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I am implementing the Black Litterman model for a few assets, in particular I am using five ETF: EFA, EEM, GLD, TLT and IYR. My questions are the following: how can I compute the market capitalisation weights for these assets? Is it sufficient to divide the capitalisation of each ETF by a market index capitalisation? S... |
I am implementing the Black Litterman model for a few assets, in particular I am using five ETF:
- EFA (EAFE stock index: developed markets outside US and Canada)
- EEM (stocks from Emerging Markets)
- GLD (Gold)
- TLT (Long Term (20+ years) US Treasury Bonds)
- IYR (US Real Estate Investments Trusts)
... |
Do not use the capitalizations of the ETFs, they hold a small and variable proportion of the securities outstanding.
Instead base your weights on the papers by Swinkels et al. which have tried to estimate the actual amounts invested by all investors (not just ETFs!) in various assets. Their latest paper is *Historic... |
Is it valid to apply GRS-test ([Gibbons, Ross and Shanken 1989][1]) on non-normal and autocorrelated residuals?
I got residuals using 10 test-assets regressed on 3-factor and carhart.
If it is valid, how should I do it?
And if it is not valid, what should we do?
[1]: https://econpapers.repec.org/article... |
Take for example this passage from "Monte Carlo Methods in Financial Engineering".
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/0YkMu.png
Is this a result of the risk neutral world or is this the real world drift as well? I've never seen the explicit distinction made and I don't s... |
Is the differential between risk free rates the drift of an exchange rate only in the risk neutral world? |
This question is in regards to incomplete markets where multiple risk-neutral measures exist. I am a little bit confused by this idea. Say we have an incomplete market with only one stochastic process X(t) with
dX(t) = μ(t, X(t))dt + σ(t, X(t))dW(t)
Now let's say this market has two risk neutral measures(P1 and ... |
This question is in regards to incomplete markets where multiple risk-neutral measures exist. I am a little bit confused by this idea. Say we have an incomplete market with only one stochastic process $X_t$ with
$$\mathrm{d}X_t = \mu(t, X_t)\mathrm{d}t + σ(t, X_t)\mathrm{d}W_t$$
Now let's say this market has two... |
Bachelier Normal Implied Vol Python Script Jaekel Not computing? |
Do not use the capitalizations of the ETFs, they hold a small and variable proportion of the securities outstanding.
Instead base your weights on the papers by Swinkels et al. which have tried to estimate the actual amounts invested by all investors (not just ETFs!) in various assets. Their latest paper is *Historic... |
I have a fairly short and straightforward question. I am running a dynamic optimization strategy and therefore need to construct the FF5 characteristics. I am using COMPUSTAT quarterly accounting data.
Now my question is why are all accounting variables such as Book-to-market ratio, size etc. constructed using at l... |
My question is regarding the European regulation on standardizing the information in the KID's for PRIIPs. (<https://esas-joint-committee.europa.eu/Publications/Technical%20Standards/JC%202017%2049%20(PRIIPs_flow_diagram_risk_reward).pdf>)
In Part 3 (calculation of MRM for category 3 PRIIP) I can´t figure out what t... |
I wish to understand how dual curve bootstrapping is done?
Lets say we want to bootstrap FF OIS curve and Libor 3 month fwd curve simultaneously. Lets also assume we don't LIBOR-OIS basis swap rates to calculate ois discount curve (post 2 years in time: Till 2 years one can use the FF Futures). Specifically I want t... |
How to do simultaneous dual curve bootstrapping? |
In section 2.5 he describes an example of arbitrage-free pricing (attached below). I have a pretty solid understanding of how we arrived at $K' = K\frac{1+d}{1+r}$, but I got a little lost when he mentioned the alternative rate $L$.
[![Section 2.5][1]][1]
If $L$ is greater than $K$ do we buy or sell yen, and ... |
Is there a reliable machine readable source of stock market holiday calendars?
