Black Scholes Formula II

Probability and Stochastics for finance

7 Feb 201642:26

EducationalLearning

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TLDRThe video script delves into the intricacies of option pricing using stochastic calculus, focusing on the evolution of option values over time. It discusses the concept of delta hedging, the application of Ito's formula, and the derivation of the Black-Scholes Partial Differential Equation (PDE). The instructor emphasizes the importance of understanding financial markets and the mathematical models that describe them, concluding with an invitation to a finance class for further exploration.

Takeaways

📈 The lecture delves into the dynamics of option pricing, focusing on the continuous evolution of option prices over time.
📚 It introduces the concept of a call option, where the buyer has the right, but not the obligation, to purchase an asset at a specified price within a certain time frame.
🏦 The script discusses the role of delta (Δ) in options trading, representing the number of shares held in a portfolio and its relation to the option's price sensitivity.
💰 The importance of the premium received from selling an option is highlighted, and how it is used to manage the portfolio's exposure to the underlying asset.
📉 The script explains the financial mechanics of a portfolio, including the purchase of stocks and the management of funds in a bank account or fixed deposit.
🧮 Ito's formula is repeatedly used to derive the stochastic differential equations that model the evolution of stock prices and option values.
🔍 The concept of 'delta hedging' is introduced, which is a strategy to manage the risk of price movements by adjusting the number of shares held in a portfolio.
🌐 The script touches on the idea of a complete market, where one can fully hedge their risk, and the implications for option pricing at expiration.
📝 The Black-Scholes Partial Differential Equation (PDE) is presented as a key tool for determining option prices, given certain boundary and terminal conditions.
⚖️ The balance between the evolution of the discounted portfolio value and the discounted option value is emphasized, aiming for their paths to align perfectly to achieve a perfect hedge.
🎓 The lecture concludes with an invitation to a finance class that will further explore these concepts, indicating the depth and complexity of the subject matter.

Q & A

What is the main topic discussed in the script?
-The script discusses the concept of option pricing, specifically focusing on the Black-Scholes model and the stochastic calculus involved in financial mathematics.
What is a call option as mentioned in the script?
-A call option is a financial contract that gives the buyer the right, but not the obligation, to buy a stock at a predetermined price within a specific time period.
What is the role of 'delta' in the context of the script?
-In the script, 'delta' represents the number of shares held in a portfolio at a given time, which is used to calculate the total worth of the stocks and plays a crucial role in the delta hedging formula.

Outlines

00:00

📈 Understanding Option Pricing Dynamics

This paragraph introduces the concept of option pricing, focusing on the seller's perspective and the role of the delta (Δ), which represents the number of shares held in a portfolio at time t. The speaker explains how the portfolio's value is calculated, including the cost of stocks and the remaining money after investment. The paragraph also touches on the idea of instantaneous buying and selling, which is a theoretical construct rather than a market reality, and the importance of the interest earned on the remaining money in the portfolio.

05:00

🔍 Discrete Time Analysis and Differentials in Finance

The speaker delves into the discrete time analysis of financial markets, explaining the concept of a binomial tree and how it can be used to understand price movements. The paragraph discusses the differential of the portfolio value and introduces the stochastic differential form, highlighting the importance of understanding the change in interest price over time. The speaker also mentions the need for a deeper understanding of discrete time analysis before fully grasping the concepts discussed.

10:00

📚 Application of Ito's Formula in Finance

This section discusses the application of Ito's formula to understand the evolution of stock prices and portfolio values. The speaker explains the process of applying Ito's formula to a function of the form f(t, x), where x represents the stock price. The paragraph covers the quadratic variation term and how it relates to the differential of the stock price. The speaker emphasizes the importance of continuous processes in mathematics for understanding financial models.

15:04

📉 The Evolution of Option Value and Hedging Risk

The paragraph explores the evolution of option value over time and the concept of hedging risk. The speaker explains the need to find a continuous function that represents the option value and how it should evolve similarly to the portfolio value for effective risk management. The discussion includes the use of Ito's formula to derive the differential of the option value and the importance of matching the evolution of the portfolio and option values.

