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Stock Index Options Chapter 9
Stock Index Option • Options on stock indices: S&P 500, S&P 100, and MMI. • Features: – Cash Settlement: when you exercise, the assigned writer pays you the difference between the closing (spot) index (S) and the exercise price; Call: S-X; Put: X-S. – Multiplier: $100. – End-of-the-day exercise
Uses • Speculation: – Bullish: long in call – Bearish: long in put – Stable market: short • Portfolio Insurance
Portfolio Insurance • An important use of stock index options is portfolio insurance. This is a hedging strategy in which an equity portfolio manager protects the future value of her fund by buying index put options. – The put options provide downside protection against a market decline, while allowing the fund to grow if the market increases.
Portfolio Insurance Example – Suppose a fund manager has a $50 M portfolio that he may have to liquidate in September. The fund is well-diversified with a beta of 1. 25. The current S&P 500 is at 1250, and there is a September S&P 500 put with an exercise price of 1250, multiplier of 100, and trading at 50. – To form a portfolio insurance position, the manager would need to buy 1. 25(50 M)/X =500 puts at a cost of $2. 5 M= (100)(50).
Put-Hedged Portfolio at T
Fiduciary Call • If the manager knew he was going to liquidate in September, he could sell the portfolio and invest the proceeds in a RF security and buy a call option. • Assume the September option expires in six months, Rf = 6%, and the put-call parity holds:
Fiduciary Call • • Strategy: Liquidate the portfolio for $50 M. Invest $50 M in RF security at 6% for 6 months. Buy 500 calls:
Pricing Index Options • The B-S model and the BOPM are defined by a replicating portfolio. For an index option, the replicating portfolio is a stock portfolio consisting of all the stocks in the index or a proxy portfolio formed with fewer stocks that are highly correlated with the index (maybe a MF). • Later we will define it in terms of a index futures position.
Proxy Portfolio • A proxy portfolio can be viewed as a position in the index. • For example, suppose you have a proxy portfolio worth Vo = $1. 5 M and the S&P 500 index is at So = 1250. • The proxy portfolio would be the equivalent of buying n = $1. 5 M/1250 = 1200 index shares at $1250 per share.
BOPM • Consider single-index BOPM with u = 1. 0227, d =. 9778, Rf =. 005, So = 310, X= 310.
Black-Scholes Model for Index Options • The most common way to price index option using the B-S model is to use the continuous dividend adjustment model where Sd is used instead of So. • Note: BAW model may be a better model for pricing index options.
Dynamic Portfolio Insurance • Because of the position limits on options, a portfolio insurance strategy formed with index put options may not be applicable for large portfolios. • An alternative is to use a dynamic portfolio insurance strategy using a stock and bond portfolio which is managed over time using a binomial framework. • Consider a portfolio which is highly correlated with a stock index that is currently at So = 150. Assume u = 1. 1, d = 1/1. 1 =. 9091, Rf =. 05, and n = 2. • As show in the figure, with portfolio insurance one could obtain portfolio values of 3. 025 M if the market increases and 2. 5 M if its stays at 150 or declines.
Dynamic Strategy Objective • Objective: Construct a bond and stock portfolio and manage it in such a way that you will have possible CFs of 3. 025 M, 2. 5 M, and 2. 5 M given the three possible spot index values at the end of period 2.
Dynamic Strategy Methodology • Define the portfolio as an investment in hypothetical shares of the stock index. • Form a bond and stock portfolio consisting of a diversified stock portfolio and an a investment in RF security (Io)
Dynamic Strategy Methodology • Period 1: • If the stock and bond portfolio is worth Vu = $2. 75 M when Su = 165, then it could be converted to all stock and reach next period’s target of being either 3. 025 M or 2. 5. • If the stock and bond portfolio is worth Vd = 2. 5 M/1. 05 = 2. 381 when Sd = 136. 3636, then the portfolio could be converted to all bonds and hit its target next period of 2. 5.
