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My current understanding of the EMA calculation with daily periods is to use as quote the closing price of each day: - Day 0 - Take the closing price as Q0. EMA = Q0 - Day 1 - Take the closing price as Q1. EMA = Q1 * k + Q0 * (1-k) - Day 2 - Take the closing price as Q2. EMA = Q2 * k + Q1 * (1-k) - ... - ...
When using periods that are not days, what is the usual quote that is used as input into the EMA calculation?
The risk free rate is $0,01$ while the risky rate follows a $2$ period binomial model and the risky rate at time $t=0$ is $1$, where $u= 1.5$ and $d=0.6$. How can I determine a swap rate of IRS with maturity 2 years whose floating leg is represented by risky rates? My idea is to create the binomial tree for the...
Assume that the price of DF stock went from a price of $104 on March 2 to 146 on April 1. With a current stock price of 146, Invest all of your amount 14,600 in the DF stock (buy 100 shares) Calculate the payoff and 6-month return for this investment alternative by assuming that the stock price is observed ...
Wondering if Morningstar financial statement data reliable? For example it doesn't have [the Balance Sheet statement for BHP](https://www.morningstar.com/stocks/xlon/bhp/financials), while Marketwatch [has it](https://www.marketwatch.com/investing/stock/bhp/financials/balance-sheet). Same for [Renault SA](https:/...
Is Morningstar data reliable?
Assume that the price of DF stock went from a price of $104 on March 2 to 146 on April 1. With a current stock price of 146, Invest all of your amount 14,600 in the DF stock (buy 100 shares) Calculate the payoff and 6-month return for this investment alternative by assuming that the stock price is observed ...
Wondering if Morningstar financial statement data reliable? For example it doesn't have [the Balance Sheet statement for BHP](https://www.morningstar.com/stocks/xlon/bhp/financials), while Marketwatch [has it](https://www.marketwatch.com/investing/stock/bhp/financials/balance-sheet). Same for [Renault SA](https:/...
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**): [![enter image description here][1]][1] The spreads are computed as follows: $$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$ where: - $T$ is the time to maturity of the bond, - $D$ is the...
**My question is** Please discuss about the following statement > “ the advantages and disadvantages of writing a call option on an existing portfolio of stocks” Note that I read an article nearly about that, and I think such a question, but I could not generate a sound idea about that. Thanks a lot! ...
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**): [![enter image description here][1]][1] The spreads are computed as follows: $$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$ where: - $T$ is the time to maturity of the bond, - $D$ is the...
Consider this equation for calculating the Present Value of Bond that pays a coupon and its face value at maturity: [![enter image description here][1]][1] C is the coupon, r is the interest rate on the bond, m is the number of times it is compounded per year, n is the number of years, and F is the face value ...
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**): [![enter image description here][1]][1] The spreads are computed as follows: $$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$ where: - $T$ is the time to maturity of the bond, - $D$ is the...
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**): [![enter image description here][1]][1] The spreads are computed as follows: $$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$ where: - $T$ is the time to maturity of the bond, - $D$ is the...
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**): [![enter image description here][1]][1] The spreads are computed as follows: $$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$ where: - $T$ is the time to maturity of the bond, - $D$ is the...
Assume that the price of DF stock went from a price of $104 on March 2 to 146 on April 1. With a current stock price of 146, there is a call option available on the DF stock with an exercise price of 146, an expiration in six months and a price of 7.30 And there are 3 different investment strategy (1) Inves...
How can I calculate returns for three investment strategy?
Are there rules of thumbs/models that you could use to develop intuition for when skew is cheap or expensive? From what I gather, volatility is a number that is backed out of price in a sense, "the wrong number to get the right answer". But then someone has to set the price, they have a sense of what the correct "skew"...
What are some heurestics you could use to judge if skew is cheap or expensive? If any?
It says assuming a no-uncertainty Weiner process that models stock price: $$ \Delta S = \mu S\Delta t $$ Can be rearranged to (after taking the limit of $\Delta t \to 0$... $$ \frac{dS}{S}=\mu dt $$ Then integrating between time 0 and T to get: $$ S_T=S_0 e^{\mu T} $$ I don't understand the last step. Ar...
Note: with options contracts, payoff is different to profit. I think something is not correct with your formula (2) Secondly, sounds like this assignment question only requires payoff and return. Return is usually simple regarding options contracts such as 1 - this is what I have now divided by this is what I had bef...