I found this source: https://eresearch.fidelity.com/eresearch/markets_sectors/global/holidayCalendar.jhtml
I could scrape it, but honestly, I have my doubts about it staying well formatted and conforming over time, or even exist at all... |
Does anyone have codes that would solve the multi-period Kyle model? |
I got extensive experience on algo trading for cash equity, FX, so just wondering any algo strategy popular for vanilla bond trading? |
Just wondering any algo strategy popular for vanilla bond trading? |
anyone can provide solution or some idea to the following question? thanks
[![Picture][1]][1]
[1]: https://i.stack.imgur.com/qRlPZ.png |
Notations: Given a binomial tree with $N$ periods and time to maturity $T,$ let $\Delta t = T / N.$
It is well-known that [CRR][1] uses the up and down multipliers as
$$u = e^{\sigma\sqrt{\Delta t}} \quad \text{and} \quad d = e^{\sigma\sqrt{\Delta t}} = \frac{1}{u}.$$
In [another post][2], mark Joshi suggested... |
Notations: Given a binomial tree with $N$ periods and time to maturity $T,$ let $\Delta t = T / N.$
It is well-known that [CRR][1] uses the up and down multipliers as
$$u = e^{\sigma\sqrt{\Delta t}} \quad \text{and} \quad d = e^{\sigma\sqrt{\Delta t}} = \frac{1}{u}.$$
In [another post][2], mark Joshi suggested... |
Notations: Given a binomial tree with $N$ periods and time to maturity $T,$ let $\Delta t = T / N.$
It is well-known that [CRR][1] uses the up and down multipliers as
$$u = e^{\sigma\sqrt{\Delta t}} \quad \text{and} \quad d = e^{\sigma\sqrt{\Delta t}} = \frac{1}{u}.$$
In [another post][2], mark Joshi suggested... |
The Fundamental Theorem of Asset Pricing states that:
\begin{align*}
\frac{X_0}{N_0} &= \mathbb{E}^N{ \left[ \frac{X_0}{N_0}|\mathcal{F}_0 \right] }
\end{align*}
The usual conditions apply (both $ N(t) $ and $ X(t) $ are traded assets, markets are complete, etc.)
**Question**: **does the equation above still... |
The Fundamental Theorem of Asset Pricing states that:
\begin{align*}
\frac{X_0}{N_0} &= \mathbb{E}^N{ \left[ \frac{X(t)}{N(t)}|\mathcal{F}_0 \right] }
\end{align*}
The usual conditions apply (both $ N(t) $ and $ X(t) $ are traded assets, markets are complete, etc.)
**Question**: **does the equation above sti... |
When implementing a Black-Scholes delta-neutral portfolio using Python to perform delta hedging, I am not sure whether I implemented it correctly or not.
Unlike coding binomial trees for European call option, we can compare Black-Scholes analytical pricing formula with the price given by binomial tree model. If they h... |
Is there any way to check my delta hedging is implemented correctly? |
Notations: Given a binomial tree with $N$ periods and time to maturity $T,$ let $\Delta t = T / N.$
It is well-known that [CRR][1] uses the up and down multipliers as
$$u = e^{\sigma\sqrt{\Delta t}} \quad \text{and} \quad d = e^{\sigma\sqrt{\Delta t}} = \frac{1}{u}.$$
In [another post][2], Mark Joshi suggested... |
Notations: Given a binomial tree with $N$ periods and time to maturity $T,$ let $\Delta t = T / N.$
It is well-known that [CRR][1] uses the up and down multipliers as
$$u = e^{\sigma\sqrt{\Delta t}} \quad \text{and} \quad d = e^{\sigma\sqrt{\Delta t}} = \frac{1}{u}.$$
In [another post][2], Mark Joshi suggested... |
I computed daily returns via $\ln(P_t)-\ln(P_{t-1})$ in order to get the volatility. But I do not want to present my results over a certain period of time (whether it's a week, a month or a year). I just want to see the daily variations. Is that meaningful?
Also, in that case, should I still compute the standard de... |
I want to consider 4 scenarios in google sheets. All deal with a periodic return over `n` periods
- Positive to Positive
- Positive to Negative
- Negative to Negative
- Negative to Positive
For instance EPS in 2010 was $1 and EPS in 2020 was $3. How to best calculate the annual return between these data ... |
I am currently reading through a study published by the Institute and Faculty of Actuaries on [hedging practices][1] within the insurance industry.
