Investing: Option Pricing Factors

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Questions and Answers

What is the upper price bound for a call option?

  • It is based on the volatility of the stock.
  • It can never exceed the strike price.
  • It is determined by the risk-free interest rate.
  • It is capped at the current stock price. (correct)

For a European put option, what is the maximum value it can reach?

  • The strike price. (correct)
  • The risk-free interest rate compounded.
  • The current stock price.
  • The present value of the strike price today. (correct)

What influences the lower boundary of a non-dividend paying European call option?

  • The current stock price only.
  • The maximum of $0 and the difference of stock price and present value of the strike price. (correct)
  • The risk-free interest rate alone.
  • The time to expiration alone.

Which of the following factors does NOT affect the price of a stock option?

<p>The historical price of the stock. (C)</p> Signup and view all the answers

What is the significance of the strike price in determining option prices?

<p>It is the price at which an option can be exercised. (B)</p> Signup and view all the answers

In the context of an American put option, what is the maximum price it can hold?

<p>The strike price. (C)</p> Signup and view all the answers

What is the risk-free interest rate's effect on option pricing?

<p>It lowers the present value of strike prices causing call options to increase. (D)</p> Signup and view all the answers

What defines the lower boundary of a European put option on a non-dividend paying stock?

<p>The maximum of $0 and the difference between strike price and adjusted stock price. (A)</p> Signup and view all the answers

What is the objective of put-call parity?

<p>To ensure options with the same strike price and time to maturity have equivalent value (C)</p> Signup and view all the answers

Which formula correctly represents the lower bound for a European call option?

<p>$max(S_0 - K e^{-rT}, 0)$ (C)</p> Signup and view all the answers

What is the implication when a European put option is overpriced compared to a European call option?

<p>Arbitrage opportunities exist (B)</p> Signup and view all the answers

When is it optimal to exercise a put option early?

<p>When the stock price decreases significantly (D)</p> Signup and view all the answers

In the context of American options, what prevents the early exercise of an American call option on a non-dividend paying stock?

<p>The time value of money and insurance against price drops (C)</p> Signup and view all the answers

What condition generally makes the early exercise of a put option less beneficial?

<p>When the stock price is increasing (C)</p> Signup and view all the answers

What is one primary reason an American call option on a dividend-paying stock might be exercised early?

<p>To maximize the cash flow from dividends (C)</p> Signup and view all the answers

Which is NOT a characteristic of the lower bound of a European put option?

<p>It is irrelevant to the strike price of the option. (C)</p> Signup and view all the answers

What does the formula $c + K e^{-rT} = p + S_0$ represent?

<p>The relationship between the values of call and put options in arbitrage (B)</p> Signup and view all the answers

What is one reason why an American put option may be exercised early?

<p>To protect against further stock price decline (B)</p> Signup and view all the answers

Flashcards

Upper Bound of a Call Option

The price of a stock option can never be higher than the current stock price. If this were not true, an arbitrageur could buy the stock, sell the option, and profit risk-free.

Upper Bound of an American Put Option

The price of an American put option can never be higher than the strike price. This is because the holder can always sell the stock for the strike price, no matter how low the stock price falls.

Upper Bound of a European Put Option

The value of a European put option can never be higher than the present value of the strike price. This is due to the fact that the option can only be exercised at maturity.

Lower Bound of a European Call Option

The lower bound for a European call option on a non-dividend-paying stock is the maximum of zero or the difference between the stock price and the present value of the strike price. This is because the option will expire worthless if the stock price is lower than the strike price.

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Lower Bound of a European Put Option

The lower bound for a European put option on a non-dividend-paying stock is the maximum of zero or the difference between the present value of the strike price and the stock price. This is because the option will expire worthless if the stock price is higher than the strike price.

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Volatility and Option Prices

Higher volatility leads to a higher price for both call and put options. This is because higher volatility increases the chances of large price movements, which can benefit both buyers and sellers of options.

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Interest Rates and Option Prices

Higher interest rates lead to higher prices for call options and lower prices for put options. This is because higher interest rates make it more expensive to borrow money, which reduces the value of put options but increases the value of call options.

