Optionpricing — Indicators and Strategies

Options Price Calculator In the team, we continue to explore and expand the boundaries of TradingView. For now, there is not much an options trader can do with options in TradingView. We wanted to change that and created a simple option pricer. You can set up in parameters a set of strikes, implied volatility, and days to expiry. The indicators will take a risk-free rate from US01Y and the underlying price from your current chart. It will compute prices and greeks for both put and call options. Thanks to @MUQWISHI for helping code it. Disclaimer Please remember that past performance may not indicate future results. Due to various factors, including changing market conditions, the strategy may no longer perform as well as in historical backtesting. This post and the script don’t provide any financial advice.

Generalized Black-Scholes-Merton w/ Analytical Greeks [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 BSM formula and its binomial counterpart may easily be the most used "probability model/tool" in everyday use — even if we con- sider all other scientific disciplines. Literally tens of thousands of people, including traders, market makers, and salespeople, use option formulas several times a day. Hardly any other area has seen such dramatic growth as the options and derivatives businesses. In this chapter we look at the various versions of the basic option formula. In 1997 Myron Scholes and Robert Merton were awarded the Nobel Prize (The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel). Unfortunately, Fischer Black died of cancer in 1995 before he also would have received the prize. It is worth mentioning that it was not the option formula itself that Myron Scholes and Robert Merton were awarded the Nobel Prize for, the formula was actually already invented, but rather for the way they derived it — the replicating portfolio argument, continuous- time dynamic delta hedging, as well as making the formula consistent with the capital asset pricing model (CAPM). The continuous dynamic replication argument is unfortunately far from robust. The popularity among traders for using option formulas heavily relies on hedging options with options and on the top of this dynamic delta hedging, see Higgins (1902), Nelson (1904), Mello and Neuhaus (1998), Derman and Taleb (2005), as well as Haug (2006) for more details on this topic. In any case, this book is about option formulas and not so much about how to derive them. Provided here are the various versions of the Black-Scholes-Merton formula presented in the literature. All formulas in this section are originally derived based on the underlying asset S follows a geometric Brownian motion dS = mu * S * dt + v * S * dz where t is the expected instantaneous rate of return on the underlying asset, a is the instantaneous volatility of the rate of return, and dz is a Wiener process. The formula derived by Black and Scholes (1973) can be used to value a European option on a stock that does not pay dividends before the option's expiration date. Letting c and p denote the price of European call and put options, respectively, the formula states that c = S * N(d1) - X * e^(-r * T) * N(d2) p = X * e^(-r * T) * N(d2) - S * N(d1) where d1 = (log(S / X) + (r + v^2 / 2) * T) / (v * T^0.5) d2 = (log(S / X) + (r - v^2 / 2) * T) / (v * T^0.5) = d1 - v * T^0.5 Inputs S = Stock price. X = Strike price of option. T = Time to expiration in years. r = Risk-free rate b = Cost of carry 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. You can adjust the text size to fit the screen

Generalized Black-Scholes-Merton Option Pricing Formula [Loxx]Generalized Black-Scholes-Merton Option Pricing Formula is an adaptation of the Black-Scholes-Merton Option Pricing Model including Numerical Greeks aka "Option Sensitivities" and implied volatility calculations. The following information is an excerpt from Espen Gaarder Haug's book "Option Pricing Formulas". Black-Scholes-Merton Option Pricing The BSM formula and its binomial counterpart may easily be the most used "probability model/tool" in everyday use — even if we con- sider all other scientific disciplines. Literally tens of thousands of people, including traders, market makers, and salespeople, use option formulas several times a day. Hardly any other area has seen such dramatic growth as the options and derivatives businesses. In this chapter we look at the various versions of the basic option formula. In 1997 Myron Scholes and Robert Merton were awarded the Nobel Prize (The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel). Unfortunately, Fischer Black died of cancer in 1995 before he also would have received the prize. It is worth mentioning that it was not the option formula itself that Myron Scholes and Robert Merton were awarded the Nobel Prize for, the formula was actually already invented, but rather for the way they derived it — the replicating portfolio argument, continuous- time dynamic delta hedging, as well as making the formula consistent with the capital asset pricing model (CAPM). The continuous dynamic replication argument is unfortunately far from robust. The popularity among traders for using option formulas heavily relies on hedging options with options and on the top of this dynamic delta hedging, see Higgins (1902), Nelson (1904), Mello and Neuhaus (1998), Derman and Taleb (2005), as well as Haug (2006) for more details on this topic. In any case, this book is about option formulas and not so much about how to derive them. Provided here are the various versions of the Black-Scholes-Merton formula presented in the literature. All formulas in this section are originally derived based on the underlying asset S follows a geometric Brownian motion dS = mu * S * dt + v * S * dz where t is the expected instantaneous rate of return on the underlying asset, a is the instantaneous volatility of the rate of return, and dz is a Wiener process. The formula derived by Black and Scholes (1973) can be used to value a European option on a stock that does not pay dividends before the option's expiration