Final - Options

F303 - Intermediate Investments

Professor Mathias S. Kruttli

Spring 2026
Topic: Options

Agenda

  • Put-call parity

  • Factors affecting the value of options

  • Option strategies

Examples of Call Options

  • Scenario 1: You buy a call option for $5 on General Motors. The strike price is $75. At expiry, the stock price is $83.   - Calculate Payoff:     - Options Payoff = Max[0, ST - X]     - Payoff = Max[0, 83 - 75] = Max[0, 8] = $8       - Options Payoff Choices: A. $8 B. $3 C. $13 D. None of the above
      - Calculate Profit:     - Profit = Payoff - Call Premium     - Profit = 8 - 5 = $3       - Profit Choices: A. $7 B. $3 C. -$2 D. None of the above

  • Scenario 2: A put option on Ford trades at $7. The strike price is $50. The current stock price is $52.   - Calculate Time Value of the Option:     - Time Value = Option Price - Intrinsic Value     - Intrinsic Value = Max[0, X - S] = Max[0, 50 - 52] = 0     - Time Value = 7 - 0 = $7       - Time Value Choices: A. $5 B. $3 C. $7 D. None of the above

Put-Call Parity

  • Definition: The relationship that links the prices of call and put options, which states that the following equation must hold:   C+PV(X)=P+S0C + PV(X) = P + S_0   - Where:     - C = Call price (premium)     - P = Put price (premium)     - X = Exercise price     - S0 = Current stock price (assumed no dividends until option maturity)

  • Portfolio Equivalence:   - Portfolio A: One call option plus cash equal to the Present Value of X.   - Portfolio B: One put option plus one share of stock.   - If both portfolios produce identical payoffs, they should cost the same; otherwise, arbitrage opportunities exist.

Payoff Structures of Portfolios

Portfolio A (Fiduciary Call)

  • Down State:   - Payoff = Max[0, ST - X] = 0

  • Up State:   - Payoff = ST - X

Portfolio B (Protective Put)

  • Down State:   - Payoff = X - ST

  • Up State:   - Payoff = ST

Example of Put-Call Parity

  • Given Values:   - Stock Price = 110   - Call Price = 14   - Put Price = 5   - Risk-Free Rate = 5%   - Maturity = 0.5 years (6 months)   - Strike Price (X) = 105

  • Determine Payoff and Arbitrage Opportunity:   - If the right side of the equation is less expensive, buy that option (RHS) and sell the more expensive one (LHS).

Factors Affecting Option Prices

  1. Stock Price, S0

  2. Exercise Price, X

  3. Time to Expiration, T - t0

  4. Volatility of Stock Price, σ

  5. Risk-Free Interest Rate, r_f

  6. Dividend Rate, d

Effects of Each Factor on Option Prices

Calls

  • Increasing Stock Price: Value increases

  • Increasing Strike Price: Value decreases

  • Increasing Time to Maturity: Value increases

  • Increasing Volatility: Value increases

  • Increasing Risk-Free Rate: Value increases

  • Increasing Dividends: Value decreases

Puts

  • Increasing Stock Price: Value decreases

  • Increasing Strike Price: Value increases

  • Increasing Time to Maturity: Value increases

  • Increasing Volatility: Value increases

  • Increasing Risk-Free Rate: Value decreases

  • Increasing Dividends: Value increases

Detailed Examination of Factors

Stock Price and Strike Price

  • Call Payoff at Expiration:   - Formula: Max[0, ST - X]
      - As stock price (S) increases, call value increases; as strike price (X) increases, call value decreases.

  • Put Payoff at Expiration:   - Formula: Max[0, X - ST]   - Reverse effect: as stock price increases, put value decreases, while as strike price increases, put value increases.

Time to Expiration

  • Impact on Option Value:   - Longer expiration time enhances potential for exercise, thus increasing value.

Volatility of Underlying Asset

  • General Insight: As volatility (σ) rises, so does the potential price range, benefiting option owners.

  • Effect on Calls: A call benefits from large positive movements in stock price, posing limited downside risk.

  • Effect on Puts: A put benefits from substantial negative movements in stock price, posing limited upside risk.

Example: Assurance Corp stock at $100

  1. Price Outcomes:    - Price could be $125 or $75 with equal probability.    - Call Payoff: Expected Payoff = (0.5 * (125-100)) + (0.5 * 0)= $12.5.    - Stock Payoff: $100.    - Put Payoff: Expected Payoff = (0.5 * (100-75)) + (0.5 * 0) = $12.5.

Altered Volatility Scenario

  • If stock varies between $150 and $50, the calculations will adjust accordingly to predict outcome changes.

Risk-Free Interest Rate

  • Influence on Option Prices: If interest rates rise, present value of future cash flows declines.   - For Puts: Value decreases.   - For Calls: Value increases.

Dividends

  • Increase in dividend rates increases the benefits of holding stocks, leading to:   - Puts: Increased value.   - Calls: Decreased value.

Early Exercise Considerations for American Options

Early Exercise of Calls

  • Insight: No incentive to exercise early if stock pays no dividends; value identical for American and European calls.

  • Rationale: A call option gains value as stock price rises; infinite growth potential makes early exercise unnecessary.

Early Exercise of Puts

  • Insight: American puts usually have greater value than European puts due to early exercise potential.

  • Rationale: Early exercise is optimal if stock is at risk of becoming worthless.

Option-Like Securities

  • Categories Include:   - Callable Bonds   - Convertible Securities   - Warrants (similar to options, but issued by the underlying firm)

Option Strategies

Protective Put

  • Description: An investor owns the stock and a put option on the stock.

  • Payoff and Profit Diagram: Illustrative representation of the combined strategy.

Covered Call

  • Description: An investor owns the stock and writes a call option on it.

  • Payoff and Profit Diagram: Visual summary of strategy outcomes.

Straddle

  • Description: An investor buys a call and put option on the same stock with the same strike price and expiration.

  • Payoff and Profit Diagram: Diagrammatic representation of risk and profit potential with a straddle strategy.