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The relationship between annual discrete and continuous compounding interest rates is given as:
1+r_d = e^r_c
My question is what are the properties of the difference between r_d and r_c?
For example, it should hold r_d>r_c because more compounding should have lower interest to arrive at the same value. Can... |
The relationship between annual discrete and continuous compounding interest rates is given as:
$$1+r_d = e^{r_c}$$
My question is what are the properties of the difference between $r_d$ and $r_c$?
For example, it should hold $r_d>r_c$ because more compounding should have lower interest to arrive at the sam... |
Consider a financial market with a filtered probability space $\left(\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb P\right)$ satisfying usual conditions equipped with a stock price process $S_t$. Suppose there exists a risk-free asset who is governed by $\mathrm{d}B_t=r_tB_t\mathrm{d}t$.
Suppose the market is free of a... |
# Discount Rate Matches the Cashflow Type #
For DCF valuation, there is a simple rule: the type of cashflow dictates the type of discount rate to use. Dividends are discounted at the CAPM rate (although one can also use other equity factor models). Free cash flow to the firm is discounted using the WACC; and, free cas... |
I don't know "what this identity intuitively means" but I can tell you an anecdote about how I encountered it. (This is a true story, though I won't mention the school and the professor).
The professor wrote on the blackboard the BS equation (with no dividends):
$$C=S N(d_1) - K e^{-rT} N(d_2)$$
and asked: wha... |
I have a question to think about. I have constructed a portfolio of AMEX/NYSE/NASDAQ firms. After some data manipulation and cleaning I am left with some 4000 firms over the complete sample (max 1600 at the same time). I have excluded firms I do not have some observations for the variable I seek to examine.
Subseque... |
I don't know "what this identity intuitively means" but I can tell you an anecdote about how I encountered it. (This is a true story, though I won't mention the school and the professor).
The professor wrote on the blackboard the BS equation (with no dividends):
$$C=S N(d_1) - K e^{-rT} N(d_2)$$
and asked: wha... |
I don't know "what this identity intuitively means" but I can tell you an anecdote about how I encountered it. (This is a true story, though I won't mention the school and the professor).
The professor wrote on the blackboard the BS equation (with no dividends):
$$C=S N(d_1) - K e^{-rT} N(d_2)$$
and asked: wha... |
I'm trying to find the best way to decompose the yield on an index of fixed residential MBS securities and want to open up the question to the community. The goal is to look at this from a quantitative perspective. I have interest rate models, but no models strictly specific to mortgages. I also have all the index spec... |
Under the assumptions of the Black--Scholes model, I read that the market price of risk of two assets $S_1$ and $S_2$ are the same, if they both follow Geometric Brownian motion driven by the same Brownian motion.
The claim is that if
\begin{align*}
dS_1(t)&=\mu_1S_1(t)dt+\sigma_1S_1(t)dW(t),\qquad\text{and} \\
d... |
I have been using AlphaVantage to get the historic data for NSE stocks. But since past 1 month the API has stopped returning data for NSE stocks. It returns data for NASDAQ stocks.
I tried searching the internet and the closest I found was [this question of StackOverflow][1] but it was also not answered. I am using ... |
Has AlphaVantage stopped NSE stock quotes data? |
If X is a $T\times N$ pandas DataFrame of multivariate asset returns, the cumulative returns can be computed in python as
> (1 + X).cumprod() - 1
How can I reverse this operation so that I go backwards from cumulative returns to the original returns matrix X? |
How to find reverse cumulative product of returns in Python? |
How to compute returns from cumulative returns in Python? |
As the topic says. I'm trying to understand the **process** behind the evaluation and execution through to post (and pre? if there is some) acquisition portfolio management? I think maybe that acquisition has something to with M&A? But I'm not sure.
Thanks in advance, |
Can someone explain how the evaluation and execution through to post-acquisition portfolio management? |
I am looking into Private Equity.
