instruction stringlengths 15 26.1k |
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I have the following system of SDEs (which you can think of as 3 different stocks)
$$dX_t^1 = \mu_t X_t^1 dt + \sigma_t X_t^1 dW_t^1$$
$$dX_t^2 = \mu_2 dt + \sigma_2 dW_t^2$$
$$dX_t^3 = \mu_3 dt + \sigma_3 dW_t^3$$
where $dW_t$ is a standard Brownian motion and correlated such that
$dW_t^1 dW_t^2 = \rho_2 dt$ and... |
I am a student trying to price volatility and variance swaps.
People who price those two products usually try to get the "fair strike", and don't seem to care about the price.
However, I have a hard time understanding what the fair strike is on those swaps. Also, if people just want to get this value, how do they g... |
What is the Fair Strike in a Var/Vol Swap and how does it relate to its price? |
I am a student trying to price volatility and variance swaps.
People who price those two products usually try to get the "fair strike", and don't seem to care about the price.
However, I have a hard time understanding what fair strike means. Also, if people just want to get this value, how do they get a price out o... |
I am a student trying to price volatility and variance swaps.
People who price those two products usually try to get the "fair strike", and don't seem to care about the price.
However, I have a hard time understanding what fair strike means. Also, if people just want to get this value, how do they get a price out o... |
I am trying to fit kappa for a ticker. I am using 5 days of data to illustrate how this can be done, which isn't that much data but I think is sufficient to show my problem. This data however appears to have a non exponential function for the (A, k) part.
I am following the method described in this post: https://qua... |
I am trying to fit kappa for a ticker. I am using 5 days of data to illustrate how this can be done, which isn't that much data but I think is sufficient to show my problem. This data however appears to have a non exponential function for the (A, k) part.
I am following the method described in this post: https://qua... |
I am trying to understand the Fama-French factor model, or any kind of CAPM extensions really. What is really puzzling me is the use of mimicking portfolios. Fama and French create mimicking portfolios (indices) of the SMB and HML factors, so similarly as the market risk-factor (index). My question is: why not simply u... |
Fama-French factor model: why mimicking portfolios? |
Does the expected return increase with variance for stocks? |
I searched the web, but could not find OCC TIMS methodology detailed description anywhere. Any ideas?
[OCC TIMS][1]
[1]: https://www.theocc.com/risk-management/customer-portfolio-margin |
Where can I find OCC TIMS methodology? |
I could find any reference restricting the sign of the volatilities in the multi-factor HJM framework.
Can someone please confirm if $\sigma_i(t,T)$ can assume negative values for some $i,t$ and $T$?
$$
df(t,T) = \left(\sum_i \sigma_i(t,T)\int_t^T \sigma_i(t,u) du \right) dt + \sum_i \sigma_i(t,T) dW_i(t)
$$ |
Can volatility assume negative values under multi-factor HJM framework? |
Let $x_i$ be a characteristic of firm $i$ (eg. the firm's book to market ratio), let $R_i$ be a random variable denoting the return of firm $i$, and let $c_i = \operatorname{Cov}(R_i, S)$ where $S$ is the excess return of some portfolio. The heart of your question is why do typical academic asset pricing models take th... |
Let $x_i$ be a characteristic of firm $i$ (eg. the firm's book to market ratio), let $R_i$ be a random variable denoting the return of firm $i$, and let $c_i = \operatorname{Cov}(R_i, S)$ where $S$ is the excess return of some portfolio. The heart of your question is why do typical academic asset pricing models take th... |
The innovation of Fama and French's Three Factor Model wasn't in finding book to market ratios forecast returns but in reconciling that empirical regularity with the standard framework of macro-finance theory which views higher expected returns as compensation for positive covariance with undiversifiable, macroeconomic... |
The innovation of Fama and French's Three Factor Model wasn't in finding book to market ratios forecast returns but in reconciling that empirical regularity with the standard framework of macro-finance theory which views higher expected returns as compensation for positive covariance with undiversifiable, macroeconomic... |
The innovation of Fama and French's Three Factor Model wasn't in finding book to market ratios forecast returns but in reconciling that empirical regularity with the standard framework of macro-finance theory which views higher expected returns as compensation for positive covariance with undiversifiable, macroeconomic... |
According to the saddlepoint approximation, if the cumulant generating function $K(t) = \log E[e^{tX}]$ of the distribution of the random variable $X$ exists and is known, then the density $f(x)$ of $X$ can be approximated as follows:
$$
f(x) \approx \sqrt{\frac{1}{2\pi K''(\hat t(x))}} \exp\{K( \hat t(x)) - \hat t(x... |
According to the saddlepoint approximation, if the cumulant generating function $K(t) = \log E[e^{tX}]$ of the distribution of the random variable $X$ exists and is known, then the density $f(x)$ of $X$ can be approximated as follows:
$$
f(x) \approx \sqrt{\frac{1}{2\pi K''(\hat t(x))}} \exp\{K( \hat t(x)) - \hat t(x... |
According to the saddlepoint approximation, if the cumulant generating function $K(t) = \log E[e^{tX}]$ of the distribution of the random variable $X$ exists and is known, then the density $f(x)$ of $X$ can be approximated as follows:
$$
f(x) \approx \sqrt{\frac{1}{2\pi K''(\hat t(x))}} \exp\{K( \hat t(x)) - \hat t(x... |
The innovation of Fama and French's Three Factor Model wasn't in finding book to market ratios forecast returns but in reconciling that empirical regularity with the standard framework of macro-finance theory which views higher expected returns as compensation for macroeconomic risk of hedging concern to investors.
