Greeks in Option Trading

In financial mathematics, "Greeks" are numerical measures that describe how the price of derivatives, such as options, change in response to changes in the underlying parameters. These measures are crucial for risk management and trading strategies. The main Greeks are Delta, Gamma, Theta, Vega, and Rho. Here are several different ways to calculate these Greeks: 1. Analytical Formulas For many standard options, like European call and put options, Greeks can be directly calculated using closed-form formulas derived from the Black-Scholes model or other models like the Bachelier model. Each Greek has its own specific formula: - Delta (Δ) measures sensitivity to changes in the underlying asset's price. - Gamma (Γ) measures the rate of change in Delta with respect to changes in the underlying price. - Theta (Θ) measures sensitivity to time decay. - Vega (ν) measures sensitivity to changes in the volatility of the underlying. - Rho (ρ) measures sensitivity to changes in the risk-free interest rate. 2. Numerical Methods When analytical formulas are not available (such as for exotic options or American-style options), numerical methods can be used: a) Finite Difference Methods: This involves approximating the derivatives by making small changes to the input parameters and observing how the output (option price) changes. This method can be used to estimate all Greeks. b) Monte Carlo Simulation: Used for estimating Greeks of complex derivatives by simulating the underlying asset's price multiple times and calculating the average effect on the option's price due to small changes in inputs. 3. Lattice Models Options can also be priced using lattice-based models like the Binomial or Trinomial tree models, which can naturally provide estimates for Greeks. These models build a discrete-time and discrete-state tree of possible future stock prices and calculate the option prices backwards from expiration to the present. Greeks are estimated by making incremental changes in the model inputs (like underlying price, volatility, etc.) and recalculating the option price. 4. The Likelihood Ratio Method (Monte Carlo) This is a sophisticated technique used in Monte Carlo simulations, which involves modifying the probability measure and applying the likelihood ratio as a weight during the simulation. This method is particularly useful for calculating sensitivities like Vega and Rho, where direct differentiation might not be straightforward.

Good morning, all,    Let us discuss about the option Greeks in simple words with examples.   ➡️ Delta (∆): Ratio of change in the price of an option to the change in the price of the stock. Delta of call is positive and put is negative. Delta of 0.7 implies that when the price of the stock increases by say 1 rupee, the price of the call option increases by 70% of this amount i.e., Re 0.7. Delta of a call has an upper bound of 1, as an option can never gain or lose value faster than the underlying asset. An option cannot move in a direction opposite to the asset and hence the lower bound of a call is zero. Put option has delta ranging between -1 and 0.   ➡️Gamma (Γ): Rate of change of the asset’s delta with respect to the price of the underlying asset. It is the second partial derivative of the portfolio with respect to asset price. If Gamma is 0.06, it implies that when stock price changes by Rs 1, delta of the option changes by 0.06 ×1. Gamma of put and call will always be equal.   ➡️Theta (ɵ): Rate of change of the value of the option with respect to the passage of time. This is the time decay of the portfolio.   ➡️Vega (υ): Rate of change of the value of the option with respect to the change in volatility of the underlying asset.   ➡️Rho (ρ): Rate of change of the value of the option with respect to the change in interest rate. The Rho of call option is positive and put option is negative.     ➡️ Stock is currently trading at Rs 200. Greeks of the option contracts on the stock are             as follows:             ∆ = 0.4763, ɵ = -34.65, Γ= 0.054, υ= 16, ρ = 11.45                         (Currently call option is trading at 5.598) ➡️ If the stock moves to Rs 201, what is the predicted new option price? What is the price if stock price comes down to 199? If S changes to 201, dS=1. Thus, the change in the option price predicted by the delta is ∆.dS=0.4763×1=0.4763 Since the original option price is 5.598, predicted new price is C=5.598+0.4763= 6.0743   If S changes to 199, dS=-1. Thus, the change in the option price predicted by the delta is ∆.-dS=0.4763×-1=-0.4763 Since the original option price is 5.598, predicted new price is C=5.598-0.4763= 5.1217   ➡️ If stock jumps to Rs 220, what is the new option price predicted by delta alone? What is the new option price predicted by delta and gamma? If there is a large If there is a large increase in the price, we must use a curvature correction, since using the delta alone will lead to large errors. In this problem, dS=20. If we used the delta alone, we would predict a change in the call price of ∆.dS=0.4763 × 20 =9.526? So the predicted new price on this basis would be 5.598+9.526 =15.124 Combined effect of delta and gamma = ∆.dS+1/2 Γ.dS2 = 0.4763 × 20 +1/2×0.054×20^2= 9.526 + 10.8 =20.326   New price is 5.598+20.326  = 25.924   #options #derivatives #financecareers #investmentbanking

💡 Options Greeks: Through the Lens of Implied Distribution Building on the derivative price representation as the integration of a payoff and an implied probability distribution, I decided to group the Greeks in a way that directly reflects this structure. Since the payoff is fixed (a contractual function), all option sensitivities must, in some way, aim to describe the behaviour of the implied distribution. And when we think in those terms, some core dynamics matter, e.g.: 1️⃣ Shifts along the underlying price axis 2️⃣ Changes in distribution shape (volatility- or time-driven) 3️⃣ Combined effects (cross-sensitivities) I’ve captured this framework in the Greeks Cheat Sheet below for most common Greeks👇. While much of it follows standard intuition, a few groupings may seem unconventional: ▪️ Rho sits in the “shift” category - since the forward price (and thus the distribution) is calibrated using both interest rate and spot. ▪️ Theta is grouped with Vega, as time decay reduces the effect of volatility, reshaping the distribution. ▪️ Vanna and Charm, often underappreciated, represent important cross-effects between variables. 🎯 Bottom line: understanding the behaviour of the implied distribution is key to managing risk whether in markets or broader applications. 🌍 In climate risk, for instance, we're still in the early stages defining a set of relevant probability distributions, and identifying their shifters and shapers..