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Project 4: Advanced Options Pricing and Risk Management

Learning Objectives:

· Apply the Monte Carlo Method and Binomial Lattice models to price call and put options.

· Compare and assess the accuracy of different pricing models relative to real market data.

· Understand the assumptions and limitations of volatility models and their impact on options pricing.

· Develop and analyze Delta Neutral Portfolios to minimize market risks using basic option strategies.

Motivation: Options pricing and risk management are crucial skills in financial markets. This project combines theoretical models with practical application to simulate real-world pricing scenarios for options. By implementing various pricing models, including the Monte Carlo and Binomial Lattice methods, and constructing risk-neutral portfolios, students will learn how to effectively manage risk and understand the factors influencing option prices. This hands-on approach will reinforce key concepts in financial modeling and help students become more proficient in interpreting market data.

Part 1: Pricing Options Using Simulations (250 points)

In this section, you will create different pricing models for a non-dividend-paying stock and explore the use of Monte Carlo simulations and Binomial Lattices.

1. Monte Carlo Simulations:

· Explanation: The Monte Carlo method simulates the possible future price paths of an underlying asset by generating random price movements based on its volatility and time to maturity. It then averages the outcomes to determine the option price.

· Tasks:

o Run 100 simulations to price a Call option.

o Run 100 simulations to price a Put option.

· Goal: Understand how random price movements affect the value of an option, and observe how averaging across many simulations improves the estimate of the option price.

2. Binomial Lattice Models:

· Explanation: The Binomial Lattice model divides the option’s time to expiration into several periods, where the price of the underlying asset moves up or down at each step. By working backward through the lattice, we can compute the option's value at each node and ultimately arrive at its current price.

· Tasks:

o Build a 25-period lattice for a Call option.

o Build a 25-period lattice for a Put option.

· Goal: Learn how the lattice model approximates option prices by simulating price changes in discrete steps over time.

3. Compare Model Prices with Market Prices:

· Use Drexel University’s Black-Scholes Merton calculator to determine the theoretical price of the Call and Put options. https://www.math.drexel.edu/~pg/fin/VanillaCalculator.html

· Compare your results from both the Monte Carlo and Binomial Lattice models to the Black-Scholes price and the actual market prices (from Yahoo Finance).

· Calculate the error between the simulated prices and the market prices using the absolute value of the differences.

Explanation: This step will help you assess the accuracy of each model. The Black-Scholes Merton model provides a widely accepted theoretical price, while market prices reflect real-world conditions. By comparing these with your simulated results, you can evaluate the strengths and limitations of each model.

4. Extended Monte Carlo Simulations:

· Run additional simulations for Call options using:

o 250 simulations

o 500 simulations

· Explanation: By increasing the number of simulations, you improve the accuracy of the Monte Carlo model. This task will help you observe how the number of simulations affects the stability and precision of the option price estimates.

Goal: Compare the results of different simulation runs to see how increasing the number of simulations impacts pricing accuracy.

Part 2: Delta Neutral Portfolio Construction (250 points)

In this section, you will construct Delta Neutral Portfolios using different options strategies.

1. Delta Neutral Portfolio:

· Explanation: A Delta Neutral Portfolio involves balancing the quantities of options and the underlying asset so that the portfolio’s value is insensitive to small changes in the price of the underlying asset. Delta measures how much an option's price is expected to move per $1 change in the price of the underlying asset. By creating a portfolio where the Deltas of the long and short positions offset each other, you can reduce risk.

· Tasks:

o Construct a Delta Neutral Portfolio for:

§ Long Call

§ Short Call

§ Long Put

§ Short Put

· Assumptions:

o Gamma = 0 (this means there’s no curvature, or non-linearity, in how Delta changes with price).

o All other Greeks (e.g., Theta, Vega) remain constant.

Goal: Learn how to hedge against price changes by neutralizing the Delta of your positions, an essential risk management technique.

Part 3: Readme and Reflections (250 points)

This section focuses on reflecting on your models and assumptions. You will answer questions that encourage critical thinking about the accuracy of models, assumptions about volatility, and the Greeks’ influence on option prices.

1. Volatility Assumptions:

· Explanation: Volatility refers to the degree of variation in the price of the underlying asset. In this project, you are making assumptions about future volatility based on historical data or implied volatility from the options market.

· Question: What assumptions are you making about volatility in your models? Are they realistic?

2. Improving Binomial Lattice Accuracy:

· Explanation: A Binomial Lattice with more time periods will provide a more refined approximation of the underlying asset’s possible future prices.

· Question: How can you adjust the number of periods in the Binomial Lattice to increase accuracy?

3. Improving Monte Carlo Accuracy:

· Explanation: More simulations in the Monte Carlo model will lead to a more accurate estimate of the option price by reducing the error due to randomness.

· Question: How does increasing the number of simulations in the Monte Carlo Method improve accuracy?

4. Impact of Increasing Simulations:

· Question: What would happen to call prices if you increased the number of simulations from 100 to 10,000, 100,000, or even 10 million simulations? Would you expect the call prices to stabilize or fluctuate more?

5. Model Accuracy Comparison:

· Question: Compare the accuracy of the Binomial Lattice, Monte Carlo, and Black-Scholes Merton models. Which model provided the most accurate results relative to the actual market price, and why?

6. Factors Affecting Call Prices:

· Explanation: The Greeks help you understand how different factors impact the price of a Call option.

· Question: How do the following factors correlate with the price of a Call option? (Hint: refer to the Greeks for your explanations)

o Stock Price

o Interest Rates

o Time to Maturity

o Volatility

7. Factors Affecting Put Prices:

· Explanation: The same Greeks also impact Put options, but in different ways compared to Calls.

· Question: How do the following factors correlate with the price of a Put option? (Hint: refer to the Greeks for your explanations)

o Stock Price

o Interest Rates

o Time to Maturity

o Volatility

8. Straddle Strategy and Volatility:

· Explanation: A long straddle involves buying both a Call and a Put option with the same strike price and expiration date. This strategy benefits from large price movements in either direction, and thus, is sensitive to volatility.

· Question: Reflect on your work with the straddle strategy from pset 3. Why does volatility have a positive correlation with the price of options, particularly in straddle strategies? Comment on how the straddle’s legs (Call and Put) are impacted by volatility.