5  Options

References
  • HULL, John. Options, futures, and other derivatives. Ninth edition. Harlow: Pearson, 2018. ISBN 978-1-292-21289-0.
    • Chapter 10. Mechanics of Options Markets
    • Chapter 11. Properties of Stock Options
  • PIRIE, Wendy L. Derivatives. Hoboken: Wiley, 2017. CFA institute investment series. ISBN 978-1-119-38181-5.
    • Chapter 1 - Derivative Markets and Instruments

Learning Outcomes:

5.1 Understanding Options

Definition

Options are financial derivatives that offer the buyer the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a predetermined price (strike price) on or before a specified date. The seller of the option, also known as the writer, receives a premium from the buyer in exchange for this right.

Types of Options

  • Call Option: Grants the holder the right to purchase an asset at the strike price by the expiration date. It is a bullish bet, with the buyer anticipating an increase in the asset’s price.
Code 
import plotly.graph_objects as go
import numpy as np

# Parameters
S = 100  # Strike price of the option
Q = 1  # Quantity of the asset
spot_prices = np.linspace(80, 120, 100)  # Range of spot prices
premium = 5  # Option premium

# Payoff calculations for call options
long_call_profit = np.maximum(spot_prices - S, 0) * Q - premium
short_call_profit = -np.maximum(spot_prices - S, 0) * Q + premium

# Create the figure
fig = go.Figure()

# Add traces for long and short call option positions
fig.add_trace(
    go.Scatter(
        x=spot_prices,
        y=long_call_profit,
        mode="lines",
        name="Long Call",
        line=dict(width=3),
        hovertemplate="Long Call<br>Spot Price: %{x:.0f}<br>Profit: %{y:.0f}<extra></extra>",
    )
)
fig.add_trace(
    go.Scatter(
        x=spot_prices,
        y=short_call_profit,
        mode="lines",
        name="Short Call",
        line=dict(width=3),
        hovertemplate="Short Call<br>Spot Price: %{x:.0f}<br>Profit: %{y:.0f}<extra></extra>",
    )
)
fig.add_hline(y=0, line_dash="solid", line_color="black", line=dict(width=0.7))

# Layout
fig.update_layout(
    title="Profit from Call Option, K = 100, c = 5",
    xaxis_title="Spot Price at Expiration",
    yaxis_title="Profit",
    legend_title="Position",
)

# Show the figure
fig.show()
  • Put Option: Provides the holder the right to sell an asset at the strike price by the expiration date. It represents a bearish outlook, where the buyer expects the asset’s price to decline.
Code 
# Parameters
S = 100  # Strike price of the option
Q = 1  # Quantity of the asset
spot_prices = np.linspace(80, 120, 100)  # Range of spot prices
premium = 5  # Option premium

# Payoff calculations for call options
long_put_profit = np.maximum(S - spot_prices, 0) * Q - premium
short_put_profit = -np.maximum(S - spot_prices, 0) * Q + premium

# Create the figure
fig = go.Figure()

# Add traces for long and short call option positions
fig.add_trace(
    go.Scatter(
        x=spot_prices,
        y=long_put_profit,
        mode="lines",
        name="Long Put",
        line=dict(width=3),
        hovertemplate="Long Put<br>Spot Price: %{x:.0f}<br>Profit: %{y:.0f}<extra></extra>",
    )
)
fig.add_trace(
    go.Scatter(
        x=spot_prices,
        y=short_put_profit,
        mode="lines",
        name="Short Put",
        line=dict(width=3),
        hovertemplate="Short Put<br>Spot Price: %{x:.0f}<br>Profit: %{y:.0f}<extra></extra>",
    )
)
fig.add_hline(y=0, line_dash="solid", line_color="black", line=dict(width=0.7))

# Layout
fig.update_layout(
    title="Profit from Put Option, K = 100, p = 5",
    xaxis_title="Spot Price at Expiration",
    yaxis_title="Profit",
    legend_title="Position",
)

# Show the figure
fig.show()

Execution Styles

  • American Option: Characterized by the flexibility it offers, an American option can be exercised at any point up until its expiration. This feature provides the holder with the opportunity to respond to market movements and exercise the option when it is most advantageous.

  • European Option: This option type can only be exercised on its expiration date, not before. The restriction on early exercise makes European options more predictable in terms of their valuation, but it limits the holder’s flexibility in responding to market changes.

