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I have estimated the term premium for the Polish Government Bonds (POLGBs) using the methods described by [Adrian et al. (2013)][1] (often referred to as ACM). The underlying yields are interpolated from the Nelson-Siegel-Svensson model based on a bootstrapped zero-coupon yield curve.
I have estimated the model for ... |
#### Giving my two cents on your questions:
- **The main motivation for the use of Vasicek single factor framework**, is that the model produce analytically tractable formulas that are easy to implement and does not require any extensive numerical simulations. The model is able to reproduce the qualitative behavi... |
I am looking for ways to express a directional bet on a commodity through futures options.
**Assume** that there's 50% probability that oil will spike up 100% in the span of 30% during the next 9 months and that the spike will not last more than 15 days.
How to plug in this opinionated view of the future distribu... |
1. Typically events like those last a quarter or some variable time period. Then your position may get stop lossed half way through. So instead of forcing an alignment, just simulate a portfolio.
2. Your strategy must start with some seed money, otherwise how can you calculate a return or do a trade? You start with ... |
- **The main motivation for the use of Vasicek single factor framework**, is that the model produce analytically tractable formulas that are easy to implement and does not require any extensive numerical simulations. The model is able to reproduce the qualitative behavior of empirical credit loss distributions, namely ... |
Why is there a square root in the formulation for the asset value $ X_i $ given the common factor $ Y $ and the idiosyncratic component $ Z_i $ below?
$$ X_i=\sqrt{\rho}\,Y+\sqrt{1-\rho}\,Z_i,$$
As you write it follows from the assumption of assets all having equal pairwise correlations $ \rho\ $ to each other an... |
I am trying to get the implied swap rate of cap on a YoYIIS. As per any other option, I wanted to use the put-call parity relationship of the B&S model.
In this case tho the underlying of the option is a swap that pays a the difference between the Year-on-Year inflation and the strike rate. My process then was to:
... |
In 2020 I wrote a MSC thesis on the hedging of exotic options using recurrent neural networks (loosely based on the paper Deep Hedging (2018)by Buehler et al.).
Since then I have been interested in all types of ML applications in the financial domain, but I haven't been reading any of the recent publications/follow... |
What are some interesting recent machine learning related developments in the QF domain? |
In 2020 I wrote a MSc thesis on the hedging of exotic options using recurrent neural networks (loosely based on the paper Deep Hedging (2018)by Buehler et al.).
Since then I have been interested in all types of ML applications in the financial domain, but I haven't been reading any of the recent publications/follow... |
What are some good books to get started with option theory? |
**Update** (Original post below): After a bit of reasearch I have found that the put call parity on caps and floors works as it follows:
cap - floor = swap
Where cap, floor and swap are expressed in their NPV values.
So, using the below model of a fixed rate bond, I could back out the **implied YoY swap ra... |
Given we know the
- Notional
- Trade Price
- Currency in which the Bond Pays Coupons
- FX Rate from Bond Currency to USD
Trying to understand if a Bond can have FX Delta Risk and how it should be computed? |
Can a Bond have FX Delta Risk? |
How does one scale the 10yr yield on a chart to visualize its relative impact to asset valuations? I.e., so that the risk-free rate moving from 1->2% shows as a much larger movement than 11->12%.
For instance, assuming stockskies have a constant equity risk premium and EPS, the risk-free rate moving from 1%->2% hits... |
How to scale t-bond yield movements on a chart to visualize its relative impact to the pricing of other assets? |
for treasury off the run bonds, in general, does the market prefer high coupon or low coupon bonds ?
How about for credit bonds ? |
I am investigating two time series where the first is the daily closing stock price changes and the other is the daily changes in the PCE index.
I want to investigate how much the PCE index explains the returns by using a rolling granger causality like this one: [![enter image description here][1]][1]
[1]: h... |
The problem I am looking at concerns the treatment of almost identical assets in portfolio construction.
