s... |
I am working in some related field and working on the data, I am curious that how did Bloomberg obtain the interest rate of a particular currency.
In particular, one of my workings is on the HK government bond interest rate, and I realized that the Bloomberg rate is different from the official one calculated by
ht... |
How does Bloomberg calculate interest rate for a government bond? |
As you know, most of the EOD data available have only OHLC price. I used to do back-testing using the Close price as both bid and ask. however, in real world, the bid and ask spread may be huge and the bid/ask quantity is also never constant. Above that, the bid/ask price changes with every shares sold/bought. Therefor... |
Suppose I want to value a (fwd) starting swap, that means I would like to calculate the fixed rate $S_{\alpha, \beta}(t)$. Note, I'm using Brigo's Notation here. We know that the discounted payoff of the cash flows are of the form (payer swap)
$$ D(t,T_i) N \tau_i(L(T_{i-1},T_i)-S_{\alpha, \beta}(t))$$
where $N$ ... |
Suppose I want to value a (fwd) starting swap, that means I would like to calculate the fixed rate $S_{\alpha, \beta}(t)$. Note, I'm using Brigo's Notation here. We know that the discounted payoff of the cash flows are of the form (payer swap)
$$ D(t,T_i) N \tau_i(L(T_{i-1},T_i)-S_{\alpha, \beta}(t))$$
where $N$ ... |
I was wondering what portfolio optimisation is used by professionals. I know about these 3:
- Mean-variance
- Black-Litterman
- Kelly Criterion
Which one is preferred? Is there some other more robust method? |
Since VIX spot is not tradable, why do the futures and spot converge @ expiration? By what mechanism does this occur if arbitrage is not one of them? |
Why do VIX spot and futures converge if there is no cash and carry arbitrage? |
https://github.com/jerryxyx/TreasuryFutureTrading/blob/master/README.pdf
On page 2 of the pdf above, the sensitivity of a bond to increases in the slope is given as $\ln(T)$, where $T$ is the maturity of the bond.
- Is this relationship exact, i.e. derived from the bond pricing equation, or simply a rule of thu... |
https://github.com/jerryxyx/TreasuryFutureTrading/blob/master/README.pdf
On page 2 of the pdf above, the sensitivity of a bond to increases in the slope of the curve, is given as $\ln(T)$, where $T$ is the maturity of the bond.
- Is this relationship exact, i.e. derived from the bond pricing equation, or simply... |
I read on the Internet that future price = spot price + carrying cost.
What if I buy a future before the goods/assets are even being manufactured.
Will I get a discount on the spot price since I commit in advance? Or it still depends on the expectation of price in the future? |
What is the relation between spot price and future price if the goods are not produced yet? |
Let's say we caught arbitrage between spot and futures, we sold futures and bought spot. The risk-free rate is 5%, and our arbitrage position profit is 5.5%. When the interest rates increase to 6% before maturity, I will lose money.(This is called basis risk, I think)
How can I hedge this risk? Which derivatives instr... |
How can I hedge basis risk? |
I have an implied vol **discrete** grid, obtained from market data.
To obtain prices from these implied vols, a dividend model with discrete and proportional dividends is used.
How can I verify if there are calendar arbitrages in this implied vol grid?
[This paper (Gatheral, Jacquier. Arbitrage free SVI volatili... |
I have an implied vol **discrete** grid, obtained from market data.
To obtain prices from these implied vols, a dividend model with discrete and proportional dividends is used.
How can I verify if there are calendar arbitrages in this implied vol grid?
[This paper (Gatheral, Jacquier. Arbitrage free SVI volatili... |
Well, how do you know if the good were already produced? In fact, probably most of soft commodities for long dated contracts weren't produced yet.
This "extra risk" is a type of credit risk, the risk of not receiving the payment or good. If it will affect the price or not is a matter of contract specificities.
If... |
The risk neutral measure is often said to reflect the risk aversion of investors. So intuitively, I would think that an asset's expected discounted value should be lower under the risk neutral measure (-> market price) than under the real world measure. In other words, a risk averse investor should always prefer a fixe... |
Is the market price of an asset always lower than the expected discounted value under the REAL WORLD measure? |
Assume the Heston Model with dynamics under the martingale measure $Q$ given by
\begin{align}
dS_t &= (r-q)S_t dt + \sqrt{v_t}S_tdW_{1,t}^Q\\
dv_t &= \kappa(\theta-v_t)dt + \sigma\sqrt{v_t}dW_{2,t}^Q.
\end{align}
How, do I find the price of an European put option, i.e.
$$e^{-rT}E^Q[(K-S_T)^+]$$.
My idea is... |
From "Mathematics of Financial Derivatives" by Wilmott, Howison and Dewynne, section 5.4, p76.
How do I start making the transformations to get to the **dimensionless** equation? I.e. we start with the standard Black-Scholes PDE:
$${\frac {\partial V}{\partial t}}
+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{... |
As in the topic. Would for example lower volatility justify lower return, (if the German funds are less volatile in the first place?).
Perhaps it's due to the German https://en.wikipedia.org/wiki/Mittelstand not being traded on the exchange? Any level materials which would help me understand this would be helpful, ... |
Why is the long-term return (3+ years) of German index funds lower than that of many developed countries? |
I have a large dataset that includes my strategy back-test run data. The dataset columns include candle date, close price and many strategy related data.
I’ve built a Mathplotlib visualization for my backtest run data. While my visualization works, working with Mathplotlib charts is tedious. I like to have somethin... |
Assume the Heston Model with dynamics under the martingale measure $Q$ given by
\begin{align}
dS_t &= (r-q)S_t dt + \sqrt{v_t}S_tdW_{1,t}^Q\\
dv_t &= \kappa(\theta-v_t)dt + \sigma\sqrt{v_t}dW_{2,t}^Q.
\end{align}
How, do I find the price of an European put option, i.e.
$$e^{-rT}E^Q[(K-S_T)^+]$$.
My idea is... |
My question is quite simple: Does the Put-Call-Parity hold for the Heston model? My textbook handels the Black-Scholes model with the Put-Call-Parity being
$$p_t = Ke^{-rt(T-t)}+c_t-S_t.$$
However, it is not very specific about the assumptions.
Does the same formula apply directly to other models (e.g. the Hesto... |
Does the put-call-parity hold for the Heston model? |
My question is quite simple: Does the Put-Call-Parity hold for the Heston model? My textbook handels the Black-Scholes model with the Put-Call-Parity being
$$p_t = Ke^{-r(T-t)}+c_t-S_t.$$
However, it is not very specific about the assumptions.
Does the same formula apply directly to other models (e.g. the Heston... |
What are the approaches to optimise a portfolio where for each security the analyst specifies expected return (10%) and downside risk (-3%)? |
I am trying to develop a pricer for Call VS Call and I'm using MonteCarlo method to do so because my stocks are correlated between each others.
Basically my inputs are ```stocks```which is a dataframe whose columns are stock name, volatility, weight for the basket, strike, spot and dividendes, ```data```which is a d... |
I am trying to get the same value of EMA as in TradingView but different number of data points are creating different EMA values. Since it is a moving average, 200 is not enough and I have through some trial and error found 535 datapoints are yielding closer results to the 200 ema indicator value in TradingView. Could ... |
How many data point on EMA indicator? |
So I coded up the solution from here: https://quant.stackexchange.com/questions/57235/do-basket-options-have-a-closed-form-valuation-formula
Which provides a good solution for equally-weighted underlyings under a Black model. The simplified Python code is here after modifying it quite a bit. I tried to add a weigh... |
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