5.2 Option Payoffs and Profits

5.2.1 Understanding Option Payoffs

  • \(S_T\): Represents the price of the underlying asset at the expiration date \(T\).
  • \(K\): Denotes the exercise or strike price of the option.

The payoff to the option buyer, whether for a call or put, depends on the relationship between the strike price \(K\) and the underlying asset’s price at expiration \(S_T\):

  • Call Option Payoff: \(c_T = \max(0, S_T - K)\)
  • Put Option Payoff: \(p_T = \max(0, K - S_T)\)
Example: Option Payoff

Given an underlying asset price at expiration \(S_T\) of $28 and a strike price \(K\) of $25, the payoffs for call and put options are calculated as follows:

  • Call Buyer Payoff: \(c_T = \max(0, \$28 - \$25) = \$3\)
  • Put Buyer Payoff: \(p_T = \max(0, \$25 - \$28) = \$0\)
  • The call option is in the money as it has a positive payoff of $3.
  • The put option is out of the money, resulting in zero payoff, indicating that exercising the option is not advantageous.

5.2.2 Calculating Option Profit

The profit from an option trade must account for the option premium paid upfront (\(c_0\) for calls and \(p_0\) for puts). Thus, profit formulas are adjusted as follows:

  • Profit for Call Buyer: \(\Pi_{call} = \max(0, S_T - K) - c_0\)
  • Profit for Put Buyer: \(\Pi_{put} = \max(0, K - S_T) - p_0\)
Example: Option Profit

Considering CBX stock options with a strike price \(K = \$30\), where the call and put premiums are $1 and $2 respectively, and the stock price at expiration \(S_T\) is $27.50:

  • Call Option Profit: \(\Pi_{call} = \max(0, \$27.5 - \$30) - \$1 = -\$1\)
  • Put Option Profit: \(\Pi_{put} = \max(0, \$30 - \$27.5) - \$2 = \$0.5\)
  • The call option buyer incurs a loss of $1, as the call’s intrinsic value does not offset the premium paid.
  • The put option buyer realizes a profit of $0.5, indicating that the intrinsic value exceeds the premium paid, making the option trade beneficial despite the underlying asset’s price movement.

5.4 Understanding the Dynamics of Option Pricing

In the realm of financial derivatives, stock options are pivotal instruments whose valuation intricately hinges on multiple underlying factors. The valuation dynamics of these options can be dissected into several key variables, each exerting a distinct influence under the ceteris paribus (all else equal) assumption. Herein, we delineate the impact of these variables on the prices of European and American options, using the following notations to denote directional influences: a positive (\(+\)) impact suggests that an increase in the variable elevates the option’s price, a negative (\(-\)) impact denotes a price decrease, and an uncertain (\(?\)) relationship indicates an ambiguous effect.

Variable European Call European Put American Call American Put
Current Stock Price (\(S_0\)) \(+\) \(-\) \(+\) \(-\)
Strike Price (\(K\)) \(-\) \(+\) \(-\) \(+\)
Time to Expiration (\(T\)) \(?\) \(?\) \(+\) \(+\)
Volatility (\(\sigma\)) \(+\) \(+\) \(+\) \(+\)
Risk-free Rate (\(r\)) \(+\) \(-\) \(+\) \(-\)
Amount of Future Dividends (\(D\)) \(-\) \(+\) \(-\) \(+\)

To ensure clarity in discourse, we employ the following notations throughout:

  • \(c\): Price of a European call option
  • \(p\): Price of a European put option
  • \(C\): Price of an American call option
  • \(P\): Price of an American put option
  • \(S_0\): Current stock price
  • \(S_T\): Stock price at option maturity
  • \(K\): Strike price
  • \(T\): Option’s life span
  • \(\sigma\): Volatility of the stock price
  • \(D\): Present value of dividends disbursed during the option’s life
  • \(r\): Risk-free interest rate with continuous compounding over maturity \(T\)

5.4.1 Comparative Analysis of American and European Options

A pivotal aspect of option theory is the intrinsic value comparison between American and European options. American options, characterized by their flexibility of exercise prior to expiration, inherently command a value that is not less than their European counterparts, which are exercisable only at maturity. This valuation principle is succinctly encapsulated in the following inequalities:

\[ C \geq c \] \[ P \geq p \]

These relations underscore a fundamental valuation floor for American options, driven by their enhanced exercise flexibility. This comparative analysis not only enriches our understanding of option pricing dynamics but also accentuates the critical role of exercise timing in option valuation.

