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**FINC4101 Investment Analysis**

Instructor: Dr. Leng Ling Topic: Introduction to options

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**Learning objectives Define call and put options.**

Understand the various features of options: exercise price, option premium, option exercise, American vs. European. Describe how options trading is organized. List the various types of option contracts. Compute the payoff and profit of call option holder, call option writer, put option holder and put option writer. Describe the composition of various option strategies. Compute the payoff and profit of various option strategies. The payoffs and profits are computed at expiration.

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**Concept Map Foreign Exchange Derivatives Market Efficiency**

Fixed Income Equity Asset Pricing Portfolio Theory FI400

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Derivative security A derivative transaction involves no actual transfer of ownership of the underlying assets at the time the contract is initiated. A derivative represents an agreement to transfer ownership of underlying assets at a specific place, price, and time specified in the contract. Its value (or price) depends on the value of the underlying assets. The underlying assets: stocks, bonds, interest rates, foreign exchanges, index, commodities, some derivatives, etc.

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What is an option? A derivative security that gives the holder the right to buy / sell an asset (the “underlying”) at a specified price (“exercise price”) on or before the option expiration date.

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**Two types of options: Call vs. Put options**

Call option Gives holder the right to buy an asset at a specified exercise price on or before a specified expiration date. Put option Gives holder the right to sell an asset at a specified exercise price on or before a specified expiration date. There are two main types of options: call and put options

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**Exercise price Exercise price**

For a call option, it is the price set for buying the underlying asset. For a put option it is the price set for selling the underlying asset. Exercise price is also called the strike price.

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Option premium Options are financial assets. If you want an option, you have to buy it from an option seller (counterparty). The purchase price or cost of an option is the option premium. The option seller earns the option premium. The option premium is an immediate expense for the buyer and an immediate return for the seller, whether or not the owner (buyer) ever exercises the option. In option markets, to sell an option is to “write” an option. An option seller is also called an “option writer”.

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Examples At March 1, XYZ stock’s spot price = $110. A trader buys a call option on XYZ at strike (exercise) price = $100/share. The right lasts until August 15, and the price (option premium) of this call option is $15/share. At March 1, ABC stock’s spot price = $100. A trader buys a put option to on ABC at strike (exercise) price = $120/share. The right lasts until August 15, and the price (option premium) of this put option is $22/share.

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**The long and short of it…**

If you buy an option, then you are “long the option” or “long option” or you have a “long position”. If you sell an option, then you are “short the option” or “short option” or you have a “short position”. Example: if you buy a call option, you are “long call”.

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Buyer (Long) Seller (Short) Call - Right to buy the underlying (i.e. to exercise the option) - Pays the premium - Obligation to sell the underlying, if buyer exercises the option - Receives the premium Put - Right to sell the underlying (i.e. to exercise the option) - Obligation to buy the underlying, if buyer exercises the option

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Options trading (1) Option contracts are traded in two types of markets: Over-the-counter (OTC) markets Exchanges, such as: Chicago Board Options Exchange (CBOE) Chicago Mercantile Exchange (CME) International Securities Exchange Option Clearing Corporation (OCC) Options contracts traded on exchanges are standardized by allowable maturity dates and exercise prices for each listed option. Each stock option contract provides the right to buy or sell 100 shares of stock (except when stock splits occur after the contract is listed and the contract is adjusted for the terms of the split). Option Clearing Corporation (OCC), the clearinghouse for options trading, is jointly owned by the exchanges on which stock options are traded. The OCC places itself between options traders, becoming the effective buyer of the option from the writer and the effective writer of the option to the buyer. All individuals, therefore, deal only with the OCC, which effectively guarantees contract performance. When an option holder exercises an option, the OCC arranges for a member firm with clients who have written that option to make good on the option obligation. The member firm selects from among its clients who have written that option to fulfill the contract. The selected client must deliver 100 shares of stock at a price equal to the exercise price for each call option contract written or must purchase 100 shares at the exercise price for each put option contract written. Because the OCC guarantees contract performance, option writers are required to post margin to guarantee that they can fulfill their contract obligations. The margin required is determined in part by (i) the amount by which the option is in the money, and (ii) whether the underlying asset is held in portfolio.

