Option Market

Options and Futures

Zhiyu Fu

Options vs. Forwards & Futures

  • Rights vs. obligations
    • Futures: Both rights and obligations for the long and short positions
      • e.g., Buyers/sellers agree to buy/sell the stock at $50 in 3 months
    • Options:
      • Buyers (long position): Right but not obligation;
        • e.g., Buyers can buy the stock at $50 in 3 months
      • Sellers (short position): Obligation
        • e.g., If the buyer exercises, the seller must sell the stock at $50 in 3 months
  • Upfront payment
    • Futures: Nothing (except margin requirements) to enter into a forward/futures contract
    • Options: Options separate rights from obligations and hence have value, there is a cost to acquire an option

What are the underlying assets?

  • Most commonly, stocks;
    • Exchange-traded products (e.g., ETF), indicies;
  • Foreign currency;
  • Futures;
  • Events (e.g., CME event contracts, ForecastEx)
    • You can trade it using many serious brokers e.g. IBKR;
    • Pure speculation not encouraged

Call Options

Call Options

A call option is a contract that gives the holder of the call option the right, but not the obligation, to buy an underlying asset by a certain date (the expiration date or maturity date T) for a certain price (the exercise price or strike price K).

  • The seller of a call option, also known as the call writer, has the obligation to sell the underlying asset by the expiration date for the exercise price.
  • The price of a call option is commonly referred to as the call premium.
  • Different types of call options:
    • European call option: Can only be exercised at the expiration date
      • Easier to analyze
    • American call option: Can be exercised at any time before the expiration date
      • More common in practice

Call Option Example

Example

An investor buys a European call option with the right to buy a certain stock at the strike price of $100 per share. The expiration date of the option is in 4 months, and the call option price is $5. Assuming 0% interest rate,

  • When can the options be exercised? Only at the expiration date.
  • What is the initial investment of getting this option? $5.

At the expiration/maturity date,

  • Scenario 1: If the stock price is $98:
    • The option will not be exercised.
    • The payoff/profit from the call option is $0/-$5.
  • Scenario 2: If the stock price is $102:
    • The option will    be      exercised.
    • The payoff/profit from the call option is $2/-$3.

Payoff&Profit Structure for a Call Option

\text{Payoff} = \max(S_T - K, 0)\\ \text{Profit at } T = \max(S_T - K, 0) - \text{C} e^{rT}

\text{Payoff} = -\max(S_T - K, 0)\\ \text{Profit at } T = - \max(S_T - K, 0) + \text{C} e^{rT}

Put Options

Put Options

A put option is a contract that gives the holder of the put option the right, but not the obligation, to sell an underlying asset by the maturity date T at the strike price K.

  • The seller of a put option, also known as the put writer, has the obligation to buy the underlying asset by the expiration date for the exercise price.

  • A long put option takes a bearish view on the underlying asset.

Put Option Example

Example

An investor buys an American put option with the right to sell a certain stock at the strike price of $100 per share. The expiration date of the option is in 4 months, and the put option price is $7. Assuming 0% interest rate,

  • When can the options be exercised? Any time before the expiration date.
  • What is the initial investment of getting this option? $7.

At the expiration/maturity date,

  • Scenario 1: If the stock price is $90:
    • The option will    be      exercised.
    • The payoff/profit from the put option is $10/$3.
  • Scenario 2: If the stock price is $102:
    • The option will not be exercised.
    • The payoff/profit from the put option is $0/-$7.

Payoff&Profit Structure for a Put Option

\text{Payoff} = \max(K - S_T, 0)\\ \text{Profit at } T = \max(K - S_T, 0) - \text{P} e^{rT}

\text{Payoff} = -\max(K - S_T, 0)\\ \text{Profit at } T = -\max(K - S_T, 0) + \text{P} e^{rT}

Summary of Call and Put Options

  • Long underlying assets:
    • Long call
    • Short put

  • Short underlying assets:
    • Short call
    • Long put

Moneyness of Options

Define S_t as the current price of the underlying asset and K as the strike price.

  • In-the-money (ITM):
    • Call: S_t > K
    • Put: S_t < K
  • At-the-money (ATM): S_t = K
  • Out-of-the-money (OTM):
    • Call: S_t < K
    • Put: S_t > K

An option is exercised only when it is in the money.

Intrinsic value of an option

Intrinsic value of an option

  • The intrinsic value of an option is defined as the maximum of zero and the value the option would have if it were exercised immediately. \text { Intrinsic Value }= \begin{cases}\max (S_t -K, 0) & \text { for a call } \\ \max (K-S_t, 0) & \text { for a put }\end{cases}

\text{Time Value} = \text{Option Price} - \text{Intrinsic Value}

  • At expiration, the value of an option is just its intrinsic value.
  • Before expiration, the value of an option is the sum of its intrinsic value and time value.
  • Assuming positive interest rates, time value is always non-negative for American options, but not necessarily for European options.

Example: Intrinsic Value of Options

Call

Consider a call option with a strike price of $40 exists with 21 days to expiration. Suppose this call is selling for $1.68. The underlying asset price is $41.12. Calculate the intrinsic value and the time value of the option.

  • Intrinsic value: \max(41.12 - 40, 0) = \$1.12
  • Time value: 1.68 - 1.12 = \$0.56

Put

Consider a put option with a strike price of $40 exists with 21 days to expiration. Suppose this put is selling for $5.68. The underlying asset price is $41.12. Calculate the intrinsic value and the time value of the option.

  • Intrinsic value: \max(40 - 41.12, 0) = \$0
  • Time value: 5.68 - 0 = \$5.68

Why Do People Trade Options?

  • Hedging:
    • Buying insurance to hedge against downside risk
  • Speculation on asset prices:
    • Speculating on the direction of the underlying assets with a limited loss
    • Create principal-protected products
    • Embedded leverage

      • e.g., SPX options on CBOE

      • Question: Should there be a margin requirement for the buyer of an option?

        • No. It’s a right but not an obligation.
        • Yes for the seller of an option.
  • Speculate on uncertainty: Options love uncertainty!

Contract Size

  • Call and put option contracts are in general written over several units of the underlying asset, such as 100 shares, but their prices are quoted per unit of the underlying asset.
  • For example, consider a call option contract on 100 shares of AAPL stock with strike $110.
    • the stock price is $122 at maturity
    • the price of an option to call one share in 3 months is $0.85
    • the contract size is 100 shares
  • the payoff and profit of a contract is:

\text{Payoff} = \max(122 - 110, 0) \times 100 = \$1200\\ \text{Profit} = 1200 - 0.85 \times 100 = \$1115

Traded Volume vs. Open Interest

  • For both call and put options, for every long position there is a corresponding short position, i.e., the contracts are in zero net-supply.
  • The total number of long positions, which is the same as the total number of short positions, is called open-interest.
  • The traded volume is the number of contracts that are bought or sold.
  • Note that a trader can buy a contract, then sell it the same day.
    • The volume for that day would be 2
    • but the open-interest would not change.
  • Put/call ratio: ratio of long puts vs. calls in the S&P 500, a common indicator for market sentiment.