5.5 Upper and Lower Bounds for Option Prices

5.5.1 Upper Boundaries for Option Prices

Call Options

For both American and European call options, the principle that an option’s value cannot exceed the current price of the underlying stock is fundamental. Mathematically, this is represented as: \[ c \leq S_0 \text{ and } C \leq S_0 \]

This ceiling on call option prices prevents the possibility of arbitrage profits that could arise from buying the stock outright and selling the call option, thereby exploiting price discrepancies.

Put Options

The valuation cap for put options varies between American and European styles due to their exercise terms.

  • American Put Options: The value is naturally capped at the strike price, \(K\), because the option grants the right but not the obligation to sell the stock at \(K\). Thus, \(P \leq K\).

  • European Put Options: The maximum value is the present value of the strike price, \(p \leq K e^{-rT}\), considering that it can only be exercised at maturity. This prevents arbitrage opportunities involving writing the option and investing the proceeds at the risk-free rate.

5.5.2 Lower Boundaries for Option Prices

European Call Options

For a European call option on a stock that does not pay dividends, the price floor is determined by the difference between the stock’s current price and the present value of the strike price: \[c \geq S_0 - K e^{-rT}\]

This lower bound highlights the intrinsic value of the option, beyond which arbitrage becomes viable.

Example

What is a lower bound for the following European call option?

  • \(S_0 = 20\), \(T = 1\), \(r = 10\%\), \(K = 18\), \(D = 0\)

\[\text{Lower bound} = S_0 - K e^{-rT} = 20 - 18 e^{-0.1} = 3.71\]

What if the European call price is $3?

  • An arbitrageur can short the stock, buy the call, and invest proceeds at 10%.

European Put Options

The lower bound for a European put option, similarly on a non-dividend-paying stock, is the present value of the strike price minus the current stock price: \[p \geq K e^{-rT} - S_0\]

This calculation ensures that the option’s price reflects its minimum economic value.

Example

What is a lower bound for the following European put option?

  • \(S_0 = 37\), \(T = 0.5\), \(r = 5\%\), \(K = 40\), \(D = 0\)

\[K e^{-rT} - S_0 = 40 e^{-0.05 \times 0.5} - 37 = 2.01\]

What if the European put price is $1?

  • An arbitrageur can borrow $38 for 6 months to buy both the put and the stock.

5.5.3 American Options and Early Exercise Decision

American Call Options on Non-dividend Paying Stocks

The conventional wisdom suggests that it is suboptimal to exercise an American call option on a non-dividend paying stock before expiration. Consider an American call option with the following parameters: \(S_0 = 100\), \(T = 0.25\), \(K = 60\), and \(D = 0\). The dilemma of whether to exercise the option immediately hinges on the anticipated utility from holding the stock versus the option.

  • If intending to hold the stock: Exercising early forfeits the call option’s time value, providing no additional benefit over holding the option.

  • If intending to close the stock position: Selling the option is preferable, capturing both its intrinsic and time value, unlike exercising, which yields only the intrinsic value.

  • Justifications Against Early Exercise:

    • Preservation of Capital: Delaying the exercise preserves liquidity by deferring the payment of the strike price.
    • Insurance Benefit: The option acts as a hedge against the stock’s depreciation below the strike price.
    • Maximization of Value: Selling the option rather than exercising it realizes both intrinsic and time value.

  • Because it is not optimal to exercise an American stock option on a non-dividend-paying stock early, the upper and lower bounds will be the same as those for European options.

American Put Options on Non-dividend Paying Stocks

Contrary to calls, American put options on non-dividend paying stocks may warrant early exercise, especially when deeply in the money, due to the immediate gain realization and the time value of money.

Imagine a scenario where the strike price is $10, and the stock price plummets near zero. Immediate exercise yields a $10 gain, maximizing the investor’s return as stock prices cannot become negative and due to the preference for current versus future value.

  • Justifications for Early Exercise:
    • Immediate Value Realization: Exercising deep in-the-money puts captures the maximum possible gain immediately.
    • Time Value of Money: Receiving proceeds today is financially more beneficial than an identical future payment due to potential investment returns.
    • Decreased Attractiveness with Stock Price Decline: As \(S_0\) diminishes, the attractiveness of early exercise increases, especially under higher interest rates and lower volatility.
  • Because it may be optimal in some cases to exercise an American put option early, the lower bound changes accordingly:

\[P \geq \max(K - S_0, 0)\]

5.5.4 Summary of Price Boundaries

The upper and lower price limits for options serve as critical indicators for arbitrage strategies and investment analysis. They are succinctly summarized as follows:

Call Options

The upper bound is the stock’s current price, while the lower bound is either the difference between the stock’s price and the discounted strike price or zero, whichever is greater.