Let us assume that we have two assets, both with an expected return $\mu=0.1$, a standard deviation $\sigma=0.2$ and a correlation matrix
$$
C = \begin{pmatrix}
1 & 0.98 \\
0.98 & 1
\end{pmatrix}.
$$
A... |
I've been trying to price some barrier options with a Heston model on Quantlib and it works quite well most of the times, but sometimes I get the error: "interpolation range is [3.4314, 4.28384] x [0, 1.07814]: extrapolation at (3.30138, 0.0939798) not allowed". Is anyone familiar with this issue? I tried to do it with... |
I've been trying to price some barrier options with a Heston model on Quantlib and it works quite well most of the times, but sometimes I get the error: "interpolation range is [3.4314, 4.28384] x [0, 1.07814]: extrapolation at (3.30138, 0.0939798) not allowed". Is anyone familiar with this issue? I tried to do it with... |
I am trying to understand the below:
Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] =some positive number be a profit or loss?
In this calculation the purchase price is not taken into consideration.
The above positive number make sense if it called portfolio value at the end of time t1.
Be... |
Say I have 100 different alphas that all have statistically significant returns in-sample.
Is the best practice to use historical covariance matrix plus Markowitz portfolio theory to create an optimal weighting for each alpha?
I have to imagine that there is a way at forecasting covariance that is better than ju... |
What is industry best practice to combine alphas? |
Question as above regarding ISDA standard initial margin model. Are there any technical limitations to calculate margin for various risk classes (i.e. wait for end of tay trading book data)? |
What is the usual timing of IM calculation (same day 4pm, COB, COB +x hours etc)? What is the range on the market? |
The current paper (June 2022) "An Investor’s Guide to Crypto" from several authors from Duke University and MAN Group (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4124576) provides a good overview of valuation methods:
- The value of a network is proportional to the square of network size (Metcalfe’s Law). H... |
**Solved**
As pointed by @dm63 in the comments, the implied swap rate can be derived by solving the caplet (floorlet) formula for the the interest rate, where you set the formula equal to the Swap NPV (due to the put-call parity).
**Update** (Original post below): After a bit of reasearch I have found that the ... |
**Added later** The first answer I came up with was embarrassingly incorrect; I have pointed out why. Please feel free to uncheck it.
<s> Technically, this should be impossible: the Spec pool is always deliverable into the TBA, so the TBA OAS is a floor for the Spec pool valuation. </s> This is plain wrong -- the ... |
I am scratching my head with an optimization problem for Avellaneda and Stoikov market-making algorithm (optimizing the risk aversion parameter), and I've come across https://github.com/im1235/ISAC
which is using SACs to optimize the gamma parameter.
----
since SAC is a model-free reinforcement learning, does ... |
can Soft Actor-Critic reinforcement learning algorithms be used in real-time trading? |
How is the simple forward rate L(0,T,T+1) calculated given the spot rate L(0,T)? |
Status of this answer: _latest update July 7, 2022_.
There is a new paper out which is quite interesting and which basically says that cryptocurrencies are indeed a new asset class, potentially useful as a diversifier of conventional asset classes:
Corbet, Shaen and Meegan, Andrew and Larkin, Charles James and Lu... |
The current paper (June 2022) "An Investor’s Guide to Crypto" from several authors from Duke University and MAN Group (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4124576) provides a good overview of valuation methods:
- The value of a network is proportional to the square of network size (Metcalfe’s Law). H... |
I have coded a machine learning algo (sklearn) in Python, that uses different 'look back periods' for training a model, which is then used to predict future prices of a stock.
