Blacksholes — Indicators and Strategies

Black-Scholes option price model & delta hedge strategy Black-Scholes Option Pricing Model Strategy The strategy is based on the Black-Scholes option pricing model and allows the calculation of option prices, various option metrics (the Greeks), and the creation of synthetic positions through delta hedging. ATTENTION! Trading derivative financial instruments involves high risks. The author of the strategy is not responsible for your financial results! The strategy is not self-sufficient for generating profit! It is created exclusively for constructing a synthetic derivative financial instrument. Also, there might be errors in the script, so use it at your own risk! I would appreciate it if you point out any mistakes in the comments! I would be even more grateful if you send the corrected code! Application Scope This strategy can be used for delta hedging short positions in sold options. For example, suppose you sold a call option on Bitcoin on the Deribit exchange with a strike price of $60,000 and an expiration date of September 27, 2024. Using this script, you can create a delta hedge to protect against the risk of loss in the option position if the price of Bitcoin rises. Another example: Suppose you use staking of altcoins in your strategies, for which options are not available. By using this strategy, you can hedge the risk of a price drop (Put option). In this case, you won't lose money if the underlying asset price increases, unlike with a short futures position. Another example: You received an airdrop, but your tokens will not be fully unlocked soon. Using this script, you can fully hedge your position and preserve their dollar value by the time the tokens are fully unlocked. And you won't fear the underlying asset price increasing, as the loss in the event of a price rise is limited to the option premium you will pay if you rebalance the portfolio. Of course, this script can also be used for simple directional trading of momentum and mean reversion strategies! Key Features and Input Parameters 1. Option settings: - Style of option: "European vanilla", "Binary", "Asian geometric". - Type of option: "Call" (bet on the rise) or "Put" (bet on the fall). - Strike price: the option contract price. - Expiration: the expiry date and time of the option contract. 2. Market statistic settings: - Type of price source: open, high, low, close, hl2, hlc3, ohlc4, hlcc4 (using hl2, hlc3, ohlc4, hlcc4 allows smoothing the price in more volatile series). - Risk-free return symbol: the risk-free rate for the market where the underlying asset is traded. For the cryptocurrency market, the return on the funding rate arbitrage strategy is accepted (a special function is written for its calculation based on the Premium Price). - Volatility calculation model: realized (standard deviation over a moving period), implied (e.g., DVOL or VIX), or custom (you can specify a specific number in the field below). For the cryptocurrency market, the calculation of implied volatility is implemented based on the product of the realized volatility ratio of the considered asset and Bitcoin to the Bitcoin implied volatility index. - User implied volatility: fixed implied volatility (used if "Custom" is selected in the "Volatility Calculation Method"). 3. Display settings: - Choose metric: what to display on the indicator scale – the price of the underlying asset, the option price, volatility, or Greeks (all are available). - Measure: bps (basis points), percent. This parameter allows choosing the unit of measurement for the displayed metric (for all except the Greeks). 4. Trading settings: - Hedge model: None (do not trade, default), Simple (just open a position for the full volume when the strike price is crossed), Synthetic option (creating a synthetic option based on the Black-Scholes model). - Position side: Long, Short. - Position size: the number of units of the underlying asset needed to create the option. - Strategy start time: the moment in time after which the strategy will start working to create a synthetic option. - Delta hedge interval: the interval in minutes for rebalancing the portfolio. For example, a value of 5 corresponds to rebalancing the portfolio every 5 minutes. Post scriptum My strategy based on the SegaRKO model. Many thanks to the author! Unfortunately, I don't have enough reputation points to include a link to the author in the description. You can find the original model via the link in the code, as well as through the search indicators on the charts by entering the name: "Black-Scholes Option Pricing Model". I have significantly improved the model: the calculation of volatility, risk-free rate and time value of the option have been reworked. The code performance has also been significantly optimized. And the most significant change is the execution, with which you can now trade using this script.

