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Options

Chapter 21. 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

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Options

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  1. Chapter 21 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). Jacoby, Stangeland and Wajeeh, 2000

  2. 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, 2000

  3. 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: • type (put or call) • style (American or European) • underlying asset (stock/bond/etc…) • unit of trade • exercise price • expiration date Jacoby, Stangeland and Wajeeh, 2000

  4. 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 ST > E=$100: Exercise the call option - pay $100 for an IBM stock with a market value of ST (e.g. ST=$105). Payoff at T:ST- E = $105-$100=$5> 0. If ST ŸE=$100: Can buy IBM stocks in the market for ST (e.g. ST=$90). Holder willnot choose to exercise (option expires worthless). Payoff at T: $0. Jacoby, Stangeland and Wajeeh, 2000

  5. Holding a European Call Conclusion: A call option holder will never lose at T (expiration), since his/her payoff is never negative: If ST ŸE=$100If ST > E=$100 Call option value at T $0 ST - E = ST - $100 Payoff at T 0 ST E=$100 Jacoby, Stangeland and Wajeeh, 2000

  6. For the Call Option Writer (Short Seller) If ST >E = $100: Holder will exercise. Writer will deliver an IBM stock with a market value of ST ($105) to the holder, in return for E dollars ($100). Payoff at T:E-ST = $100-$105= - $5 < 0. If ST ŸE = $100: Holder will not exercise. Payoff at T: $0. Jacoby, Stangeland and Wajeeh, 2000

  7. 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 ST ŸE=$100If ST > E=$100 Call option value at T $0 E - ST = $100 - ST Payoff at T 0 ST E=$100 Jacoby, Stangeland and Wajeeh, 2000

  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 ST < E=$100: Exercise the put option - receive $100 for an IBM stock with a market value of ST (e.g. ST=$90). Payoff at T:E - ST = $100-$90=$10> 0. If ST ¦E=$100: Can sell IBM stocks in the market for ST (e.g. ST=$105). Holder will not choose to exercise (option expires worthless). Payoff at T: $0.

  9. Holding A European Put Conclusion: A put option holder will never lose at T, since his/her payoff isnever negative: If ST ¦E=$100If ST < E=$100 Put option value at T $0 E - ST = 100 - ST Payoff at T $100 0 ST E=$100 Jacoby, Stangeland and Wajeeh, 2000

  10. For the Put Option Writer (Short Seller) If ST<E = $100: Holder will exercise. Writer will payE dollars ($100) in return for an IBM stock (worth ST = $90). Payoff at T:ST - E = $90-$100 = -$10< 0. If ST ¦E = $100: Holder will not exercise. Payoff at T: $0. Jacoby, Stangeland and Wajeeh, 2000

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

  12. 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): Buy Stock Short Sell Call Combination Payoff at T Payoff at T Payoff at T $85 0 0 0 ST ST ST E=$85 E=$85 Same as Short Sell Put & Buy T-bill Jacoby, Stangeland and Wajeeh, 2000

  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 ST = $100 If ST = $80 Stock (S) Put (P) Call(C) Total (Certain) Payoff at T: To calculate the PV of thecertain payoff ($85=E) today, we use the risk-free rate: *Only for European Options

  14. The Put-Call Parity Portfolio 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): Buy Stock Buy Put Short Sell Call Combination Payoff at T $85 E=$85 0 ST ST ST E=$85 E=$85 E=$85 A Certain Payoff of E=$85 Jacoby, Stangeland and Wajeeh, 2000

  15. Using Put-Call-Parity (PCP) to Replicate Securities A synthetic security • Definition - A portfolio of other securities which will pay the same future cash flows as the security being replicated. • 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. • 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. • 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: So = S, Po = P, and Co = C

  16. How Do We Replicate Securities? • The Put-Call-Parity (PCP) Relationship: This is a risk free T-bill that pays E dollars in T years • Recall that the PCP portfolio was created by: Long one stock (+S), Long one put (+P), and Short one call (-C) • We saw that this is equivalent to: Long a T-bill (+Ee-Trf) • Thus, we replicated a long position in a T-bill with: long stock, long put and short call. • For security replication purposes, use PCP with the following rule: Long is “+” Short is “-” Jacoby, Stangeland and Wajeeh, 2000

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

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

  19. A Synthetic Put • We first rearrange the PCP equation to isolate P: • According to the above replication rule: Long one put (+P) = Long a T-bill (+Ee-Trf) & Long one call (+C) & Short one stock (-S) • The payoff (cash flow) at maturity: ST < EST > E Long T-bill Long Call Short Stock Total Replicated Payoff:E - ST $0 • 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: • below the difference b/w the stock price (S) and the exercise price (E). • If C < S - E: investors will pocket an arbitrage profit. • Example:S = $100, E = $90, C = $8 => C = 8 < 10 = S – E • Arbitrage Strategy: • Buy 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: -8-90+100= $2 • Excess demand will force C to rise to $10 • As long as there is time to expiration, we will have C > $10 =S - E • Above the value of the underlying stock (S) • If it is, buy the stock directly Boundary Conditions value of the American call will be here Payoff at t 450 0 St E=$100

  21. Determinants of American Option Pricing 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) (-) (+) (+) (+) (-)

  22. Determinants of American Option Pricing Determinants of Relation to Relation to Option Pricing Call Option Put Option Stock price Positive Negative Strike price Negative Positive Risk-free rate Positive Negative Volatility of the stock Positive Positive Time to expiration date Positive Positive Jacoby, Stangeland and Wajeeh, 2000

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