date. Letting c and p denote the price of European call and put options, respectively, the formula states that c = S * N(d1) - X * e^(-r * T) * N(d2) p = X * e^(-r * T) * N(d2) - S * N(d1) where d1 = (log(S / X) + (r + v^2 / 2) * T) / (v * T^0.5) d2 = (log(S / X) + (r - v^2 / 2) * T) / (v * T^0.5) = d1 - v * T^0.5 Inputs S = Stock price. X = Strike price of option. T = Time to expiration in years. r = Risk-free rate b = Cost of carry 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). Numerical Greeks or Greeks by Finite Difference Analytical Greeks are the standard approach to estimating Delta, Gamma etc... That is what we typically use when we can derive from closed form solutions. Normally, these are well-defined and available in text books. Previously, we relied on closed form solutions for the call or put formulae differentiated with respect to the Black Scholes parameters. When Greeks formulae are difficult to develop or tease out, we can alternatively employ numerical Greeks - sometimes referred to finite difference approximations. A key advantage of numerical Greeks relates to their estimation independent of deriving mathematical Greeks. This could be important when we examine American options where there may not technically exist an exact closed form solution that is straightforward to work with. (via VinegarHill FinanceLabs) Things to know Only works on the daily timeframe and for the current source price. You can adjust the text size to fit the screen

Sprenkle 1964 Option Pricing Model w/ Num. Greeks [Loxx]Sprenkle 1964 Option Pricing Model w/ Num. Greeks is an adaptation of the Sprenkle 1964 Option Pricing Model in Pine Script. The following information is an except from Espen Gaarder Haug's book "Option Pricing Formulas". The Sprenkle Model Sprenkle (1964) assumed the stock price was log-normally distributed and thus that the asset price followed a geometric Brownian motion, just as in the Black and Scholes (1973) analysis. In this way he ruled out the possibility of negative stock prices, consistent with limited liability. Sprenkle moreover allowed for a drift in the asset price, thus allowing positive interest rates and risk aversion (Smith, 1976). Sprenkle assumed today's value was equal to the expected value at maturity. c = S * e^(rho*T) * N(d1) - (1 - k) * X * N(d2) d1 = (log(S/X) + (rho + v^2 / 2) * T) / (v * T^0.5) d2 = d1 - (v * T^0.5) Inputs S = Stock price. X = Strike price of option. T = Time to expiration in years. r = Risk-free rate v = Volatility of the underlying asset price k = Market risk aversion adjustment rho = Average growth rate share cnd (x) = The cumulative normal distribution function nd(x) = The standard normal density function nd(x) = The standard normal density function convertingToCCRate(r, cmp) = Rate compounder Numerical Greeks or Greeks by Finite Difference Analytical Greeks are the standard approach to estimating Delta, Gamma etc... That is what we typically use when we can derive from closed form solutions. Normally, these are well-defined and available in text books. Previously, we relied on closed form solutions for the call or put formulae differentiated with respect to the Black Scholes parameters. When Greeks formulae are difficult to develop or tease out, we can alternatively employ numerical Greeks - sometimes referred to finite difference approximations. A key advantage of numerical Greeks relates to their estimation independent of deriving mathematical Greeks. This could be important when we examine American options where there may not technically exist an exact closed form solution that is straightforward to work with. (via VinegarHill FinanceLabs) Things to know Only works on the daily timeframe and for the current source price. You can adjust the text size to fit the screen

Cox-Ross-Rubinstein Binomial Tree Options Pricing Model [Loxx]Cox-Ross-Rubinstein Binomial Tree Options Pricing Model is an options pricing panel calculated using an N-iteration (limited to 300 in Pine Script due to matrices size limits) "discrete-time" (lattice based) method to approximate the closed-form Black–Scholes formula. Joshi (2008) outlined varying binomial options pricing model furnishes a numerical approach for the valuation of options. Significantly, the American analogue can be estimated using the binomial tree. This indicator is the complex calculation for Binomial option pricing. Most folks take a shortcut and only calculate 2 iterations. I've coded this to allow for up to 300 iterations. This can be used to price American Puts/Calls and European Puts/Calls. I'll be updating this indicator will be updated with additional features over time. If you would like to learn more about options, I suggest you check out the book textbook Options, Futures and other Derivative by John C Hull. ***This indicator only works on the daily timeframe!*** A quick graphic of what this all means: In the graphic, "n" are the steps, in this case we can do up to 300, in production we'd need to do 5-15K. That's a lot of steps! You can see here how the binomial tree fans out. As I said previously, most folks only calculate 2 steps, here we are calculating up to 300. Want to learn more about Simple Introduction to Cox, Ross Rubinstein (1979) ? Watch this short series "Introduction to Basic Cox, Ross and Rubinstein (1979) model." Limitations of Black Scholes options pricing model This is a widely used and well-known options pricing model, factors in current stock price, options strike price, time until expiration (denoted as a percent of a year), and risk-free interest rates. The Black-Scholes Model is quick in calculating any number of option prices. But the model cannot accurately calculate American options, since it only considers the price at an option's expiration date. American options are those that the owner may exercise at any time up to and including the expiration day. What are Binomial Trees in options pricing? A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram representing different possible paths that might be followed by the stock price over the life of an option. The underlying assumption is