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Options trading (2) OTC Option contract can be customized to needs of trader. Difficult to trade. Secondary market illiquid. Exchanges Option contracts are standardized by maturity dates and exercise price. Easy to trade. Secondary market is liquid. If you need a customized option and don’t anticipate trading it, then OTC is suitable. If you desire the ability to trade quickly and cheaply and you don’t need customization then Exchanges are suitable. The OTC market offers the advantage that the terms of the option contract – the exercise price, maturity date, and number of shares committed – can be tailored to the needs of the traders. The costs of establishing an OTC option contract, however, are relative high. Options contracts traded on exchanges are standardized by allowable maturity dates and exercise prices for each listed option. Standardization of the terms of listed option contracts means all market participants trade in a limited and uniform set of securities. This increases the depth of trading in any particular option, which lowers trading costs and results in a more competitive market. Exchanges offer two benefits: ease of trading, which flows from a central marketplace where buyers and sellers or their representative congregate, and a liquid secondary market where buyers and sellers of options can transact quickly and cheaply.

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**Options on IBM June 7, 2004 Source: Wall Street Journal Online Edition, June 8, 2004.**

This figure shows both call and put options listed for each exercise price and expiration date. The three sets of columns for each option report closing price, trading volume in contracts (1 contract = 100 shares of stock), and open interest (number of outstanding contracts).

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**Underlying asset Individual stocks Stock market indexes Futures**

S&P 100, S&P 500, DJIA, Nikkei 225, FTSE 100 etc. Futures Foreign currency Treasury bonds, Treasury notes And many others. Index options: An index option is a call or put based on a stock market index such as the S&P500 or the NYSE index. Index options are traded on several broad-based indexes as well as on several industry-specific indexes. Futures options: give holders the right to buy or sell a specified futures contract, using as a futures price the exercise price of the option. The terms of futures options contracts are designed in effect to allow the option to be written on the futures price itself. Foreign currency options: a currency option offers the right to buy or sell a quantity of foreign currency for a specified amount of domestic currency. Currency option contracts call for purchase or sale of the currency in exchange for a specified number of U.S. dollars. Contracts are quoted in cents or fractions of a cent per unit of foreign currency. Interest rate options: options on treasury notes, bonds, bills, CDs, GNMA pass-through certificates. Options on several interest rate futures also are traded.

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Option exercise (1) To “exercise a call option” means to use the option to buy the underlying asset at the exercise price. To “exercise a put option” means to use the option to sell the underlying asset at the exercise price. This is not about physical education.

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**Option exercise (2) Question: When do you exercise an option?**

Answer: Simple. Only when it’s optimal to do so. That is, when you are better off exercising the option. “Buy low, sell high” Question: What if exercising the option does not make me better off? Answer: Simple. Don’t exercise. After all, it’s just an option. Tell student that I will explain what “better off” means later. Roughly speaking, “better off” means that you are able to buy low and sell high, i.e., your buying/purchase price is lower than your selling price. Just remember the rule, “buy low, sell high”.

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**American vs. European options**

American option: Holder has the right to exercise the option on or before the expiration date. European option: Holder has the right to exercise the option only on the expiration date. Most traded options in the US are American-style. Exceptions: foreign currency options, some stock index options.

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**Payoff of Long Bond Position at Expiration**

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**Payoffs of a Call Option**

Long Call at $20 Short Call at $20

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**Profit/Loss of a Call Option**

Long Call at $20 Short Call at $20

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**Profit/Loss of Long and Short on Call Option**

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Payoffs of a Put Option Long Put at $20 Short Put at $20

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**Profit/Loss of a Put Option**

Long Put at $20 Short Put at $20

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**Profit/Loss of Long and Short on Put Option**

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**Call Option’s Payoff/Profit at Expiration**

Payoff for a Long Call: Profit for a Long Call: payoff - option premium Payoff for a Short Call: Profit for a Short Call: option premium + payoff

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**Put Option’s Payoff/Profit at Expiration**

Payoff for a Long Put: Profit for a Long Put: payoff – option premium Payoff for a Short Put: Profit for a Short Put: option premium + payoff

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Example A trader short a Call at X=20 with a premium of $5. At maturity, the stock price is 30. What is the profit/loss to this trader? Profit/Loss = 5 + [-(30-20)] = = -5 A trader long a Put at X=30 with a premium of $5. At maturity, the stock price is 15. What is the profit/loss to this trader? Profit/Loss = (30-15) - 5 = = 10

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**Call option: Payoff & Profit at expiration (1)**

Consider a call option on a share of IBM stock with an exercise price of $80 per share. Suppose this call option expires on July 16, Suppose today is the expiration date. The call option price (premium) was $5. We look at payoff (value) and profit at expiration.

