Black-Scholes PDE

Introduction

The Black-Scholes Partial Differential Equation (PDE) is a fundamental concept in computational finance and modeling. It provides a mathematical framework for pricing options and understanding the dynamics of financial markets. In this topic, we will explore the key concepts and principles associated with the Black-Scholes PDE, including European calls and puts, put-call parity, and its applications in real-world scenarios.

Key Concepts and Principles

Black-Scholes PDE

The Black-Scholes PDE is a partial differential equation that describes the evolution of the price of a financial derivative, such as an option, over time. It was developed by economists Fischer Black and Myron Scholes in 1973 and has become a cornerstone of option pricing theory.

Definition and Derivation

The Black-Scholes PDE is derived under certain assumptions, including:

1. The underlying asset follows a geometric Brownian motion.
2. The market is efficient and there are no arbitrage opportunities.
3. There are no transaction costs or taxes.

The Black-Scholes PDE is given by:

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$

where:

• $$V$$ is the price of the option as a function of time $$t$$ and the underlying asset price $$S$$.
• $$\sigma$$ is the volatility of the underlying asset.
• $$r$$ is the risk-free interest rate.

Assumptions and Limitations

The Black-Scholes model makes several assumptions that may not hold in real-world scenarios. These assumptions include:

1. Constant volatility: The model assumes that the volatility of the underlying asset remains constant over time. In reality, volatility can change significantly.
2. Efficient markets: The model assumes that markets are efficient and there are no arbitrage opportunities. In practice, markets may not always be perfectly efficient.
3. No transaction costs: The model ignores transaction costs and taxes, which can have a significant impact on option prices.

European Calls and Puts

European call and put options are two common types of financial derivatives. Understanding their characteristics and pricing is essential in the context of the Black-Scholes PDE.

Definition and Characteristics

A European call option gives the holder the right, but not the obligation, to buy an underlying asset at a predetermined price (strike price) on or before the expiration date. A European put option gives the holder the right, but not the obligation, to sell an underlying asset at a predetermined price (strike price) on or before the expiration date.

European options differ from American options in that they can only be exercised at expiration. This simplifies the pricing process, as there is no need to consider early exercise.

Pricing European Options using the Black-Scholes PDE

The Black-Scholes PDE provides a mathematical formula for pricing European options. The formula is derived by assuming a risk-neutral probability distribution for the underlying asset price at expiration.

For a European call option, the Black-Scholes formula is given by:

$$C = S_0e^{-qt}N(d_1) - Xe^{-rt}N(d_2)$$

where:

• $$C$$ is the price of the call option.
• $$S_0$$ is the current price of the underlying asset.
• $$X$$ is the strike price of the option.
• $$t$$ is the time to expiration in years.
• $$r$$ is the risk-free interest rate.
• $$q$$ is the continuous dividend yield of the underlying asset.
• $$N()$$ is the cumulative standard normal distribution function.
• $$d_1$$ and $$d_2$$ are calculated as follows:

$$d_1 = \frac{\ln\left(\frac{S_0}{X}\right) + (r - q + \frac{\sigma^2}{2})t}{\sigma\sqrt{t}}$$

$$d_2 = d_1 - \sigma\sqrt{t}$$

The formula for a European put option is similar, with the following modification:

$$P = Xe^{-rt}N(-d_2) - S_0e^{-qt}N(-d_1)$$

Put-Call Parity

Put-call parity is a relationship between the prices of European put and call options with the same strike price and expiration date. It states that the difference between the prices of a call option and a put option is equal to the difference between the current price of the underlying asset and the present value of the strike price.

Mathematically, put-call parity can be expressed as:

$$C - P = S_0 - Xe^{-rt}$$

Put-call parity is a useful concept in option pricing, as it allows for the calculation of the price of one option given the price of another option.

Discontinuous Payoffs - Binary and Digital Options

In addition to European options, the Black-Scholes PDE can also be used to price options with discontinuous payoffs, such as binary options and digital options.

Binary Options

Binary options are a type of option where the payoff is either a fixed amount of cash or nothing at all. They are often used for speculative purposes or as a form of hedging.

Definition and Characteristics

A binary option pays a fixed amount of cash if a specified condition is met at expiration. Otherwise, it pays nothing. For example, a binary call option may pay $100 if the underlying asset price is above a certain level at expiration, and $0 otherwise.

Binary options can have different types of payoff conditions, such as being above or below a certain price level, or being within a specified range.

Pricing Binary Options using the Black-Scholes PDE

Binary options can be priced using the Black-Scholes PDE by modifying the payoff function. The key is to determine the probability of the specified condition being met at expiration.

Digital Options

Digital options are similar to binary options in that they have discontinuous payoffs. However, digital options have a fixed payout amount, rather than a variable payout.

Definition and Characteristics

A digital option pays a fixed amount of cash if a specified condition is met at expiration. The amount of the payout is predetermined and does not depend on the degree to which the condition is met.

Digital options can have different types of payoff conditions, such as being above or below a certain price level, or being within a specified range.

Pricing Digital Options using the Black-Scholes PDE

Digital options can also be priced using the Black-Scholes PDE by modifying the payoff function. The key is to determine the probability of the specified condition being met at expiration.

Step-by-Step Walkthrough of Typical Problems and Solutions

To further understand the application of the Black-Scholes PDE, let's walk through two example problems and their solutions.

