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I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion.
My python script is below, but it contains an error because of which the end result is always zero ```0.0```. I cannot figure out where the error is.
The characteristics of this option are as follows... |
I was trying to compute the turnover when you have the initial monthly weights. Is not so easy to figure out the logic in the package used in the previous answer.
Here I show how to compute the DeMiguel and Nogales (Portfolio Selection with Robust Estimation, Operations Research 57(3), pp. 560–577, 2009) turnover.
... |
I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion.
My python script is below, but it contains an error because of which the end result is always zero ```0.0```. I cannot figure out where the error is.
The characteristics of this option are as follows... |
The change in a call portfolio (f), deriving it by Itô's Lemma, is:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/722C7.png
Where:
df/dt: Theta.
df/dS: Delta
d^f/dS^2: Gamma
So Gamma's PnL would be = 1/2 * Gamma * (volatility^2) * (dS^2),
where dS is the underlying... |
This is more of a theoretical question.
I have been working on some mean-variance / Black-Litterman models and played around with Ledoit/Wolf's covariance shrinkage method (sklearn function in Python). If I shrink the covariance matrix for mean variance portfolios, I get really nice portfolios along the efficient fr... |
The change in a call portfolio ($f$), derived from Itô's Lemma, is:
$$ \frac{\partial f}{\partial t}+rS\frac{\partial f}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2} -rf=0,$$
where $\frac{\partial f}{\partial t}$ denotes theta, $\frac{\partial f}{\partial S}$ denotes delta and $\frac{\par... |
For someone desperate (and you seem to qualify), the three ways to manufacture vols I know of are:
1. Find a related underlying and, even if only few options trade on it, 'borrow' its *implied vols*.
2. Compute *statistical vol* from historical underlying prices (not strike dependent, still useful to know).
3. ... |
The change in a call portfolio ($f$), derived from Itô's Lemma, is:
\begin{align*}
\left( \frac{\partial f}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2}\right)\mathrm{d}t &=r\left( f-rS\frac{\partial f}{\partial S}\right)\mathrm{d} t, \\
\implies\frac{\partial f}{\partial t}+rS\frac{\partial ... |
The three ways to manufacture quasi-implied vols I know of are:
1. Find a related underlying and, even if only few options trade on it, 'borrow' its *implied vols*.
2. Compute *statistical vol* from historical underlying prices (not strike dependent, still useful to know).
3. Compute *breakeven vol*, still base... |
The three ways to manufacture pseudo-implied vols I know of are:
1. Find a related underlying and, even if only few options trade on it, 'borrow' its *implied vols*.
2. Compute *statistical vol* from historical underlying prices (not strike dependent, still useful to know).
3. Compute *breakeven vol*, still bas... |
I'm interviewing for some quantitative researcher positions at some hedge funds, and I've been told that there will be one interview session focused on stats, and one focused on ML, among others. This made me realize that I have a hard time distinguishing between stats and ML because there's such a great deal of overla... |
What are important statistical concepts used as a quant? |
For some research purposes, I need the official s&p500 companies list. I cannot use any data from Wikipedia or any other such kind of websites. I can use data from yahoo finance or s&p500 official websites. I got historical data from yahoo finance but I couldn't find any data for S&P500 companies list there. Can anybod... |
Where can I find official s&p500 companies list? |
I'm trying to come up with a reasonable and mostly mathematical way to trade signals between two people with interests in collaboration but still wary and skeptical. The idea being that you start with trading the signals with weakest alpha.
However, it's not as simple as that, as the signal might not be appropria... |
I interpret your question to be asking about curve fitting techniques (for constructing fitted par/zero curves), since a term structure model (HW, LMM, etc.) can always be constructed to fit a yield curve perfectly.
In an institutional setting, there really hasn't been any new models being proposed, because the exi... |
I'm trying to come up with a reasonable and mostly mathematical way to trade signals between two people with interests in collaboration but still wary and skeptical. The idea being that you start with trading the signals with weakest alpha.
However, it's not as simple as that, as the signal might not be appropria... |
I interpret your question to be asking about curve fitting techniques (for constructing fitted par/zero curves), since a term structure model (HW, LMM, etc.) can always be constructed to fit a given yield curve perfectly.
In an institutional setting, there really hasn't been any new models being proposed, because t... |
I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion.