I don't think the accepted answer fully answers the question. In particular simulating a non-stationary VAR is not guaranteed to give you a cointegrated system, and we may be interested in simulating $n>2$ time series. To be specific, the question is asking how to simulate a vector time series $Y_t \in \mathbb{R}^n$ wi...
My current understanding of the EMA calculation with daily periods is to use as quote the closing price of each day: - Day 0 - Take the closing price as Q0. EMA = Q0 - Day 1 - Take the closing price as Q1. EMA = Q1 * k + Q0 * (1-k) - Day 2 - Take the closing price as Q2. EMA = Q2 * k + Q1 * (1-k) - ... - ...
This might be a stupid or basic concept for some of you, I'm new to the concept of hedging with interest rate derivatives, I understand how to hedge an equity portfolio but i'm struggling with the concept of hedging when it comes to liability hedging and DV01 hedging using Interest rate derivatives for example using Sw...
My current understanding of the EMA calculation with daily periods is to use as quote the closing price of each day: - Day 0 - Take the closing price as Q0. EMA = Q0 - Day 1 - Take the closing price as Q1. EMA = Q1 * k + Q0 * (1-k) - Day 2 - Take the closing price as Q2. EMA = Q2 * k + Q1 * (1-k) - ... - ...
I've searched thru dozens of papers and did not find in any of them satisfying and enough theoretical answers to my concerns. So I've combined everything what I found below. Please indicate if my understanding of the topic is proper and correct me if necessary. Beside theoretical side I've found also an obstacle in pr...
I'm new to the concept of Swaps, I'm having hard time understanding what each curve relationship with a swap derivative traded in the market, can someone please explain this ? thank you
For european options, we know the delta of a synthetic forward position (long call short put with the same strikes and maturities) because of the put-call parity. However the P-C parity does not apply to american options. Is there a nice formula for the delta of a synthetic forward of american options?
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be a Brownian motion with respect to $\mathbb{F}$, in the given probability space. We have the following theorem (*Sto...
Are all change of measure operations Dol\'{e}ans-Dade exponentials?
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be a Brownian motion with respect to $\mathbb{F}$, in the given probability space. We have the following theorem (*Sto...
Are all change of measure operations Doléans-Dade exponentials?
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be a Brownian motion with respect to $\mathbb{F}$, in the given probability space. We have the following theorem (*Sto...
A portfolio is built systematically by calculating scores and rebalanced each month to invest only in the 80 best scores. Scores change frequently and therefore the portfolio changes each month, sometimes significantly. The benchmark is MSCI World. At each month end I need to know what portion of the return is due t...
Performance attribution and monthly rebalance: Is a month enough data to calculate Beta and Alpha?
I'm new to the concept of swaps, I'm having hard time understanding what each curve relationship with a swap derivative traded in the market. Can someone please explain this? Thank you.
I will work through a detailed example. I hope it helps. Suppose for simplicity that you are trying to hedge the interest rate risk of one simplistic debt instrument. Suppose that the obligor owes you USD 10 million and promises to pay fixed 10% annual coupon, and to repay the principal in equal installments in year...
I will work through a detailed example. I hope it helps. Suppose for simplicity that you are trying to hedge the interest rate risk of one simplistic debt instrument. Suppose that the obligor owes you USD 10 million and promises to pay fixed 10% annual coupon, and to repay the principal in equal installments in year...
Are all change of measure operations between equivalent porbability measures Doléans-Dade exponentials?
Are all change of measure operations between equivalent probability measures Doléans-Dade exponentials?
At close 3/13/20 SPY was at 270.2, by close 3/16 it dropped to 239.41 ~ 8.8% drop... I'm looking at how to capitalize on these big swings with options. I'm backtesting option strategies and plotted the gains for various PUTS by strike from 3/13 close to 3/16 close. I see the 'peak' gain of 600% is at a strike price ...
Most profitable PUT strike price in these times of high volatility?
ARIMA and GARCH are old news for predicting volatility time series of asset returns. How does the performances of machine learning algorithms compare (such as random forest, support vector regression (SVR), gradient boosting, K nearest neighbors, etc) at forecasting volatility for horizons farther ahead than just 1...
Heuristic portfolio construction techniques include the **equally-weighted portfolio (1/N)** and the **inverse volatility portfolio (IVP)**, which is based on the low-volatility effect. They can be assembled much more efficiently than Markwoitz' mean-variance model that uses optimization of the asset returns covariance...
How good is the inverse-volatility portfolio?