As the topic says. I'm trying to understand the **process** behind the evaluation and execution through to post (and pre? if there is some) acquisition portfolio management? I think maybe that acquisition has something to with M&A? But I'm not sure.
|
I am currently looking through some actuarial study materials (CM2, formerly CT8) in which models of asset returns are being discussed. One such model is the *market model* (A.K.A The *single-index model*), which is defined to be of the form
$$
R_i = \alpha_i + \beta_i R_M + \varepsilon_i
$$
where
- $R_i$ is th... |
Why is diversifiable risk unrewarded? |
I am currently looking through some actuarial study materials (CM2, formerly CT8) in which models of asset returns are being discussed. One such model is the *market model* (A.K.A The *single-index model*), which is defined to be of the form
$$
R_i = \alpha_i + \beta_i R_M + \varepsilon_i
$$
where
- $R_i$ is th... |
# Equal-weighted Portfolio #
The typical way to answer this is to consider an equally-weighted portfolio of $n$ assets. An asset $i$ thus has portfolio weight $w_i=\frac{1}{n}$ and excess returns $R_i$ with all assets having the same volatility $\sigma_i=\sigma$ and correlation $\rho_{ij}=\rho$ with another asset $j$.... |
I recently just got asked the below question. Please help.
"You are about to execute a zero fixed rate vs. Float rate swap under daily cash margining with a client in a normal swap rate curve environment. The Bank is receiving fixed. Just before execution, you get an opportunity to either change the collateral rate ... |
I'd like to understand why $\Sigma$ is the same under both measures $\mathbb{P}$ and $\mathbb{Q}$.
Is it an assumption or a general fact based on theoretical concepts?
|
I have been using AlphaVantage to get the historic data for NSE stocks (NSE= National Stock Exchange of India Ltd.). But since past 1 month the API has stopped returning data for NSE stocks. It returns data for NASDAQ stocks.
I tried searching the internet and the closest I found was [this question of StackOverflow]... |
Is there any way to simulate buy an option at a pric of x when that price is not available to buy ? If x-3 and x+3 exists, how do I buy it at price x? |
I have bene researching on the best data structure to implement a limit order book. Some of the most common implementations include arrays and balanced trees. [This link](https://quant.stackexchange.com/questions/3783/what-is-an-efficient-data-structure-to-model-order-book) has a good set of references.
Most of th... |
How much would you sell an option whose expected payoff equals 0? |
Under the assumptions of the Black--Scholes model, I read that the market price of risk of two assets $S_1$ and $S_2$ are the same, if they both follow Geometric Brownian motion driven by the same Brownian motion.
The claim is that if
\begin{align*}
dS_1(t)&=\mu_1S_1(t)dt+\sigma_1S_1(t)dW(t),\qquad\text{and} \\
d... |
I am asking this question because I want to research some variables. An example is the RSI where the current RSI is updated every tick. This means that the value of the RSI is fluctuating a lot inside an individual candle. Lets say I want to research what happens after the RSI reaches a value. But if I only take the cl... |
Is there a noticeable difference in making scatter plots and regression models with tick-data or with candle data? |
If on $(\Omega, \mathcal{F},\mathbb{P})$, $r>0$ is a constant and $Z_t =\sum_{i=1}^{N_t} Y_i$ where $Y_i$ are i.i.d with $E[Y_i]=L$ denotes the size of the jump and can have distributions like normal, exponential, etc; $N_t$ is the Poisson process with intensity $\lambda$. If $X_t$ is given by
$$dX_t = X_t(rdt -g(X(t)... |
If I have data for IV for the previous day the next da and the stock price in intraday format , can i calculate the option price in intrday format? |
May I ask what's the difference between At-the-money volatility vs At-the-money foward Vol? |
What's the difference between ATM Vol vs ATMF Vol? |
I am reading this [article][1], and I am wondering how comments like
> there may be a 50/50 chance that the underlying asset price can
> increase or decrease by 30 percent in one period.
are justified in practice. Is it a statistical assumption? Meaning, you would study the historic behavior of the underlying a... |
I am reading this [article][1], and I am wondering how comments like
> there may be a 50/50 chance that the underlying asset price can
> increase or decrease by 30 percent in one period.
are reconciled with obviously non-zero and potentially asymmetric typical historical return distributions in the underlying? ... |
Nobody is saying that diversifiable risks are not rewarded (or punished) in practice.