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Dividends and Option Prices

Dividends paid by the underlying stock reduce the value of call options and increase the value of put options. This is because dividends reduce the value of the stock, making call options less valuable and put options more valuable.

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Lower bound for a put option

The minimum price of a put option. It's the greater of zero or the difference between the discounted strike price, the present value of the strike price, minus the current stock price.

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Put-Call Parity

A principle that establishes a relationship between European call and put options with the same strike price and maturity date. It states that the difference in value between a call and a put option is equal to the difference between the current stock price and the present value of the strike price.

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Valuing a Put option using Put-Call Parity

The price of a put option is determined by rearranging the Put-Call Parity formula to solve for 'p'.

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Valuing a Call option using Put-Call Parity

The price of a call option is determined by rearranging the Put-Call Parity formula to solve for 'c'.

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Early Exercise of American Call (Non-Dividend Paying)

It is never optimal to exercise an American call option early on a non-dividend paying stock. This is because the value of an American call is always greater than its intrinsic value.

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Lower Bound of American Call (Non-Dividend Paying)

The minimum value of an American call option on a non-dividend paying stock is equal to its intrinsic value, which is the difference between the current stock price and the strike price.

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Early Exercise of American Put

In contrast to call options, exercising an American put option early might be optimal, particularly if it's deep in the money. This is because the strike price is realized immediately.

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Lower Bound of American Put

The lower bound for an American put option is the maximum of zero and the difference between the strike price and the current stock price. This is the intrinsic value.

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Lower Bound of Calls and Puts with Dividends

Dividends affect the lower bounds of both calls and puts. The adjusted lower bound for a call takes into account the present value of future dividends.

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Early Exercise of American Call with Dividends

Exercising an American call option immediately before an ex-dividend date can be optimal, as the holder receives the dividend and then exercises, maximizing profit. It is generally not optimal to exercise at other times.

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Study Notes

Factors Affecting Option Prices

  • Six factors influence stock option prices:
    • Current stock price (S₀)
    • Strike price (K)
    • Time to expiration (T)
    • Stock price volatility
    • Risk-free interest rate (r)
    • Expected dividends

Upper Price Bounds

  • Call Options: A call option's price (C) cannot exceed the stock price (S₀).
    • C ≤ S₀; c ≤ So
    • Otherwise, arbitrage opportunities exist.
  • Put Options (American): A put option's value (P) cannot exceed the strike price (K).
    • P ≤ K
  • Put Options (European): A European put option's value cannot exceed the present value of the strike price (K).
    • P ≤ Ke⁻rT

Lower Bounds (Non-Dividend Paying Stocks)

  • Call Options: A call option's value (c) cannot be negative.
    • c ≥ max(S₀ - K e⁻rT, 0)
    • The worst case is expiration with zero value.
  • Put Options (European):
    • p ≥ max(K e⁻rT - S₀, 0)
  • Example Calculations: Provided examples illustrate how to determine lower bounds using the formulas provided for various stock prices, strike prices, time to maturity, and risk-free interest rates.

Put-Call Parity

  • Relationship: A theory stating the relationship between the prices of a call and put option with the same strike price and time to maturity:
    • c + Ke⁻rT = p + S₀
  • Formula rearrangements are used to evaluate options:
    • p = c + Ke⁻rT - S₀
    • c = p + S₀ - Ke⁻rT
  • Arbitrage: Mispricing between these options indicates arbitrage opportunities for profit.

American Options on Dividend-Paying Stocks

  • Impact of Dividends: Dividends affect the option's lower bounds.
  • American Calls: Generally, American calls are never optimally exercised before expiration on non-dividend paying stocks, valuing them like European calls.
    • Lower Bound: c ≥ max(S₀ - D - Ke⁻rT, 0)
  • American Puts: American puts may be exercised early.
    • Lower Bound: p ≥ max(K - S₀, 0)
    • Early exercise is more attractive as S₀ decreases, r increases, and volatility decreases.
  • Optimal Exercise Conditions: When dividends are involved, it may be optimal to exercise early on American options.
    • Specific conditions, including ex-dividend dates, may make early exercise optimal for American calls. This differs from some scenarios where this is not the case.

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