## Understanding Delta Hedging and Greeks

## Understanding Option Greeks & Delta Hedging

The problem with Greeks is that the topic is so out there for most students and non-practitioners. Who would actually care about the second moment (Gamma) or the third (Delta of Gamma) for that matter in the non-trading desk world. Plus by the time you actually get to a level that you can talk intelligently about the subject you are so short of oxygen that there is nobody left to talk to. Try our sample posts below to see how different our approach is to teaching Greek behavior and intuition.

**Understanding Greeks – Introduction****Understanding Greeks – Analyzing Delta & Gamma****Understanding Greeks – The Guide to delta hedging using Monte Carlo Simulation****Dynamic Delta Hedging Simulation – Cash PnL calculations****Using Dynamic Delta Hedging Simulation as a learning tool****Understanding Greeks – Quick Reference Guide to Delta, Gamma, Vega, Theta & Rho**

The 73 pages are primarily a guide to building a delta hedging spreadsheet for European Call and Put Options. Nassim Taleb emphasizes the use of a generator function to build trader intuition and we have included one. We then use the Delta hedging sheet to think about Delta, Gamma, Vega & Rho by asking questions that help reinforce that intuition. For Greeks we include an Excel calculation spread sheets as well as over thirty graphs that analyze Greeks behavior across changing Spot, Strikes, Volatility, Time & Interest rates. To get the most out of the package, we recommend that you follow the book and the templates to build your Excel spreadsheets from scratch.

## Table of Content

**Dynamic Delta Hedging – Monte Carlo Simulation – Greeks – Downloadable Excel File Guide**

This product contains 3 EXCEL files.

#### 1. Option Greeks Calculation & Graphs

- Calculation of the Black Scholes option price for a European Call and a European Put option
- Calculation of Greeks- Delta, Gamma, Vega, Theta & Rho- for a European Call and a European Put option
- Data table that captures the Black Scholes risk adjusted probabilities and option premium across a series of volatilities
- Graphical representation of Black Scholes risk adjusted probabilities and option premium against volatilities
- Data tables that capture the sensitivity of the Greeks against Spot, Strike, Time to maturity, Volatility and the Risk Free Rate respectively
- Graphical representation of the sensitivities of the various Greeks against Spot, Strike, Time to maturity, volatility and risk free rate respectively

#### 2. Dynamic Delta Hedging – Call Option – Monte Carlo Simulation – Cash PnL

- Calculation of a 12-step Monte Carlo simulation model that generates the underlying stock price series
- Calculation of theoretical option values using the Black Scholes call option price formula
- Calculation of call option deltas at each rebalancing interval
- Calculation of a replicating portfolio that consists of a long position in Delta times the stock and a short position in the amount borrowed (net of the option premium received at inception) to fund the initial & subsequent incremental purchases
- Graphical representation of the theoretical option value and the replicating portfolio value over the life of the option
- Calculation of a tracking error for the difference between the value of the replicating portfolio and the theoretical value of the option
- Graphical representation of the tracking error across the life of the option
- Determination of the per period interest and principal portions of the amount borrowed
- Determination of the Gain (Loss) on sale of portions of the stock
- Setting up a Cash Accounting P&L that shows cash inflows from option premium received and strike received in the event the option is exercise and cash outflows from interest and principal repayment on the amount borrowed
- A choice of including of excluding the option premium in determining the amount borrowed at inception. In this case the Principal repaid will equal the gain (loss) if the option is not exercised.
- 100 simulated runs including a graphical depiction of the results showing the Net P&L, Amount borrowed (principal & interest) and Gain/ Losses; and averages across the 100 runs for each of these items

#### 3. Dynamic Delta Hedging – Put Option – Monte Carlo Simulation – Cash PnL

- Calculation of a 12-step Monte Carlo simulation model that generates the underlying stock price series
- Calculation of theoretical option values using the Black Scholes put option price formula
- Calculation of put option deltas at each rebalancing interval
- Calculation of a replicating portfolio that consists of a short sale of Delta times the stock and lending of the initial (net of the option premium received at inception) & subsequent incremental short sales proceeds
- Graphical representation of the theoretical option value and the replicating portfolio value over the life of the option
- Calculation of a tracking error for the difference between the value of the replicating portfolio and the theoretical value of the option
- Graphical representation of the tracking error across the life of the option
- Determination of the per period interest and principal portions of the amount lent
- Determination of the Gain (Loss) on closing of short sale positions
- Setting up a Cash Accounting P&L that shows cash inflows from option premium received, interest earned on amount lent and sales proceeds from short sales and cash outflows from strike paid if the option is exercised
- A choice of including or excluding the option premium in determining the amount borrowed at inception. In this case the sales proceeds from short sales will equal the gain (loss) if the option is not exercised.
- 100 simulated runs including a graphical depiction of the results showing the Net P&L, Proceeds from Short Sales, Interest Earned and Gain/ Losses; and averages across the 100 runs for each of these items