20:05

📝 The Black-Scholes Partial Differential Equation

This section introduces the Black-Scholes partial differential equation (PDE), which is key to finding the option price at any given time. The speaker explains the process of equating the differentials of the discounted portfolio value and the discounted option value to derive the Black-Scholes PDE. The paragraph also discusses the terminal condition of the PDE and how the option price at expiration is determined.

25:41

🛡️ Delta Hedging and the Greeks in Options Trading

The speaker introduces the concept of delta hedging, one of the Greeks used in options trading, which helps determine the number of stock shares to buy or sell to hedge an option position. The paragraph explains the delta hedging formula and its practical application in finance. The discussion also covers the process of equating different parts of the equations to derive the delta value and

Mindmap

Keywords

💡Option Price

Option price refers to the premium paid for an option contract, which gives the buyer the right, but not the obligation, to buy or sell an underlying asset at a specified price within a certain period. In the video, the theme revolves around understanding the dynamics of option pricing and how it evolves over time, especially in relation to stock prices and market conditions.

💡Call Option

A call option is a financial contract that gives the buyer the right to purchase an asset at a specified price within a certain time frame. The script discusses the perspective of both the seller and buyer of the call option, emphasizing the buyer's right to purchase the stock at a predetermined price.

💡Delta Hedging

Delta hedging is a strategy used to manage the risk of price changes in an option by adjusting the number of shares held in a portfolio. The script explains that delta (Δ) represents the sensitivity of an option's price to changes in the underlying asset's price, and the delta hedging formula is crucial for determining the appropriate number of shares to hold.

💡Portfolio Value

Portfolio value refers to the total worth of a collection of financial assets, such as stocks and bonds. The video script discusses how the value of a portfolio, which includes stocks and a money market account, changes over time and how it is influenced by the buying and selling of stocks.

💡Interest Rate

The interest rate in the context of the video is the percentage at which money can be borrowed or lent, impacting the earnings from a money market account or fixed deposit. The script describes how the interest rate affects the growth of the money left in the bank after purchasing stocks.

💡Ito's Formula

Ito's formula is a result in stochastic calculus used to find the differential of a function of a stochastic process. The script frequently refers to Ito's formula in the process of deriving the differential equations that describe the evolution of stock prices and option values.

💡Stochastic Differential Equation

A stochastic differential equation is an equation in which one or more of the terms is a stochastic process, incorporating a level of randomness. The video script uses these equations to model the changes in stock prices and option values over time, incorporating elements of uncertainty.

💡Black-Scholes Equation

The Black-Scholes equation, also known as the Black-Scholes PDE, is a partial differential equation that describes the price of a financial derivative, such as an option. The script culminates in the introduction of this equation, which is key to understanding how option prices evolve and are calculated.

💡Risk Neutral

Risk neutrality is a state where investors are indifferent to risk, valuing all assets based on expected return alone. The script mentions the concept of a risk-neutral measure, which is used in financial mathematics to simplify the pricing of derivatives by assuming all investors are risk-neutral.

💡Girsanov's Theorem

Girsanov's theorem is a result in the theory of stochastic processes, which allows for the change of measure between two probability measures, often used to transform a real-world probability measure into a risk-neutral measure. The script suggests that this theorem can be used to compute option prices without directly solving the Black-Scholes PDE.

💡Martingale

In probability theory and finance, a martingale is a model of a fair game where the expected value of the next step is always the same, regardless of the previous steps' outcomes. The script mentions that the discounted stock price is a martingale, which is a key concept in understanding the behavior of stock prices in a risk-neutral world.

Highlights

Introduction to option pricing and the role of the seller and buyer in a call option contract.

Explanation of delta (Δ) as the number of shares held in

Transcripts

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Related Tags

Option Pricing Stochastic Calculus Portfolio Management Financial Modeling Delta Hedging Black-Scholes Risk Management Ito's Formula Interest Rate Stock Market