It has a 52% accuracy in predicting the future price. I now wish to build a simple strategy around this for backtesting. I am specifically in... |
I would like to ask about discount rates calculation algorithm by Bloomberg terminal.[![Zero Curve from Bloomberg Terminal][1]][1]
In the image above is possible to notice the discount rate for each term. The short end, instruments from 1 DY up to 18 MO, is composed by zero coupon swaps. The long end, instruments fr... |
I'm having an hard time understanding how cointegration works. Basically i'm trying to find cointegrated pairs in the crypto market, so i do the following:
1) Get OHLC data for the two markets (i'm getting 5k candles on the 5m timeframe)
2) Get the log returns for the two markets
3) Check the p-value when i look f... |
I thought that the standard mean-variance dynamic problem is actually time inconsistent. See for example: [Basak and Chabakauri (2010)][1]
[1]: https://www.jstor.org/stable/40782974?seq=1 |
A lot of the literature in dynamic mean-variance problem states that the dynamic mean-variance problem is time-inconsistent. Now I was not able to find an example of why the problem is time inconsistent that is easy enough to digest. So the question is: why is the dynamic mean-variance problem time-inconsistent? |
Why is the dynamic mean-variance problem time-inconsistent? |
in the notes about arbitrage arguments I am reading, I notice the statement
> We can also see that
> $$C^E_t>(S_t-K\mathrm{e}^{-r(T-t)})^+$$
> Notice that the inequality holds STRICTLY!
I don't particularly understand why the inequality must be strict. What arbitrage can occur when equality occurs? What exactly... |
Why is this inequality strict for arbitrage argument for European call? |
Synthetix project provides the system where different assets like USD, BTC, stocks are emulated by minting tokens representing them (sUSD, sBTC) collateralised by SNX token. Prices are defined via oracles.
Docs https://docs.synthetix.io/
The system looks like a more fragile system than DAI as the collateral imple... |
Below is an extract from Steven Shreve’s BK 2, Chapt 10: Term Structure models. [LINK][1]
I am trying to understand Stochastic Calculus from the above book with the help of a Pure Math PhD student.
Despite trying to wade through resource in Public domain (Google Search) and a few of the other books I bought, we cou... |
So, I've just started looking into financial mathematics and the following question keeps bugging me. From what I understood, if the market is arbitrage-free and a given contingent claim of value $h$ is attainable, then there is a measure $Q$ such that the discounted asset prices $(\tilde{S}_t)$ form a martingale. Cons... |
Should we use the conditional expectation to write the value of an option? |
Stochastic solutions are an unavoidable property of stochastic methods, in particular optimisation methods. See for instance section 3 in [A Review of Heuristic Optimization Methods in Econometrics][1]. In general, you cannot get rid of randomness; you
need to analyse it, by looking at and analysing distributions (e... |
On Prof. French's web site we have the following
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_developed.html
>>WML is the equal-weight average of the returns for the two winner portfolios for a region minus the average of the returns for the two loser portfolios,
>>
>>WML = 1/2 (Small... |
The current Fed Funds Rate is 1.75% whereas the 1 Month Treasury Rate is at 1.28%. I would have expected the Fed Funds Rate to be lower than Treasury rate on short maturities, what is the reason of the spread? |
I have a set of portfolio weights $w$. I'm using `cvxpy` to optimise the portfolio sharpe, subject to a set of constraints.