Within the executive summary, under 'Key Risks', it is stated that
> Equity, credit and interest rates are the big three dominant risks
> out of our respondents
... |
Why do (life) insurance companies face equity risk? |
We have two **uncorrelated** Stock price processes and the classical Money-Market (MM) account. Under the MM Numeraire, both stocks are Martingales when discounted by the MM, as usual.
**Question**: **I would like to change the Numeraire to one of the stocks and apply the Fundamental Theorem of Asset pricing to sho... |
We have two **uncorrelated** Stock price processes and the classical Money-Market (MM) account. Under the MM Numeraire, both stocks are Martingales when discounted by the MM, as usual.
**Question**: **I would like to change the Numeraire to one of the stocks and apply the Fundamental Theorem of Asset pricing to sho... |
I want to consider 4 scenarios in google sheets. All deal with a periodic return over `n` periods
- Positive to Positive
- Positive to Negative
- Negative to Negative
- Negative to Positive
For instance EPS in 2010 was \$1 and EPS in 2020 was \$3. How to best calculate the annual return between these dat... |
I'm currently studying the Black&Scholes model and I'm not sure about the following thing: the log return, say r, doesn't evolve in time? I mean, dr/dt = 0, it's derivative is zero? Does only its average evolve in time, that is, d average(r)/dt = something? Thank you. |
>how do you handle the covariance matrix, are you allowed to do the same thing you do with equities? I.e. just calculate the variance and covariance as the sample variance and covariance from the time series of the underlying interest rates for each time to maturity
Many people do exactly what you wrote: they collec... |
I'm currently studying the Black&Scholes model and I'm not sure about the following thing: the log return, say r, doesn't evolve in time? I mean, dr/dt = 0, its derivative is zero? Does only its average evolve in time, that is, d average(r)/dt = something? Thank you. |
The objective of hedging is to reduce the variance of the (position+hedge) portfolio. So which of these two solutions gives a smaller variance?
You could calculate it numerically and compare the variances. However, in general ... the answer is going to be: whichever of commodity 1 or commodity 2 has higher correlat... |
The publication is made by the UK institute of actuaries so I'm answering from the perspective of the insurance industry in the European union.
## Is it allowed?
For European Insurance companies EIOPA gathers a [number of statistics][1], among which [asset exposures][2]. The table below shows figures as of 2019Q4... |
I would like to add to @DaneelOlivaw answer.
Your question: **"How does one go about finding the right measure for a product?"**
Answer: **One should chose any measure that will make it easy and convenient to compute the pricing at hand.**
You are free to use whichever measure you would like. For example, it... |
I try to find the proof of the theorem of Foreign-Domestic Symmetry for FX options, but I did not found, Some paper or book, where it is explained?
Theorem 2.1 in this paper:
https://arxiv.org/pdf/1912.01387.pdf
Thx. |
Let me try to answer. In the Black-Scholes model, we have the following dynamics for a stock Price $S_t$:
$$S(t)=S(0)+\int^{t}_{0}r S(h)dh+\int^{t}_{0}\sigma S(h)dW(h)$$
The short-hand notation for the above would be:
$$dS_t= r S_t dt+\sigma S_tdW_t$$
The two equations are the same thing (just two different... |
This is an easy question, I hope.
Suppose we have a swap A with a long position, which, originally, has a weight of 30%. Over time, it has a positive performance of 3%, meaning we have a multiplier of 1.03 relative to its original value.
Hence the resulting weight would be 30% * 1.03 = 39% (as a % over original por... |
How does negative performance of a portfolio constituent affect its weight? |
Suppose i have the following OIS Swap rates:
1 year OIS Swap: 0.36%
2 year OIS Swap: o.37%
3 year OIS Swap: 0.38%
4 year OIS Swap: 0.40%
From these, how do I get the OIS Discounting factors for these respective years?
Someone please explain with a proper formula that can be applied for this? |
It depends:
Does the exchange support Stop orders? Some do, some don't. You can find it in exchange's documentation. If the answer is "no" but your broker offers it, then Stop orders are managed either by your broker (on the "server side") or maybe by your trading platform (on the "client side"). If the exchange sup... |
This is an easy question, I hope.
Suppose we have a swap A with a long position, which, originally, has a weight of 30%. Over time, it has a positive performance of 3%, meaning we have a multiplier of 1.03 relative to its original value.
