# lec04 - Lecture 4 Trading Strategies and Slope/Convexity...

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Lecture 4: Trading Strategies and Slope/Convexity Restrictions This lecture studies elementary options trading strategies. In the process, we derive no-arbitrage restrictions for options that are identical except for their strike price. We restrict how quickly the option price can change with the strike price (slope restrictions) and how quickly this slope can change with the strike price (convexity restrictions). I. Motivation II. Definitions and Notation III. Trading Strategies A. Hedges B. European Spreads C. American Spreads D. European Butterflys E. American Butterflys F. Other Combinations

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Trading Strategies and Slope/Convexity Restrictions I. Motivation Suppose a stock is trading at \$100. SI You see four month European calls priced at \$8 for K D \$ 100 , and at \$19 for K D \$ 90 . SI The simple 4 month risk-free is 5.26%. I.e., B(t, t C 1 = 3) D \$ 95 , and Questions: SI Does the \$8 price of the K D \$ 100 call satisfy the no-arbitrage bounds from Lecture 3? SI Does the \$19 price of the K D \$ 90 call satisfy the no-arbitrage bounds from Lecture 3? SI Does this mean that we cannot have arbitrage? How do recognize that there is an arbitrage opportunity? ETX Pricing “restrictions” How can we exploit it? ETX I.e., with what trading strategy? Bus 35100 Page 2 Robert Novy-Marx
Trading Strategies and Slope/Convexity Restrictions II. Definitions and Notation Traders have names for common options portfolios. SI These portfolios are typically specific “bets” on what will happen to the prices and/or volatilities of the underlying securities. SI We use portfolios of options to illustrate further no- arbitrage restrictions and to generate profits when these restrictions are violated. We describe a portfolio of options by the equation for the current price of the portfolio. However, we drop subscripts which are common to all securities. Examples: SI c(K 1 ) NUL c(K 2 ) is a portfolio of a bought call with exercise price K 1 and a written call with exercise price K 2 and with the same maturity date. SI p(T 1 ) NUL p(T 2 ) is a portfolio of a bought put with maturity T 1 and a written put with maturity T 2 and with the same exercise price. Bus 35100 Page 3 Robert Novy-Marx

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Trading Strategies and Slope/Convexity Restrictions III. Trading Strategies A. Hedges: Combine options + underlying; protects the underlying against a loss, or vice-versa. Example: A covered call – long stock / short a call on the stock S(T) Profit K Covered Call Payoff is always positive, so no-arbitrage H) price of the covered call is positive, i.e., S(t) NUL c(K) NAK 0 or c(K) DC4 S(t) Bus 35100 Page 4 Robert Novy-Marx
Trading Strategies and Slope/Convexity Restrictions Another Example: A protective put – long stock / long a put on the stock * the put provides insurance * so you must pay an “insurance premium” S(T) Profit Protected Put K Bus 35100 Page 5 Robert Novy-Marx

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Trading Strategies and Slope/Convexity Restrictions B. Spreads Combine options of the same type, with – different strikes (a “vertical spread”), or – different maturities (a “horizontal spread”) 1. Example:
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