Heuristic portfolio construction techniques include the **equally-weighted portfolio (1/N)** and the **inverse volatility portfolio (IVP)**, which is based on the low-volatility effect. They can be assembled much more efficiently than Markwoitz' mean-variance model that uses optimization of the asset returns covariance...
Heuristic portfolio construction techniques include the **equally-weighted portfolio (1/N)** and the **inverse volatility portfolio (IVP)**, which is based on the low-volatility effect. They can be assembled much more efficiently than Markwoitz' mean-variance model that uses optimization of the asset returns covariance...
I was recently reading (and very much struggling to understand) the CBOE white paper on their Skew Index, I thought it might be useful as I'm trying to better understand volatility skews. I'm having a lot of difficulty and I was wondering if someone who understands could clarify for me. 1. It seems like the CBOE ca...
I am working on a dataset with aim to predict the MF ratings. There are cols like, 10 yr, 7 yr, 5 yr etc returns. I also have commencement date of MFs, the question is there are MFs with commencements dates only 3 yrs back, so would it be prudent to impute the returns for them for 10 yrs and 7 yrs period? In my opin...
Ito's Lemma gives that for a function $F$ of a stochastic variable $X$, $dF = \frac{dF}{dX}dX + \frac{1}{2}\frac{d^2F}{dX^2}dt$ Given a stochastic differential equation $dS = a(S) dt + b(S) dX$ and a function $V(S)$, the differential equation satisfied by $V(S)$ is stated to be $dV = \frac{dV}{dS}dS + \frac{1}{2}b^2...
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be a Brownian motion with respect to $\mathbb{F}$, in the given probability space. We have the following theorem (*Sto...
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be a Brownian motion with respect to $\mathbb{F}$, in the given probability space. We have the following theorem (*Sto...
Can someone explain the difference between zero coupon inflation swap rate and breakeven rate? For example, currently, US 10y zero coupon inflation swap rate is about 1.4%, while US breakeven 10y rate is about 1.3%. Is liquidity the main driver for the difference? Thanks
I am wondering how I can find the vector of Lagrange multipliers $\mu$ for the non-negativity constraint of the following problem: $$ L(w,\lambda, \mu) = w^{T}\Sigma w - \lambda(w -1) + \mu w $$ So far what I cam up with is, that $\mu$ can be isolated with first order condition $ \frac{\partial L }{\partial w} $:...
Can someone explain mě the difference between zero coupon inflation swap rate and breakeven rate? For example, currently, US 10y zero coupon inflation swap rate is about 1.4%, while US breakeven 10y rate is about 1.3%. Is liquidity the main driver for the difference? Thanks
Can someone explain me the difference between zero coupon inflation swap rate and breakeven rate? For example, currently, US 10y zero coupon inflation swap rate is about 1.4%, while US breakeven 10y rate is about 1.3%. Is liquidity the main driver for the difference? Thanks
I was recently reading (and very much struggling to understand) the CBOE white paper on their Skew Index ([CBOE Link][1]), I thought it might be useful as I'm trying to better understand volatility skews. I'm having a lot of difficulty and I was wondering if someone who understands could clarify for me. 1. It seems...
One bitcoin exchange isn't directly connected to another. So price in bitcoin can vary from one exchange to another. The price differences are not usually huge. One bitcoin exchange is only weakly connected with another bitcoin exchange. If price in one exchange is higher than price at another, people can buy where...
How are forex prices maintain it's value accross various exchanges?
One bitcoin exchange isn't directly connected to another. So price in bitcoin can vary from one exchange to another. The price differences are not usually huge. One bitcoin exchange is only weakly connected with another bitcoin exchange. If price in one exchange is higher than price at another, people can buy where...
What does it mean in if something isor not?
I have been reading a lot of posts on this topic, but still cannot find a definitive answer though. If there is a CDS on a corporate reference entity in a currency denomination, does that imply there is a reference obligation issued by that corporate in the same currency denomination? In other words, can you have a ...
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**): [![enter image description here][1]][1] The spreads are computed as follows: $$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$ where: - $T$ is the time to maturity of the bond, - $D$ is the...
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**): [![enter image description here][1]][1] The spreads are computed as follows: $$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$ where: - $T$ is the time to maturity of the bond, - $D$ is the...
Let $X_T,Y_T$ be the terminal values of two price processes following Continuous Gaussian Motion (I.E.) let us assume no jumps. Further assume the correct forwards/futures price is given by $F^X_{t,T} = E_t[X_T],F^Y_{t,T} = E_t[Y_T]$. We know then the following equation is True: $cov_t(X_T,Y_T) = E_t[X_TY_T] - F...