Hence the resulting weight would be 30% * 1.03 = 30.9% (as a % over original p... |
The only true order book is the order book which consists of orders, i.e. order-by-order or market-by-order (MBO) full-depth order book. It's the order book managed internally by absolutely all exchanges (stocks, futures, crypto/digital, etc). The market-by-price (MBP) is an aggregated view of the order book. You manag... |
The only true order book is the order book which consists of orders, i.e. order-by-order or market-by-order (MBO) full-depth order book. It's the order book managed internally by absolutely all exchanges (stocks, futures, crypto/digital, etc). The market-by-price (MBP) is an aggregated view of the order book. You manag... |
Let me try to answer. Option price is proportional to the IV. In fact, liquid options are quoted NOT in terms of prices, but in terms of IV.
**(A)** **simple strategies involving options** (long or short): the statement that **"Buying options when IV is 55 and selling when it is 30 is a sure way to lose money"** ma... |
Could someone please help me explain the concept of Digital Height clearly in the options world? Thank you |
Let me try to answer. Option price is proportional to the IV. In fact, liquid options are quoted NOT in terms of prices, but in terms of IV.
**(A)** **Simple strategies involving options** (long or short):
The statement that **"Buying options when IV is 55 and selling when it is 30 is a sure way to lose money"*... |
You have a conversion ratio of $100/12.1 \approx 8.26$, so the convertibility is an option on about 8 underlying stocks and the delta scales accordingly. I'm not familiar about the way it's quoted, though. Did you expect it to be the delta for one unit of stock? |
I was recently speaking to someone who works at a UK life insurer which offers defined benefit pension scheme buy-outs. He mentioned that the company employs traders (of bonds and swaps, mostly) and has a hedging team who prescribes limits to the traders on a weekly basis.
He explained that the hedging team consider... |
Why might a bulk annuity provider hedge their exposure to risks such as inflation, interest rates, and exchange rates on a weekly basis? |
I am really upset about an exercise, because i don't get it. Maybe someone can find my error?
We are talking about a single period model. Therefore our Asset is denoted by $S(0)=1$, $U=0.05$, $D=-0.05$, $R=0$ and $K=1$. Question seems easy: How much is the premium for the option $C(0)$?
I know: $C(0)=x_{C}S(0)+y... |
We have two **uncorrelated** Stock price processes and the classical Money-Market (MM) account. Under the MM Numeraire, both stocks are Martingales when discounted by the MM, as usual.
**Question**: **I would like to change the Numeraire to one of the stocks and apply the Fundamental Theorem of Asset pricing to sho... |
Suppose I have the following OIS Swap rates:
1 year OIS Swap: 0.36%
2 year OIS Swap: 0.37%
3 year OIS Swap: 0.38%
4 year OIS Swap: 0.40%
From these, how do I get the OIS Discounting factors for these respective years?
Could someone please explain with a proper formula that can be applied for this? |
The C++ version of QuantLib has a function called [years][1]:
Real years(const &Period p);
I don't see this in QuantLib-Python version 1.18. Is it supposed to be there or was it not SWIGed?
[1]: http://cogitolearning.co.uk/2012/10/quantlib-periods-time-units-and-frequencies/ |
QuantLib-Python: where is the function to take Period to years? |
Presuppose that
1. on May 1 2020, you bought 1 BA 90P 6/19/2020.
2. BA rocketed back to [$205](https://finance.yahoo.com/quote/BA?p=BA&.tsrc=fin-srch) as at 6/7/2020, and you no longer expect BA will drop back to \$90. Now you solely desire to minimize your loss, even if you must limit your gains.
What are t... |
Can't losses from Long Puts always be reduced, by turning them into Put Credit Spreads? |
I am really upset about an exercise, because i don't get it. Maybe someone can find my error?
We are talking about a single period model. Therefore our Asset is denoted by $S(0)=1$, $U=0.05$, $D=-0.05$, $R=0$ and $K=1$. Question seems easy: How much is the premium for the option $C(0)$?
