Option Pricing and Hedging Strategies Project

This project investigates option pricing and hedging strategies based on WBA stock. It covers key concepts including Put-Call Parity, Implied Volatility, Delta Hedging, and Volatility Spreads.

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1. Put-Call Parity

Put-Call Parity is a fundamental concept in finance that defines the relationship between European call and put options with the same expiration date and strike price. When violated, it provides arbitrage opportunities.

Key Findings:

• Equation 1: ( S + P > C + Ke^{-rT} )
  Strategy: Buy a call, sell a put, and short the stock or lend money at the risk-free rate.
• Equation 2: ( S + P < C + Ke^{-rT} )
  Strategy: Buy a put, sell a call, buy the stock, or borrow money at the risk-free rate.

Risks to Estimates:

• Risk-Free Rate Assumptions: May not reflect actual rates during the option's lifetime.
• Time to Expiration: Small errors could distort discount factors.
• Transaction Costs: Ignoring commissions, bid-ask spreads, etc., could reduce or eliminate profits.
• Liquidity: Illiquid options might hinder execution.
• Model Limitations: Assumes European-style options in a perfect market.

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2. Implied Volatility

Implied Volatility (IV) represents the market's expectation of future price volatility. It is forward-looking compared to historical volatility, which is backward-looking.

Methodology:

1. Used the Black-Scholes Formula to calculate theoretical option prices.
2. Employed the Goal Seek function to adjust IV until the theoretical price matched the trading price.

Observations:

• Volatility Smile:
  • For put options, IV is lowest at at-the-money (ATM) strikes and increases for in-the-money (ITM) and out-of-the-money (OTM) options.
  • Similar behavior was observed for call options.
  • The downward slope for high IV and upward slope for low IV reflects mean reversion.

Inconsistencies in IV:

Factors like put-call parity violations, market performance, and investor sentiment contributed to variations in IV.

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3. Delta Hedging

Delta Hedging is an options trading strategy to mitigate price change risks in the underlying asset.

Methodology:

1. Calculated Delta (( \Delta )) using ( \Delta_{Call} = N(d1) ).
2. Adjusted the portfolio daily to remain delta-neutral by buying or selling shares as Delta changed.
3. Compared two approaches:
   • Using Implied Volatility
   • Using Historical Volatility (22-day)

Costs Considered:

• Cost of Shares: Price paid or received for hedging.
• Cumulative Costs: Includes interest (4.7%) for borrowed shares.

Key Insights:

• Daily rebalancing ensured minimal risk.
• The two approaches yielded different profit and loss results, highlighting the impact of volatility estimation.

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4. Volatility Trade

A volatility trade was constructed by comparing historical volatility (22-day) with option implied volatility.

Steps:

1. Constructed a volatility spread using both calls and puts.
2. Plotted the payoff function for the spread.
3. Calculated the payoff on the option's expiry.

Observations:

• Call Spreads and Put Spreads were compared for cost-effectiveness.

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Conclusion

This project provided insights into the application of advanced financial concepts for real-world trading scenarios. Key learnings included:

• Identifying arbitrage opportunities through Put-Call Parity.
• Using Implied Volatility to understand market sentiment.
• Implementing dynamic Delta Hedging to minimize risk.
• Constructing effective Volatility Spreads for profitable trades.

By addressing risks and model limitations, the analysis demonstrates the complexity and challenges of financial modeling in live markets.

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