Properties of Stock Options Chapter 10 Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

The Concepts Underlying Black-Scholes The option price and the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty This involves maintaining a delta neutral portfolio The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

The Black-Scholes Formulas (See page 309) Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

The N(x) Function N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x See tables at the end of the book Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

The Probabilities The delta of a European call on a non-dividend paying stock is N (d 1) The variable N (d 2) is the probability of exercise 0.72*10-0.6(10) =1.2 Same as (12-10)*0.6 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

The Costs in Delta Hedging continued Delta hedging a written option involves a “buy high, sell low” trading rule Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

First Scenario for the Example: Table 18.2 page 384 Week Stock price Delta Shares purchased Cost (‘$000) Cumulative Cost ($000) Interest 49.00 0.522 52,200 2,557.8 2.5 1 48.12 0.458 (6,400) (308.0) 2,252.3 2.2 2 47.37 0.400 (5,800) (274.7) 1,979.8 1.9 ....... 19 55.87 1.000 1,000 55.9 5,258.2 5.1 20 57.25 5263.3 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

Properties of Black-Scholes Formula As S0 becomes very large c tends to S0 – Ke-rT and p tends to zero As S0 becomes very small c tends to zero and p tends to Ke-rT – S0 What happens as s becomes very large? What happens as T becomes very large? Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

Risk-Neutral Valuation The variable m (expected appreciation in the stock value) does not appear in the Black-Scholes equation The equation is independent of all variables affected by risk preference This is consistent with the risk-neutral valuation principle When we use Monte Carlo to value an option we typically assume risk neutrality Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

Applying Risk-Neutral Valuation Assume that the expected return from an asset is the risk-free rate Calculate the expected payoff from the derivative Discount at the risk-free rate Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

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

Lower Bound for European Call Option Prices; No Dividends (Equation 10 Lower Bound for European Call Option Prices; No Dividends (Equation 10.4, page 238) c  max(S0 – Ke –rT, 0) Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

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) Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

Put-Call Parity; No Dividends Consider the following 2 portfolios: Portfolio A: European call on a stock + zero-coupon bond that pays K at time T Portfolio C: European put on the stock + the stock Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

Values of Portfolios ST > K ST < K Portfolio A Call option Zero-coupon bond K Total ST Portfolio C Put Option K− ST Share Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

The Put-Call Parity Result (Equation 10.6, page 241) Both are worth max(ST , K ) at the maturity of the options They must therefore be worth the same today. This means that c + Ke -rT = p + S0 Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

Put-Call Parity Results: Summary Fundamentals of Futures and Options Markets, 8th Ed, Ch 16, Copyright © John C. Hull 2013

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 Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

An Extreme Situation For an American call option: S0 = 100; T = 0.25; K = 60; D = 0 Should you exercise immediately? What should you do if You want to hold the stock for the next 3 months? You do not feel that the stock is worth holding for the next 3 months? Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

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 Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

Bounds for European or American Call Options (No Dividends) Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013

Bounds for European and American Put Options (No Dividends) Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013