that the stock price follows a random walk. In each time step, it has a certain probability of moving up by a certain percentage amount and a certain probability of moving down by a certain percentage amount. In the limit, as the time step becomes smaller, this model is the same as the Black–Scholes–Merton model. What is the Binomial options pricing model ? This model uses a tree diagram with volatility factored in at each level to show all possible paths an option's price can take, then works backward to determine one price. The benefit of the Binomial Model is that you can revisit it at any point for the possibility of early exercise. Early exercise is executing the contract's actions at its strike price before the contract's expiration. Early exercise only happens in American-style options. However, the calculations involved in this model take a long time to determine, so this model isn't the best in rushed situations. What is the Cox-Ross-Rubinstein Model? The Cox-Ross-Rubinstein binomial model can be used to price European and American options on stocks without dividends, stocks and stock indexes paying a continuous dividend yield, futures, and currency options. Option pricing is done by working backwards, starting at the terminal date. Here we know all the possible values of the underlying price. For each of these, we calculate the payoffs from the derivative, and find what the set of possible derivative prices is one period before. Given these, we can find the option one period before this again, and so on. Working ones way down to the root of the tree, the option price is found as the derivative price in the first node. Inputs Spot price: select from 33 different types of price inputs Calculation Steps: how many iterations to be used in the Binomial model. In practice, this number would be anywhere from 5000 to 15000, for our purposes here, this is limited to 300 Strike Price: the strike price of the option you're wishing to model % Implied Volatility: here you can manually enter implied volatility Historical Volatility Period: the input period for historical volatility; historical volatility isn't used in the CRRBT process, this is to serve as a sort of benchmark for the implied volatility, Historical Volatility Type: choose from various types of implied volatility, search my indicators for details on each of these Option Base Currency: this is to calculate the risk-free rate, this is used if you wish to automatically calculate the risk-free rate instead of using the manual input. this uses the 10 year bold yield of the corresponding country % Manual Risk-free Rate: here you can manually enter the risk-free rate Use manual input for Risk-free Rate? : choose manual or automatic for risk-free rate % Manual Yearly Dividend Yield: here you can manually enter the yearly dividend yield Adjust for Dividends?: choose if you even want to use use dividends Automatically Calculate Yearly Dividend Yield? choose if you want to use automatic vs manual dividend yield calculation Time Now Type: choose how you want to calculate time right now, see the tool tip Days in Year: choose how many days in the year, 365 for all days, 252 for trading days, etc Hours Per Day: how many hours per day? 24, 8 working hours, or 6.5 trading hours Expiry date settings: here you can specify the exact time the option expires Take notes: Futures don't risk free yields. If you are pricing options of futures, then the risk-free rate is zero. Dividend yields are calculated using TradingView's internal dividend values This indicator only works on the daily timeframe Included Option pricing panel Loxx's Expanded Source Types

Black-Scholes Model for American Options This model uses Black's Approximation to price American Options. Black's Approximation is an extension of the traditional Black-Scholes model that allows the price of American Options to be approximated within the Black-Scholes Framework. This is necessary because the traditional Black-Scholes model only works on options that are exercised at expiry, not before; like American Options can be. Black's Approximation approximates the value of an American option by: 1st. Calculating the theoretical price of a european call or put based on the strike price (K), spot price (S), annual return (sigma), time until expiry (T), times until the next 2 ex-dividend dates (t1 & t2), and the dividend paid out at times t1 and t2 (D1 and D2). 2nd. The theoretical price of an option expiring on the second ex-dividend date (t2) is calculated. This replicates exercising the option early. 3rd. Finally, the highest price of the two theoretical prices calculated in steps 1 & 2 is chosen as the approximated price. How to use this: 1st. Input your strike price. 2nd. Input the risk-free-rate of the currency the option is based in. 3rd. Input the dividend yield for the next ex-dividend date. For example AAPL's dividend yield is 0.82 and will be paid out on August 7,2020. 4th. Input the time until the next ex-dividend date. For example AAPL's next ex-dividend date is August 7,2020, which is 61 days away. So you'd input 61 (this includes weekends and holidays). 5th. Input the dividend yield for the ex-dividend date after the next one. For example AAPL's dividend yield after the next one is 0.82 and will be paid out on November 6, 2020. 6th. Input the time until the next furthest ex-dividend date. For example AAPL's next ex-dividend date after Aug 7th, is on November 6, 2020, which is 152 days away. So you'd input 152 (this includes weekends and holidays). 7th. Input your time until expiry. You can do so in terms of days, hours, and minutes. 8th. Input your chart time-frame in term of minutes. For example, if you're using the 1 min time-frame enter 1, 4hr time-frame enter 480, daily time-frame enter 1440. 9th. Lastly, pick what type of option you want data for: Long Call or Long Put. *Disclaimer, because Black's Approximation is mostly geared towards stocks, this will only work for stocks. Also, the time variables: time until expiry and time until the ex-dividend dates; don't automatically update. So you will have to update them each day.