$$
\text{maximize} \hspace{10mm} \mu^Tw - w^T\Sigma w
$$
$$
\text{subject to:} \hspace{5mm} 1^Tw = 0 \hspace{5mm} \text{(Long/Short cash neutral)}
$$
$$
||w||_1 \leq L_{max} \hspace{5mm}... |
When / how do I vol-scale portfolio weights when optimizing the portfolio? |
ERROR: type should be string, got "https://qoppac.blogspot.com/2017/02/can-you-eat-geometric-returns.html#:~:text=We%20can%20also%20use%20geometric,costs%20and%20still%20be%20ahead.&text=This%20explains%20the%20weird%20title,them%20to%20pay%20higher%20costs.\r\n\r\nThe blog post above discusses the use of geometric means to evaluate investments. The section \"The consequences of using geometric returns\" gives the following example.\r\n\r\nAssuming correlation of 0.85:\r\n\r\n - 1 asset: arithmetic mean 5%, geometric mean 1.3% \r\n - 5 assets: arithmetic mean 5%, geometric mean 1.8%\r\n\r\nUnfortunately the calculation is not given so could someone run me through how this was calculated?\r\n\r\nI want to understand the intuition as to why the geometric mean is improved by the positive correlation what would happen if the correlation was zero or negative? \r\n\r\nThanks\r\n\r\nBaz\r\n\r\n" |
Why is the performance of a portfolio based on geometric means boosted by positive correlation? |
ERROR: type should be string, got "https://qoppac.blogspot.com/2017/02/can-you-eat-geometric-returns.html\r\n\r\nThe blog post above by Rob Carver discusses the use of geometric means to evaluate investments. The section \"The consequences of using geometric returns\" gives the following example.\r\n\r\nAssuming correlation of 0.85:\r\n\r\n - 1 asset: arithmetic mean 5%, geometric mean 1.3% \r\n - 5 assets: arithmetic mean 5%, geometric mean 1.8%\r\n\r\nUnfortunately the calculation is not given so could someone run me through how this was calculated?\r\n\r\nI want to understand the intuition as to why the geometric mean is improved by the positive correlation what would happen if the correlation was zero or negative? \r\n\r\nThanks\r\n\r\nBaz\r\n\r\n" |
[Brooks - *Introductory Econometrics for Finance*][1] is a very comprehensive, accessible overview of all the mainstream models for capturing non-normality, serial correlation, non-stationarity and other properties of financial returns, data, and volatility, bundled with plenty of reproducible examples.
[Mills and ... |
This might be silly, but I’m seeking to use QuantLib to price vanilla American call and put options using a Black-Scholes-Merton process and the Monte Carlo pricing engine based on the Longstaff Schwartz algorithm.
My question is: Am I confined to Gaussian pseudorandom numbers in this engine? Or can I use pseudo RN... |
Can I use the QuantLib Monte Carlo engine to price American options using heavy/fat tailed-distributed random numbers? |
In their paper, Francis Longstaff and Eduardo Schwartz found that using Laguerre, Hermite, Legendre or simple powers made very little difference in the results. Some time ago, I also played around with the various choices of polynomials and came to the same conclusion.
There are many more significant choices in the ... |
On the 16th of March 2020, the Polish Central Bank announced its first-ever round of Quantitative Easing. I am conducting an event study on how this announcement impacted the term structure. The major issue is that in the same press announcement the central bank also lowered its policy rate by 50 bps. My goal is to get... |
Let's consider there is an instrument N traded on a single venue (centralized anonymous limit orderbook). Let's say that most taker orders are tiny, therefore the one who stays at the best bid/offer gets most turnover. A market maker, who knows this, pursues the following strategy:
> if inventory < X, place a buy or... |
How can a top-of-the book market maker protect itself from exploiting? |
I am trying to determine how CME calculated their Implied Repo Rates in table 3 on the penultimate page of the Understanding Treasury Futures Document:
https://www.cmegroup.com/education/files/understanding-treasury-futures.pdf
The quotes for the ten year note contract are presented from October 10, 2017, with fu... |
Does implied volatility vary for calls vs puts? |
On the 16th of March 2020, the Polish Central Bank announced its first-ever round of Quantitative Easing. I am conducting an event study on how this announcement impacted the term structure.
The main obstacle is the fact that in the same press announcement the central bank also lowered its policy rate by 50 bps. My... |
There is a famous formula for the variance swap strike that reads
$$
K_{var}^2 = \int_{-\infty}^\infty dz\, n(z) I^2(z)
$$
where $I(z)$ is the Black-Scholes implied volatility function,
$$
n(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac12 z^2}
$$
and $z$ is the Black-Scholes `$d_2$' function
$$
z = \frac{\log S_t/K}{... |
There is a famous formula for the variance swap strike that reads
$$
K_{var}^2 = \int_{-\infty}^\infty dz\, n(z) I^2(z)
$$
where $I(z)$ is the Black-Scholes implied volatility function,
$$
n(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac12 z^2}
$$
and $z$ is the Black-Scholes `$d_2$' function
$$
z = \frac{\log S_t/K}{... |
There is a famous formula for the variance swap strike that reads
$$
K_{var}^2 = \int_{-\infty}^\infty dz\, n(z) I^2(z)
$$
where $I(z)$ is the Black-Scholes implied volatility function,
$$
n(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac12 z^2}
$$
and $z$ is the Black-Scholes `$d_2$' function
$$
z = \frac{\log S_t/K}{... |
I have started to look deeply in the hedging and I have created some simulations to simulate delta hedging strategies. I use BS model to calculate delta. The only issue was, which **Volatility** should I use?
