Options Chapters 8, 9, and 10 Fin 288
Option Terminology • Call Option – the right to buy an asset at some point in the future for a designated price. • Put Option – the right to sell an asset at some point in the future at a given price
Review of Option Terminology Expiration Date The last day the option can be exercised (American Option) also called the strike date, maturity, and exercise date Exercise Price The price specified in the contract American Option Can be exercised at any time up to the expiration date European Option Can be exercised only on the expiration date
Review of Option Terminology • Long position: Buying an option • Long Call: Bought the right to buy the asset • Long Put: Bought the right to sell the asset • Short Position: Writing or Selling the option • Short Call Agreed to sell the other party the right to buy the underlying asset, if the other party exercises the option you deliver the asset. • Short Put - Agreed to buy the underlying asset from the other party if they decide to exercise the option.
Review of Terminology • In - the - money options • when the spot price of the underlying asset for a call (put) is greater (less) than the exercise price • Out - of - the - money options • when the spot price of the underlying asset for a call (put) is less (greater) than the exercise price • At the money options • when the exercise price and spot price are equal.
Underlying Assets • Stock Options • Traded on: CBOE, Philadelphia Stock Exchange, Pacific Exchange, American Stock Exchange, International Stock Exchange • Traded in 100 share blocks • Foreign Currency Options • Mainly over the counter and on Philadelphia Exchange • Index Options • One contract is for 100 times the index (S&P for example)
Other underlying Assets • Interest Rate Options • Traded on CBOE on treasury securities. • Interest rate Options are traded on 13 Week, 5 year, 10 year and 30 year treasury securities • Futures Options • Traded on the same exchange as the futures contract
Options on Futures • Options on futures are as popular or even more popular than on the actual asset. • Options on futures do not require payments for accrued interest. • The likelihood of delivery squeezes is less. • Current prices for futures are readily available, they are more difficult to find for bonds.
Futures Options • Call option holder will own a long futures position if the option is exercised. • The writer of the call option accepts the corresponding short position at the exercise price.
Mechanics of Options on Futures • Call Option Example Exercise price = $85 Current Futures price = $95 Upon exercise both the long position and the short owned by the writer of the short option is set to $85. When marked to market the holder of the long makes $10, the holder of the short looses $10. The holders of the short and long position then face the same risks as any other holder of the futures contract.
Buyer Margin Requirements on Futures Options • The buyer of the call option is not required to place any margin deposits. The most that could be lost is the cost of the option.
Margins on Stock Options • Stock purchases are allowed to be purchased on margin. • An option on must be purchased with cash since they already leverage the underlying asset. • When an option is written the writer must maintain a margin account to help prevent default.
Writing Naked Options • Naked Option- No offsetting position in the underlying stock • Written Naked Call Options – Margin is the greater of: • 100% of the proceeds of the sale plus 20% of the underlying share price minus any amount by which it is out of the money • 100% of the option proceeds plus 10% of the underlying share price
Margin on Put Options • Written Naked Put Options • 100% of the proceeds of the sale plus 20% of the underlying share price minus any amount by which it is out of the money • 100% of the option proceeds plus 10% of the underlying exercise price
Covered Calls • Covered Calls – writing an option when you also own the underlying stock. • No margin is required if the options is out of the money or in the money • If it is in the money the share price is reduced by the amount it is in the money for the purpose of calculating the equity position.
Call Option on Futures Writer Margin Requirements • The writer of the call option accepts all of the risk since the buyer will not exercise if there would be a loss. • The writer is required to deposit the original margin that would be required on the futures contract and the option price that is received for writing the option. The writer is also required to deposit variation margin as the contract is marked to market.
Position Limits and Exercise Limits • CBOE and most other exchanges have a limit on the number of positions on the same side of the market. Long Calls and Short Puts would be considered to be on the same side of the market. • Large cap stocks 75,000 contracts, small cap stocks 60,000, 31,500, 22,500 or 13,500 contracts • Exercise limit is often equal to the position limit
Basic Call Option Profit • Call option – as the price of the asset increases the option is more profitable. • Once the price is above the exercise price (strike price) the option will be exercised • If the price of the underlying asset is below the exercise price it won’t be exercised – you only loose the cost of the option. • The Profit earned is equal to the gain or loss on the option minus the initial cost.
Profit Diagram Call Option(Long Call Position) Profit Spot Price Cost S-X-C S X
Call Option Intrinsic Value • The intrinsic value of a call option is equal to the current value of the underlying asset minus the exercise price if exercised or 0 if not exercised. • In other words, it is the payoff to the investor at that point in time (ignoring the initial cost) the intrinsic value is equal to max(0, S-X)
Payoff Diagram Call Option Payoff Spot Price S-X X S
Example: Naked Call Option • Assume that you can purchase a call option on an 8% coupon bond with a par value of $100 and 20 years to maturity. The option expires in one month and has an exercise price of $100. • Assume that the option is currently at the money (the bond is selling at par) and selling for $3. • What are the possible payoffs if you bought the bond and held it until maturity of the option?
Five possible results • The price of the bond at maturity of the option is $100. The buyer looses the entire purchase price, no reason to exercise. • The price of the bond at maturity is less than $100 (the YTM is > 8%). The buyer looses the $3 option price and does not exercise the option.