Why is the Geometric Brownian Motion defined as $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for stock prices? $S(t)$ has a lognormal distribution which is right skewed. Another problem is that while this cannot go negative, it also cannot change trend - always goes up to infinity or down to zero. Even a...
Why is $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for prices?
1) Prices are assumed to be lognormal and skewed to accommodate the observation that under normal circumstances returns are normal (which they’re not, but it’s the simplest model we can fit). That’s why the “geometric” part 2) Stock prices by definition can only go down till zero(in case of bankruptcy) so we don’t n...
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**): [![enter image description here][1]][1] The spreads are computed as follows: $$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$ where: - $T$ is the time to maturity of the bond, - $D$ is the...
I am working on a g2++ model in a dualcurve setup for both Euribor and EONIA. I have the model built, but have some issues in calibrating it - I get a perfect fit with a Nelder-Mead algorithm, but it seems like it is shifted downwards compared to the ZC curve (for both of the g2++ models). Can anybody help me to as...
I'm getting to know the Black Scholes model, which apparently is no longer suitable for pricing options on the market. I would like to write a Master's thesis on option pricing but I do not know what specific topic related to option pricing may be interesting, what model are you using now. Someone will tell me what to ...
# We don't model the prices, we model the returns. The stock prices aren't explicitly modelled as log-normal, but rather this is a consequence of the actual model used to describe the returns. The core of the model used in the Black-Scholes model is to assume a geometric Brownian motion for the change in the price $...
I'm reading in Radu Tunaru's book "Model Risk in Financial Markets" and he argues that two parties in the financial contract do not have the same magnitude of exposure to model risk. The reason for that is the following: > "call option contracts have no downside and variation comes from the > upside and also becaus...
//! base class for the one day deposit ICE %USD %LIBOR indexes class DailyTenorUSDLibor : public DailyTenorLibor { public: DailyTenorUSDLibor(Natural settlementDays, const Handle<YieldTermStructure>& h = Handle<YieldTermStructure>...
I consider the version of Girsanov's theorem presented in [this question](https://quant.stackexchange.com/questions/53265/are-all-change-of-measure-operations-between-equivalent-probability-measures-dol/53270#53270). Let us take the particular case that $\mathbb{F}$ is the filtration **generated** by standard Brown...
If I have a portfolio with 3 initial assets on day 1 (say, stock 1 with beginning market value of $100, stock 2 $150 and stock 3 $175) and after 10 days the stock 2 is sold for $200, how can I calculate the return of the portfolio after 30 days, assuming that stock 1 is now costing $150 and stock 3 $175?
If I have a portfolio with 3 initial assets on day 1 (say, stock 1 with beginning market value of \$100, stock 2 \$150 and stock 3 \$175) and after 10 days the stock 2 is sold for \$200, how can I calculate the return of the portfolio after 30 days, assuming that stock 1 is now costing \$150 and stock 3 \$175?
If I have a portfolio with 3 initial assets on day 1 (say, stock 1 with beginning market value of \$100, stock 2 \$150 and stock 3 \$175) and after 10 days the stock 2 is sold for \$200, how can I calculate the return of the portfolio after 30 days, assuming that stock 1 is now costing \$150 and stock 3 \$200?
I consider the version of Girsanov's theorem presented in [this question](https://quant.stackexchange.com/questions/53265/are-all-change-of-measure-operations-between-equivalent-probability-measures-dol/53270#53270). Let us take the particular case that $\mathbb{F}$ is the filtration **generated** by standard Brown...
Nowadays, most cross currency basis spread (against USD) is negative, while AUD, NZD basis spread against USD is positive. Can someone explain why these two are positive, unlike the rest?
I'm just starting to learn how to trade options and as part of an algorithmic options market making simulation I have risk limits for the greeks (gamma, vega, delta, and theta). There are 9 strikes total for the same underlying. I know to delta hedge with the underlying, but how should I go about managing the others...
I started learning about the stock market during the Pandemic. I don't necessarily think to be a quant but I'd like to atleast use my skills in programming in python, data visualization, (i don't think machine learning). Is there like some guides or lectures that shows how i can leverage the above mentioned skills? T...
How Can I Use Python and Data Science in my Personal Trading?