... |
At which Lag the AutoCorrelation of Rolling 10 Year Returns is not significant? |
The innovation of Fama and French's Three Factor Model wasn't in finding book to market ratios forecast returns but in reconciling that empirical regularity with the standard framework of macro-finance theory which views higher expected returns as compensation for macroeconomic risk of hedging concern to investors.
... |
Suppose the interest rate is zero. A stock with price $S(t)$ at time $t$ pays only one dividend at time $s$ such that $S(s_+)=S(s_-)q$ where $q\in[0,1]$ is a constant. Consider a European call and an American call with the same strike $K$ and expiry $T>s>0$ at time $0$. Denote the price of the former $C_{\text{E}}(S)$ ... |
Suppose the interest rate is zero. A stock with price $S(t)$ at time $t$ pays only one dividend at time $s$ such that $S(s_+)=S(s_-)q$ where $q\in[0,1]$ is a constant. Consider a European call and an American call with the same strike $K$ and expiry $T>s>0$ at time $0$. Denote the price of the former $C_{\text{E}}(S)$ ... |
In the academic literature - often "momentum" and "positive feedback" traders are used interchangeably. Like "Most researchers have found that institutional investors are momentum traders (also referred to as positive feedback traders and trend followers)"
Is the crucial difference - that momentum is computed as an... |
When constructing curves, are there any generic and quantitative metrics that can be computed for any kind curve (government, corporate, swap, etc)? |
Imagine a trading house that trades options, and is looking for simple but effective metrics over which trading option limits will be set. Which ones should it choose?
Some random thoughts:
1) VaR is not ideal, since the biggest concern is what happens 1% of the time.
2) Because the focus is to keep it simple,... |
The innovation of Fama and French's Three Factor Model wasn't in finding book to market ratios forecast returns but in reconciling that empirical regularity with the standard framework of macro-finance theory which views higher expected returns as compensation for macroeconomic risk of hedging concern to investors.
... |
I often hear people talking about the skew of the volatility surface, model, etc... but it appears to me that there isn't a clear standard definition unanimously used by practitioners.
**So here is my question:** Does anyone have a clear and unifying definition that can be stated in mathematical terms of what skew i... |
I have a portfolio composed of $ N $ assets.
I know the one-year beta of these assets, I also know the past (ex-post) beta ($\beta$) of my portfolio.