I know: $C(0)=x_{C}S(0)+y... |
The Fundamental Theorem of Asset Pricing states that:
\begin{align*}
\frac{X_0}{N_0} &= \mathbb{E}^N{ \left[ \frac{X(t)}{N(t)}|\mathcal{F}_0 \right] }
\end{align*}
The usual conditions apply (both $ N(t) $ and $ X(t) $ are traded assets, markets are complete, etc.)
**Question**: **does the equation above sti... |
What are some asset pricing textbooks that give a solid introduction into the field?
I suggest one textbook per answer with a list of its pros and cons. |
I know that is possible to download the market capitalisation os stocks using the "pandas_datareader" library: ```data.get_quote_yahoo```. However, this command gives me the last data of capitalisations for the tickers I provide while I need the whole time series of stock capitalisations.
There's a way to get it dir... |
The Fundamental Theorem of Asset Pricing states that:
\begin{align*}
\frac{X_0}{N_0} &= \mathbb{E}^N{ \left[ \frac{X(t)}{N(t)}|\mathcal{F}_0 \right] }
\end{align*}
The usual conditions apply (both $ N(t) $ and $ X(t) $ are traded assets, markets are complete, etc.)
**Question**: **does the equation above sti... |
**Edit**: **is there even any work published on Numeraire being correlated to one of the traded assets? I haven't found a single paper online on this topic. Could anyone please share links to any resources on this topic?**
The Fundamental Theorem of Asset Pricing states that:
\begin{align*}
\frac{X_0}{N_0} &= \m... |
Lets say I want to estimate a replicating portfolio by doing a linear regression between the returns of a long-only portfolio and several long-short factors like **Fama-French 5-factor** or **Betting Against Beta** (CAPM framework).
I discover that I can replicate and explain about 60% of the variance in my **long-... |
I am struggling with developing a binomial tree with jumps. although there are models such as CRR, could you suggest a book or have any idea to proceed?
Thanks,
Amir |
anyone can provide solution or some idea to the following question? thanks
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/s5fFa.png |
**I'm trying to find Flow of Funds and Money Trail using python**
**but i'm stuck in middle of it not understanding what to do next**
**My Main goal is to finding Flow of Funds i.e flow of transaction from one account to another**
**Or there is any library that will be helpful ?**
*data.csv file*
[![da... |
What is the meaning of the difference between the quantile of prob=0.05 and mean for a sample form a specific distribution?
In other words, I would like to understand the relationship between quantile and mean?
Thanks a lot.
[![enter image description here][1]][1]
Question:
[![enter image description... |
the relationship between VaR(0.05) and mean? |
**Edit**: **is there any work published on Numeraire being correlated to one of the traded assets? I haven't found a single paper online on this topic. If anyone has links to any resources on this topic, could you please share?**
The Fundamental Theorem of Asset Pricing states that:
\begin{align*}
\frac{X_0}{N_0... |
I am just trying to understand that if commodity forwards are available, what is the use of a commodity swap. If a farmer wants to hedge his risk, he can do it via entering into a commodity forward, why is the commodity swap required at all ? |
What is the need of commodity swap if commodity forwards are available? |
Let $\mathscr{T}$ be the set of stopping times with values in $[0, T]$. Note that, for any $\tau \in \mathscr{T}$, $\lambda_1\ge 0$, $\lambda_2 \ge 0$, and $\lambda_1+\lambda_2 =1$,
\begin{align*}
&\ \max(\lambda_1 K_1+\lambda_2 K_2 -S_{\tau}, \, 0) \\
=&\ \max\big(\lambda_1 (K_1-S_{\tau})+\lambda_2 (K_2 -S_{\tau})... |
Let $\mathscr{T}$ be the set of stopping times with values in $[0, T]$. Note that, for any $\tau \in \mathscr{T}$, $\lambda_1\ge 0$, $\lambda_2 \ge 0$, and $\lambda_1+\lambda_2 =1$,
\begin{align*}
\small
&\ \max(\lambda_1 K_1+\lambda_2 K_2 -S_{\tau}, \, 0) \\
=&\ \max\big(\lambda_1 (K_1-S_{\tau})+\lambda_2 (K_2 -S... |
Let $\mathscr{T}$ be the set of stopping times with values in $[0, T]$. Note that, for any $\tau \in \mathscr{T}$, $\lambda_1\ge 0$, $\lambda_2 \ge 0$, and $\lambda_1+\lambda_2 =1$,
\begin{align*}
&\ \max(\lambda_1 K_1+\lambda_2 K_2 -S_{\tau}, 0) \\
=&\ \max\big(\lambda_1 (K_1-S_{\tau})+\lambda_2 (K_2 -S_{\tau}), ... |
Does anyone know of a Python library that includes the calculation of historical stock volatility using the Yang Zhang estimator? I have tried and failed to find one but would expect this to have been implemented in one of the Python libraries used by quants. |
Where can I find a Python module for Stock volatility estimators using Yang Zhang method? |
Does Alpha Vantage provides real-time data for free? |
For the fitting and forecasting of time-series data on stock price, the most frequent model I have heard of is ARIMA. ARIMA is actually conducting a regression of stock prices and residuals of stock prices in the few days prior to the current date. ARCH model and its derivatives model variances on the data in the previ... |
What are some good models for stock price predictions? |
This is an easy question, I hope.