Binomial Option Pricing Model A binomial option pricing model is an option pricing model that calculates an option's price using binomial trees. The BOPM method of calculating option prices is different from the Black-Scholes Model because it provides more flexibility in the type of options you want to price. The BOPM, unlike the BS model typically used for European style options, allows you to price options which have the ability to exercise early, such as American or Bermudan options. Although you can use the BOPM for any option style. This specific model allows you to price both American and European vanilla options. The way the BOPM calculates option prices is by: First, dividing up the time until expiry into equal parts called steps. This specific model presented only uses 2 steps. For example, say you have an option with an expiry of 60 days, and your binomial tree has only two steps. Then each step will contain 30 days. Second, the model will project the expected price of the underlying at the end of each step, called a node. The expected price is calculated by using the underlying's volatility and projecting what the price of the underlying would be if it were to rise and fall. This step is repeated until the terminal node, aka the end of the tree, is reached. Third, once the terminal node's expected underlying prices are calculated, their expected option prices must be calculated. Finally, after calculating the terminal option prices, backwards induction must be used to calculate the option prices at the previous nodes, until you reach Node 0, aka the current option price. In order to use this model: 1st. Enter your option's strike price. 2nd. Enter the risk-free-rate of the currency the option is based in. 3rd. Enter the dividend yield of the underlying if it's a stock, or the foreign risk-free-rate if it's an FX option. *For example, if you were trading an AAPL stock option, in the risk-free-rate box mentioned in step 2, you would enter the US risk-free-rate because AAPL options are traded in US dollars. In the dividend yield box mentioned in step 3, you would enter the stock's dividend yield, which for AAPL is 0.82. *If you were, for example, trading an option on the EUR/JPY currency pair, the risk-free-rate mentioned in step 2, would be the Japanese risk-free-rate. Then in the the dividend yield box from step 3, you'd input the Eurozone risk-free-rate. *If you were trading an options on futures contract, the risk-free-rate mentioned in step 2, would be the risk-free-rate for whatever currency the futures contract is denominated in. For example EUR futures are denominated in USD, so you would input the US risk-free-rate. Meanwhile, something like FTSE futures are denominated in GBP, so you would input the British risk-free-rate. As for the dividend yield box mentioned in step 3, for all options on futures, enter 0. 4th. Pick what type of underlying the option is based on: stock, FX, or futures. 5th. Pick the style of option: American or European. 6th. Pick the type of option: Long Call or Long Put. 7th. Input your time until expiry. You can express this in terms of days, hours, and minutes. 8th. Lastly, input your chart time-frame in term of minutes. For example, if you're using the 1 min time-frame enter 1, 4hr time-frame enter 480, daily time-frame enter 1440. *Disclaimer, because this particular model only uses 2 steps, it won't work on stocks with high prices (over $100). If you want to use this on stocks with prices greater than $100, you would need to add more steps to the code, shown below. The model in its current form should work for stocks below $100.

