Pair Creation🙏🏻 The one and only pair construction tech you need, unlike others:
Applies one consistent operation to all the data features (not only prices). Then, the script outputs these, so you can apply other calculations on these outputs.
calculates a very fast and native volatility based hedge ratio, that also takes into account point value (think SPY vs ES) so you can easily use it in position sizing
Has built-in forward pricing aka cost of carry model , so you can de-drift pairs from cost of carry, discover spot price of oil based on futures, and ofc find arbitrage opportunities
Also allows to make a pair as a product of 2 series, useful for triangular arbitrage
This script can make a pair in 2 ways:
Ratio, by dividing leg 1 by leg 2
Product, by multiplying leg 1 by leg 2
The real mathematically right way to construct a pair is a ratio/product (Spreads are in fact = 2 legged portfolio, but I ain't told ya that ok). Why? Because a pair of 2 entities has a mathematically unique beauty, it allows direct comparisons and relationship analysis, smth you can't do directly with 3 and more components.
Multiplication (think inversions like (EURUSD -> USDEUR), and use cases for triangular arbitrage) is useful sometimes too.
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Quickguide:
First, "Legs" are pair components: make a pair of related assets. Don’t be guided exclusively by clustering, cointegrations, mutual information etc. Common sense and exogenous info can easily made them all Forward pricing model: is useful when u work with spot vs futures pairs. Otherwise: put financing, storage and yield all on zeros, this way u will turn it off and have a pure ratio/product of 2 legs.
Look at the 2 numbers on the script’s status line: the first one would always be 1), and the second one is a variable.
First number (always 1) is multiplier for your position size on leg 1
The second number is the multiplier for your position size on leg 2 in the opposite direction.
If both legs are related, trading your sizes with these multipliers makes you do statistical arbitrage -> trading ~ volatility in risk free mode, while the relationship between the assets is still in place.
Also guys srsly, nobody ‘ever’ made a universal law that somewhy somehow for whatever secret conspiracy reason one shall only trade pairs in mean reverting style xd. You can do whatever you want:
Tilt hedge ratio significantly based on relative strength of legs
Trade the pair in momentum style
Ignore hedge ratio all together
And more and more, the limit is your imagination, e.g.:
Anticipate hedge ratio changes based on exogenous info and act accordingly
Scalp a pair just like any other asset
Make a pair out of 2 pairs
Like I mean it, whatever you desire
About forward pricing model:
It’s applied only to leg 2;
Direct: takes spot price and finds out implied futures price
Inverse: takes futures price and finds out implied spot price (try on oil)
Pls read online how to choose parameters, it’s open access reliable info
About the hedge ratio I use:
You prolly noticed the way I prefer to use inferred volumes vs the “real” ones. In pairs it’s especially meaningful, because real volumes lose sense in pair creation. And while volumes are closely tied to volatility, the inferred volumes ‘Are’ volatility irl (and later can be converted to currency space by using point value, allowing direct comparisons symbol vs symbol).
This hedge ratio is a good example of how discovering the real nature of entities beats making 100s of inventions, why domain knowledge and proper feature engineering beats difficult bulky models, neural networks etc. How simple data understanding & operations on it is all you need.
This script simply does this:
Takes inferred volume delta of both assets, makes a ratio, normalizes it by tick sizes and points values of both legs, calculates a typical value of this series.
That’s it, no step 2, we’re done. No Kalman filters, no TLS regression, no vine copulas, or whatever new fancy keywords you can come up with etc.
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^^ comparing real ES prices vs theoretical ones by forward-pricing model. Financing: 0.04, yield 0.0175
^^ EURUSD, 6E futures with theoretical futures price calculated with interest rate differential 0.02 (4% USD - 2% EUR interest rates)
^^4 different pairs (RTY/ES, YM/ES, NQ/ES, ES/ZN) each with different plot style (pick one you like in script's Style settings)
^^ YM/RTY pair, each plot represents ratio of different features: ratio of prices, ratio of inferred volume deltas, ratio of inferred volumes, ratio of inferred tick counts (also can be turned on/off in Style settings)
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How can u upgrade it and make a step forward yourself:
On tradingview missing values are automatically fixed by backfilling, and this never becomes a thing until you hit high frequency data. You can do better and use Kalman filter for filling missing values.
Script contains the functions I use everywhere to calculate inferred volume delta, inferred volume, and inferred tick count.
