Week 9 Options PDF Lecture Notes

Week 9 Options 1 © 2024 ICMA Centre, Henley Business School  Options Introduction to Options Option Pricing Basics Portfolio of...

Week 9 Options 1 © 2024 ICMA Centre, Henley Business School  Options Introduction to Options Option Pricing Basics Portfolio of Options 2 © 2024 ICMA Centre, Henley Business School  What is an Option? An option is a contract giving the holder a choice that affects the cash flows of the contract A call option gives the holder the right, but not the obligation, to buy a specified asset (the underlying) at a specified date (the expiration or maturity date) for a specified price (the strike or exercise price) – The seller has the obligation to sell the asset to the buyer for the specified price if the buyer demands it A put option gives the holder the right, but not the obligation, to sell a specified asset (the underlying) at a specified date (the expiration or maturity date) for a specified price (the strike or exercise price) – The seller has the obligation to buy the asset from the buyer for the specified price if the buyer demands it European options can only be exercised at the expiration date American options can be exercised at any time before or at expiration 3 © 2024 ICMA Centre, Henley Business School  Other Option Examples Holders of convertible preferred stocks and bonds have the option to convert their securities into common stocks Many companies offer some compensation in the form of employee stock options Most corporate bonds are callable Issuing firm can “call" its debt by paying the strike price (usually slightly above the par value) Mortgage loans usually have a refinancing option – Borrower can pay off existing loan with proceeds of new loan “Real options" e.g. option to choose timing of investment 4 © 2024 ICMA Centre, Henley Business School  “Money-ness” At-the-money option (St = K for a call or a put) In-the-money option (St > K for a call; St < K for a put) Out-of-the-money option (St < K for a call; St > K for a put) 5 © 2024 ICMA Centre, Henley Business School  Payoffs from Options These option payoffs (at expiration date T) can be written in the form: – Long Call: Max[0, ST - K] or Short Call: Min[0, K - ST] – Long Put: Max[0, K - ST] or Short Put: Min[0, ST - K] 6 © 2024 ICMA Centre, Henley Business School  Long Call Option (Buyer) Profit from buying an Intel European October call option – option price c = $1.15, K = $22.50, T = 5 months Call option allows stock purchase at $22.50 Buyer has an option to exercise the call option when stock price is higher than $22.50: ST – K - c Buyer pays premium (option price $1.15) to enjoy positive payoffs: - c 7 © 2024 ICMA Centre, Henley Business School  Short Call Option (Seller) Profit from writing (selling) an Intel European October call option – option price c = $1.15, K = $22.50, T = 5 months Call option forces stock sale at $22.50 Seller has obligation to sell the stock at $22.50 when stock price is higher than $22.50: c + K - ST Seller earns premium (option price $1.15) to offset negative payoffs: c 8 © 2024 ICMA Centre, Henley Business School  Long Put Option (Buyer) Profit from buying an Intel European July put option – option price p = $1.60, K = $20.00, T = 2 months Put option allows stock sale at $20 Buyer has an option to sell the stock at $20 when stock price is lower than $20: K – ST - p Buyer pays premium (option price $1.6) to enjoy positive payoffs: - p 9 © 2024 ICMA Centre, Henley Business School  Short Put Option (Seller) Profit from writing an Intel European July put option – option price p = $1.60, K = $20.00, T = 2 months Put option forces stock purchase at $20 Seller has obligation to buy the stock at $20 when stock price is lower than $20: p + ST - K Seller earns premium (option price $1.6) to offset negative payoffs: p 10 © 2024 ICMA Centre, Henley Business School  Option Exercise Strategy at Expiration Date Rights and Obligations – An option holder (long) has the right, not obligation, to exercise it – An option writer (short) has the obligation to pay if the option is exercised The current market price of a stock is $100. You hold an expiring option. Should you exercise it, if... – It is a call option with exercise price $90? Payoff? Yes; $10 – It is a put option, with exercise price also $90? Payoff? No – It is a call option, with exercise price $120? Payoff? No – It is a put option, with exercise price also $120? Payoff? Yes; $20 11 © 2024 ICMA Centre, Henley Business School  Notation c: European call option price p: European put option price C: American Call option price P: American Put option price S0: Stock price today ST: Stock price at time T (St, 0 ≤ t ≤ T) K: Strike price T: Life of option σ: Volatility of stock price D: Present value of