Lecture on Market Implied Volatility

Introduction

• Speaker: Roman
• Topic: Market Implied Volatility
• Goal: Unpack the implied volatility surface in its most basic form

Key Concepts

Implied Volatility

• Implied volatility can get complicated quickly due to its relationship with prices (market price vs. model price).
• Aim: Provide a fundamental understanding of the implied volatility surface.

Option Contracts

• European Style Payoff: Can only exercise at expiration.
• Objective: Determine theoretical value of the option.

The Black-Scholes Framework

Assumptions

• Geometric Brownian Motion: Underlying equity follows this model.
• Equation: ds/s = μdt + σdW_t
  • μ (Mu): Expected return
  • σ (Sigma): Shock to expected return (volatility)

Parameters in Black-Scholes Model

• Risk-free rate (r)
• Strike price (K)
• Spot price (S)
• Time to maturity (T)
• Volatility (σ)

Closed Form Solution

• Mapping: Five parameters (R^5) to price (R^1)
• Parameters Recap:
  • Risk-free rate: Market proxies available
  • Strike price: Pre-defined by contract
  • Spot price: Obtainable from places like Yahoo Finance
  • Time to maturity: Pre-defined
  • Volatility: Not directly available; cannot be looked up

Market Price vs. Theoretical Price

• Market Reality: Only have four parameters + observed option price.
• Difference: Black-Scholes provides theoretical price, market provides actual price.

Example

• Yahoo Finance: Checking Apple’s options gives the market price.
• Illustration: Gather market prices and compare with theoretical prices.
• Observation: Differences highlight the need for implied volatility.

Implied Volatility Explained

• Definition: Volatility that, when input into Black-Scholes, maps to the current market price.
• Implied Volatility Surface: Quoted in terms of Black-Scholes volatility.

Practical Demonstration

Using Python and qfin

• Library: qfin for quantitative finance
• Example Parameters:
  • Spot price (S): 145.38
  • Time to maturity (T): 6/252 (6 days)
  • Strike price (K): 145
  • Risk-free rate (r): ~0.1%
  • Market option price: $3

Coding Process

• Use Black-Scholes class from qfin to input parameters and guess initial volatility.
• Optimize using scipy.optimize.least_squares to minimize the difference between theoretical and observed market prices.
• Result: Implied volatility of ~31.38%

Conclusion

• Summary: The goal is to understand and compute implied volatility.
• Implied Volatility Surface: Represents Black-Scholes volatilities that fit market prices.
• Next Steps: Apply similar methods to other financial models.

Upcoming

• Course Announcement: Introduction to Python heavily oriented towards application, releasing in June.

Actions:

• Watch previous videos on geometric brownian motion.
• Comments/questions/suggestions.
• Stay tuned for the upcoming Python course.