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Costofcarry
BSM OPM 1973 w/ Continuous Dividend Yield [Loxx]Generalized Black-Scholes-Merton w/ Analytical Greeks is an adaptation of the Black-Scholes-Merton Option Pricing Model including Analytical Greeks and implied volatility calculations. The following information is an excerpt from Espen Gaarder Haug's book "Option Pricing Formulas". The options sensitivities (Greeks) are the partial derivatives of the Black-Scholes-Merton ( BSM ) formula. Analytical Greeks for our purposes here are broken down into various categories:
Delta Greeks: Delta, DDeltaDvol, Elasticity
Gamma Greeks: Gamma, GammaP, DGammaDSpot/speed, DGammaDvol/Zomma
Vega Greeks: Vega , DVegaDvol/Vomma, VegaP
Theta Greeks: Theta
Rate/Carry Greeks: Rho, Rho futures option, Carry Rho, Phi/Rho2
Probability Greeks: StrikeDelta, Risk Neutral Density
(See the code for more details)
Black-Scholes-Merton Option Pricing
The Black-Scholes-Merton model can be "generalized" by incorporating a cost-of-carry rate b. This model can be used to price European options on stocks, stocks paying a continuous dividend yield, options on futures, and currency options:
c = S * e^((b - r) * T) * N(d1) - X * e^(-r * T) * N(d2)
p = X * e^(-r * T) * N(-d2) - S * e^((b - r) * T) * N(-d1)
where
d1 = (log(S / X) + (b + v^2 / 2) * T) / (v * T^0.5)
d2 = d1 - v * T^0.5
b = r ... gives the Black and Scholes (1973) stock option model.
b = r — q ... gives the Merton (1973) stock option model with continuous dividend yield q. <== this is the one used for this indicator!
b = 0 ... gives the Black (1976) futures option model.
b = 0 and r = 0 ... gives the Asay (1982) margined futures option model.
b = r — rf ... gives the Garman and Kohlhagen (1983) currency option model.
Inputs
S = Stock price.
X = Strike price of option.
T = Time to expiration in years.
r = Risk-free rate
d = dividend yield
v = Volatility of the underlying asset price
cnd (x) = The cumulative normal distribution function
nd(x) = The standard normal density function
convertingToCCRate(r, cmp ) = Rate compounder
gImpliedVolatilityNR(string CallPutFlag, float S, float x, float T, float r, float b, float cm , float epsilon) = Implied volatility via Newton Raphson
gBlackScholesImpVolBisection(string CallPutFlag, float S, float x, float T, float r, float b, float cm ) = implied volatility via bisection
Implied Volatility: The Bisection Method
The Newton-Raphson method requires knowledge of the partial derivative of the option pricing formula with respect to volatility ( vega ) when searching for the implied volatility . For some options (exotic and American options in particular), vega is not known analytically. The bisection method is an even simpler method to estimate implied volatility when vega is unknown. The bisection method requires two initial volatility estimates (seed values):
1. A "low" estimate of the implied volatility , al, corresponding to an option value, CL
2. A "high" volatility estimate, aH, corresponding to an option value, CH
The option market price, Cm , lies between CL and cH . The bisection estimate is given as the linear interpolation between the two estimates:
v(i + 1) = v(L) + (c(m) - c(L)) * (v(H) - v(L)) / (c(H) - c(L))
Replace v(L) with v(i + 1) if c(v(i + 1)) < c(m), or else replace v(H) with v(i + 1) if c(v(i + 1)) > c(m) until |c(m) - c(v(i + 1))| <= E, at which point v(i + 1) is the implied volatility and E is the desired degree of accuracy.
Implied Volatility: Newton-Raphson Method
The Newton-Raphson method is an efficient way to find the implied volatility of an option contract. It is nothing more than a simple iteration technique for solving one-dimensional nonlinear equations (any introductory textbook in calculus will offer an intuitive explanation). The method seldom uses more than two to three iterations before it converges to the implied volatility . Let
v(i + 1) = v(i) + (c(v(i)) - c(m)) / (dc / dv (i))
until |c(m) - c(v(i + 1))| <= E at which point v(i + 1) is the implied volatility , E is the desired degree of accuracy, c(m) is the market price of the option, and dc/ dv (i) is the vega of the option evaluaated at v(i) (the sensitivity of the option value for a small change in volatility ).
Things to know
Only works on the daily timeframe and for the current source price.
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