My portfolio changes allocation every month. So I want to calculate the beta ex-ante at each rebalancing date in order to check if my allocation model chooses to increa... |
How to calculate the ex-ante beta of a portfolio between several rebalancing? |
Suppose the interest rate is zero. A stock with price $S(t)$ at time $t$ pays only one dividend at time $s$ such that $S(s_+)=S(s_-)q$ where $q\in[0,1]$ is a constant. Consider a European call and an American call with the same strike $K$ and expiry $T>s>0$ at time $0$. Denote the price of the former $C_{\text{E}}(S(0)... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). For some reason, the Finite Differences Gamma calculation does not agree with the Autograd AAD results. I believe I have properly called the Heaviside Approxima... |
It is known that a variance swap can be replicated by a strip of options. However, it is costly too trade that many OTM options and there is not enogh liquidity to trade the wings in the quantity that is necessary. So I suppose banks do not actually trade the entire strip. The question is then, how do they hedge this p... |
How do banks and dealers effectively hedge a variance swap? |
Suppose the interest rate is zero. A stock with price $S(t)$ at time $t$ pays only one dividend at time $s$ such that $S(s_+)=S(s_-)q$ where $q\in[0,1]$ is a constant. Consider a European call and an American call with the same strike $K$ and expiry $T>s>0$ at time $0$. Denote the price of the former $C_{\text{E}}(S(0)... |
Suppose the interest rate is zero. A stock with price $S(t)$ at time $t$ pays only one dividend at time $s$ such that $S(s_+)=S(s_-)q$ where $q\in[0,1]$ is a constant. Consider a European call and an American call with the same strike $K$ and expiry $T>s>0$ at time $0$. Define
$$C_{\text{E}}(S_t,t):=\mathbf E\big[(S(T... |
It is known that a variance swap can be replicated by a strip of options. However, it is costly to trade that many OTM options and there is not enough liquidity to trade the wings in the quantity that is necessary. So I suppose banks do not actually trade the entire strip. The question is then, how do they hedge this p... |
I'm looking for a way to introduce naive diversification bias in a mean variance framework and had the idea to model it as some sort of "aversion to extreme portfolio weights" of the following form:
$$
(1-\delta)\omega^{'} \Sigma\omega+\delta\omega^{'}\omega \rightarrow Min!\\
s.t.1^{'}\omega=1
$$
Here, $\ome... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. For some reason, the Finite Differences Gamma cal... |
How does one scale bond yields 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 stocks have a constant equity risk premium and EPS, the risk-free rate moving from 1%->2% impacts CAP... |
[3-month CORRA futures][1] are traded on the Montreal Exchange.
The Exchange gives an [example][2] of calculating the probability of a rate change based on these future prices.
In the example, the prediction is made from only 12 days away.
Suppose we are six months away with a few BoC announcements between "toda... |
I recently came to know about TAS: [Trade at Settlement](https://www.cmegroup.com/trading/trading-at-settlement.html). In that context, as a layman, I'm tyring to make sense of why the settlement price matters in a trade.
Perhaps these betray my ignorance:
1. What's the relevance of the settlement price? I.e., as... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. For some reason, the Finite Differences Gamma cal... |
__The Settlement Price__
When you buy a futures contract, you're not actually paying the price of the underlying asset upfront. Instead, you're entering into an agreement to buy or sell the asset at a specific price (the contract price) on a future date.
To determine the difference between the contract price and ... |
__The Settlement Price__
When you buy a futures contract, you're not actually paying the price of the underlying asset upfront. Instead, you're entering into an agreement to buy or sell the asset at a specific price (the contract price) on a future date.
To determine the difference between the contract price and ... |
__The Settlement Price__
When you buy a futures contract, you're not actually paying the price of the underlying asset upfront. Instead, you're entering into an agreement to buy or sell the asset at a specific price (the contract price) on a future date.
To determine the difference between the contract price and ... |
__The Settlement Price__
When you buy a futures contract, you're not actually paying the price of the underlying asset upfront. Instead, you're entering into an agreement to buy or sell the asset at a specific price (the contract price) on a future date.
To determine the difference between the contract price and ... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. For some reason, the Finite Differences Gamma cal... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. For some reason, the Finite Differences Gamma cal... |
Earlier I asked a general question about how probabilities are derived from futures prices for derivatives related to the Bank of Canada's policy rate.
I have been advised the Overnight Index Swaps (OIS) are the best vehicle to use, in this case three month CORRA futures.
[3-month CORRA futures][1] are traded on... |
Is Nassim Taleb wrong about his DdeltaDvol dynamics in his Dynamic Hedging book? |
[![enter image description here][1]][1]
if you're long OTM calls, an increase in vol would increase your delta (converges to ATM) and if you're long ITM calls, increase in vol would decrease your delta (converges to ATM as well). So OTM calls have positive DdeltaDvol (Vanna) and ITM calls have negative Vannas. Follo... |
In Excel, I have the monthly stock price data for the past few years for Asset A and Asset B. I have calculated the monthly returns, mean returns, variances, and standard deviations for both stocks as well as the covariance and correlation.
Then I calculated the weighted portfolios of Asset A and Asset B calculating... |
https://www.daytrading.com/greeks-options-pricing
so the green matrixx is missing the ultimate lost greek..
looks like Conspiracy?.. similar to gamma for delta..
Xeva-Xeveta would identify exact time point when
long putcalls better be sold back huh..
> https://quant.stackexchange.com/questions/48958/what... |
The Lost freaky Greek -- Xeva -- dTheta/dt2 --- 2nd deriva. How to calc & bsm formula? |
https://www.daytrading.com/greeks-options-pricing
so the green matrixx is missing the ultimate lost greek..
looks like Conspiracy?.. similar to gamma for delta..
Xeva-Xeveta would identify exact time point when
long putcalls better be sold back. just
before jumping off that cliff huh..