The question is whether it's reasonable to expect compensation for taking this kind of risk. This would then implicitly assume that someone else should reasonably expect to take a loss on the other side of the same risk that they ... |
Under the assumptions of the Black--Scholes model, I read that the market price of risk of two assets $S_1$ and $S_2$ are the same, if they both follow Geometric Brownian motion driven by the same Brownian motion.
The claim is that if
\begin{align*}
dS_1(t)&=\mu_1S_1(t)dt+\sigma_1S_1(t)dW(t),\qquad\text{and} \\
d... |
According to my portfolio analysis program (pyfolio), the alpha of the following strategy is .17 (I am assuming 17%). [Based on pyfolio documentation, alpha here is the "annualized alpha".]
However, the cumulative returns of the benchmark are about 10-fold higher than that of the strategy (~12% vs ~125%), see graph.... |
Can alpha be positive if cumulative returns underperform the benchmark? |
I below are my codes using QuantLib to build a term structure
What I would like to do is use that to price any hypothetical bond lets say
startdate : 8 Feb 2016
end date : 8 Feb 2021
coupons : 10% semi
How do I do that?
```
import matplotlib
matplotlib.use('macosx')
import matplotlib.pyplot as plt
impo... |
The Wikipedia article on LIBOR says:
> In 1986, the Libor initially fixed rates for three currencies. These
> were the US dollar, British pound sterling and the Deutsche Mark. Over
> time this grew to **sixteen currencies**. After a number of these
> currencies in 2000 merged into the euro, there remained ten
> ... |
What currencies has LIBOR historically been offered in? |
The [Wikipedia article][1] on LIBOR says:
> In 1986, the Libor initially fixed rates for three currencies. These
> were the US dollar, British pound sterling and the Deutsche Mark. Over
> time this grew to **sixteen currencies**. After a number of these
> currencies in 2000 merged into the euro, there remained te... |
According to my portfolio analysis program (pyfolio), the alpha of the following strategy is .17 (I am assuming 17%). [Based on pyfolio documentation, alpha here is the "annualized alpha".]
However, the cumulative returns of the benchmark are about 10-fold higher than that of the strategy (~12% vs ~125%), see graph.... |
> Does the integrated CIR process itself by any chance has such a representation in the form of the sum of independent random variables?
I think the answer to this is clearly "no." The CIR process is (as @DavidAddison points out in comments above) like an Ornstein-Uhlenbeck process. The mean-reverting property of th... |
> How would you simulate a covariance matrix of 1,000 stocks where each pair has nonzero correlation?
I have literally no idea how to start with this.
Any suggestions? |
I'm trying to understand what Gordy [1] added to Vasicek [2] model (the core of the IRB formula of Basel Accords).
Is it correct to say the Vasicek shows that the portfolio loss _conditional on $Y$_ converges, by the law of large numbers, to its expectation $$p(Y)=\Phi\left(\frac{\Phi^{-1}(p)-\sqrt{\rho}\,Y}{\sqrt{1... |
As I understand it (correct me if I'm wrong) the theoretical price of an asset should be the present value of all future cash-flows that it is expected to yield, discounted at the risk-free rate.
I am currently reading some notes on the CAPM which quote the key result of this model to be the equation of the security... |
How is CAPM used to price an asset once it has been used to derive the assets expected return? |
What does 'simulate a covariance matrix' mean?
- If the question means, generate an arbitrary correlation matrix for 1000 stocks, then we can choose any symmetric matrix with all 1s down the diagonal, so long as every element is between -1 and 1 and the matrix is [positive semi-definite][1]. The large size of the m... |
Suppose
$$
dS = \sigma S dW
$$
and
$$
d\sigma = a(\sigma,t) dt + b (\sigma,t) dZ
$$
with $dW dZ = \rho dt$.