Capital Asset Pricing Model (CAPM) [Loxx]Capital Asset Pricing Model (CAPM) demonstrates how to calculate the Cost of Equity for an underlying asset using Pine Script. This script will only work on the monthly timeframe. While you can change the default inputs, you should study what CAPM is and how this works before doing so. This indicator pulls various types of data from SPY from various timeframes to calculate risk-free rates, market premiums, and log returns. Alpha and Beta are computed using the regression between underlying asset and SPY. This indicator only calculates on the most recent data. If you wish to change this, you'll have to save the script and make adjustments. A few examples where CAPM is used: Used as the mu factor Geometric Brownian Motion models for options pricing and forecasting price ranges and decay Calculating the Weighted Average Cost of Capital Asset pricing Efficient frontier Risk and diversification Security market line Discounted Cashflow Analysis Investment bankers use CAPM to value deals Account firms use CAPM to verify asset prices and assumptions Real estate firms use variations of CAPM to value properties ... and more Details of the calculations used here Rm is calculated using yearly simple returns data from SPY, typically this is just hard coded as 10%. Rf is pulled from US 10 year bond yields Beta and Alpha are pulled form monthly returns data of the asset and SPY In the past, typically this data is purchased from investments banks whose research arms produce values for beta, alpha, risk free rate, and risk premiums. In 2022 ,you can find free estimates for each parameter but these values might not reflect the most current data or research. History The CAPM was introduced by Jack Treynor (1961, 1962), William F. Sharpe (1964), John Lintner (1965) and Jan Mossin (1966) independently, building on the earlier work of Harry Markowitz on diversification and modern portfolio theory. Sharpe, Markowitz and Merton Miller jointly received the 1990 Nobel Memorial Prize in Economics for this contribution to the field of financial economics. Fischer Black (1972) developed another version of CAPM, called Black CAPM or zero-beta CAPM, that does not assume the existence of a riskless asset. This version was more robust against empirical testing and was influential in the widespread adoption of the CAPM. Usage The CAPM is used to calculate the amount of return that investors need to realize to compensate for a particular level of risk. It subtracts the risk-free rate from the expected rate and weighs it with a factor – beta – to get the risk premium. It then adds the risk premium to the risk-free rate of return to get the rate of return an investor expects as compensation for the risk. The CAPM formula is expressed as follows: r = Rf + beta (Rm – Rf) + Alpha Therefore, Alpha = R – Rf – beta (Rm-Rf) Where: R represents the portfolio return Rf represents the risk-free rate of return Beta represents the systematic risk of a portfolio Rm represents the market return, per a benchmark For example, assuming that the actual return of the fund is 30, the risk-free rate is 8%, beta is 1.1, and the benchmark index return is 20%, alpha is calculated as: Alpha = (0.30-0.08) – 1.1 (0.20-0.08) = 0.088 or 8.8% The result shows that the investment in this example outperformed the benchmark index by 8.8%. The alpha of a portfolio is the excess return it produces compared to a benchmark index. Investors in mutual funds or ETFs often look for a fund with a high alpha in hopes of getting a superior return on investment (ROI). The alpha ratio is often used along with the beta coefficient, which is a measure of the volatility of an investment. The two ratios are both used in the Capital Assets Pricing Model (CAPM) to analyze a portfolio of investments and assess its theoretical performance. To see CAPM in action in terms of calculate WACC, see here for an example: finbox.com Further reading en.wikipedia.org

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. You can adjust the text size to fit the screen

Implied Volatility Estimator using Black Scholes [Loxx]Implied Volatility Estimator using Black Scholes derives a estimation of implied volatility using the Black Scholes options pricing model. The Bisection algorithm is used for our purposes here. This includes the ability to adjust for dividends. Implied Volatility The implied volatility (IV) of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model (such as Black–Scholes), will return a theoretical value equal to the current market price of that option. The VIX , in contrast, is a model-free estimate of Implied Volatility. The latter is viewed as being important because it represents a measure of risk for the underlying asset. Elevated Implied Volatility suggests that risks to underlying are also elevated. Ordinarily, to estimate implied volatility we rely upon Black-Scholes (1973). This implies that we are prepared to accept the assumptions of Black Scholes (1973). Inputs Spot price: select from 33 different types of price inputs Strike Price: the strike price of the option you're wishing to model Market Price: this is the market price of the option; choose, last, bid, or ask to see different results Historical Volatility Period: the input period for historical volatility ; historical volatility isn't used in the Bisection algo, this is to serve as a comparison, even though historical volatility is from price movement of the underlying asset where as implied volatility is the volatility of the option 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 *** the algorithm inputs for low and high aren't to be changed unless you're working through the mathematics of how Bisection works. Included Option pricing panel Loxx's Expanded Source Types Related Indicators Cox-Ross-Rubinstein Binomial Tree Options Pricing Model