My python script is below, but it contains an error because of which the end result is always zero ```0.0```. I cannot figure out where the error is. Also, I am not sure how to model the leverage factor... |
I am trying to take position in future as per the delta position of short put. My strike is 13794 for short put option, spot 10305.3 and volatility is 20.153 then I am getting 5890 position to buy and delta of 0.7222. Now when my spot increased to 10311.2 for same strike and volatility changed to 20.03, I am getting 59... |
Why my delta position is increasing with increase in spot? |
I'm testing certain asset pricing factor models (e.g. Fama and French 3 factor model) and want to check if the alphas of my time series regressions are jointly zero.
Most papers use the Gibbons, Ross, Shanken (1989) Test to do so.
I buildt the test by myself in R, but I want to do it with robust residuals. How d... |
### In all brevity
1. What is the termination condition used in QuantLib's curve bootstrapping?
2. Can I modify this setup to my needs, e.g. can I tune this to a higher accuracy?
### Background
When bootstrapping discounting and projection curves, a necessary condition is for the implied curves to price the **... |
I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion.
My python script is below. I am not sure how to correctly model the increasing barrier ```B``` and leverage factor that multiplies the payoff when the stock price goes up.
The characteristics of this... |
I have just finished my thesis on a heterogeneous dividend expectations model applied to the COVID-19 crisis! However after receiving some feedback there is one last issue I want to resolve. I'm using a *dynamic Gordon model* where estimated AR(1) coefficients (rho and tau) determine variable price-dividend ratios inst... |
I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion.
My python script is below. I am not sure how to correctly model the increasing barrier ```B``` and leverage factor that multiplies the payoff when the stock price goes up.
The characteristics of this... |
### In all brevity
1. What is the termination condition used in QuantLib's curve bootstrapping?
2. Can I modify this setup to my needs, e.g. can I tune this to a higher accuracy?
### Background
When bootstrapping discounting and projection curves, a necessary condition is for the implied curves to price the **... |
I'm testing certain asset pricing factor models (e.g. Fama and French 3 factor model) and want to check if the alphas of my time series regressions are jointly zero.
Most papers use the Gibbons, Ross, Shanken (1989) Test to do so.
I buildt the test by myself in R, but I want to do it with robust residuals. How d... |
Why are my Neural Network predictions “correct”, but offset from true value? Not using any past lagged values |
(**Edited:** sorry, I totally mis-understood the question initially.)
For concreteness, let us look at markets like Bolivia and Paraguay, where the market observables are spot FX and government bonds. There are no observable interest rate swaps or FX forwards. The yield of a government bond is your best information... |
In the asset management industry, many reports contain liquidity metrics such as the **no. of days to liquidate** 95% of a position, based on a certain participation rate.
If that position is a stock or a bond, this is straightforward.
How would we calculate this number of days with a position that's a **deriva... |
How do you best incorporate the weekends in the calculation of the Black Scholes historical volatility? (Of course historical volatility serves as approximation, if the market price of the options is not available).
In the book "Options, Futures and other Derivatives" by John C. Hull, it is described that over the ... |
I am using a simulation approach to compare the performances achievable by simple risk management and portfolio optimization for portfolio selection. My problem is that my results indicate that simple risk management outperforms portfolio optimization. Is this possible or is there most likely an error in my simulation?... |
Can simple risk management outperform portfolio optimization like this, or is there most likely an error? |
Let's **set the spot price as 1** (spot price of underlying security) and express each option contract as a point in 3D space $$ \{ x, y, z \} = \{ tenor, moneyness, premium \} $$ where the premium is also relative, like moneyness.
Past volatility, interest rates and in-the-money options are ignored, we are interest... |
Let's **set the spot price as 1** (spot price of underlying security) and express each option contract as a point in 3D space $$ \{ x, y, z \} = \{ tenor, moneyness, premium \} $$ where the premium is also relative, like moneyness.
Past volatility, interest rates and in-the-money options are ignored, we are interest... |
I am asked to comment on the Funds Transfer Pricing methodology used by our Treasury to assign a Cost of Funds to a Loan. This is the current methodology:
Let us say there is a 2 year loan with an annual payment and each payment pays down half of the principal (+ some interest).
[![enter image description here][1]]... |
I am pricing derivatives under a $T$-forward measure and as such I need to discount the expected payoff with a bond price. I have available to me overnight-indexed swap (OIS) rates, how do I reconstruct bond prices from this data?
When I have done this before I have had access to LIBOR rates, and I know that since L... |
I came across these questions below in somewhere and was quite interested to know the answer.