  • Upper bound for European and American call options:

    \[c \leq S_0 \text{ and } C \leq S_0\]

  • Lower bound for European and American call option:

\[c \geq \max(S_0 - K e^{-rT}, 0)\] \[C \geq \max(S_0 - K e^{-rT}, 0)\]

Put Options

The upper bound for European options is the discounted strike price, while for American options, it’s the strike price itself. The lower bound is the greater of the discounted strike price minus the stock’s price or zero.

  • Upper bound for European and American put options:

\[p \leq K e^{-rT} \text{ and } P \leq K\]

  • Lower bound for European and American put option:

\[p \geq \max(K e^{-rT} - S_0, 0)\]

\[P \geq \max(K - S_0, 0)\]

5.6 Put-Call Parity

The principle of put-call parity is a cornerstone in the theoretical framework of financial derivatives, providing a fundamental relationship between the prices of European call and put options. This parity underlines the equilibrium that must exist between these options when they have identical strike prices and expiration dates. Below, we delve deeper into the concept, illustrating its implications for both non-dividend and dividend-paying stocks, and extend the discussion to American options.

5.6.1 Put-Call Parity for Non-dividend Paying Stocks

Consider two distinct portfolios, A and B, each composed of different financial instruments but constructed to have equivalent values at the expiration of the options involved:

  • Portfolio A consists of a European call option and a zero-coupon bond paying \(K\) (the strike price) at time \(T\) (expiration).
  • Portfolio B includes a European put option and the underlying stock itself.

The table below presents the values of these portfolios at option expiration under two scenarios: when the stock price at expiration, \(S_T\), is above or below the strike price, \(K\).

\(S_T > K\) (Above Strike) \(S_T < K\) (Below Strike)
Portfolio A Call option \(S_T - K\) \(0\)
Zero-coupon bond \(K\) \(K\)
Total \(S_T\) \(K\)
Portfolio B Put Option \(0\) \(K - S_T\)
Share \(S_T\) \(S_T\)
Total \(S_T\) \(K\)

Given their identical payouts at maturity, both portfolios must have the same present value, leading to the foundational put-call parity equation:

\[c + K e^{-rT} = p + S_0\]

5.6.2 Arbitrage Opportunities via Put-Call Parity

Consider an example where \(S_0 = \$31\), \(r = 10\%\), the call option price \(c = \$3\), and the strike price \(K = \$30\). The table outlines potential arbitrage strategies based on discrepancies in put option pricing, illustrating the mechanism for securing risk-free profits by leveraging the put-call parity principle.

Three-month put price = $2.25 Three-month put price = $1
Action now: Action now:
Buy call for $3 Borrow $29 for 3 months
Short put to realize $2.25 Short call to realize $3
Short the stock to realize $31 Buy put for $1
Invest $30.25 for 3 months Buy the stock for $31
…
Action in 3 months if \(S_T > 30\): Action in 3 months if \(S_T > 30\):
Receive $31.02 from investment Call exercised: sell stock for $30
Exercise call to buy stock for $30 Use $29.73 to repay loan
Net profit = $1.02 Net profit = $0.27
…
Action in 3 months if \(S_T < 30\): Action in 3 months if \(S_T < 30\):
Receive $31.02 from investment Exercise put to sell stock for $30
Put exercised: buy stock for $30 Use $29.73 to repay loan
Net profit = $1.02 Net profit = $0.27

5.6.3 Extension to American Options

While put-call parity directly applies to European options, its principles offer insight into the valuation boundaries of American options, which can be exercised at any time before expiration. In the absence of dividends, the relationship between American call and put prices can be expressed as:

\[ S_0 - K \leq C - P \leq S_0 - Ke^{-rT} \]

This range provides a basis for evaluating the relative pricing of American calls and puts, emphasizing the influence of early exercise rights and the absence of dividend payments on option valuation.

5.7 Effect of Dividends on Options

The presence of dividends within the lifespan of an option introduces significant nuances to the valuation and strategic exercise decisions of options. This section delves into how known dividends impact the valuation of options, adjusting the traditional models to accommodate the dividend factor.