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**Call option: Payoff & Profit at expiration (2)**

Are you better off exercising the option? What is the payoff from the option exercise? What is the profit from the option exercise? Answer these questions if IBM’s stock price is (a) 95, (b) 76 and (c) 81. What is the breakeven point for this call option? Breakeven point is the stock price at which profit is zero. Stock price = 95 1) Are you better off exercising the option? If you exercise, you can buy the stock at $80 and simultaneously sell the stock at the market price of $95. Your anticipated gain from exercise is = 95 – 80 = 15. Now, if you don’t exercise, you let the option expire worthless. You get nothing, i.e., zero. Comparing the two, clearly, you are BETTER OFF exercising the option. So, the answer is yes, you are better off exercising the option. 2) What is the payoff from the option exercise? For a call option, the payoff is either the gain you receive (when you are better off exercising the option) or zero (when you are not better off exercising the option and you do nothing). This goes back to the previous slides. When it’s optimal to exercise, you obviously will realize a gain (i.e., the payoff). If it’s not optimal, you don’t exercise, and you get nothing. In this case, you are better off exercising the option. So your payoff = gain = stock price – exercise price = 95 – 80 = 15 3) The profit is = gain – option premium = 15 – 5 = 10. Now, suppose IBM’s stock price is $76 If you exercise, you buy the stock at $80 when you can actually get it at a lower price of $76. If you buy the stock at 80 and simultaneously sell it at the market price of $76, your ‘gain’ = 76 – 80 = - 4, i.e., a loss. So, you are not better off exercising the option. In fact, you are better off doing nothing, i.e., don’t exercise. 2) In this case, you are better off not exercising the call option. So your payoff = 0. 3) Profit = 0 – 5 = -5 Now, suppose IBM’s stock price is $81 1) Now, you are better off exercising the option because you can buy the stock at 80 and sell at 81. 2) Payoff = 81 – 80 = 1. 3) Profit = 1 – 5 = -4. Even though the profit is -4, it is still optimal to exercise. To see, consider what happens if you don’t exercise. Then your payoff = 0 and your profit = - 5. If you don’t exercise, you get a bigger loss. So exercising minimizes your loss. The breakeven point is the stock price at which profit = 0. For this call option, let C be option premium (ST – X) – C = 0 ST – 80 – 5 = 0 ST = 85

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**Payoff & profit diagram of call option holder at expiration**

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**Payoff and profit of call option writer**

Compute payoff, profit and breakeven point, if the stock price at expiration is (a) 95, (b) 76 and (c) 81. Note that – (ST – X) is just the negative of the call holder’s payoff. Payoff: The general idea is that the call holder’s gain is the call writer’s loss, vice versa. The payoff of the call option writer is the negative of the call holder’s payoff. The call writer incurs losses if the stock price is high. In that case, the writer will receive a call and will be obligated to deliver a stock worth ST for only X dollars (buy high, sell low). The call writer, who is exposed to losses if IBM increases in price, is willing to bear this risk in return for the option premium. The profit of the call option writer is = the premium income less call holder’s payoff. The profit equation is written like this because the max profit is the premium income. If the option is exercise, the profit is the premium income reduced by the payoff to the call holder, since the call holder’s payoff is the loss incurred by the call writer. a) Stock price = 95 Payoff = - (95 – 80) = - 15 Profit = 5 – 15 = -10 b) Stock price = 76 Payoff = 0 Profit = 5 c) Stock price = 81 Payoff = -(81 – 80) = -1 Profit = 5 – 1 = 4. Breakeven stock price 5 – (ST – 80) = 0 5 – ST + 80 = 0 ST = 85

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**Payoff & profit diagram of call option writer at expiration**

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Call Review Which of the following statements about the value (i.e., payoff) of a call option at expiration is false? A short position in a call option will result in a loss if the stock price exceeds the exercise price. The value of a long position equals zero or the stock price minus the exercise price, whichever is higher. The value of a long position equals zero or the exercise price minus the stock price, whichever is higher. A short position in a call option has a zero value for all stock prices equal to or less than the exercise price. Bkm chp 14 q1 Answer: C. This statement describes the payoff of the long put option, not the long call option.

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**Put option: Payoff & Profit at expiration (1)**

Consider a put option on a share of IBM stock with an exercise price of $80 per share. Suppose this put option expires on July 16, Suppose today is the expiration date. The put option price (premium) was $3.