Given an ARMA(1) equation: x<sub>t</sub> = a + bx<sub>t-1</sub> + e<sub>t</sub> + θe<sub>t-1</sub>, how can we:
1. Find the conditional variance, i.e. Var<sub>t-1</sub>(x<sub>t</sub>)
2. Find the unconditional va... |
How to calculate conditional and unconditional variance, autocovariance and autocorrelation of an ARMA process? |
Given an ARMA(1,1) process $x_t = a + bx_{t-1} + \varepsilon_t + \theta\varepsilon_{t-1}$, how can we
1. find the conditional variance, i.e. $Var_{t-1}(x_t)$,
2. find the unconditional variance, i.e. $Var(x_t)$,
3. find the autocovariance and autocorrelation for the lags 1 and 2?
I would appreciate a detaile... |
How would you price \begin{equation*}
\mathbb{E}^{Q} \left[ e^{-\int_{0}^{T}r_{s}ds} f \left( S_{T_f}^1, S_{T_f}^2 \right) | \mathcal{F}_{0} \right]\end{equation*} with $T_{f} \le T$ and $S^{1}, S^{2}$ the assets.
by static replication , is using a copula the only way ?
|
An investor typically decides a portfolio objective and sticks with that objective for every rebalance date in the portfolio's life. Common characteristic portfolios that the investor chooses are:
- minimum variance portfolio
- maximum Sharpe (tangency) portfolio
- inverse volatility portfolio
- equally-weigh... |
I am trying to value an FRA in quantlib Python using the below code:
import QuantLib as ql
calendar = ql.UnitedStates()
todaysDate = ql.Date(7, ql.May, 2017)
#
ql.Settings.instance().evaluationDate = todaysDate
spotDates = [ql.Date(7, 5, 2017)+ql.Period(i*6, ql.Mo... |
I have a dispersion question I need help with...An index SPY has only 2 components - AAPL and AMZN. We are given the implied volatility of SPY and AAPL as well as the correlation between the 2 assets.
SPY IV = 20%
AAPL IV = 25%
Correlation SPY/AAPL = 85%.
We then use Margrabe formula to back out the fair value i... |
I've been backtesting on a spread mean reversion strategy on certain stock pairs.
I observe the stationarity via scatterplot and plotting a histogram.
Then I verify it using Augmented Dickey Fuller test.
Example: Whiting Petroleum Corporation (WLL) vs Approach Resources, Inc. (AREXQ)
Time period: 01 Jan 2012 - ... |
I've been backtesting on a spread mean reversion strategy on certain stock pairs.
I observe the stationarity via scatterplot and plotting a histogram.
Then I verify it using Augmented Dickey Fuller test.
Example: Whiting Petroleum Corporation (WLL) vs Approach Resources, Inc. (AREXQ)
Time period: 01 Jan 2012 - ... |
Is there a scientific significance to Fibonacci numbers in economics? |
I have just finished my thesis on a heterogeneous dividend expectations model applied to the COVID-19 crisis. However after receiving some feedback there is one last issue I want to resolve. I'm using a *dynamic Gordon model* where estimated AR(1) coefficients (rho and tau) determine variable price-dividend ratios inst... |
Pay in currency : cur
The FX is : $FX^{cur_2/cur_1}$
European options on the FX (and itself) are quoted in currency cur 1.
I'm looking for the price of
\begin{equation*}
\mathbb{E}^{Q} \left[ e^{-\int_{0}^{T}r_{s}^{cur}ds} f \left( FX_{T_f}^{cur_2/cur_1} \right) | \mathcal{F}_{0} \right] = ?
\end{equation*... |
It is interesting that there is no thorough discussion and clear derivation on this per my search. I know TIPS are complex (compared to nominal bonds). The naive use of simple spot/forward yield difference seems to be the "carry/rolldown" in real terms but not nominal.
My best hunch is that: assuming we ignore flo... |
From Joshi's Quant Interview Questions and Answers:
> What is riskier: a call option or the underlying? (Consider a one day time horizon and compute which has bigger Delta as a fraction of value).
I find the answer he has given very confusing. Does anyone have a better answer?
|
Which is riskier: a call option or the underlying? |
From Joshi's Quant Interview Questions and Answers:
> What is riskier: a call option or the underlying? (Consider a one day time horizon and compute which has bigger Delta as a fraction of value).
I find the answer he has given very confusing and not coherent. He is claiming the delta is as a fraction of the valu... |
Pay in currency : cur
The FX is : $FX^{cur_2/cur_1}$
European options on the FX (and itself) are quoted in currency cur 1.