dividends during options life d: Dividend rate with continuous compounding r : Risk-free rate for maturity T with continuous compounding 12 © 2024 ICMA Centre, Henley Business School  Payoff and Intrinsic Value Option Value = Intrinsic Value + Time Value – Intrinsic value is the value if the option is exercised immediately (i.e. Max[St – K, 0] for a call, where 0 ≤ t ≤ T ) – Time value is the extra value if option is held rather than exercised immediately Suppose you hold a call option on Lotus, with $60 strike price and one year maturity. – Price of option = $1.75 But St = $55 < K = $60, so the payoff from exercising the option today (t) is $0, i.e. you do not exercise. This is an out of the money option. The current payoff of the option is $0, so why is its price positive? – Stock price could increase before expiration – The likelihood of positive payoff at T is positive – $1.75 (Option Value) = $0 (Intrinsic Value) + $1.75 (Time Value) 13 © 2024 ICMA Centre, Henley Business School  Payoff and Intrinsic Value Suppose you hold a call option on AT&T, with $50 strike price and one- year maturity. – Price of option = $6.375 But St = $55 > K = $50, so the payoff from exercising the option today (t) is $5, i.e. you would exercise. This is an in-the-money option Why is the current price of the option greater than $5? – Stock price could increase further – The likelihood of even greater payoff at T is positive – $6.375 (Option Value) = $5 (Intrinsic Value) + $1.375 (Time Value) Suppose you have a call option on ABC, currently trading at $35, with a strike of $30. The price of the 1-year call option on ABC is $7. Can you guess the price of the put option on the same stock, with the same strike price and expiry date? 14 © 2024 ICMA Centre, Henley Business School  Options Monitor 15 © 2024 ICMA Centre, Henley Business School  Basic Assumptions of Option Pricing There are no transaction costs All trading profits (net of trading losses) are subject to the same tax rate Borrowing and lending at the risk-free rate is possible Option prices are arbitrage-free 16 © 2024 ICMA Centre, Henley Business School  Put-Call Parity for European Options; No Dividends Consider two portfolios: Both are worth Max[ST, K] at maturity T They must, therefore, be worth the same at time 0: c + Ke-rT = p + S0 17 © 2024 ICMA Centre, Henley Business School  Upper Bounds An American option is worth at least as much as the corresponding European option For call options we must have otherwise we can profit from buying the stock and selling the call Similarly, for put options we must have otherwise we can profit from writing the put For a European put, at time T, the option will not be worth more than K, so 18 © 2024 ICMA Centre, Henley Business School  Lower Bound on European Call For a lower bound on a European call on a stock option with no dividends, we compare two portfolios: At time T: the payoff of Portfolio 2 the payoff of Portfolio 1 To avoid arbitrage, this relationship must also hold at time 0: 19 © 2024 ICMA Centre, Henley Business School  Lower Bound on European Put For a lower bound on a European put on a stock option with no dividends, we will also compare two portfolios: At time T: the payoff of Portfolio 2 the payoff of Portfolio 1 To avoid arbitrage, this relationship must also hold at time 0: 20 © 2024 ICMA Centre, Henley Business School  American vs. European Options The only (but important) difference is that American options can be exercised early – American option cannot be worth less than European option – C = c + value of right to exercise immediately – P = p + value of right to exercise immediately Compare American call option to European call option: 21 © 2024 ICMA Centre, Henley Business School  Never Early Exercise American Call Option (No Dividends) For an American call on a non-dividend paying stock, immediate exercise at t yields: St - K, where 0 < t < T Given the lower bound and upper bound of European call option, we have – Note, replace time 0 by t, and T by T - t in the European call option lower bound table. Subscript t represents value at t Because r > 0 and T- t > 0 For non-dividend stocks, it is not optimal to exercise the American call early: C = c 22 © 2024 ICMA Centre, Henley Business School  American Call vs. European Call (with dividends) Assume dividend of D paid shortly after 0, or D is the present value of all dividends between 0 and T Therefore – At time t: – Immediate exercise yields: St - K for 0 < t < T – Early exercise possibly optimal (especially for high dividends) 23 © 2024 ICMA Centre, Henley Business School  Early Exercise for American Put Options? Put Options may be exercised early if deep in the money – Main argument: time value of money – Receipt of K - St (> 0) at time t generates