# Delta Hedging and Greeks

### Study Guide + EXCEL Value Pack

This course is based on a four part executive MBA course on derivative pricing and risk management that Jawwad taught in Dubai and Singapore at the SP Jain School of Management.

It includes one PDF study guide with six Excel files. Themes covered understanding the difference between N(d1) and N(d2), the Black Scholes model, Greeks (Delta, Gamma, Vega, Theta and Rho), Hedging higher order Greeks using Solver, Volatility Surfaces, Delta Hedging and P&L distributions for a vanilla call and put options.

The value pack includes the content of the Hedging higher order Greeks and Delta Hedging study guides.

Primary lessons have been compressed into short bite sized pieces. There are some equations, but much time is not spent on them or their derivations. Ground rules, behaviour and intuition however are discussed. For instance, a trader is more likely to ask you about how Gamma is going to behave under a given scenario and how that is different from Vega’s reaction rather than ask for the formula. He or she is more interested in how you think, your ability to grasp a concept and your intuition than your memory.

The methodology followed, therefore, is to present EXCEL models with the text – read the chapter, dissect the model, load up on intuition. Repeat. Deriving numbers and plotting graphs ensures that the intuition behind how Greeks work is retained.

This study guide walks through option price sensitivities and Greeks behaviour and plots the same in EXCEL. It presents a dynamic Delta hedging simulation and uses a Cash P&L simulation to dissect the minor Greeks.

The course includes 6 EXCEL files that demonstrate the calculation of Greeks and Delta Hedging for European call & put options.

The “**Greeks**” EXCEL file contains the:

- 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

The “**Delta Hedging – Call Option**” EXCEL file presents the:

- 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

The “**Delta Hedging – Put Option**” EXCEL file outlines the:

- 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

The three other files include Hedging Higher Order Greeks (Gamma and Vega) using Solver. Building a volatility surface for NVIDIA using Dupire’s formula and Understanding N(d1) and N(d2) a simulation based approach to understanding the difference between the two probabilities.

After taking this course you will be able to:

- Calculate the prices of European call and put Options using the Black Scholes formula
- Calculate the Greeks for European call and put options
- Graph Black Scholes risk adjusted probabilities and option premium against varying volatilities
- Calculate and graph the sensitivity of Greeks to changing Spot, Strike, Time to maturity, Volatility and the Risk Free Rate respectively
- Understand the intuition behind how Greeks work under various scenarios
- Construct a simulation model to generate the underlying stock price
- Build a delta hedging model for European call and put options
- Create replicating portfolios for European call and put options
- Calculate tracking errors between replicating portfolio and theoretical option values
- Set up a Cash Accounting P&L for the option and the hedging portfolio
- Present the results of the simulation runs for Net P&L, Amount borrowed/Proceeds from Short sales, Gains and Losses, etc

Some familiarity with basic derivative products and their pricing, Monte Carlo simulation, basic mathematics, statistics, probability and EXCEL.

The course is aimed at professionals who deal with pricing, valuation and risk issues related to structured fixed income and foreign exchange transactions.