I'm looking for the price of
\begin{equation*}
\mathbb{E}^{Q} \left[ e^{-\int_{0}^{T}r_{s}^{cur}ds} f \left( FX_{T_f}^{cur_2/cur_1} \right) | \mathcal{F}_{0} \right] = ?
\end{equation*... |
Is it more common to hedge a mortgage bond portfolio with other bonds as opposed to Interest Rate Swaps? |
Were there any known high-frequency firms suffer from it?
As the brent price stays normal, I believe that many algorithms would have recognize that price different as an arbitrage opportunity.
For a simple example, the brent price and WTI price almost never differ more than 50%, but now WTI price is 10 while the... |
How could the WTI future price be as low as -40 on Apr 20? |
>Suppose two assets in Black Scholes world have the same volatility, but one has downward jumps at random times. How does this affect the option prices?
I would have thought that downward jumps would decrease the value of the call option because you have more chance of being out of the money (ie below the strike). A... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/jxaQ0.png
Here is a graph of Price vs Spot from Joshi's Quant Interviews book,
The first line is a down-and-out barrier option and the other one is a down-and-in barrier option. The strike is 100 and the barrier is at 95.
Why does ... |
Are there any Portfolio Optimization techniques (Markowitz, Prospect Theory, etc...), valuations and derivatives hedging techniques? What is a typical workflow and what should one master? |
What is the typical work-flow and skills/techniques required for someone working on Structuring Products? |
I was just comparing two daily returns series and noted that the correlation between them is a lot higher if they are cumulated (about .95 for cumulative returns, vs .15 for non-cumulative). I feel that there should be a simple intuitive explanation for why that is. Is it because these returns behave more similarly ove... |
I was just comparing two daily returns series and noted that the correlation between them is a lot higher if they are cumulated (about .95 for cumulative returns, vs .15 for non-cumulative). I feel that there should be a simple intuitive explanation for why that is. Is it because these returns behave more similarly ove... |
I am subscribed to Level 2 data from a single exchange. There are more exchanges offering Level 2 data subscription. However, with my single subscription I can see ask and bid sizes from all routing destinations, so I was wondering if there is any advantage in subscribing to Level 2 data feed from other exchanges as we... |
Is one Level 2 data subscription sufficient? |
fist of all I do apologize if my question is not fit for this forum, but after much research I didn't find a better place to ask this question. I am a PhD student in mathematics. I'm considering applying to quant positions but I do not have any background or knowledge of finance. I do know some ML and specifically rece... |
> Develop a formula for the price of a derivative paying
>
>$$\max(S_T(S_T-K))$$
>
> in the Black Scholes model.
Apparently the trick to this question is to compute the expectation under the stock measure. So,
$$\frac{C_0}{S_0} = \mathbb{E}[\frac{S_T\max{(S_T-K,0)}}{N_T}]$$
and taking $N_T = S_T$. We can ... |
> Compute the price of a derivative which has pays $\log(S_T)S_T$, you can assume that the Black Scholes model is valid.
Using the stock measure we can write the expectation as
$$D(0) = S_0 \mathbb{E}_S(\log S_T)$$
with the expectation in the stock measure. In this measure,
$$dS_t = (r + \sigma^2)S_t dt + \... |
I am looking for a metric to track the performance of investments in a security over time. I know what I have in mind but I am not sure how to map it onto a known metric. I want to answer the question, “Given a series of securities purchases and dividends, what annual interest rate would have given the same return?” ... |
How to Compute the Return for a Series of Investments with Payouts? |
I am subscribed to Level 2 data from a single exchange. There are more exchanges offering Level 2 data subscription. However, with my single subscription I can see ask and bid sizes from all routing destinations, so I was wondering if there is any advantage in subscribing to Level 2 data feed from other exchanges as we... |
I am subscribed to Level 2 data from a single exchange. There are more exchanges offering Level 2 data subscription. However, with my single subscription I can see ask and bid sizes from all routing destinations, so I was wondering if there is any advantage in subscribing to Level 2 data feed from other exchanges as we... |
Implicit finite differnce method always guarentees positive and stable price of derivative? |
Is there a free or cheap (<15$/month) data source for European Options?
Something like Yahoo Finance option page, with option chain contract prices. It's ok if it's partially incomplete and slightly outdated.