I had read this [https://quant.stackexchange.com/questions/1640/where-to-download-list-of-all-common-stocks-traded-on-nyse-nasdaq-and-amex][1]. I know the `ETF` field of `nasdaqlisted.txt` indicate whether it is an ETF, but it is not always correct. For example: `DGLD` - `VelocityShares 3x Inverse Gold ETN Linke...
How to check if a symbol is a Common Stock (Equity)?
I was reading up on variance swaps and encounter the notion of Dollar Gamma, which is defined as the change in dollar value of the Dollar Delta (Δ * S) for a 1% change in spot (S). The formula for Dollar Gamma is then given as $Γ = Γ/100 * S^2. However, as I tried to prove it, I ran into an issue. The proof is belo...
In the COS method (http://ta.twi.tudelft.nl/mf/users/oosterlee/oosterlee/COS.pdf), they say that [![enter image description here][1]][1] What do they mean that this holds for "Levy processes"? Surely this holds for all processes? Here, $x = log(S_0/K)$, and $S_t = S_0 e^{L_t}$, and $\phi(\omega;x)$ is the ...
On average, there should be about 253 trading days per year, though some people will use 252. If we assume $\sigma^2 = N_{\text{trading days}} \sigma_{\text{daily}}^2$, then obviously we will have $\sigma = \sqrt{N_{\text{trading days}}} \sigma_{\text{daily}}$. Now, why 253? You could look up the data from NYSE or N...
Well, if you are willing to learn a little bit about option pricing, you could take advantage of the large cross-section of derivatives written on some equity securities, such as the stock market index, to inform your bets. In particular, Ian Martin proposed a way to relate the price of a small portfolio of European op...
In posts regarding the $\mathbb{P}$ vs $\mathbb{Q}$ debate (see [1], [2], [3] or [4]), most answers conclude that historical-based forecast are better suited than risk-neutral models for financial predictions. However, this conclusion seems to be based on sentimental beliefs. In fact, empirical analyses comparing h...
Many people get confused by this, including on derivatives trading desks... what you want is to know how much dollar delta you will have to rebalance after a small move to get an idea of the quantum of monetary risk you’re running and be able to compare across stocks with differents spots (your _gamma shares_ will not ...
we have set of historical data of different price forecasts and the real prices. When assessing "which forecasts are best", what parts of the analysis should one never miss out? We are at a very start of our investigation, using forecast error mean, forecast error standard deviation, forecast error 90% percentiles, ...
A global equity portfolio has for objective to outperform a benchmark (MSCI World). I hedge the sensitivity of the portfolio to MSCI World (the beta) so that only the alpha remains unhedged. The ex-ante beta is calculated by looking at the covariance and volatility of the portfolio and MSCI World for the past 2 year...
Hedge backtesting: ex-ante Beta vs observed Beta (is this even possible?)
Hi: There are a couple of different issues in what you're doing. A) One key question is what does "series" contain ? B) 1) and 3) are always going to differ and it's never clear which is correct ( it's one of the pitfalls of the EG test ) I would do it both ways and see if the adf test result is consistent. Do...
I am not sure if this question is appropriate here, but I am just going to give a shot. It is related to an empirical observation about the price of a call option. Specifically, I was looking at a very out-of-the-money call on AMZN, namely AMZN Jan 15 '21 $3000 call. I noticed that it follows the price of the stock (AM...
I am not sure if this question is appropriate here, but I am just going to give a shot. It is related to an empirical observation about the price of a call option. Specifically, I was looking at a very out-of-the-money call on AMZN, namely AMZN Jan 15 '21 $3000 call. I noticed that it follows the price of the stock (AM...
I was recently asked: *Given a random time series of 1s and -1s. Eg of a sample = [1, 1, 1, -1, -1, 1, -1,..]. The autocorrelation of this series is Z. What can you say about the probability of a 1(or -1) followed by 1 (or -1 respectively)?* One thing is for clear, if Z is -1, the probability of 1(-1) followed by...
I am extremely new to the field of high-frequency trading. I have been trying to read a ton of materials out there to understand the general workflow. I have very basic knowledge of different concepts like FIX, Direct Market Access, Order Management System, DataFeed engines etc. However, I have not been able to bui...
I'm trying to find the FX Forward Points for 3M, the same as in the table. However, in the conventional way (Forward points = Spot x (USD Rate - EUR Rate) x 90/360) I get a different result. Can anyone help me with this? [![enter image description here][1]][1] [1]: https://i.stack.imgur.com/XEMlm.png