Black Scholes Option Pricing Model w/ Greeks [Loxx]

Sep 30, 2022

The Black Scholes Merton model
If you are new to options I strongly advise you to profit from Robert Shiller's lecture on same. It combines practical market insights with a strong authoritative grasp of key models in option theory. He explains many of the areas covered below and in the following pages with a lot intuition and relatable anecdotage. We start here with Black Scholes Merton which is probably the most popular option pricing framework, due largely to its simplicity and ease in terms of implementation. The closed-form solution is efficient in terms of speed and always compares favorably relative to any numerical technique. The Black–Scholes–Merton model is a mathematical go-to model for estimating the value of European calls and puts. In the early 1970’s, Myron Scholes, and Fisher Black made an important breakthrough in the pricing of complex financial instruments. Robert Merton simultaneously was working on the same problem and applied the term Black-Scholes model to describe new generation of pricing. The Black Scholes (1973) contribution developed insights originally proposed by Bachelier 70 years before. In 1997, Myron Scholes and Robert Merton received the Nobel Prize for Economics. Tragically, Fisher Black died in 1995. The Black–Scholes formula presents a theoretical estimate (or model estimate) of the price of European-style options independently of the risk of the underlying security. Future payoffs from options can be discounted using the risk-neutral rate. Earlier academic work on options (e.g., Malkiel and Quandt 1968, 1969) had contemplated using either empirical, econometric analyses or elaborate theoretical models that possessed parameters whose values could not be calibrated directly. In contrast, Black, Scholes, and Merton’s parameters were at their core simple and did not involve references to utility or to the shifting risk appetite of investors. Below, we present a standard type formula, where: c = Call option value, p = Put option value, S=Current stock (or other underlying) price, K or X=Strike price, r=Risk-free interest rate, q = dividend yield, T=Time to maturity and N denotes taking the normal cumulative probability. b = (r - q) = cost of carry. (via VinegarHill-Financelab)

Things to know

• This can only be used on the daily timeframe
  
• You must select the option type and the greeks you wish to show
  
• This indicator is a work in process, functions may be updated in the future. I will also be adding additional greeks as I code them or they become available in finance literature. This indictor contains 18 greeks. Many more will be added later.
  

Inputs

• Spot price: select from 33 different types of price inputs
  
• Calculation Steps: how many iterations to be used in the BS 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 BS 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
  

The Black Scholes Greeks
The Option Greek formulae express the change in the option price with respect to a parameter change taking as fixed all the other inputs. (Haug explores multiple parameter changes at once.) One significant use of Greek measures is to calibrate risk exposure. A market-making financial institution with a portfolio of options, for instance, would want a snap shot of its exposure to asset price, interest rates, dividend fluctuations. It would try to establish impacts of volatility and time decay. In the formulae below, the Greeks merely evaluate change to only one input at a time. In reality, we might expect a conflagration of changes in interest rates and stock prices etc. (via VigengarHill-Financelab)

First-order Greeks
Delta: Delta measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price. Delta is the first derivative of the value

Vega: Vegameasures sensitivity to volatility. Vega is the derivative of the option value with respect to the volatility of the underlying asset.

Theta: Theta measures the sensitivity of the value of the derivative to the passage of time (see Option time value): the "time decay."

Rho: Rho measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk free interest rate (for the relevant outstanding term).

Lambda: Lambda, Omega, or elasticity is the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing.

Epsilon: Epsilon, also known as psi, is the percentage change in option value per percentage change in the underlying dividend yield, a measure of the dividend risk. The dividend yield impact is in practice determined using a 10% increase in those yields. Obviously, this sensitivity can only be applied to derivative instruments of equity products.

Second-order Greeks
Gamma: Measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative of the value function with respect to the underlying price.

Vanna: Vanna, also referred to as DvegaDspot and DdeltaDvol, is a second order derivative of the option value, once to the underlying spot price and once to volatility. It is mathematically equivalent to DdeltaDvol, the sensitivity of the option delta with respect to change in volatility; or alternatively, the partial of vega with respect to the underlying instrument's price. Vanna can be a useful sensitivity to monitor when maintaining a delta- or vega-hedged portfolio as vanna will help the trader to anticipate changes to the effectiveness of a delta-hedge as volatility changes or the effectiveness of a vega-hedge against change in the underlying spot price.

Charm: Charm or delta decay measures the instantaneous rate of change of delta over the passage of time.

Vomma: Vomma, volga, vega convexity, or DvegaDvol measures second order sensitivity to volatility. Vomma is the second derivative of the option value with respect to the volatility, or, stated another way, vomma measures the rate of change to vega as volatility changes.

Veta: Veta or DvegaDtime measures the rate of change in the vega with respect to the passage of time. Veta is the second derivative of the value function; once to volatility and once to time.

Vera: Vera (sometimes rhova) measures the rate of change in rho with respect to volatility. Vera is the second derivative of the value function; once to volatility and once to interest rate.

Third-order Greeks
Speed: Speed measures the rate of change in Gamma with respect to changes in the underlying price.

Zomma: Zomma measures the rate of change of gamma with respect to changes in volatility.

Color: Color, gamma decay or DgammaDtime measures the rate of change of gamma over the passage of time.

Ultima: Ultima measures the sensitivity of the option vomma with respect to change in volatility.

Dual Delta: Dual Delta determines how the option price changes in relation to the change in the option strike price; it is the first derivative of the option price relative to the option strike price

Dual Gamma: Dual Gamma determines by how much the coefficient will changedual delta when the option strike price changes; it is the second derivative of the option price relative to the option strike price.

Related Indicators
Cox-Ross-Rubinstein Binomial Tree Options Pricing Model [Loxx]

Implied Volatility Estimator using Black Scholes [Loxx]

Boyle Trinomial Options Pricing Model [Loxx]

Sep 30, 2022

Release Notes

Updated formatting of formulas

Oct 2, 2022

Release Notes

Error corrections with dividend calculation; updated formulas.

Oct 8, 2022

Release Notes

Small error correction. in requesting Risk free rate data.