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**Put option: Payoff & Profit at expiration (2)**

Are you better off exercising the option? What is the payoff from the option exercise? What is the profit from the option exercise? Answer these questions if IBM’s stock price is (a) 89, (b) 70 and (c) 79. What is the breakeven point for this put option? Breakeven point is the stock price at which profit is zero. Stock price = 89 1) Are you better off exercising the option? If you exercise, you would buy the stock at $89 and simultaneously deliver (sell) the stock at $80 to the put option holder. Anticipated gain from exercise is = 80 – 89 = -9. Now, if you don’t exercise, you let the option expire worthless. You get nothing, i.e., zero. Comparing the two, clearly, you are BETTER OFF by not exercising the option. So, the answer is no, you are better off by letting the option exercise worthless. 2) What is the payoff from the option exercise? For a put option, the payoff is either the gain you receive (when you are better off exercising the option) or zero (when you are not better off exercising the option and you do nothing). This goes back to the previous slides. When it’s optimal to exercise, you obviously will realize a gain (i.e., the payoff). If it’s not optimal, you don’t exercise, and you get nothing. In this case, you are better off by not exercising the option. So your payoff = zero 3) The profit is = gain – option premium = 0 – 3 = -3. Now, suppose IBM’s stock price is $70 If you exercise, you can buy the stock at $70 and simultaneously sell the stock to the put option writer at $80. Your ‘gain’ = = So, you are better off exercising the option. In this case, you are better off exercising the call option. So your payoff = 10. 3) Profit = 10– 3 = 7 Now, suppose IBM’s stock price is $79 1) Now, you are better off exercising the option because you can sell the stock at 80 and buy it at 79. 2) Payoff = 80 – 79 = 1. 3) Profit = 1 – 3 = -2. Even though the profit is -2, it is still optimal to exercise. To see, consider what happens if you don’t exercise. Then your payoff = 0 and your profit = - 3. If you don’t exercise, you get a bigger loss. So exercising minimizes your loss. Below $80 (exercise price), the put value at expiration increases by $1 for each dollar the stock price falls. The breakeven point is the stock price at which profit = 0. For this put option, let P be option premium (X - ST) – P = 0 80 - ST – 3 = 0 ST = 77

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**Payoff & profit diagram of put option holder at expiration**

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**Payoff and profit of put option writer**

Compute payoff, profit and breakeven point, if the stock price at expiration is (a) 89, (b) 70 and (c) 79. Note that – (X – ST) is just the negative of the put holder’s payoff. Recall that the put holder’s gain is the put writer’s loss. Compute payoff, profit and breakeven point, if the stock price at expiration is (a) 89, (b) 70 and (c) 79. Stock price = 89 Payoff = 0, since stock price > exercise price and so put holder will not exercise. Profit = 3 – 0 = 3. Stock price = 70 Now, put holder will exercise, so put writer’s payoff = - (80 – 70) = - 10 Profit = 3 – 10 = -7 Stock price = 79 Again put holder will exercise. So put writer’s payoff = - (80 – 79) = -1 Profit = 3 – 1 = 2. Break even is the stock price such that profit = 0. Premium - (80 – ST) = 3 – (80 – ST) = 0 -77 = - ST ST = 77

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**Payoff & profit diagram of put option writer at expiration**

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Put Review Consider a put option written on ABC Inc.’s stock. The put option’s exercise price is $80. Which of the following statements about the value (payoff) of the put option at expiration is true? The value of the short position in the put is $4 if the stock price is $76. The value of the long position in the put is -$4 if the stock price is $76. The long put has value when the stock price is below the $80 exercise price. The value of the short position in the put is zero for stock prices equaling or exceeding $76. BKM Chp 14 q2 Answer: C, just by applying the formula for payoff. A is wrong because the short position’s value is -$4 since the put will be exercised. Thus the writer will buy (take delivery) at $80 and sell at $76. Value = 76 – 80 = -4. B is wrong because the long position’s value is $4 since the put will be exercised. Thus the put holder will exercise the option, buy the stock at $76 and simultaneously deliver (sell) the stock at $80 to the put writer. The payoff = 80 – 76 = 4. D is wrong because the payoff of the short position is negative between $76 and $80 and only equal zero for $80 or more.

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Practice 9 Chapter 15: 4, 5, 6.