> https://qua... |
**-- i'm looking for the pure math glyph formula?**
& use might be tremendous as far as we can see..
---------------------------------------
_https://www.daytrading.com/greeks-options-pricing_
so the green matrixx is missing the ultimate lost greek..
looks like Conspiracy?.. similar to gamma for delta... |
_oh i may as well be looking for the 3rd deriva instead.._
**-- i'm looking for the pure math glyph formula?**
& use might be tremendous as far as we can see..
---------------------------------------
https://www.daytrading.com/greeks-options-pricing
so the green matrixx is missing the ultimate lost g... |
how to solve it for max Theta Curvature?
**i'm looking for the pure math glyph formula..**
_it may be related to actually_
_3rd deriva & curvature function.._ |
I'm looking for a way to introduce naive diversification bias in a mean variance framework and had the idea to model it as some sort of "aversion to extreme portfolio weights" of the following form:
$$
(1-\delta)\omega^{'} \Sigma\omega+\delta\omega^{'}\omega \rightarrow Min!\\
s.t.1^{'}\omega=1
$$
Here, $\ome... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. For some reason, the Finite Differences Gamma cal... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. For some reason, the Finite Differences Gamma cal... |
Let us assume we have two FX rates: $ 1 EUR = S_t^{(1)} USD$ and $ 1 GBP=S_t^{(2)} USD $. Let $K_1>0, K_2>0$ be strictly positive values and a payoff at some time $ T>0 $ (called maturity) defined by: $ V_T=1_{ \{ \kappa_1\cdot S_T^{(1)}<\kappa_1\cdot K_1 \} } \cdot 1_{\{\kappa_2\cdot S_T^{(2)}<\kappa_2\cdot K_2\}} , \... |
Suppose the interest rate is zero. A stock with price $S(t)$ at time $t$ pays only one dividend at time $t_1$ such that $S(s_+)=S(t_1^-)q$ where $q\in[0,1]$ is a constant. Consider a European call and an American call with the same strike $K$ and expiry $T>t_1>0$ at time $0$. Define
$$C_{\text{E}}(S_t,t):=\mathbf E\bi... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. For some reason, the Finite Differences Gamma cal... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. For some reason, the Finite Differences Gamma cal... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. For some reason, the Finite Differences Gamma cal... |
r > 0. I understand that money today is worth more than money tomorrow. So if volatility is 0, it's better to take the money today. But I don't understand how to square away the following:
- Time value for American Options is always nonnegative
- Value = Time Value + Intrinsic Value
How is it ever optimal to th... |
If American Options always have positive time value, how can it be optimal to exercise an American Put early? |
“How is it ever optimal to throw away positive time value?”
Answer: when the thing you are throwing away is less than what you are gaining through the early exercise . What is that ? Interest you receive on the strike price by receiving it earlier. |
I'm looking for a way to introduce naive diversification bias in a mean variance framework and had the idea to model it as some sort of "aversion to extreme portfolio weights" of the following form:
$$
(1-\delta)\omega^{'} \Sigma\omega+\delta\omega^{'}\omega \rightarrow Min!\\
s.t.1^{'}\omega=1
$$
Here, $\ome... |
I'm looking for a way to introduce naive diversification bias in a mean variance framework and had the idea to model it as some sort of "aversion to extreme portfolio weights" of the following form:
$$
(1-\delta)\omega^{'} \Sigma\omega+\delta\omega^{'}\omega \rightarrow Min!\\
s.t.1^{'}\omega=1
$$
Here, $\ome... |
I'm looking for a way to introduce naive diversification bias in a mean variance framework and had the idea to model it as some sort of "aversion to extreme portfolio weights" of the following form:
$$
(1-\delta)\omega^{'} \Sigma\omega+\delta\omega^{'}\omega \rightarrow Min!\\
s.t.1^{'}\omega=1
$$
Here, $\ome... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. I believe I have properly called the Heaviside Ap... |
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function vs. a max function, so the payoff is continuous rather than discrete. I believe I have properly called the Heaviside Ap... |
Could somebody help me finding the python equivalent to Matlab's:
IRDataCurve.bootstrap('Zero', quote_date, instrument_types, instruments,'InterpMethod','spline','Basis',2,'Compounding',-1,'IRBootstrapOptions', IRBootstrapOptions('ConvexityAdjustment',@(t) .5*sigma^2.*t.^2));
Is it possible to replicate?
