# Binary call option delta formulas

First-order Greeks are in blue, second-order Greeks are in binary call option delta formulas, and third-order Greeks are in yellow. The Greeks are vital tools in risk management.

Scholes model are relatively easy to calculate, a desirable property of financial models, and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. The most common of the Greeks are the first order derivatives: delta, vega, theta and rho as well as gamma, a second-order derivative of the value function. For a vanilla option, delta will be a number between 0. For example, if a portfolio of 100 American call options on XYZ each have a delta of 0.

Since the delta of underlying asset is always 1. Delta is close to, but not identical with, the percent moneyness of an option, i. If the value of delta for an option is known, one can calculate the value of the delta of the option of the same strike price, underlying and maturity but opposite right by subtracting 1 from a known call delta or adding 1 to a known put delta. For example, if the delta of a call is 0. 42 then one can compute the delta of the corresponding put at the same strike price by 0. 58 and add 1 to get 0. Vega is the derivative of the option value with respect to the volatility of the underlying asset.

Vega is not the name of any Greek letter. Presumably the name vega was adopted because the Greek letter nu looked like a Latin vee, and vega was derived from vee by analogy with how beta, eta, and theta are pronounced in American English. Vega can be an important Greek to monitor for an option trader, especially in volatile markets, since the value of some option strategies can be particularly sensitive to changes in volatility. The value of an option straddle, for example, is extremely dependent on changes to volatility. By convention, it is usual to divide the result by the number of days in a year, to arrive at the amount an option’s price will drop, in relation to the underlying stock’s price. An exception is a deep in-the-money European put.

The value of an option can be analysed into two parts: the intrinsic value and the time value. 10, whereas the corresponding put would have zero intrinsic value. Except under extreme circumstances, the value of an option is less sensitive to changes in the risk free interest rate than to changes in other parameters. For this reason, rho is the least used of the first-order Greeks.

Rho is typically expressed as the amount of money, per share of the underlying, that the value of the option will gain or lose as the risk free interest rate rises or falls by 1. Obviously, this sensitivity can only be applied to derivative instruments of equity products. Gamma is the second derivative of the value function with respect to the underlying price. Most long options have positive gamma and most short options have negative gamma. Long option delta, underlying price, and gamma. Gamma is important because it corrects for the convexity of value.

When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio’s gamma, as this will ensure that the hedge will be effective over a wider range of underlying price movements. Charm or delta decay measures the instantaneous rate of change of delta over the passage of time. It is often useful to divide this by the number of days per year to arrive at the delta decay per day. This use is fairly accurate when the number of days remaining until option expiration is large. When an option nears expiration, charm itself may change quickly, rendering full day estimates of delta decay inaccurate. It is common practice to divide the mathematical result of veta by 100 times the number of days per year to reduce the value to the percentage change in vega per one day.

Vera can be used to assess the impact of volatility change on rho-hedging. Speed measures the rate of change in Gamma with respect to changes in the underlying price. Speed is the third derivative of the value function with respect to the underlying spot price. Speed can be important to monitor when delta-hedging or gamma-hedging a portfolio.

Zomma measures the rate of change of gamma with respect to changes in volatility. Zomma is the third derivative of the option value, twice to underlying asset price and once to volatility. It is often useful to divide this by the number of days per year to arrive at the change in gamma per day. When an option nears expiration, color itself may change quickly, rendering full day estimates of gamma change inaccurate. Ultima measures the sensitivity of the option vomma with respect to change in volatility. Ultima is a third-order derivative of the option value to volatility.

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