Options, Put-Call Parity, Bounds, and Option Portfolios Notes

Options

  • Call Option:

    • A call option grants the buyer the right, but not the obligation, to purchase an underlying asset at a predetermined strike price K on or before a specified expiration date T.

    • The seller, however, is obligated to sell the asset at the strike price if the buyer chooses to exercise the option.

    • Payoff at Maturity: The payoff for the call option buyer at maturity is calculated as max(ST - K, 0), where ST represents the asset's price at time T. This ensures the buyer profits only if the asset's market price exceeds the strike price at expiration; otherwise, the option expires worthless.

  • Put Option:

    • A put option provides the buyer with the right to sell an underlying asset at a strike price K by a specific date T.

    • This is particularly valuable in declining markets, offering a safety net against potential losses.

    • Payoff at Maturity: The payoff for the put option buyer at maturity is determined by max(K - S_T, 0). This allows the buyer to profit when the asset's price falls below the strike price, effectively capping potential losses.

  • Option Types:

    • European Option: These options can be exercised only on the expiration date, offering a clear-cut, end-of-term settlement. Their valuation is generally simpler due to this restriction.

    • American Option: These options offer greater flexibility, allowing exercise at any point before or on the expiration date. This feature adds complexity to their valuation but provides more tactical opportunities.

Option States

  • In-the-Money (ITM): An option is ITM if exercising it immediately would yield a profit.

    • Call Option: This occurs when S_t > K, indicating the asset's current price exceeds the strike price.

    • Put Option: This happens when S_t < K, meaning the asset's current price is below the strike price.

  • Out-of-the-Money (OTM): An option is OTM if immediate exercise would result in a loss.

    • Call Option: This is when S_t < K, showing the asset's price is less than the strike price.

    • Put Option: This is when S_t > K, indicating the asset's price is higher than the strike price.

  • At-the-Money (ATM): An option is ATM if immediate exercise would neither yield a profit nor a loss.

    • Condition: This is when S_t = K, meaning the asset's price equals the strike price.

Option Payoffs and Profits

  • Initial Fee: At the time the option is initiated, the buyer pays a fee to the option seller.

  • Value of Option Fee at Maturity: v = option_fee \times e^{(T-t) \times r_{t,T}}. This reflects the time value of money and the risk-free rate's impact on the initial fee.

  • Option Buyer's Profit at Maturity: Payoff - v. This is the net gain after accounting for the initial fee paid.

  • Option Seller's Profit at Maturity: Payoff + v. This represents the income from selling the option, adjusted for the eventual payoff.

Put-Call Parity

  • Definition: For European call (ct) and put (pt) options on the same asset, with the same strike price K and maturity T, where the asset pays no cash flows until maturity: ct - pt = St - K \times e^{-(T-t) \times r{t,T}}

    • Where St is the price of the underlying asset at time t, and r{t,T} is the continuously-compounded risk-free interest rate at time t with maturity at time T. This equation is vital for arbitrage pricing.

Derivation of Put-Call Parity

  • Portfolio Construction: Consider a portfolio: buy one European call and sell one European put with the same strike K and maturity T.

  • Value of the Portfolio at Time t: ct - pt

  • Payoff of the Portfolio at Time T: S_T - K (same as a long forward contract with forward price K).

  • Current Value of Such a Forward Contract: St - K \times e^{-(T-t) \times r{t,T}}.

  • Parity Equation: Therefore, ct - pt = St - K \times e^{-(T-t) \times r{t,T}}.

Bounds on Call Options (Non-Dividend-Paying Stock)

  • For European (ct) and American (Ct) call options:

    • max(St - Ke^{-(T-t) \times r{t,T}}, 0) \leq ct \leq St

    • max(St - Ke^{-(T-t) \times r{t,T}}, 0) \leq Ct \leq St

Derivation of Lower Bound max(St - Ke^{-(T-t) \times r{t,T}}, 0) \leq c_t

  • Portfolio Construction: Consider a portfolio: a European call option plus a zero-coupon bond paying K at time T.

  • Payoff of this Portfolio at Time T: either ST (if ST > K) or K (if S_T \leq K).

  • The Payoff: The payoff is at least as high as the stock itself, S_T.

  • Inequality: Therefore, ct + Ke^{-(T-t) \times r{t,T}} \geq St, or ct \geq St - Ke^{-(T-t) \times r{t,T}}.

  • Non-Negative Option Value: Since the option value cannot be negative, c_t \geq 0.