I f... |
I did a fama-macbeth regression and got alpha of over 2.3%. Dependent variable is monthly returns and independent variables are ln(size) and ln(book-to-market). I was just wondering what does the intercept measure? |
What does Fama-Macbeth regression model alpha (intercept) measure? |
I know it's possible to [efficiently estimate bid/ask spreads from OHLC prices][1] and there are several methods to do so. However I have a data source that contains only bids, asks and close prices, no High, Low or Open prices. Is it possible to do the inverse, estimate the Open, High and Low prices from bids, asks an... |
Could somebody help me finding the python equivalent to Matlab's:
IRDataCurve.bootstrap('Zero', quote_date, instrument_types, instruments,'InterpMethod','spline','Basis',2,'Compounding',-1,'IRBootstrapOptions', IRBootstrapOptions('ConvexityAdjustment',@(t) .5*sigma^2.*t.^2));
Is it possible to replicate?
I f... |
I'm doing some statistics in order to evaluate the Forex market profitability.
I first define what I call "regeneration of the market".
For example, the following fictive order-book of EURUSD:
ASK:
20M 1.10010
8M 1.10005
2M 1.10002
BID:
1M 1.99999
9M 1.99995
1... |
How fast is the forex market regenerated? |
I'm doing some statistics in order to evaluate the Forex market profitability.
I first define what I call "regeneration of the market".
For example, the following fictive order-book of EURUSD:
ASK:
20M 1.10010
8M 1.10005
2M 1.10002
BID:
1M 1.99999
9M 1.99995
1... |
I'm also in the process of learning and will just give you my thoughts here. Might not be fully right so please correct me or enter a discussion with me.
I think it depends on the information you are given. If you know that the other party knows the true value and you have only an estimate, there is information in t... |
I want to bootstrap the yield curve of multiple currencies using 3M futures. I have it implemented in Matlab, but need to transfer to Python.
Could somebody help me finding the python equivalent to Matlab's:
IRDataCurve.bootstrap('Zero', quote_date, instrument_types, instruments,'InterpMethod','spline','Basis',2,... |
I want to bootstrap the yield curve of multiple currencies using 3M futures. I have it implemented in Matlab, but need to transfer to Python.
Could somebody help me finding the python equivalent to Matlab's:
IRDataCurve.bootstrap('Zero', quote_date, instrument_types, instruments,'InterpMethod','spline','Basis',2,... |
I want to bootstrap the yield curve of multiple currencies using 3M futures. I have it implemented in Matlab, but need to transfer to Python. I have a vector of dates and a vector of future prices.
Could somebody help me finding the python equivalent to Matlab's:
IRDataCurve.bootstrap('Zero', quote_date, instrume... |
lets say party A and party B are the two parties in a derivative contract.
If party B posts USD cash as collateral, it would expect party A to return the collateral with interest at settlement. How about if party B posts a US treasury as collateral? would it expect to receive anything on top of the security at sett... |
I get from the time value of money that receiving $1 today is worth more than receiving $1 tomorrow. But what about paying money, as opposed to getting money. Do i prefer to pay $1 tomorrow than today, because that same $1 tomorrow would be worth <$1 today? Or do i prefer benefits and costs today rather than tomorrow? |
Is it better to pay $1 today or $1 tomorrow? |
I get from the time value of money that receiving \$1today is worth more than receiving \$1 tomorrow. But what about paying money, as opposed to getting money. Do i prefer to pay \$1 tomorrow than today, because that same \$1 tomorrow would be worth <\$1 today? Or do i prefer benefits and costs today rather than tomorr... |
I get from the time value of money that receiving \$1 today is worth more than receiving \$1 tomorrow. But what about paying money, as opposed to getting money. Do i prefer to pay \$1 tomorrow than today, because that same \$1 tomorrow would be worth <\$1 today? Or do i prefer benefits and costs today rather than tomor... |
Is it better to pay \$1 today or \$1 tomorrow? |
I get from the time value of money that receiving \$1 today is worth more than receiving \$1 tomorrow. But what about paying money, as opposed to getting money. Do i prefer to pay \$1 tomorrow than today, because that same \$1 tomorrow would be worth <\$1 today? Or do i prefer benefits and costs today rather than tomor... |
I need to bootstrap a yieldcurve with 3M futures, using a cubic spline if possible.
Using, for example 3M Euribor, how do I bootstrap the yield curve using python?
I have a vector of dates and a vector of future prices.
I found about the QuantLib library, and more specifically, the ```ql.PiecewiseCubicZero```.... |
I want to bootstrap the yield curve of multiple currencies using 3M futures. I have it implemented in Matlab, but need to transfer to Python. I have a vector of dates and a vector of future prices.
Could somebody help me finding the python equivalent to Matlab's:
```IRDataCurve.bootstrap('Zero', quote_date, instr... |
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