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 $T$? In other words that the slope... |
Suppose
$$
dS = \sigma S dW
$$
and
$$
d\sigma = a(\sigma,t) dt + b (\sigma,t) dZ
$$
with $dW dZ = \rho dt$.
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 $T$? In other words that the slope... |
According to this Bloomberg article, Universa was up 3,600% in March 2020, by hedging with extremely out-of-the-money puts: [https://www.bloomberg.com/news/articles/2020-04-08/taleb-advised-universa-tail-risk-fund-returned-3-600-in-march][1]
Near the end of the article, and according to a chart from the fund, it's d... |
What put options would the Universa Tail Fund have bought? |
# Estimating the Discount Rate #
As indicated in the comments, you would use the CAPM (or another equity factor model) to model stock returns $r_{it}$ beyond the risk-free rate $r_f$ as a function of returns on a market index $r_{Mt}$:
$$ r_{it} - r_f = \alpha_i + \beta_i (r_{Mt} - r_f) + \epsilon_{it}. $$
Once we... |
In Stochastic Volatility Modelling, Chapter 2, the author derived the Dupire equation
$$\mathbb{E}[\sigma_T^2|S_T = K] = 2\frac{\frac{dC}{dT} + qC +(r-q)K\frac{dC}{dK}}{K^2 \frac{d^2C}{dK^2}}.$$
The author discussed its denominator: he linked the denominator to a butterfly strategy. Then I cannot understand the fo... |
I am trying to price of a Down-and-Out Barrier call option with leverage. When the price of the underlying asset hits a certain barrier (B), the option becomes worthless. The issuer of these options indicates that their price is calculated as follows:
$$P = \frac{S - F}{ratio} $$
(Note: the ratio is used in case th... |
Can a down-and-out barrier call option be priced using a closed form formula or should it be approximated? |
Can a down-and-out barrier call option be priced using the Black & Scholes formula or should it be approximated? |
I am trying to price of a Down-and-Out Barrier call option with leverage. When the price of the underlying asset hits a certain barrier (B), the option becomes worthless. The issuer of these options indicates that their price is calculated as follows:
$$P = \frac{S - F}{ratio} $$
(Note: the ratio is used in case th... |
How would you simulate a covariance matrix of 1,000 stocks where each pair has nonzero correlation?
I have literally no idea how to start with this.
Any suggestions? |
When a option's market maker receives a quote from a broker, usually the underlying spot prices is locked with a reference.
Let's suppose the following example:
**Broker:** "*Buy 10k call 2800 of ABC Index for August. Ref: 2670*"
How the market maker is suppose to quote this option?
I believe that this is... |
I have a question for Black-Scholes. It is a continuous approach, but the real market closes every day. So for the Black-Scholes, how do we count the time effect of during the time when the market is closed? or do we just ignore that time and count the business day time? |
The mean-variance model outputs a portfolio weight vector whose elements are individual asset weights that sum to 1. Regardless of which portfolio along the efficient frontier is being solved, the individual weights within the portfolio weight vector can take on values that belong to the real number set, but are they r... |
Are mean-variance efficient portfolio weights random variables with probability distributions? |
It is just whatever the option is worth assuming the underlying is currently trading at the reference rate and assuming the option carries the amount of deltas the brokers says it has. Usually there is some negotiating on what the fair delta is - since different people use different models/assumptions and have differe... |
There are ways that you might think of portfolio weights as estimates and thus random variables. If you are working with the optimizer, you may be able to get the inverse Hessian out of it. If so, that can be used to get you estimates of the standard errors of your portfolio weights.
One big caveat though: any weigh... |
The e-ratio is simply a tool to measure the quality of entries and it says nothing about the overall performance of a trading strategy. It answers the question "Is this a good entry technique?" and perhaps "What is the optimum average holding length?"
It is a strategy development tool, not a strategy assessment too... |
It is known (see for example Joshi-Chan "Fast and Accureate Long Stepping Simulation of the Heston SV Model" available at SSRN) that for a CIR process defined as :
$$dY_t= \kappa(\theta -Y_t)dt+ \omega \sqrt{Y_t}dW_t$$
$Y_0=Y$ together with a correct set of constraints on parameters' value.
