OM-W3

Mechanics of Options Markets

Outline

• 1. Definition

• 2. Payoffs

• 3. Mechanics

• 4. Other option-type products

Definitions and Terminologies

• Option: An option grants the holder the right (but not the obligation) to buy or sell a security to the seller (option writer) for a pre-specified price (the strike price, noted as K) at or before a specified future date (the expiry date).

• Value of an Option: An option has positive value, unlike a forward contract which has zero value at inception.

Types of Options

• Call Option: Grants the holder the right to buy a security.

  • Payoff:

  • The payoff at maturity is given by $(S<em>T - K)^+$, where $ST$ is the spot price at expiration.

• Put Option: Grants the holder the right to sell a security.

  • Payoff:

  • The payoff at maturity is given by $(K - S_T)^+$.

• American Options: Can be exercised at any time before expiry.

• European Options: Can only be exercised at expiry.

More Terminologies

• Moneyness: The relationship of the strike price relative to the current spot or forward level.

  • In-the-money:

  • For call options: S_t > K

  • For put options: S_t < K

  • Intrinsic Value: The option's immediate exercise value.

  • For call options: $(S_t - K)^+$

  • For put options: $(K - S_t)^+$

  • Out-of-the-money:

  • For call options: S_t < K

  • For put options: S_t > K

  • At-the-money: When $K$ is equal to the spot price or forward price.

• Forward Price Relation: Also defines moneyness using forward price, where:

  • For a call: If F_t > K it's in-the-money;

  • For a put: If F_t < K it's in-the-money.

• Time to Maturity ( au): Defined as $au = T - t$, where $T$ is expiry and $t$ is current time.

• Option Value: Determined by the factors:

  • Current spot or forward price ($S<em>t$ or $F</em>t$)

  • Strike price $K$

  • Time to maturity $au$

  • Type of option (Call/Put, American/European)

  • Dynamics of the underlying security (e.g., volatility)

• Option Components: The total value of an option can be decomposed into:

  • $ext{Option Value} = ext{Intrinsic Value} + ext{Time Value}$

Payoffs versus P&L

• Terminal Payoff for European options can be expressed as:

  • For calls: $(S_T - K)^+$

  • For puts: $(K - S_T)^+$

• P&L Calculation:

  • The P&L from an option investment is the difference between terminal payoff and the initial option price.

  • P&L vs Payoffs:

  • P&L: Reflects potential earnings under varying scenarios based on current trading

  • Payoffs: Reflects future payoff structures and necessary options/positions to achieve them.

Example: Call Option on a Stock Index

• Parameters:

  • Current index level ($S_t$) = 100

  • Strike ($K$) = 90

  • Time to maturity ($T - t$) = 1 year

  • Current option price ($c_t$) = 14

• Analysis:

  • Determine if in-the-money or out-of-the-money concerning the current spot price.

  • Calculate intrinsic value:

  • Intrinsic Value = $(100 - 90)^+ = 10$

  • Time Value:

  • Time Value = Current Price - Intrinsic Value = $c_t - ext{Intrinsic Value} = 14 - 10 = 4$

• Terminal Payoff Evaluation:

  • At expiry with index level at:

  • $S_T = 100$: Payoff = $(100 - 90)^+ = 10$, P&L = 10 - 14 = -4.

  • $S_T = 90$: Payoff = $(90 - 90)^+ = 0$, P&L = 0 - 14 = -14.

  • $S_T = 80$: Payoff = $(80 - 90)^+ = 0$, P&L still -14.

• Short Position Payoff and P&L: Similarly calculated for writing the option.

Payoffs and P&Ls from Long/Short a Call Option

• Spot at Expiry vs Payoff and P&L charts:

• The relationship shows:

  • Long a call pays off as $(S_T - K)^+$, anticipating an increase in the index price.

  • Conversely, shorting a call anticipates a decrease in index price.

Example: Put Option on an Exchange Rate

• Parameters:

  • Current spot exchange rate ($S_t$) = $1.6285/pound

  • Strike ($K$) = $1.61

  • One-year forward price ($F_{t,T}$) = $1.61

  • Dollar continuously compounding interest rate ($r_d$) = 5%

  • Current option price ($p_t$) = $0.0489

• Analysis:

  • Calculate the continuously compounding interest rate for the pound ($r_f$).

  • Use the forward pricing formula:

  • $F<em>{t,T} = S</em>t e^{(r<em>d - r</em>f)(T - t)}$

  • Rearranged to compute $(r_f)$:

  • rf = rd - rac{1}{T - t} ext{ln}igg( rac{F{t,T}}{St}igg)

Payoffs and P&Ls from Long/Short a Put Option

• Spot at Expiry vs Payoff and P&L charts: Similar to previous call option analysis,

  • Long a put option pays off $(K - S_T)^+$, whereas shorting a put bets on the appreciation of the pound.

What Derivative Positions Generate the Following Payoff?

• Graphical representation of possible payoffs for varying values of $S_T$.

Assets Underlying Exchanged-Traded Options

• Types of Underlying Assets:

  • Stocks

  • Stock indices

  • Index return variances

  • Exchange rates

  • Futures

Specification of Exchange-Traded Options

• Key Specifications:

  • Expiration date (T)

  • Strike price (K)

  • Option class (European or American, Call or Put)

Options Market Making

• Market Makers: Required to provide bid and ask quotes complying with a minimum bid-ask spread.

  • Risks and costs associated with market making.

  • Adjustments in quotes for multiple options on a single stock.

  • Modern market making utilizes automated systems for quote updates and hedging.

Margins

• Margin Requirements:

  • For naked option writing, margin is the greater of:

  1. 100% of proceeds + 20% of the underlying share price (minus any out-of-the-money amount).

  2. 100% of proceeds + 10% of the underlying share price.

• Special rules apply for different trading strategies.

Dividends and Stock Splits

• Impact on Options:

  • No adjustments for cash dividends.

  • For stock splits:

  • Strike price adjusts to $( rac{mK}{n})$ where n is the split ratio and increases option count to $( rac{nN}{m})$.

  • Similar treatment for stock dividends.

Other Option-Type Products

• Warrants: Options issued by a corporation result in new stock issuance when exercised.

• Executive Stock Options: Remuneration to executives; typically at-the-money when issued.

  • Vesting period: 1 to 4 years; cannot be sold immediately.

• Convertible Bonds: Bonds convertible to equity, often callable to compel earlier conversion.

• Stocks: Can be viewed as call options on firm value, expressed as $( ext{Firm Value} - ext{Debt})^+$.

Summary

• Basic terminologies and mechanisms of options trading, including payoffs and P&Ls, market making, margin requirements, and underlying assets in options markets. Understanding payoff structures from different derivative positions.