When dividends are anticipated during the option’s term, they must be factored into the option’s present value. We denote the present value of expected dividends during the option’s life as \(D\). The ex-dividend date marks the occasion for these adjustments, impacting the strategic exercise decisions for American call options.

The prospect of dividends alters the conventional wisdom that American call options should not be exercised early. Specifically, it may become optimal to exercise these options just before the ex-dividend date to capture the dividend payout.

Crucial Point: Apart from the period just before the ex-dividend date, early exercise of a call option remains suboptimal.

5.7.1 Adjusting Lower Bound Valuations for Dividends

The introduction of dividends necessitates a revision of the lower bound calculations for both call and put option prices:

  • For Call Options: The lower bound formula adjusts to account for the dividend’s negative impact on the option’s value, reflecting the loss of dividend income upon early exercise: \[c \geq S_0 - D - K e^{-rT}\]

  • For Put Options: Conversely, the lower bound for put options incorporates dividends positively, indicating an increase in value due to the potential decrease in the underlying stock’s price upon dividend payout: \[p \geq D + K e^{-rT} - S_0\]

5.7.2 Revising Put-Call Parity with Dividends

The presence of dividends also modifies the put-call parity relationship, a fundamental principle in options pricing:

  • European Options with Dividends: The parity formula integrates \(D\) to balance the equation, highlighting the direct impact of dividends on the call option’s lower valuation compared to its put counterpart: \[c + D + K e^{-rT} = p + S_0\]

  • American Options with Dividends: When dividends are present, the valuation bounds for American options adjust to reflect the diminished value of the call option due to potential early exercise for dividend capture: \[S_0 - D - K < C - P < S_0 - K e^{-rT}\]

Dividends play a pivotal role in the strategic exercise and valuation of options, particularly affecting American call options’ early exercise decisions. Adjusting option valuation models to account for dividends is essential for accurate pricing and effective investment strategy formulation.

5.8 Practice Questions and Problems

5.8.1 Option Profitability and Exercise Conditions

  1. Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a long position in the option depends on the stock price at maturity of the option.
  2. An investor sells a European call on a share for $4. The stock price is $47 and the strike price is $50. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the investor’s profit with the stock price at the maturity of the option.
  3. An investor buys a European put on a share for $3. The stock price is $42 and the strike price is $40. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the investor’s profit with the stock price at the maturity of the option.
  4. Suppose that a European put option to sell a share for $60 costs $8 and is held until maturity. Under what circumstances will the seller of the option (the party with the short position) make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a short position in the option depends on the stock price at maturity of the option.

5.8.2 Margin Requirements, Market Choices, and Contract Adjustments

  1. Explain why margin accounts are required when clients write options but not when they buy options.
  2. A corporate treasurer is designing a hedging program involving foreign currency options. What are the pros and cons of using (a) the NASDAQ OMX and (b) the over-the-counter market for trading?
  3. The treasurer of a corporation is trying to choose between options and forward contracts to hedge the corporation’s foreign exchange risk. Discuss the advantages and disadvantages of each.
  4. Consider an exchange-traded call option contract to buy 500 shares with a strike price of $40 and maturity in four months. Explain how the terms of the option contract change when there is
    1. A 10% stock dividend
    2. A 10% cash dividend
    3. A 4-for-1 stock split

5.8.3 Option Pricing Bounds

  1. Explain why an American option is always worth at least as much as a European option on the same asset with the same strike price and exercise date.
  2. Explain why an American option is always worth at least as much as its intrinsic value.
  3. What is a lower bound for the price of a four-month call option on a non-dividend-paying stock when the stock price is $28, the strike price is $25, and the risk-free interest rate is 8% per annum?
  4. What is a lower bound for the price of a one-month European put option on a non-dividend paying stock when the stock price is $12, the strike price is $15, and the risk-free interest rate is 6% per annum?

5.8.4 Early Exercise and Put-Call Parity

  1. Give at least two reasons that the early exercise of an American call option on a non-dividend-paying stock is not optimal.
  2. The early exercise of an American put is a trade-off between the time value of money and the insurance value of a put. Explain this statement.
  3. The price of a non-dividend paying stock is $19 and the price of a three-month European call option on the stock with a strike price of $20 is $1. The risk-free rate is 4% per annum. What is the price of a three-month European put option with a strike price of $20?
  4. List and explain the six factors affecting stock option prices.