Then the law of $\in... |
**Background**
I am trying to implement Monte Carlo Simulation with Stratified Sampling for barrier option under Black Scholes Model. I understand there is an analytic formula for this instrument and we can directly simulate the integration from time 0 to maturity because we have the distribution of stock price unde... |
Hey how do you calculate the dividend rate if you want to price your stock options eg apple? Just take the dividends paid last year and divide by today's share price? This page reports 0.85% (https://finance.yahoo.com/quote/AAPL?p=AAPL) |
Does the volatility of a Currency Pair depend on the currency in which the premium is paid? For example- will the Volatility of USDJPY change if the premium is paid in USD instead of JPY. Is there any mathematical formulation for this? |
following Cochrane (2005) and other insights, we know that under Arbitrage Pricing Theory (Ross, 1976), if investors believe returns follow a linear multifactor structure of the form
$x^i=r^f+\sum_{j=1}^{M}\beta_{ij}f_j+\epsilon_i$
where $x^i$ is a the asset $i$ return, $r^f$ is the return on a risk-free asset... |
Does the volatility of a Currency Pair (for option pricing purposes) depend on the currency in which the premium is paid? For example- will the Volatility of USDJPY change if the premium is paid in USD instead of JPY. Is there any mathematical formulation for this? |
Note that yahoo is posting the *forward dividend yield*. Other yields, trailing (that you seem to describe) and indicated, are described (Investopedia) [here][1] and (Wikipedia) [here][2]. See also [this][3], [this][4], and related links on Stack Exchange.
In general, for (estimated) *discrete dividends* $ (D_i)_{1\... |
Does the volatility of a Currency Pair depend on the currency in which the premium is paid? For example- will the Volatility of USDJPY change if the premium is paid in USD instead of JPY. Is there any mathematical formulation for this? |
Why there is a "market price of volatility risk" variable in the PDE of Heston Model and no such variable in Monte Carlo Simulation?
Do we obtain the same price from both methods? |
Note that yahoo is posting the *forward dividend yield*. Other yields, trailing (that you seem to describe) and indicated, are described (Investopedia) [here][1] and (Wikipedia) [here][2]. See also [this][3], [this][4], and related links on Stack Exchange.
In general, for (estimated) *discrete dividends* $ (D_i)_{1\... |
Borrowing from another department or borrowing from the market face the same issues:
- You have to get capital from somewhere;
- raising that capital has a cost; and,
- other uses of that capital may offer more appealing return per unit of risk given the firm's current holdings (so opportunity cost matters).
It i... |
I am looking for some materials for profiling all `Options sensitivities` for `Asian options` for different maturities.
Is there any place to look into?
Any pointer will be highly appreciated.
Thanks, |
I am looking for some materials for profiling all `Options sensitivities` for `Asian options` for different maturities.
The underlying price $S_t$ follows a standard `GBM`. I am looking for both `Geometric average` and `arithmetic average` for Options payoff.
Is there any place to look into?
Any pointer will ... |
I am looking for some materials for profiling all options sensitivities for Asian options with both geometric averaging and arithmetic averaging . The underlying price $S_t$ follows a standard GBM.
Is there any place to look into? |
There are a few approaches to use trade prices and quotes to classify the aggressor as "buy" or "sell." Also, many of these methods have historically had to deal with unsynchronized data streams.
# Classification of Approaches #
We can largely break trade classification methods into four types:
- **Tick tests**, a... |
In his 1985 paper, Kyle presents 3 versions of the same model: a single period model, a multiple period model and the continuous time limit of the multiple period model.
When he formalizes the equilbrium problem for the discrete time multiple period model, he restricts himself to recursive linear pricing (P) and dem... |
Here are a few points that might help you get a better sense of all the bells and whistles in your question:
**1) Structure of Sell Side Desks**
From a Sell Side perspective (Market Maker) its common to have Risk Specialization/Segmentation across desks, this is important because it determines which factor you're... |
Which stock are you looking for? Try checking out [IEX Cloud][1]. They have a wide range of core and premium data delivered through APIs. If you're looking at stocks in the US markets, [AlgoSeek][2] is also an option.
