Lesson 9.3: Options and Arbitrage-Free Pricing

Introduction

In this lesson, we will explore the fundamental concepts surrounding options in financial markets, particularly focusing on the ideas of option payoffs, intrinsic and time value, put-call parity, and the logic of arbitrage-free pricing. Throughout this section, our objective will be to build an understanding of how options are valued and how they can be used strategically in trading.

By the end of this lesson, students will be able to:

Describe option payoffs, intrinsic and time value, and put-call parity.
Identify the determinants of option value and the logic underlying arbitrage-free pricing.
Diagram option payoffs and distinguish between intrinsic value and time value.
Apply the concept of put-call parity in different scenarios.

Hook

Imagine you have the opportunity to buy a stock at a fixed price anytime within the next three months. What if the stock price soars, and you can buy it for much less? This is the power of options—they provide strategic advantages that can significantly affect investment returns.

Understanding Options

Options are financial derivatives that give the buyer the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a specified price (the strike price) before or at expiry. Let's break down the essential components of options.

Option Payoffs

The payoff of an option at expiration is determined by the relationship between the underlying asset's market price and the option's strike price.

Call Option Payoff: The payoff occurs when the market price of the underlying asset exceeds the strike price. The formula is given by:

$\text{Payoff}_{\text{call}} = \max(0, S_T - K)$

where:

$S_T$ is the underlying asset price at expiration.
$K$ is the strike price.

Put Option Payoff: Conversely, a put option is profitable when the market price is below the strike price. The formula is:

$\text{Payoff}_{\text{put}} = \max(0, K - S_T)$

Worked Example 1: Call Option Payoff

Suppose an investor holds a call option with a strike price of $50, and at expiration, the stock is trading at $70. The payoff would be:

$$\text{Payoff}_{\text{call}} = \max(0, 70 - 50) = 20$$

This means the option's holder gains $20 per share.

Worked Example 2: Put Option Payoff

Consider a put option with a strike price of $50, and at expiration, the stock price is $30. The payoff would be:

$$\text{Payoff}_{\text{put}} = \max(0, 50 - 30) = 20$$

The option holder enjoys a gain of $20 per share.

Intrinsic and Time Value of Options

Options have two components to their value: intrinsic value and time value.

Intrinsic Value: This is the measurable value of an option if exercised immediately. For a call option, it is the difference between the underlying asset price and the strike price ($S_T - K$ if positive). For a put option, it is the difference between the strike price and the underlying asset price ($K - S_T$ if positive).

Time Value: Represents the additional amount an investor is willing to pay for the option above its intrinsic value, based on the time remaining until expiration. Time value diminishes as the expiration date approaches.

$$\text{Time Value} = \text{Option Price} - \text{Intrinsic Value}$$

Worked Example 3: Calculating Total Option Value

Suppose a call option is priced at $10 when the underlying asset is trading at $60, with a strike price of $50. The intrinsic value would be:

$$\text{Intrinsic Value} = 60 - 50 = 10$$

Since the market price of the option is also $10:

$$\text{Time Value} = 10 - 10 = 0$$

This indicates that the market is not valuing the time left for the option, possibly due to low volatility or nearing expiration.

Put-Call Parity

Put-call parity is a fundamental principle in options pricing that states the relationship between the prices of European call and put options with the same strike price and expiration date. The formula is given by:

$$C - P = S_0 - \frac{K}{(1+r)^T}$$

where:

$C$ = Price of the call option
$P$ = Price of the put option
$S_0$ = Current stock price
$K$ = Strike price
$r$ = Risk-free interest rate
$T$ = Time to expiration (in years)

Worked Example 4: Applying Put-Call Parity

Consider a stock currently priced at $100, with a call option priced at $5 and a put option priced at $3, both having a strike price of $95 and an expiration in one year at a risk-free rate of 5%.

Using put-call parity:

$$5 - 3 = 100 - \frac{95}{(1 + 0.05)^1}$$

Calculating the right-hand side:

$$5 = 100 - \frac{95}{1.05} = 100 - 90.48 = 9.52$$

Since $5

eq 9.52, this indicates that there could be an arbitrage opportunity, prompting traders to act to achieve balance between the derivative and underlying asset prices.

Determinants of Option Value

Several factors influence option value. These include:

Stock Price: As the stock price increases, call option prices tend to rise, while put option prices usually decrease.
Strike Price: The relationship between the strike price and the current stock price directly affects option values.
Time to Expiration: Longer expiration times typically result in higher option values due to greater uncertainty and more time for the option to become profitable.
Volatility: Higher volatility increases the likelihood of stock price swings, raising the value of both call and put options.
Interest Rates: Rising interest rates often increase call option prices (as it’s cheaper to hold off on purchasing the underlying asset), while making puts less attractive.
Dividends: Expected dividends can decrease call option values and increase put option values, as dividends generally reduce the stock price when they are paid out.

Conclusion

In conclusion, options are versatile financial instruments whose payoffs depend on various factors and market conditions. Understanding how to properly evaluate option payoffs, intrinsic and time value, along with put-call parity, is essential for making informed investment decisions. By mastering these concepts, students will be better prepared to engage with derivatives effectively and recognize potential arbitrage opportunities.

Study Notes

Options provide the right to buy or sell an asset at a predetermined price.
Call option payoff is profitable when the market price exceeds the strike price.
Put option payoff is profitable when the market price is below the strike price.
Intrinsic value is the immediate exercise value of an option.
Time value reflects the potential for future value beyond intrinsic value.
Put-call parity connects call and put prices for European options, indicating possible arbitrage opportunities.
Factors influencing option value include stock price, strike price, time to expiration, volatility, interest rates, and dividends.