Dynamic Strategy Methodology • Mathematically, our problem is to solve for the n and Io where:
Dynamic Strategy Methodology • Solution:
Foreign Currency Options Chapter 10
Foreign Currency Options • Currency options are traded on the Philadelphia exchange (PHLX) and on exchanges in Toronto, Montreal, and Amsterdam. There is also a Dealer’s Market that is part of the Interbank FC market. • PHLX offers trading in: BP, DM, JY, SF, FF, AD, and CD. See JG, pp. 305 -307 for features of FC options.
FC Put-Call Parity • Conversion:
FC Put-Call Parity • At T position will be worth X.
FC BOPM • BOPM for FC is similar to that for stock except that a foreign interest rate is included. • The replicating portfolio consists of buying Ho units of FC at Eo and borrowing Bo dollars. The Ho units of FC can be invested for the period in a foreign RF security.
FC BOPM • Single Period
FC BOPM • Solve for Ho and Bo where:
FC: B-S Model • Use Ed instead of Eo:
FC Options Use • Speculation on Exchange rates • Hedging future FC payment or receipt positions
Hedging Currency Positions – Suppose a U. S. investment fund is expecting 625, 000 DM in September from its Eurobonds. – The Fund expects the exchange rate to increase from its current level of Eo =$. 40/DM, suggesting greater dollar revenues than $250, 000, but it is not confident. To benefit from an increase in Eo, while hedging against an Eo decrease, the fund buys 10 DM put contracts with X=$. 40, P=$. 01, size=62500 DM.
Hedge DM Revenue at T
Hedging Currency Positions – Suppose a U. S. Company has a 625, 000 DM debt to be paid in September. – The company expects the exchange rate to decrease from its current level of Eo =$. 40/DM, suggesting lower dollar cost than $250, 000, but it is not confident. To benefit from a decrease in Eo, while hedging against an Eo increase, the company buys 10 DM call contracts with X=$. 40, C=$. 01, size=62500 DM.
Hedge DM Cost at T
Options Issued by Corporations Chapter 11
Warrant – A warrant is a call option issued by a Corporation. – The contractual features of a warrant and a call are the same. – The primary difference between a warrant and a call is that the writer of the warrant is the issuing corporation. – If a warrant is exercised, the corporation receives cash and creates new shares -- dilution effect.
Warrants Example – Example: LM Co. is a $100, 000 oil well company; all equity, with 100 shares outstanding and with each share worth $1000; has 4 shareholders, A, B, C, and D, each with 25 shares. – Consider alternatives: Shareholder D sells a call option to investor E, giving her the right to buy 25 shares at X = $1100/share; the LM Co. sells a warrant with the same features.
Oil Well Increases in Value • Oil Well increases in value to $120, 000, causing LM stock to go to $1200. • Call Exercised: – When Investor E exercises, shareholder D will simply turn over his 25 shares for $1100/share. – Exercising has no impact on the value of LM. – IVc = $1200 -$1100 = $100.
Warrant Exercised – When E exercises her warrant, the LM Co. will have to print 25 new shares (Nw) and sell them to E at X = $1100/share. – The company will receive cash, but the number of shares will increase: • 100 to 125 shares (dilution) • ($1100) (25) = $27, 500 – In this case, the exercise of the warrant lowers the value of the stock from $1200 to $1180.
Warrant Value Stock and Warrant Values:
Warrant and Call Relation • Relation: The value of the warrant is equal to the value of the call times the dilution factor:
Rights – Definition: A right is call option issued by a corporation to existing shareholders, giving the holder the right to buy new shares at a specified price (subscription price). – Corporations use rights to ensure they adhere to preemptive rights. – A right is like a warrant: when it is exercised, new shares are created and the company receives cash.
Rights Example ABC company is planning to raise $20 M in equity to finance a capital acquisition. The company is worth $200 M, has no debt, and has 1 million shares of stock outstanding, with each share currently trading at $200. ABC plans to finance its $20 M investment with a rights offering in which it gives shareholders the right to buy new shares at $160, with each holder receiving one right for each share owned.
Value of the Right Number of new shares = Number of Rights for 1 Share = Stock Value IV of Right