04 Properties of Financial Options

Call and put options

• A call option gives the holder the right (not the obligation) to buy an asset at a pre-specified time for a pre-specified price.

• A put option gives the holder the right (not the obligation) to sell an asset at a pre-specified time for a pre-specified price.

• European options can be exercised only at maturity

• American options can be exercised anytime up to maturity

• Stocks, bonds, commodities, currencies

• Collections of assets, like stock indices

• Other derivative instruments such as futures

Payoff function & profit/loss

• : Value of European (American) call option

• : Value of European (American) put option

• : Value of underlying asset at time 0 (T)

• : The pre-specified strike price

• : The pre-specified time-to-maturity

• : The volatility of the underlying asset return

• : Present value (PV) of dividends paid during the life of option

• : The T-period risk-free rate of return for maturity (continuously compounded)

• It refers to the expiry date

• It assumes that the investor keeps the position open up to expiry

• It does not take into account the initial cost of the option

• It depends on the option type (call/put) and on the position (long/short)

• Long call:

• Short call:

• Long put:

• Short put:

• Long call:

• Short call:

• Long put:

• Short put:

Other Issues

• Long call:
• Bullish expectations
• In a bear market the potential losses are limited
• BUT in a bull market the call buyers gains less than a holder of the underlying asset (cost for pretection in a bear market)

• Short call
• Bearish expectations
• Receives the premium
• Hedging a long position in the underlying asset (covered call writing)

• Long put
• Bearish expectations
• In a bull market, the potential losses are limited
• BUT in a bear market the put buyer gains less than a seller of the underlying asset (cost for protection in a bull market)
• Hedging a long position in the underlying asset (protective)

• Short put
• Bullish expectations
• Receives the premium

• In-the-money (ITM): Immediate exercise yields a positive payoff

• At-the-money (ATM): Immediate exercise yields a zero payoff

• Out-of-the-money (OTM): Immediate exercise yields a negative payoff

• The price of an option consists of two parts
• Intrinsic value and time value

• The American option price consists of two parts:
• The revenues from exercising immediately the option → intrinsic value
• The added value the option will have by postponing its exercise → time value

• Max(payoff, 0)

• Holds even if the option cannot be exercised at this point in time

• ITM (OTM) options have positive (negative) payoff and hence, a positive (zero) intrinsic value

• Value of the option associated with the possibility that the option could move further ITM as time passes.

• Time value is defined as: Time value = Option price - intrinsic value

• The time value is non-negative and is highest for ATM options (where the ITM start to increase)

• For America options, the less the time value, the more likely the options are exercised earlier.

Institutional setting

• Strike price (usually scattered around the current stock price)

• Delivery date (e.g. the 3 rd Friday during the month)

• Contract size, etc

• Market makers must quote bid and ask prices when requested

• They do not know whether the counter party intends to buy or sell

• The seller of the option must post margin as a guarantee that the payoff on the option will be made (seller has the obligation if the buyer want to exercise their rights)

• The buyer of the option usually pays for the option upfront and so no margin is required

Corporate events

• Exchange traded options are NOT adjusted for cash dividends
• An exception is sometimes made for large cash dividends
• Example: You own a call option on IBM stock () with . IBM suddenly decides to pay a dividend over per share. In the absense of taxes, IBM’s share price drops to . And the former ITM call option is suddenly OTM

• The strike price is adjusted for stock splits
• Do not change the assets or earning ability of the company
• A n-for-m split leads to a new strike price of and the number of options changes to
• Example: IBM decide a 2-for-1 split, the share price drops to 60 (120/2). The strike price is also reduced to half its former value, namely to 50. Each option is exchange for two new ones.

Factors Affecting Option Prices

• Call option:
• Payoff =
• Becomes more valuable as increases

• Put option:
• Payoff =
• Becomes less valuable as increases

• Call option:
• Payoff =
• Becomes less valuable as increases

• Put option:
• Payoff =
• Becomes more valuable as increases

• American call and put options
• Both become more valuable (or can exercise earlier)

• European call and put options
• Usually both become more valuable as T increases

• Call (put) holders benefit from price increases (decreases), but only have limited loss (the option premiun) when goes the other side

• Higher interest rates increase the expected return required by investors from the underlying asset

• Higer interest rates increase discount rates and hence, decrease the present value of future cash flows received by the option holder.

• Hence, call (put) options become more (less) valuable as increases

Option Price Bounds

• Call options price is lower than the share itself, or there are arbitrage opportunities.

• When , the arbitrage strategy is to buy the stock and sell the call option

• Put options price is lower than the strike price , or there are arbitrage opportunities. That is and (because American put option can be exercised earlier)

• When , the arbitrage strategy is to sell the put option and invest the proceeds at the risk-free rate

• Construct two portfolios:
• A: One European call + a zero-coupon bond providing a payoff of at time .
• B: One share of the stock

• The cash flow at time T is

• Construct two portfolios:
• C: One European put + one share
• D: A zero-coupon bond providing a payoff of at time

• The cash flow at time T is

Put-Call Parity

• No transaction costs

• All trading profits are subject to the same tax rate

• Possible to borrow or lend at the risk-free rate

• No arbitrage opportunities

• European calls and puts

• Same strike price

• Same time-to-maturity

• The underlying asset does not pay any dividends

• A: One European call + a zero-coupon bond providing a payoff of at time .

• B: One European put + one share of the stock.

Exercise American Options before Maturity?

• No income (dividends) are sacrificed

• Delay the payment of the strike price (one dollar today is worth more than that in the future)

• Holding the call provides insurance against low

• Holding the put (in conjunction with the underlying asset) provides insurance against low

• It may be optimal to forgo this insurance to realize the strike price immediately (one dollar today is worth more than one dollar tomorrow…)

Trading Strategies with Options

• Tabulating and plotting the payoff/profit patterns of the single assets

• Adding them up

• Buy a European call with a strike price

• Sell a European call with a strike price

• Both calls have the same expiration date

• Speculate that the underlying asset value will increase

• Cheaper than purchasing call because you give up some of the up-side potential

• Gives downward protection compared to forwards (for speculator)

• Sell a European put with a strike price

• Buy a European put with a strike pirce

• Both puts have the same expiration date

• Speculate that the underlying asset value will decrease

• Cheaper than purchasing a put because you give up some of the up-side potential

• Gives downward protection compared to forward (for speculators)

• Buy a European call with a strike price

• Buy a European put with a strike price

• Both options have the same expiration date and the same strike price

• Speculate that the underlying asset will move by a large amount (either up or down)

• To profit from this strategy, your forecast of volatility must be higher than that of the market