Example Problem 1: Pricing a European Call Option

Suppose we want to price a European call option using the Black-Scholes PDE. We have the following information:

• Current price of the underlying asset: $50
• Strike price of the option: $55
• Time to expiration: 1 year
• Risk-free interest rate: 5%
• Volatility of the underlying asset: 20%

Calculation of Option Price, Delta, Gamma, and Theta

Using the Black-Scholes formula for European call options, we can calculate the option price as follows:

$$C = S_0e^{-qt}N(d_1) - Xe^{-rt}N(d_2)$$

Substituting the given values into the formula, we have:

$$C = 50e^{-0.05\times 1}N(d_1) - 55e^{-0.05\times 1}N(d_2)$$

where $$d_1$$ and $$d_2$$ are calculated as follows:

$$d_1 = \frac{\ln\left(\frac{S_0}{X}\right) + (r - q + \frac{\sigma^2}{2})t}{\sigma\sqrt{t}}$$

$$d_2 = d_1 - \sigma\sqrt{t}$$

Once we have the option price, we can calculate the option's delta, gamma, and theta using the following formulas:

• Delta ($$\Delta$$): The sensitivity of the option price to changes in the underlying asset price.

$$\Delta = N(d_1)$$

• Gamma ($$\Gamma$$): The sensitivity of the option's delta to changes in the underlying asset price.

$$\Gamma = \frac{N'(d_1)}{S_0\sigma\sqrt{t}}$$

• Theta ($$\Theta$$): The sensitivity of the option price to the passage of time.

$$\Theta = -\frac{S_0\sigma N'(d_1)}{2\sqrt{t}} - rXe^{-rt}N(d_2) + qS_0e^{-qt}N(d_1)$$

Example Problem 2: Pricing a Binary Option

Suppose we want to price a binary option using the Black-Scholes PDE. We have the following information:

• Current price of the underlying asset: $100
• Strike price of the option: $95
• Time to expiration: 6 months
• Risk-free interest rate: 3%
• Volatility of the underlying asset: 25%

Calculation of Option Price and Probability of Payout

To price the binary option, we need to determine the probability of the specified condition being met at expiration. Let's assume that the condition is that the underlying asset price will be above $95 at expiration.

Using the Black-Scholes formula for binary options, the option price can be calculated as follows:

$$B = e^{-rt}N(d_2)$$

where $$d_2$$ is calculated as follows:

$$d_2 = \frac{\ln\left(\frac{S_0}{X}\right) + (r - q + \frac{\sigma^2}{2})t}{\sigma\sqrt{t}}$$

Once we have the option price, we can calculate the probability of payout as follows:

$$P(\text{Payout}) = 1 - N(d_2)$$

Real-World Applications and Examples

The Black-Scholes PDE has numerous real-world applications in the field of finance. Two common applications are commodity options pricing and currency options pricing.

Application of Black-Scholes PDE in Commodity Options Pricing

Commodity options are financial derivatives that give the holder the right, but not the obligation, to buy or sell a commodity at a predetermined price on or before the expiration date.

The Black-Scholes PDE can be used to price commodity options by applying the same principles used for pricing European options. The key is to determine the appropriate parameters, such as the volatility of the commodity price and the risk-free interest rate.

Application of Black-Scholes PDE in Currency Options Pricing

Currency options are financial derivatives that give the holder the right, but not the obligation, to buy or sell a currency at a predetermined exchange rate on or before the expiration date.

The Black-Scholes PDE can also be used to price currency options by applying the same principles used for pricing European options. The key is to determine the appropriate parameters, such as the volatility of the exchange rate and the risk-free interest rate.

Advantages and Disadvantages of Black-Scholes PDE

The Black-Scholes PDE has several advantages and disadvantages that are important to consider when using it for option pricing.

Advantages

1. Provides a mathematical framework for option pricing: The Black-Scholes PDE provides a rigorous mathematical model for pricing options, which allows for consistent and efficient valuation.

2. Allows for efficient calculation of option prices and sensitivities: The Black-Scholes formula provides a closed-form solution for option prices, which can be calculated quickly and accurately. It also allows for the calculation of option sensitivities, such as delta, gamma, and theta, which are important for risk management and trading strategies.

Disadvantages

1. Relies on assumptions that may not hold in real-world scenarios: The Black-Scholes model makes several simplifying assumptions, such as constant volatility and efficient markets, which may not accurately reflect real-world conditions. This can lead to inaccurate option prices.

2. Does not account for market frictions and transaction costs: The Black-Scholes model ignores market frictions, such as bid-ask spreads and transaction costs, which can have a significant impact on option prices. In practice, these costs can erode the profitability of option trading strategies.

Summary

The Black-Scholes PDE is a fundamental concept in computational finance and modeling. It provides a mathematical framework for pricing options and understanding the dynamics of financial markets. Key concepts and principles associated with the Black-Scholes PDE include the definition and derivation of the PDE, European calls and puts, put-call parity, and its applications in real-world scenarios. The Black-Scholes PDE has advantages, such as providing a mathematical framework for option pricing and allowing for efficient calculation of option prices and sensitivities. However, it also has limitations, such as relying on assumptions that may not hold in real-world scenarios and not accounting for market frictions and transaction costs.

Analogy

The Black-Scholes PDE is like a mathematical compass that helps navigate the complex world of option pricing. Just as a compass provides direction and guidance, the Black-Scholes PDE provides a framework for understanding and valuing options. It allows traders and investors to navigate the uncertainties of financial markets and make informed decisions.