[1]: https://iexcloud.io
[2]: https://algoseek.com |
First, it helps to remember that warrants are like options... except they also slightly dilute the stock price because they cause new shares to be issued.
You have a couple of choices:
- wait and exercise the warrants if they are in-the-money;
- do nothing and possibly let them expire despite being valuable; or,
... |
ARIMA and GARCH are old news for predicting volatility time series of asset returns.
How do the performances of the following, and other, machine learning algorithms compare at forecasting volatility for horizons longer than just 1-day/step ahead?
- random forest
- support vector regression (SVR)
- gradien... |
Got the following notice.
What Can I do with AIG+ WARRANT I have?
What will happen on 01/12/2021?
What will happen on 01/14/2021? Will I get $45.088365 PER WARRANT ?
What will happen If I do not sell them or do nothing, will I lose money?
Br. Instruct. Cutoff Date: Tuesday, January 12, 2021 5:00 PM
... |
ARIMA and GARCH are old news for predicting volatility time series of asset returns.
How do the performances of the following, and other, machine learning algorithms compare with one another and GARCH at forecasting volatility for horizons longer than just 1-day/step ahead?
- random forest
- support vector r... |
ARIMA and GARCH are old news for predicting volatility time series of asset returns. I am aware of papers that replace ARIMA and GARCH with machine learning algorithms to predict financial volatility more accurately, and so this question is a reference request for a survey of what's out there:
How do the performance... |
Try this API endpoint to lookup tickers https://rapidapi.com/logicione/api/stock-ticker-security-and-company-search-database?endpoint=apiendpoint_aac710aa-f501-497e-8cc7-8c552d40f955
For AAPL it returns:
{
"security_name":"Apple Inc. - Common Stock",
"ticker":"AAPL",
"exchange":"NASDAQ",
"... |
I'm studying this paper about CVA https://www.google.com/url?sa=t&source=web&rct=j&url=https://pdfs.semanticscholar.org/394f/edc22aa872453fefe1456fbccbcf17d1b9b0.pdf&ved=2ahUKEwiv7L2r3vnqAhWIDewKHZ6xDdUQFjAMegQIBhAC&usg=AOvVaw3CndwE97CwQ8UHnBpWJWdg&cshid=1596275695838
but I'm not able to understand the flow chart to es... |
I wish to calculate below 3 expectations of a typical Wiener process -
1. $E \left[ \int\limits_{0}^{T} tdW_t \right]$
2. $E \left[ \left( \int\limits_{0}^{T} tdW_t \right)^2 \right]$
3. $E \left[W_t \int\limits_{0}^{T} tdW_t \right]$
How should approach them? |
I wish to calculate below 3 expectations of a typical Wiener process -
1. $E \left[ \int\limits_{0}^{T} tdW_t \right]$
2. $E \left[ \left( \int\limits_{0}^{T} tdW_t \right)^2 \right]$
3. $E \left[W_T \int\limits_{0}^{T} tdW_t \right]$
How should approach them? |
Just based on my understanding of the ML models themselves, I have a hard time believing KNN or RF are useful in anyway. They wouldn't be the first models I try and tend to just be ML models taught in class for who knows what reason honestly - maybe because they are easy to understand? From what I have read about ML ... |
Just based on my understanding of the ML models themselves, I have a hard time believing KNN or RF are useful in anyway. They wouldn't be the first models I try and tend to just be ML models taught in class for who knows what reason honestly - maybe because they are easy to understand? From what I have read about ML ... |
Pin risk is the risk that an underlier price will tend toward the strike price of an option (or warrant or convertible bond) as the option approaches its expiry date. In some cases, the underlier price will *avoid* the strike price of an option approaches its expiry date. The reason for these dynamics is delta hedging ... |
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