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 $c<em>t \leq S</em>t$ and $C<em>t \leq S</em>t$

• 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 $max(Ke^{-(T-t) \times r<em>{t,T}} - S</em>t, 0) \leq p_t$

• 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