- Implied volatility
- Realised volatility/Actual volatility (the actual volatility, which I can not pred... |
I have started to look deeply in the hedging and I have created some simulations to simulate delta hedging strategies. I use BS model to calculate delta. The only issue was, which **Volatility** should I use?
- Implied volatility
- Realised volatility/Actual volatility (the actual volatility, which I can not pred... |
I have been reading the paper "Martingales and Arbitrage in Multiperiod Securities Markets".
The paper works in the probability space $(\Omega, F, \mathbf{P})$. $X$ is defined as the set of all random variables on $(\Omega, F)$. $M$ is a subspace of $X$.
The paper defines a consumption bundle $(r, x) \in (\mathb... |
I have been reading the paper "Martingales and Arbitrage in Multiperiod Securities Markets".
The paper works in the probability space $(\Omega, F, \mathbf{P})$. $X$ is defined as the set of all random variables on $(\Omega, F)$. $M$ is a subspace of $X$.
The paper defines a consumption bundle $(r, x) \in (\mathb... |
I have come across the statement that the Vasicek model cannot be used to price skew / smile sensitive products: i.e. it cannot be calibrated to replicate a skew or smile. Why is that?
My guess is this: the instantaneous short rate can potentially have a time-varying volatility function (say time homogenous, if not ... |
There is a famous formula for the variance swap strike that reads
$$
K_{var}^2 = \int_{-\infty}^\infty dz\, n(z) I^2(z)
$$
where $I(z)$ is the Black-Scholes implied volatility function,
$$
n(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac12 z^2}
$$
and $z$ is the Black-Scholes `$d_2$' function
$$
z = \frac{\log S_t/K}{... |
I’m trying to get data from an index on Bloomberg terminal into excel with BDH command. Specifically I am trying to get all of the closing prices for each company in the Russell 2000 index for every day for 10 years. I’ve managed to export all of the company names into excel (all of the members of Russell 2000) and usi... |
How do you extract only prices for an index from Bloomberg? |
I am new to the topic but I would like to read papers/books/anything interesting to learn more how ML and data science is used in buy side Fixed income Asset management firms. Factor investing/signals/etc? I have no idea how it can be used but I would love to know more! I would be interested in actual projects with imp... |
Is there a common approach to measure how a forward contract is performing?
Here's what I'm thinking, each day you would price your forward with the next formula.
$$
F_t = S_0 e^{-r_f T}-Ke^{-r T} = (S_0 e^{(r-r_f) T}-K)e^{-r T}=(FWD_t -K)e^{-r T}
$$
With $K$ being the forward price originally agreed.
Now... |
Forward returns measurment? |
A common strategy in trading is to use a bollinger band system. Simply put, we bet on reversion to the mean and take the opposite trade to the current movement under the assumption a move is overdone.
However implicit in this is the idea of standard deviation which really only applies to symmetrical distributions. ... |
What is the gold standard econometric model for volume? Base model for price changes is the autoregressive (AR) model and GARCH(1,1) for volatility. Is there any survey about econometric models used in volume modeling ? |
Is Risk Adjusted CAGR formula from Investopedia correct? |
I see many definitions online for z spread with formulas written for a bond, how do they change if the bond is callable? Or is z spread calculated to maturity in this case? |
Suppose
$$
dS = \sigma S \left(\rho dW + \sqrt{1-\rho^2} dZ \right)
$$
and
$$
d\sigma = a(\sigma,t) dt + b (\sigma,t) dW
$$
with $dW dZ = 0$.
