Jacoby, Stangeland and Wajeeh, Options A European Call Option gives the holder the right to buy the underlying asset for a prescribed price (exercise/strike price), on a prescribed date (expiry date). A European Put Option gives the holder the right to sell the underlying asset for a prescribed price (exercise/strike price), on a prescribed date (expiry date). American Options exercise is permitted at any time during the life of the option (call or put). Chapter 21

Jacoby, Stangeland and Wajeeh, Underlying Asset (S) The specific asset on which an option contract is based (e.g. stock, bond, real-estate, etc.). For traded Stock Options: one call (put) option contract represents the right to buy (sell) 100 shares of the underlying stock. Strike/Exercise Price (E) The specified asset price at which the asset can be bought (sold) by the holder of a call (put) if s/he exercised his/her right. Expiration Date (T) The last day an option exists.

Jacoby, Stangeland and Wajeeh, Writer: Seller of an option (takes a short position in the option). Holder: Buyer of the option (takes a long position in the option). Elements of an option contract: u type (put or call) u style (American or European) u underlying asset (stock/bond/etc…) u unit of trade u exercise price u expiration date

Jacoby, Stangeland and Wajeeh, Holding A European Call Option Contract- An Example European style IBM corp. September 100 call: entitles the buyer (holder) to purchase 100 shares of IBM common stock at $100 per share (E), at the options expiration date in September (T). At the options expiration date (T): For the Call Option Holder If S T > E=$100: Exercise the call option - pay $100 for an IBM stock with a market value of S T (e.g. S T =$105). Payoff at T: S T - E = $105-$100=$5 > 0. If S T Ÿ E=$100: Can buy IBM stocks in the market for S T (e.g. S T =$90). Holder will not choose to exercise (option expires worthless). Payoff at T: $0.

Jacoby, Stangeland and Wajeeh, Holding a European Call Conclusion: A call option holder will never lose at T (expiration), since his/her payoff is never negative: If S T Ÿ E=$100 If S T > E=$100 Call option value at T$0 S T - E = S T - $100 Payoff at T 0 E=$100 STST

Jacoby, Stangeland and Wajeeh, For the Call Option Writer (Short Seller) If S T > E = $100: Holder will exercise. Writer will deliver an IBM stock with a market value of S T ($105) to the holder, in return for E dollars ($100). Payoff at T: E-S T = $100-$105= - $5 < 0. If S T Ÿ E = $100: Holder will not exercise. Payoff at T: $0.

Jacoby, Stangeland and Wajeeh, For the Call Option Writer (Short Seller) Conclusion: A call option writer will never gain at T (expiration), since his/her payoff is never positive: If S T Ÿ E=$100 If S T > E=$100 Call option value at T$0 E - S T = $100 - S T Payoff at T 0 E=$100 STST

8 Holding A European Put Option Contract- An Example European style IBM corp. September 100 put entitles the buyer (holder) to sell 100 shares of IBM corp. common stock at $100 per share (E), at the option’s expiration date in September (T). At the options expiration date (T): For the Put Option Holder If S T < E=$100: Exercise the put option - receive $100 for an IBM stock with a market value of S T (e.g. S T =$90). Payoff at T: E - S T = $100-$90=$10 > 0. If S T ¦ E=$100: Can sell IBM stocks in the market for S T (e.g. S T =$105). Holder will not choose to exercise (option expires worthless). Payoff at T: $0.

Jacoby, Stangeland and Wajeeh, Holding A European Put Conclusion: A put option holder will never lose at T, since his/her payoff is never negative: If S T ¦ E=$100 If S T < E=$100 Put option value at T$0 E - S T = S T Payoff at T 0STST $100 E=$100

Jacoby, Stangeland and Wajeeh, For the Put Option Writer (Short Seller) If S T <E = $100: Holder will exercise. Writer will pay E dollars ($100) in return for an IBM stock (worth S T = $90). Payoff at T: S T - E = $90-$100 = -$10< 0. If S T ¦ E = $100 : Holder will not exercise. Payoff at T: $0.

11 Conclusion: A put option writer will never gain at T, since his/her payoff is never positive: If S T ¦ E=$100 If S T < E=$100 Put option value at T$0 S T - E = S T Payoff at T 0 E=$100 STST - $100 For the Put Option Writer (Short Seller)

Jacoby, Stangeland and Wajeeh, E=$85 Combinations of Options You purchase a BCE stock, and simultaneously write (short sell) the July $85 European call option. Your payoff diagram at expiration in July (T): Payoff at T 0 STST Buy Stock Payoff at T 0 STST Short Sell Call Payoff at T 0 STST Combination E=$85 Same as Short Sell Put & Buy T-bill $85

