Properties of Stock Options Chapter 10

Notation p : European put option price S0 : Stock price today c : European call option price p : European put option price S0 : Stock price today K : Strike price T : Life of option : Volatility of stock price C : American Call option price P : American Put option price ST :Stock price at option maturity D : Present value of dividends during option’s life r : Risk-free rate for maturity T with cont. comp. Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

Assumptions In this chapter we will assume the following are true in the finance environment; There are no transactions costs. All trading profits (net of trading losses) are subject to the same tax rate. Borrowing and lending are possible at the risk-free interest rate.

American vs European Options An American option is worth at least as much as the corresponding European option C  c P  p Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

Upper Bounds for Call Options European Call; 𝑐≤ 𝑆 0 American Call; 𝐶≤ 𝑆 𝑜

Upper Bounds for Put Options European Put; 𝑝≤ 𝐾𝑒 −𝑟𝑡 American Put; 𝑃≤𝐾

Lower Bound for European Call Option Prices; No Dividends c  max(S0 – Ke –rT, 0)

Calls: An Arbitrage Opportunity? Suppose that c = 3 S0 = 20 T = 1 r = 10% K = 18 D = 0 Is there an arbitrage opportunity?

Steps to follow at Call Options Check the lower bound of the option. See if the option is correctly priced or not. If c < max(S0 – Ke –rT, 0) then arbitrage exists and you can continue with following steps. If c ³ max(S0 – Ke –rT, 0) then there is no need to follow any more steps you can conclude with there is no arbitrage possibility Buy the option and short the stock Invest the proceeds until maturity is reached. At the end of maturity if 𝑆 𝑇 ≥𝐾 exercise the option and do not exercise the option if 𝑆 𝑇 <𝐾

Lower Bound for European Put Prices; No Dividends (Equation 10 Lower Bound for European Put Prices; No Dividends (Equation 10.5, page 240) p  max(Ke –rT – S0, 0)

Puts: An Arbitrage Opportunity? Suppose that p= 1 S0 = 37 T = 0.5 r =5% K = 40 D = 0 Is there an arbitrage opportunity?

Steps to follow at Put Options Check the lower bound of the option. See if the option is correctly priced or not. If p < max(Ke –rT – S0, 0)then arbitrage exists and you can continue with following steps. If p  max(Ke –rT – S0, 0)then there is no need to follow any more steps you can conclude with there is no arbitrage possibility Borrow an amount that is equal to 𝑝+ 𝑆 0 . Buy the Stock. At the end of maturity if 𝑆 𝑇 ≤𝐾 exercise the option and do not exercise the option if 𝑆 𝑇 >𝐾 and use the proceeds to pay back the loan

Put-Call Parity; No Dividends It shows that the value of a European call with a certain exercise price and exercise date can be deduced from the value of a European put with the same exercise price and exercise date, and vice versa. c + Ke -rT = p + S0

Arbitrage Opportunities Suppose that c = 3 S0= 31 T = 0.25 r = 10% K =30 D = 0 What are the arbitrage possibilities when p = 2.25 ? p = 1 ?

Factors Affecting Option Prices There are six factors affecting the price of a stock option: 1. The current stock price, 𝑆 0 2. The strike price, 𝐾 3. The time to expiration, 𝑇 4. The volatility of the stock price, 𝜎 5. The risk-free interest rate, 𝑟 6. The dividends that are expected to be paid, 𝐷

Effect of Variables on Option Pricing (Table 10.1, page 233) K T  r D c p C P – – + + – – + + + + ? + + + + – – + + – – + +

Early Exercise Usually there is some chance that an American option will be exercised early An exception is an American call on a non-dividend paying stock, which should never be exercised early

Reasons For Not Exercising a Call Early (No Dividends) No income is sacrificed You delay paying the strike price Holding the call provides insurance against stock price falling below strike price

The Impact of Dividends on Lower Bounds to Option Prices (Equations 10 The Impact of Dividends on Lower Bounds to Option Prices (Equations 10.8 and 10.9, pages 248-249)

Extensions of Put-Call Parity American options; D = 0 S0 - K < C - P < S0 - Ke -rT Equation 10.7 p. 243 European options; D > 0 c + D + Ke -rT = p + S0 Equation 10.10 p. 249 American options; D > 0 S0 - D - K < C - P < S0 - Ke -rT Equation 10.11 p. 249