What are the conditions necessary such that the implied volatility skew of vanilla options attains a (global) minimum or maximum value for each maturity date ... |
I am looking for more details on Piecewise Cubic Zero for bootstrapping/interpolating treasury curve ? Does quantlib uses Cubic interpolation or Cubic Spline interpolation ? If Cubic interpolation what is the methodology? |
Suppose
$$
dS = \sigma S \left(\rho dW + \sqrt{1-\rho^2} dZ \right)
$$
and
$$
d\sigma = a(\sigma,t) dt + b (\sigma,t) dW
$$
with $dW dZ = 0$.
What are the conditions necessary such that the implied volatility skew of vanilla options attains a (global/local) minimum or maximum value for each maturity... |
Should we include constant in linear regression while calculating hedge ratio for pairs trading strategy? |
Should we include constant in linear regression in pairs trading? |
I would like to ask if we could simulate stock price paths with different methods/techniques.
What I mean is : say we have a specific stock price hence we can extract historical mean and standard deviation. Then one could use the GBM to simulate different trajectories under the normal distribution assumption.
I... |
I would like to know a way to compute returns when the total PnL of a strategy can become negative. For example:
Total PnL day 1: 100
Total PnL on day 2: -20
Total PnL on day 3: 30
I have that kind of PnL when implementing systematic selling of straddle/strangle on FX options. As I would like to perform a statist... |
How to calculate daily returns when the cumulative PnL can become negative? |
Please excuse me in advance, as I suspect my question is slightly off compared to the other questions on this website, but I am currently taking a class in International Finance and I have a few following questions about how to properly read currency pairs as I feel the slides provided by my professor are really confus... |
The quoting conventions are sometimes illogical for legacy/historical reasons.
"CC1/CC2" is usually supposed to mean the amount of CC2 that you'd need to pay to get 1 unit of CC1. CC's are usually [ISO 4217][1] 3-letter codes. Ignoring any bid-ask, EUR/USD=1.25 means that you'd pay USD 1.25 to receive EUR 1, and the... |
If anyone could help me to understand if it is possible calculate the quantile function for fBm?
I’ve checked several papers([1],[2],[3]), and although several works stated that it is centralised Gaussian, this is not clear for me if the quantile function is applicable here.
Based on point 7 here: https://www.ran... |
I'm trying to build an implied vol surface from some listed options. In particular I have data for calls and puts for different strikes and expiries. I'm not looking to price on the interpolated vols any exotic payoffs, just other vanilla options. To keep things very simple, I'm thinking to use the following approach:
... |
Why don't you use an absolute return series, e.g. 100, -120 and 50, based on your given figures? White's reality check, permutation checks etc. will work with this type of return series just as well as with log returns (which I presume is the thrust of your question, i.e. you can't take the log of a negative number). |
When pricing a bond via a short rate model on a tree, it seems natural to include additional intermediate time steps in addition to those corresponding to cashflow dates (i.e. for bonds with American style embedded options).
One thing which isn't clear to me is how to handle the case of floating coupons when there a... |
Consider the hybrid model given by
$$dS=(r-q) S dt + \sqrt{v} S dZ_1$$
$$dv = \kappa_v (\theta_v - v) dt + \sigma_v \sqrt{v} dZ_2$$
$$dr = \kappa_r (\theta_r - r) dt + \sigma_r r^p dZ_3$$
with correlation values $\rho_{S,v}$, $\rho_{S,r}$ and $\rho_{v,r}$, where $p=0$ for the Heston-Hull-White (HHW) and $p=1/2$ f... |
Can somebody post Graeme West's VBA code for monotone convex interpolation if you have? I was struggling to find it. |
Local volatility can be computed in terms of call prices using Dupire's formula. Assume we have a rectangle call price surface, let's say $I = [30,60]\times[1 day, 1year]$.
For interpolation, should we interpolate the derivatives of call prices and compute the local volatility instead of directly interpolate the loc... |
how to interpolate and extrapolate the local volatility surface? |
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