Lesson 9.2: Pricing and Valuation of Swaps and Options

Introduction

In this lesson, students will explore the pricing and valuation mechanisms of swaps and options. With derivatives representing a significant portion of various markets, understanding these concepts is essential for effective financial management. By the end of this lesson, students should be able to:

• Value interest rate, currency, and equity swaps.
• Understand the binomial and Black-Scholes-Merton frameworks for option pricing.
• Price and value swaps through replication and discounting techniques.
• Calculate option values using the binomial and Black-Scholes-Merton models.
• Explain key ideas and terminology behind the pricing and valuation of swaps and options.

Understanding Swaps

What is a Swap?

A swap is a derivative contract through which two parties exchange financial instruments, typically cash flows based on different financial metrics. The most common types of swaps are:

• Interest Rate Swaps: Exchange of interest payment streams, one fixed and one floating.
• Currency Swaps: Exchange of principal and interest in one currency for the same in another currency.
• Equity Swaps: Exchange of cash flows related to stock indices or equity returns.

Valuing Swaps

To value a swap, we look at the present value of expected cash flows. Each cash flow in a swap must be discounted back to present value using the appropriate discount rate. For instance, in an interest rate swap:

1. Identify the cash flows on both sides of the swap.
2. Use a yield curve to derive discount factors.
3. Calculate the present value of cash flows from both parties.
4. The value of the swap is defined as the net present value (NPV) of these cash flows.

Example 1: Valuing an Interest Rate Swap

Consider a 5-year interest rate swap where Party A pays a fixed rate of 3% and Party B pays a floating rate indexed to LIBOR. Assume that the notional amount is $1,000,000.

1. Calculate the fixed leg cash flows:

Cash Flow = Notional Fixed Rate = $1,000,000 3\% = $30,000 per year.

Total Fixed Cash Flow over 5 years = $30,000 * 5 = $150,000

1. Assume LIBOR is expected to average 2% over the period.

Cash Flow from Party B = Notional Floating Rate = $1,000,000 2\% = $20,000 per year.

Total Floating Cash Flow over 5 years = $20,000 * 5 = $100,000

1. Discount the cash flows to present value:

Present Value (PV) of Fixed Cash Flows = $30,000 * (PV factor for each year)

Present Value of Floating Cash Flows = $20,000 * (PV factor for each year)

1. Subtract PV of Floating Cash Flows from PV of Fixed Cash Flows to determine swap value.

Common Misconceptions

One common misconception is that swaps are "free" when they are created because cash flows may not be exchanged upfront. However, the present value analysis shows that actual cash flows depend on future interest rate movements.

Option Pricing Basics

What is an Option?

An option is a contract that gives the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price (the strike price) before or at the expiration date. There are two main types of options:

• Call Options: Grant the holder the right to buy the underlying asset.
• Put Options: Grant the holder the right to sell the underlying asset.

Understanding Option Value

The value of an option is primarily determined by several factors: the current price of the underlying asset, the strike price, time to expiration, volatility of the underlying asset, and the risk-free interest rate. For example, an option becomes more valuable as the underlying asset price increases for a call option or decreases for a put option.

Binomial Option Pricing Model

The binomial model provides a discrete time framework for pricing options. It involves creating a binomial tree to represent possible paths for the underlying asset.

1. Build a binomial tree for the price at each node.
2. Calculate the option payoff at expiration (e.g., if the option finishes in-the-money).
3. Work backward through the tree to calculate the option value by discounting expected future payoffs at the risk-free rate.

Example 2: Using the Binomial Model

Assume a stock $ S $ is priced at 50, and we have a one-period call option with a strike price of $55. The stock can either increase by 20% or decrease by 10% over the period.

1. Possible stock prices at expiration:

• Up scenario: $50 * 1.2 = $60
• Down scenario: $50 * 0.9 = $45

1. Payoff calculations:

• Call Payoff (Up) = max(0, $60 - $55) = $5
• Call Payoff (Down) = max(0, $45 - $55) = $0

1. Determine the expected payoff and discount:

$ C = e^{-r} \cdot p \cdot 5 + e^{-r} \cdot (1 - p) \cdot 0 $

Where $ r $ is the risk-free rate and $ p $ is the probability of an increase.

Misconceptions Around Options

Many believe that options must be exercised to realize their value. However, this is not true as options can be traded before expiration, and their value can fluctuate based on market conditions.

Black-Scholes-Merton Model

Overview of the Model

The Black-Scholes-Merton model is a continuous-time model used to determine the theoretical price of European-style options. The price of a call option, according to the model, is calculated using the formula:

$$ C = S_0 N(d_1) - X e^{-rT} N(d_2) $$

Where:

• $ C $ = Call option price
• $ S_0 $ = Current price of the underlying asset
• $ X $ = Strike price of the option
• $ r $ = Risk-free interest rate
• $ T $ = Time until expiration (in years)
• $ N(d) $ = Cumulative distribution function of the standard normal distribution
• $ d_1 = \frac{\ln(S_0 / X) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}} $
• $ d_2 = d_1 - \sigma \sqrt{T} $
• $ \sigma $ = Volatility of the underlying asset

Example 3: Applying the Black-Scholes-Merton Model

Consider a stock priced at $100, a strike price of $95, an annual risk-free rate of 5%, 1 year until expiration, and a stock volatility of 20%. Calculate the call option price.

1. Calculate $ d_1 $ and $ d_2 $:

$ d_1 = \frac{\ln(100 / 95) + (0.05 + \frac{0.2^2}{2}) \cdot 1}{0.2 \cdot \sqrt{1}} $

$ d_2 = d_1 - 0.2 \cdot \sqrt{1} $

1. Evaluate $ N(d_1) $ and $ N(d_2) $ using a standard normal distribution table.
2. Calculate $ C = 100 \cdot N(d_1) - 95 e^{-0.05} N(d_2) $.

Common Misconceptions

Some may incorrectly assume that the Black-Scholes-Merton model is universally applicable. While it provides valuable insights, it relies on assumptions of market efficiency, constant volatility, and the absence of dividends, which can limit its applicability in real-world scenarios.

Conclusion

In this lesson, students has delved into the essential concepts of swaps and options, focusing on their valuation and pricing methods. By understanding the core principles behind the valuation processes, including the use of important models like the binomial and Black-Scholes-Merton frameworks, students has gained insights into how to apply these methods in various financial circumstances. Swaps and options are powerful financial tools that, when properly valued, can drive effective risk management and investment strategies.

Study Notes

• Swaps are contracts for exchanging cash flows; key types include interest rate, currency, and equity swaps.
• Valuing swaps involves calculating the present value of expected cash flows.
• Options provide the right, but not the obligation, to buy or sell an asset at a predetermined price.
• The Binomial Model gives a discrete approach to option pricing through tree structures.
• The Black-Scholes-Merton Model calculates theoretical option prices using underlying asset parameters.
• It is crucial to understand assumptions behind pricing models to evaluate their applicability correctly.