additional interest income until T Lower bound for European put: – at 0 – or in general at t Immediate exercise yields: Therefore, P > p 24 © 2024 ICMA Centre, Henley Business School  Extension of Option Prices with Dividends When dividends are paid we have the following lower bounds on option prices: c ≥ S0 – D - Ke-rT p ≥ D + Ke-rT - S0 Extensions of Put-Call Parity –American options, with no dividends S0 - K < C - P < S0 - Ke-rT European options, with dividends c + D + Ke-rT = p + S0 American options, with dividends S0 – D - K < C - P < S0 – Ke-rT 25 © 2024 ICMA Centre, Henley Business School  Factors affecting Option Prices The six factors are: – S0 : Current stock price – K : Strike price – T : Time to expiration – σ: Volatility of stock price – r : Risk-free interest rate – D : Present value of dividends during options life And in real life there are also: – Stock Borrowing and Lending costs – Taxes 26 © 2024 ICMA Centre, Henley Business School  Effect of Variables on Option Pricing c ≥ S0 – D – Ke-rT p ≥ D + Ke-rT - S0 27 © 2024 ICMA Centre, Henley Business School  The Greeks at a glance DELTA: The sensitivity of an option’s price to the movements of the underlying asset GAMMA: The sensitivity of an option’s price to the movements in Delta THETA: The sensitivity of an option’s price to the passage of time VEGA: The sensitivity of an option’s price to the changes in the volatility of returns of the underlying asset RHO: The sensitivity of an option’s price to changes in interest rates 28 © 2024 ICMA Centre, Henley Business School  Option Strategies Take a position in the option and the underlying Take a position in two or more options of the same type (spread) Take a position in a mixture of calls and puts (combination) All strategies presented in the next slides are assumed to be kept until expiration. The positions are not hedged or sold before expiry 29 © 2024 ICMA Centre, Henley Business School  Positions in the Option and Underlying (a) long stock: ST - S0, short call: Min {c, c + K - ST} (b) short stock: S0 -ST, long call: Max {- c, ST - K - c} (c) long stock: ST - S0, long put: Max {-p, K - ST - p} (d) short stock: S0 -ST , short put: Min {p, ST - K + p} 30 © 2024 ICMA Centre, Henley Business School  Bull Spread Using Calls Long a call with strike price K1 and short a call with strike price K2. Both calls have the same underlying and maturity – Option 1 price is higher than option 2 price – It is used when a moderate rise in the price of the underlying asset is expected – Maximum profit: K2 - K1 - (c1 - c2) – Maximum loss: c1 - c2 31 © 2024 ICMA Centre, Henley Business School  Bull Spread Using Puts Long a put with strike price K1 and short a put with strike price K2. Both puts have the same underlying and maturity – Option 2 price is higher than option 1 price – When stock price is higher than K2, both puts are not exercised – Maximum profit: p2 - p1 – Maximum loss: K2 - K1 - (p2 - p1) 32 © 2024 ICMA Centre, Henley Business School  Bear Spread Using Calls Long call with strike price K2 and short a call with strike price K1.Both calls have the same underlying and maturity – c1 > c2 – A strategy used when a moderate decline in the price of the underlying asset is expected – Maximum profit: c1 - c2 33 © 2024 ICMA Centre, Henley Business School  Bear Spread Using Puts Long a put with strike price K2 and short a put with strike price K1. Both puts have the same underlying and maturity – p 2 > p1 – Maximum profit: K2 - K1 - (p2 - p1) 34 © 2024 ICMA Centre, Henley Business School  Butterfly Spread Using Calls Long a call with strike price K1 , short two calls with strike price K2 and long a call with strike price K3. All calls have the same underlying and maturity – Maximum loss: c1 + c3 - 2c2 – Maximum return (K2 - K1) is achieved, when the price of the underlying asset price ST = K2 35 © 2024 ICMA Centre, Henley Business School  Butterfly Spread Using Puts Long a put with strike price K1, short two puts with strike price K2 and long a put with strike price K3. All puts have the same underlying and maturity 36 © 2024 ICMA Centre, Henley Business School  Straddle Long a call and a put with strike price K. Both options have the same underlying and maturity – A straddle is used if an investor believes that a stock's price will move significantly, but is unsure as to which direction – Maximum return is achieved, when the price of the underlying asset to be away from K Profit K ST 37 © 2024 ICMA Centre, Henley Business School  Strangle Long a put with strike price K1 and long a call with strike price K2. Both options have the same underlying and maturity – Cheaper than straddle, but a higher underlying move is needed to make a profit Profit K1 K2 ST 38 © 2024 ICMA Centre, Henley Business School