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Homework 8 You purchased one IBM July 100 call contract for a premium of $4.00. Assuming that the stock price on the expiration date is $105. What is the payoff and net profit/loss? What is the break even point? Draw the payoff and net profit/loss lines on the diagram.

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**Moneyness (1)—intrinsic value**

An option (call or put) is: In the money (ITM) if exercising it produces a positive payoff to the holder At the money (ATM) if the asset price and exercise price are equal. Out of the money (OTM) if exercising it produces a negative payoff to the holder. We can describe an option in terms of its moneyness.

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**Moneyness (2) ST < X ST = X ST > X Call option Out of the money**

At the money In the money Put option

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**Moneyness questions (1)**

Consider two call options written on ABC Inc.’s stock. The first call, C1, has an exercise price of $50. The second call, C2, has an exercise price of $70. Both calls have the same expiration date. Today is the expiration date. C1 is in the money while C2 is out of the money. Which of the following is true about ST, the stock price on the expiration date? ST > $50 ST > $70 $70 > ST > $50 ST < $50 Use the information about moneyness to find the statement which best fits the moneyness of the two options. Within this range, C1 has a positive payoff (sell at more than $50, but buy at $50) while C2 has a negative payoff (sell at less than $70 and buy at $70). Thus C1 is ITM and C2 is OTM. E.g., suppose ST = 60. C1’s payoff = 60 – 50 = 10 > 0 so ITM. C2’s payoff = 60 – 70 = -10 < 0 so OTM. Answer: C. This explains why C1 is ITM and C2 is OTM. is wrong because it does not explain while C2 is OTM. is wrong because C2 is OTM so stock price cannot be greater than $70. D) Is wrong because C1 is ITM, so stock price must exceed $50.

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**Moneyness questions (2)**

Consider two put options written on XYZ Inc.’s stock. The first put, P1, has an exercise price of $20. The second put, P2, has an exercise price of $35. Both puts have the same expiration date. Today is the expiration date. P1 is out of the money while P2 is in the money. Which of the following is true about ST, the stock price on the expiration date? ST < $20 ST < $35 $20 < ST < $35 ST > $35 Use the information about moneyness to find the statement which best fits the moneyness of the two options. Answer: C. Within this range, P1 has a negative payoff (sell at $20, but at more than $20) while P2 has a positive payoff (sell at $35 and but at less than $35). Thus P1 is OTM and P2 is ITM. E.g., suppose ST = 25. P1’s payoff = = -5 < 0 so OTM. P2’s payoff = 35 – 25 = 10 > 0 so ITM. Is wrong because P1 is OTM. Is wrong because it does not explain why P1 is OTM. So it’s not complete. d) Is wrong because P2 is ITM, so ST must be less than $35.

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**Option vs. Stock Investment (1)**

Compared to a stock investment, options offer 1) Leverage Pure option investment magnifies gains and losses compared to pure stock investment. 2) Insurance Combining options with T-bills (money market fund) limits losses compared to pure stock investment. Consider the following… Consider the following example from BKM Chp14Q5 The purpose of the example is to show these two features of options.

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**Option vs. Stock Investment (2)**

Suppose you think Wal-Mart stock is going to appreciate substantially in value in the next year. The stock’s current price, S0, is $100, and the call option expiring in one year has an exercise price, X, of $100 and is selling at a price, C, of $10. With $10,000 to invest, you are considering three alternatives: Invest all $10,000 in the stock, buying 100 shares. Invest all $10,000 in 1,000 options (10 contracts) Buy 100 options (one contract) for $1,000 and invest the remaining $9,000 in a money market fund paying 4% interest annually.

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**Option vs. Stock Investment (3)**

Compute the rate of return for each alternative for four stock prices one year from now: $80 $100 $110 $120

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**Option vs. Stock Investment (4)**

Price of stock 1 year from now Stock price: $80 $100 $110 $120 a) All stocks (100 shares) -20% 0% 10% 20% b) All options (1,000 shares) -100% 100% c) Money market fund options -6.4% 3.6% 13.6% Just show how to compute rate of return for stock price = $80, and price = $120 Remember that rate of return is just HPR. Stock price = $80 All stocks: Original Investment = 100 x 100 = 10,000 Return = (80 x 100 – 10,000)/10,000 = -0.2 or – 20% All options: Original investment = 10 x 1000 = 10,000 When stock price = $80, payoff = 0 since it’s not profitable to exercise. Therefore, the options portfolio has zero value at the end of the year. Return = (0 – 10,000)/10,000 = -1 or -100% Money market fund options Money market investment = 9000 x (1.04) = 9360. Payoff is 0 since it’s not profitable to exercise. Therefore, the options have zero value. Return = (9360 – 10000)/10000 = or -6.4% Stock price = $120 Return = (120 x 100 – 10,000)/10,000 = 0.2 or 20% When stock price = $120, payoff = 120 – 100 = 20 since it’s profitable to exercise. Therefore, the options portfolio value at year’s end = 20 x 1000 = 20,000. Return = (20,000 – 10,000)/10,000 = 1 or 100% Payoff = 120 – 100 = 20. Therefore, the options’ value =20 x 100 = 2000 Return = ( – 10000)/10000 = or 13.6%