13 The Put-Call Parity Relationship* You purchase the BCE stock, the July $85 put option, and short sell the July $85 call option (both options are European). Your payoff at expiration in July (T): If S T = $100If S T = $80 Stock (S) Put (P) Call(C) Total (Certain) Payoff at T: To calculate the PV of the certain payoff ($85=E) today, we use the risk-free rate: *Only for European Options

Jacoby, Stangeland and Wajeeh, You purchase the BCE stock, the July $85 European put option, and short sell the July $85 European call option Your payoff at expiration in July (T): The Put-Call Parity Portfolio Payoff at T 0 STST Buy Stock STST Short Sell Call STST Combination E=$85 A Certain Payoff of E=$85 E=$85 Buy Put E=$85 $85

15 Using Put-Call-Parity (PCP) to Replicate Securities A synthetic security l Definition - A portfolio of other securities which will pay the same future cash flows as the security being replicated. l Since payoffs at expiration (cash flows) are the same for the synthetic security and the original security under all states of the world, their current prices must be identical. l Otherwise, if one is currently cheaper than the other, an arbitrage opportunity will exist: buy (long) the cheaper security today for the lower price, and simultaneously short sell the expensive security for the higher price. This results in a positive initial cash flow. l This positive cash flow is an arbitrage profit (“free lunch”), since at expiration, the cash flows from both positions will offset each other, and the total cash flow at expiration will be zero. From this point forward we notate: S o = S, P o = P, and C o = C

Jacoby, Stangeland and Wajeeh, uThe Put-Call-Parity (PCP) Relationship: This is a risk free T-bill that pays E dollars in T years uRecall that the PCP portfolio was created by: Long one stock (+S), Long one put (+P), and Short one call (-C) uWe saw that this is equivalent to: Long a T-bill (+Ee - Tr f ) uThus, we replicated a long position in a T-bill with: long stock, long put and short call. uFor security replication purposes, use PCP with the following rule: Long is “+” Short is “-” How Do We Replicate Securities?

17 A Synthetic Stock u We first rearrange the PCP equation to isolate S: u According to the above replication rule: Long one stock (+S) = Long a T-bill (+Ee - Tr f ) & Long one call (+C) & Short one put (-P), u The payoff (cash flow) at maturity: S T E Long T-bill Long Call Short Put Total Replicated Payoff: +S T +S T u Conclusion - holding the replicated portfolio is the same as holding the stock

18 A Synthetic Call u We first rearrange the PCP equation to isolate C: u According to the above replication rule: Long one call (+C) = Long one stock (+S) & Long one put (+P) & Short a T-bill (-Ee - Tr f ) u The payoff (cash flow) at maturity: S T E Long Stock Long Put Short T-bill Total Replicated Payoff: $0 S T - E u Conclusion - holding the replicated portfolio is the same as holding a call

19 A Synthetic Put u We first rearrange the PCP equation to isolate P: u According to the above replication rule: Long one put (+P) = Long a T-bill (+Ee - Tr f ) & Long one call (+C) & Short one stock (-S) u The payoff (cash flow) at maturity: S T E Long T-bill Long Call Short Stock Total Replicated Payoff: E - S T $0 u Conclusion - holding the replicated portfolio is the same as holding a put

20 Bounding The Value of An American Call The value of an American call can never be: ubelow the difference b/w the stock price (S) and the exercise price (E). u If C < S - E: investors will pocket an arbitrage profit. u Example: S = $100, E = $90, C = $8=>C = 8 < 10 = S – E u Arbitrage Strategy: uBuy the call for $8, and exercise it immediately by paying the exercise price ($90)to get the stock (worth $100). This results in an immediate arbitrage profit (i.e. “free lunch”) of: = $2 uExcess demand will force C to rise to $10 u As long as there is time to expiration, we will have C > $10 = S - E uAbove the value of the underlying stock (S) u If it is, buy the stock directly Boundary Conditions Payoff at t 0 E=$100 StSt 45 0 value of the American call will be here

21 For an American Call: C = C (S, E, T, , r) (+) (-) (+) (+) (+) S - The higher the share price now, the higher the profit from exercising. Thus the higher the option price will be. E - The higher the exercise price now, the more it needs to be paid on exercise. Thus, the lower the option value will be. T - The more time there is to expiration, the higher the chance that the stock price will be higher at T, and the higher the option value will be.  - The larger the volatility, the more probable a profitable outcome, thus the higher C is. r - The higher the interest rate, the P.V. of the future exercise price decreases. The call price will increase. For an American Put: P = P(S, E, T, , r) (-) (+) (+) (+) (-) Determinants of American Option Pricing

Jacoby, Stangeland and Wajeeh, Determinants of American Option Pricing Determinants ofRelation toRelation to Option PricingCall OptionPut Option Stock pricePositiveNegative Strike price NegativePositive Risk-free rate PositiveNegative Volatility of the stockPositivePositive Time to expiration datePositivePositive