  • Combined Inequality: Combining these, max(St - Ke^{-(T-t) \times r{t,T}}, 0) \leq c_t.

Derivation of Upper Bounds ct \leq St and Ct \leq St

  • Principle: A call option (European or American) can never be worth more than the underlying stock.

  • American vs. European: An American option is always worth at least as much as a European option: Ct \geq ct.

  • Implication: This, together with max(St - Ke^{-(T-t) \times r{t,T}}, 0) \leq ct, implies max(St - Ke^{-(T-t) \times r{t,T}}, 0) \leq Ct.

Early Exercise of American Call Options

  • Optimality: It's generally not optimal to exercise an American call option on a non-dividend-paying stock before maturity.

    • Immediate Profit from Exercising at Time t: S_t - K

    • Lower Bound Consideration: St - Ke^{-(T-t) \times r{t,T}} \leq Ct, which implies St - K \leq C_t. Therefore, selling the option typically yields more profit than exercising it.

Bounds on European Put Options (Non-Dividend-Paying Stock)

  • For a European put option with price pt: max(Ke^{-(T-t) \times r{t,T}} - St, 0) \leq pt \leq Ke^{-(T-t) \times r_{t,T}}

Derivation of Lower Bound max(Ke^{-(T-t) \times r{t,T}} - St, 0) \leq p_t

  • Portfolio Construction: Consider a portfolio: a European put option plus one share of stock.

  • Payoff of this Portfolio at Time T: either ST (if ST > K) or K (if S_T \leq K).

  • Payoff: Payoff is at least as high as K.

  • Inequality: Therefore, pt + St \geq Ke^{-(T-t) \times r{t,T}}, or pt \geq Ke^{-(T-t) \times r{t,T}} - St.

  • Non-Negative Option Value: Since the option value cannot be negative, p_t \geq 0.

  • Combined Inequality: Combining these, max(Ke^{-(T-t) \times r{t,T}} - St, 0) \leq p_t.

Derivation of Upper Bound pt \leq Ke^{-(T-t) \times r{t,T}}

  • Payoff: Payoff of the put option at time T is max(K - S_T, 0), which is less than or equal to K.

  • Upper Bound: Therefore, pt \leq Ke^{-(T-t) \times r{t,T}}.

Bounds on American Put Options (Non-Dividend-Paying Stock)

  • For an American put option with price Pt: max(K - St, 0) \leq P_t \leq K

Derivation of Lower Bound max(K - St, 0) \leq Pt

  • Analogous to the European put option lower bound derivation.

Derivation of Upper Bound P_t \leq K

  • Analogous to the European put option upper bound derivation.

Option Portfolios

  • Combining different options (long/short, calls/puts) with varying strikes and maturities allows for tailored risk and return profiles.

Bull Spread

  • This strategy is designed to profit from a moderate increase in the price of an asset.

    • Long call with K1 + short call with K2, where K1 < K2. The investor buys a call option with a lower strike price and sells a call option with a higher strike price.

    • Long put with K1 + short put with K2, where K1 < K2. Here, one buys a put option with a lower strike price and sells a put option with a higher strike price.

Bear Spread

  • This is implemented to take advantage of an expected moderate decline in the price of an asset.

    • Short call with K1 + long call with K2, where K1 < K2. This involves selling a call option with a lower strike price and buying a call option with a higher strike price.

    • Short put with K1 + long put with K2, where K1 < K2. The strategy includes selling a put option with a lower strike price and purchasing a put option with a higher strike price.

Butterfly Spread

  • A neutral strategy designed for situations where minimal price movement is expected. It combines both bullish and bearish elements.

    • Long call with K1 + 2 short calls with K2 + long call with K3, where K1 < K3 and K2 = \frac{K1+K3}{2}. This involves buying call options at a lower and higher strike price and selling two call options at a strike price in between.

    • Long put with K1 + 2 short puts with K2 + long put with K3, where K1 < K3 and K2 = \frac{K1+K3}{2}. In this case, one buys put options at a lower and higher strike price and sells two put options at a strike price in the middle.

Straddle

  • Ideal for when significant price movement is expected but the direction is uncertain.

    • Long call with K + long put with K. This consists of buying both a call and a put option with the same strike price and expiration date.

Strip and Strap

  • Variations on the straddle, adjusting the weighting of calls and puts to reflect different biases.

    • Strip (one call + two puts). Used when a larger price move downwards is anticipated.

    • Strap (two calls + one put). Applied when a larger price move upwards is expected.

Strangle

  • Similar to a straddle but involves buying options that are out-of-the-money, reducing the