Five Possible Results continued • The price of the bond at maturity is greater than $100, but less than $103. The buyer will exercise the option and recover a portion of the option cost. • The price of the bond is equal to $103. The buyer will exercise the option and recover the cost of the option. • The price of the bond is greater than $103. The buyer will make a profit of S-$100-$3.
Profit Diagram Call Option(Long Call Position) Profit Spot Price -3 S-100-3 S 103 100
Price vs. Rate • Note buying a call on the price of the bond is equivalent to buying a put on the interest rate paid by the bond. • As the rate decreases, the price increases because of the time value of money.
Profit Diagram Call Option(Short Call Position) Profit Spot Price S X C+X-S
Put option payoffs • The writer of the put option will profit if the option is not exercised or if it is exercised and the spot price is less than the exercise price plus cost of the option. • In the previous example the writer will profit as long as the spot price is less than $103. • What if the spot price is equal to $103?
Put Option Profits • Put option – as the price of the asset decreases the option is more profitable. • Once the price is below the exercise price (strike price) the option will be exercised • If the price of the underlying asset is above the exercise price it won’t be exercised – you only loose the cost of the option.
Profit Diagram Put Option Profit Spot Price Cost X-S-C S X
Put Option Intrinsic Value • The intrinsic value of a put option is equal to exercise price minus the current value of the underlying asset if exercised or 0 if not exercised. • In other words, it is the payoff to the investor at that point in time (ignoring the initial cost) the intrinsic value is equal to max(X-S, 0)
Payoff Diagram Put Option Profit Spot Price Cost X-S X S
Profit Diagram Put OptionShort Put Profit Spot Price S X S-X+C
Pricing an Option • Arbitrage arguments • Black Scholes • Binomial Tree Models
PV and FV in continuous time • e = 2.71828 y = lnx x = ey FV = PV (1+k)n for yearly compounding FV = PV(1+k/m)nm for m compounding periods per year As m increases this becomes FV = PVern =PVert let t =n rearranging for PV PV = FVe-rt
Lower Bound of Call Option Price • Assume that you have an asset that does not pay a cash income (A non dividend paying stock for example) • Consider the case of an option as it expires. • In this case, regardless of whether it is an American or European option it will be worth its intrinsic value (max(S-X,0)). • Assuming a positive value the lower bound is given by: S - X
Formal Argument • Consider two portfolios • A: One European call option on the stock of Widget Inc. plus cash equal to Xe-rT • B: One share of stock in Widget Inc. • Note: If the cash in portfolio A is invested at r, it will grow to be worth X at time T.
Portfolio A • There are two possible outcomes at time T depending upon the value of S at time T • ST > X Exercise the option and purchase the asset with a current value of ST (The value of portfolio A at time T is ST ). • ST < X Do not exercise the option, The portfolio is then worth the value of the cash, X. Therefore the portfolio is worth: max(ST,X)
Portfolio B • The value of portfolio B is simply the value of the stock at time T, ST.
Comparing A to B • Combining the two results it is easy to see that portfolio A (the option and the cash) is always worth at least as much as portfolio B (owning the stock), and sometimes it is worth more than B.Without arbitrage, the same relationship should be true today as well as at time T in the future.
Equal value of the portfolios today Let c be the call price (value of option) today. Then the value of portfolio A is: c + Xe-rT The value of portfolio B is: S Since the value of A is always worth as much as B and sometimes it is worth more: c + Xe-rT >S or rearranging c > S - Xe-rT
Final result The worst outcome to buying a call option is that it expires worthless, so the option is worth either nothing or S-Xe-rT Therefore: c > max(S - Xe-rT,0)
Put Option • Similar to a call option the put option should always have a positive value. • Considering the case of an option as it expires (either an American or European Option), the value of the option should be equal to its intrinsic value. The lower bound is therefore: X - S
European Put Option • Again, in the case of a European option prior to maturity this equation will not hold and it is necessary to account for the time value of money. In this case the lower bound for the option is given by: Xe-rT - S
Formal Argument: • Consider two Portfolios C: One European put option plus one share and D: An amount of cash equal to Xe-rT
Portfolio C: • There are two possibilities: • ST < X Exercise the option at time T and the portfolio is worth X. • ST > X The option expires and the portfolio is worth STPortfolio C is therefore worth max(ST,X)
Portfolio D • Investing the amount at a rate equal to r, the portfolio will be worth X at Time T. • Combining the two arguments it is easy to see that portfolio C is always worth at least the same amount as portfolio D and sometimes it is worth more.
Comparing the two Let p = the value (price) of the put option Without arbitrage opportunities:p + S > Xe-rT or rearranging p > Xe-rT - S the value of the put option is then given as p > max(Xe-rT-S,0)
Put Call Parity • Consider portfolio A and C above A: One European call option plus an amount of cash equal to Xe-rTC: One European put option plus one share • Both portfolios are have a value of max(X,ST) at the expiration of the options. If no arbitrage opportunities exist, they should also have the same value today which implies: c + Xe-rT = p + S
Put Call Parity • In other words, the value of a European call with a given exercise date can be deduced from the value of a European put with the same exercise date and exercise price.