Suppose we have a swap A with a long position, which, originally, has a weight of 30%. Over time, it has a positive performance of 3%, meaning we have a multiplier of 1.03 relative to its original value.
Hence the resulting weight would be 30% * 1.03 = 30.9% (as a % over original p... |
I've fit a GARCH(1,1) model in R and would like to create a plot similar to the one in this question: https://quant.stackexchange.com/questions/43545/is-this-the-correct-way-to-forecast-stock-price-volatility-using-garch
Could someone direct me to a guide or literature that I could use to achieve this? |
Suppose I have a USD holding of 1,000,000 in my portfolio and I want to convert it into EUR in a month's time.
I enter into a FX forward contract of the same amount USD 1,000,000, meaning that I have a hedge ratio of 100%.
If the portfolio holding is now worth $990,000, is my hedge ratio 990,000/1,000,000? |
How do I calculate FX forward hedge ratio? |
Is there a vix-like index available for s&p 400 mid-cap instruments? i.e., implied vol based on s&p 400 put and call options? also, an iv for the s&p 600? thank you
|
According to most books the ATM option is the option with a delta of 0.50. However, this is only the case when the distribution is normal. The more positively skewed the distribution, the further the 0.50 delta option is out-of-the-money (for calls). According to the following [article](https://moontowermeta.com/lesson... |
I have a portfolio optimization problem similar to [this question][1] here, with a V-shape transaction costs such that we pay a fee proportionally to the sum of absolute rebalancing:
$$TC(\omega) = \frac{1}{2} |\omega-\omega_\text{old}|' \cdot \gamma$$
where $$\omega : \text{target portfolio to optimize},$$
$$\omega... |
Suppose someone buys $4bn of a particular stock over the period of a few weeks. Depending on how much that stock is being traded, you would expect that the price goes up in a visible way compared to if the purchase hadn't been made. I think there is no *exact* way to calculate the stock price after such a transaction u... |
How can we estimate new stock price after a large purchase? |
How do I compute the optimal strike / expiry / when to close or roll for a covered call strategy on a highly liquid underlying (e.g. SPY)?
|
I have a portfolio optimization problem similar to [this question][1] here, with a V-shape transaction costs such that we pay a fee proportionally to the sum of absolute rebalancing:
$$TC(\omega) = \frac{1}{2} |\omega-\omega_\text{old}|' \cdot \gamma$$
where $$\omega : \text{target portfolio to optimize},$$
$$\omega... |
I have a portfolio optimization problem similar to [this question][1] here, with a V-shape transaction costs such that we pay a fee proportionally to the sum of absolute rebalancing:
$$TC(\omega) = \frac{1}{2} |\omega-\omega_\text{old}|' \cdot \gamma$$
where $$\omega : \text{target portfolio to optimize},$$
$$\omega... |
I have a portfolio optimization problem similar to [this question][1] here, with a V-shape transaction costs such that we pay a fee proportionally to the sum of absolute rebalancing:
$$TC(\omega) = \frac{1}{2} |\omega-\omega_\text{old}|' \cdot \gamma$$
where $$\omega : \text{target portfolio to optimize},$$
$$\omega... |
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