Black-Scholes Options Pricing Model This is an updated version of my "Black-Scholes Model and Greeks for European Options" indicator, that i previously published. I decided to make this updated version open-source, so people can tweak and improve it. The Black-Scholes model is a mathematical model used for pricing options. From this model you can derive the theoretical fair value of an options contract. Additionally, you can derive various risk parameters called Greeks. This indicator includes three types of data: Theoretical Option Price (blue), the Greeks (green), and implied volatility (red); their values are presented in that order. 1) Theoretical Option Price: This first value gives only the theoretical fair value of an option with a given strike based on the Black-Scholes framework. Remember this is a model and does not reflect actual option prices, just the theoretical price based on the Black-Scholes model and its parameters and assumptions. 2)Greeks (all of the Greeks included in this indicator are listed below): a)Delta is the rate of change of the theoretical option price with respect to the change in the underlying's price. This can also be used to approximate the probability of your option expiring in the money. For example, if you have an option with a delta of 0.62, then it has about a 62% chance of expiring in-the-money. This number runs from 0 to 1 for Calls, and 0 to -1 for Puts. b)Gamma is the rate of change of delta with respect to the change in the underlying's price. c)Theta, aka "time decay", is the rate of change in the theoretical option price with respect to the change in time. Theta tells you how much an option will lose its value day by day. d) Vega is the rate of change in the theoretical option price with respect to change in implied volatility . e)Rho is the rate of change in the theoretical option price with respect to change in the risk-free rate. Rho is rarely used because it is the parameter that options are least effected by, it is more useful for longer term options, like LEAPs. f)Vanna is the sensitivity of delta to changes in implied volatility . Vanna is useful for checking the effectiveness of delta-hedged and vega-hedged portfolios. g)Charm, aka "delta decay", is the instantaneous rate of change of delta over time. Charm is useful for monitoring delta-hedged positions. h)Vomma measures the sensitivity of vega to changes in implied volatility . i)Veta measures the rate of change in vega with respect to time. j)Vera measures the rate of change of rho with respect to implied volatility . k)Speed measures the rate of change in gamma with respect to changes in the underlying's price. Speed can be used when evaluating delta-hedged and gamma hedged portfolios. l)Zomma measures the rate of change in gamma with respect to changes in implied volatility . Zomma can be used to evaluate the effectiveness of a gamma-hedged portfolio. m)Color, aka "gamma decay", measures the rate of change of gamma over time. This can also be used to evaluate the effectiveness of a gamma-hedged portfolio. n)Ultima measures the rate of change in vomma with respect to implied volatility . o)Probability of Touch, is not a Greek, but a metric that I included, which tells you the probability of price touching your strike price before expiry. 3) Implied Volatility: This is the market's forecast of future volatility . Implied volatility is directionless, it cannot be used to forecast future direction. All it tells you is the forecast for future volatility. How to use this indicator: 1st. Input the strike price of your option. If you input a strike that is more than 3 standard deviations away from the current price, the model will return a value of n/a. 2nd. Input the current risk-free rate.(Including this is optional, because the risk-free rate is so small, you can just leave this number at zero.) 3rd. Input the time until expiry. You can enter this in terms of days, hours, and minutes. 4th.Input the chart time frame you are using in terms of minutes. For example if you're using the 1min time frame input 1, 4 hr time frame input 480, daily time frame input 1440, etc. 5th. Pick what style of option you want data for, European Vanilla or Binary. 6th. Pick what type of option you want data for, Long Call or Long Put. 7th . Finally, pick which Greek you want displayed from the drop-down list. *Remember the Option price presented, and the Greeks presented, are theoretical in nature, and not based upon actual option prices. Also, remember the Black-Scholes model is just a model based upon various parameters, it is not an actual representation of reality, only a theoretical one. *Note 1. If you choose binary, only data for Long Binary Calls will be presented. All of the Greeks for Long Binary Calls are available, except for rho and vera because they are negligible. *Note 2. Unlike vanilla european options, the delta of a binary option cannot be used to approximate the probability of the option expiring in-the-money. For binary options, if you want to approximate the probability of the binary option expiring in-the-money, use the price. The price of a binary option can be used to approximate its probability of expiring in-the-money. So if a binary option has a price of $40, then it has approximately a 40% chance of expiring in-the-money. *Note 3. As time goes on you will have to update the expiry, this model does not do that automatically. So for example, if you originally have an option with 30 days to expiry, tomorrow you would have to manually update that to 29 days, then the next day manually update the expiry to 28, and so on and so forth. There are various formulas that you can use to calculate the Greeks. I specifically chose the formulations included in this indicator because the Greeks that it presents are the closest to actual options data. I compared the Greeks given by this indicator to brokerage option data on a variety of asset classes from equity index future options to FX options and more. Because the indicator does not use actual option prices, its Greeks do not match the brokerage data exactly, but are close enough. I may try to make future updates that include data for Long Binary Puts, American Options, Asian Options, etc.