For European companies like Bayer AG, Siemens, Volkswagen, Daimler etc. |
Free or cheap data source for European Options? |
If you combine the compound Poisson process with the Brownian motion you obtain the simplest case
of a Jump-diffusion. Let’s define
Xt = µt + σWt + Jt
where Wt is a Wiener process and Jt is a compound Poisson process. In what sense is possible to write
an SDE to represent the d... |
Any portfolio models not based on asset return moments? |
Financial models by default use **time bars** of prices/returns for input data. I use time bars to refer to both **intraday** (high frequency) and **interday** (low frequency) data since the sampling occurs at fixed intervals of time for both intraday and interday time bars.
Instead of time bars, the alternatives a... |
In finance, it is well-known that the expected value of asset returns, $\mu$, otherwise known as the average return or mean or first statistical moment, is difficult to predict. I think it was Mandelbrot or Merton who first showed proof of this.
Could someone summarize how, and what is the procedure, for evaluating ... |
How to evaluate prediction(s) made of the asset return mean? |
If you combine the compound Poisson process with the Brownian motion you obtain the simplest case
of a Jump-diffusion. Let’s define
$$X_t = \mu t + \sigma W_t + J_t$$
where $W_t$ is a Wiener process and $J_t$ is a compound Poisson process. In what sense is possible to write an SDE to represent the dynamics of $X_t$?... |
Some of the standard tricks are mentioned in this paper, *Finite Difference Schemes with Exact Recovery of Vanilla Option Prices*
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3530561
which also shows how to set up the finite difference scheme so that all vanillas with strikes and expiries on the grid are m... |
Why are interest supposed deterministic for equity? |
In finance, it is well-known that the expected value of asset returns, $\mu$, otherwise known as the average return or mean or first statistical moment, is difficult to predict. I think it was Mandelbrot or Merton who first showed proof of this.
Could someone summarise how, and what is the procedure, for evaluating ... |
One of the main assumption in the CAPM model is that all the investors are rational and they hold the most efficient portfolio for a given level of risk. What difference does this assumption make? What would change if the above assumption is not held for the CAPM model? |
Why is the efficient portfolio assumption necessary for the CAPM model? |
I'm valuing a [rainbow option][1] numerically with a monte-carlo simulation and I'm getting some unexpected results. The price of a best-of call option is decreasing in price as correlation up to a certain point where it starts to increase, contrary to my intuition. The worst-of option case is much more well behaved wi... |
This is really a career advice question, which doesn't belong here.
But if it were rephrased to ask for ideas for a cool / impressive NLP school project, I'd suggest:
* parse a financial derivative term sheet, decide whether it is a "vanilla" trade that we know how to book, or it may have some exotic features tha... |
Good evening guys
I am looking into the effects of covariance shrinkage on the diversification of asset weights for different portfolio optimisations. Initially, I was interested to see how it affects classic mean-variance, but I now digged into risk-parity and Black-Litterman.
Long story short: where should I ap... |
I hear that there are many extensions to the black scholes model to make it more realistic, however, GBM does not account for volatile swings. Is there any sort of alternative approach to use instead? |
What are some alternatives to Geometric Brownian motion that can be used in the Black-Scholes? |
Can anyone help to explain why when the square root market impact model is used in the standard mean-variance optimization, the exponent becomes 5/3 in the objective function? I suspect this has to do with expressing market impact in different units but couldn't find a clear explanation. |
Implicit finite difference method always guarantees positive and stable price of derivative? |
Basically the title. There's a course [STATS 242: Algorithmic Trading and Quantitative Strategies][1] offered in Stanford a few years ago. I searched on google a bit for the course website to see the assignments and lecture slides but found nothing. Was wondering if anyone here has an idea about it, also any reviews of... |
Has anyone done the course STATS242: Algorithmic Trading and Quantitative Strategies. Where Can I find the assignments and other resources? |
In my tick feed I get real-time bid and ask prices like below from oanda.
My question is this. Lets assume I build a time series model to buy eur/usd in the Oanda UI manually. If I were to build that time series model do I use bids or asks? If I were to build a backtester do I evaluate on bid or ask? If I sell t... |
Is there a free or cheap (<15$/month) data source for the **current** (not historical) for European Options?
Something like Yahoo Finance option page, with option chain contract prices. It's ok if it's partially incomplete and slightly outdated.
For European companies like Bayer AG, Siemens, Volkswagen, Daimler e... |
Free or cheap data source for the current European Options prices? |
Suppose I have given a finite number of grid values belonging to a probability density function. Moreover, I have the associated values of the density support. For instance:
```
support density value
0.06 0.07
-0.04 0.11
-0.02 0.52
0.00 1.56
0.02 7.87
0.04 19.18
0.06 13.66
0.08 3.40
0.10 0.98
0.13 0.3... |
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