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**Rate of return to strategies**

Leverage: Comparing the returns of portfolios B (all options) and A (all stocks), when Walmart’s stock falls, ending anywhere below $100, the value of portfolio B falls dramatically to zero – a rate of return of negative 100%. Conversely, modest increases in the rate of return on the stock result in disproportionate increases in the option rate of return. E.g., a 10% increase in the rate of return on the stock from 10% to 20% would increase the rate of return on the option portfolio from 0% to 100%! In this sense, calls are a levered investment on the stock. Their values respond more than proportionately to changes in the stock value. To see this graphically, notice that for prices above $100, the slope of the all option portfolio is far steeper than that of the all stock portfolio, reflecting its greater proportional sensitivity to the value of the underlying security. The leverage factor is the reason that investors (illegally) exploiting inside information commonly choose options as their investment vechicle. Insurance feature: Look at portfolio C. This portfolio cannot be 9,360 after 1 year as the option can always expire worthless and you are just left with the money market fund investment. The worst possible rate of return on portfolio C is -6.4% compared to a (theoretically) worst possible rate of return of Walmart stock of -100% if the company goes bankrupt. This insurance comes at a price: when walmart does well, portfolio C doesn’t perform as well as portfolio A. For stock prices above $100, portfolio C underperforms portfolio A by about 6.4%.

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**Option strategies An option strategy is:**

A portfolio of options (and possibly the underlying asset) designed to produce a particular payoff pattern. We look at the following strategies: Protective put (Insurance on portfolio) Covered call Straddle There are many option strategies which traders can employ. Since this is an introduction to options and because of time constraints, we focus on the most well-known strategies.

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Protective put Portfolio consisting of a put option and the underlying asset. Guarantees that minimum portfolio value (payoff) is equal to the put’s exercise price. Rationale: you want to maintain the value of the portfolio at a certain minimum level. The payoff shows that whatever happens to the stock price, you are guaranteed a minimum value equal to the put option’s exercise price because the put gives you the right to sell the share for the exercise price even if the stock price is below that value.

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**Protective put: Payoff & profit at expiration**

S0 = initial asset price, and P = put option premium. Cost of the position = asset price + put premium = S0 + P ST ≤ X ST > X Payoff of stock ST Payoff of put X – ST Total payoff X Profit X – (S0+P) ST – (S0 + P) Explain how you derive the payoffs and profits.

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**Payoff & profit of protective put position at expiration**

The figure illustrates the payoff and profit to the protective put strategy. The solid line is the part C is the total payoff. The dashed line is displaced downward by the cost of establishing the position, S0 + P. notice that the potential losses are limited. Profit = payoff – cost of position = Payoff - (S0 + P) Notice that loss is limited to X – (S0 + P).

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**Protective put problem**

You establish a protective put position on ABC stock today by buying 100 shares of ABC stock at $40 per share and buying a 3-month put option contract on the same stock. Each put option has a strike price of $40 and a premium of $8. At the end of 3 months, compute the profit from the position if ABC’s stock price is $30. What is the breakeven stock price for this strategy? Q1) Remember that each option contract is for 100 shares. use the payoff/profit structure for protective puts. Exercise price x no. of options = 40 x 100 = 4000 Original stock investment = 40 x 100 = 4000 Cost of puts = 100 x 8 = 800. Profit = 4000 – ( ) = -$800 You lost $800 although you maintain your portfolio value at $4000. Compare this to the loss you would have suffered without the put. If you just hold the 100 shares, your loss = (40 – 30) x 100 = 1000 So, the protective put strategy helps to limit losses compared to a pure stock investment. Q2) Breakeven stock price is the ST such that profit = 0. From the payoff diagram you know that breakeven price must be greater than exercise price. So look at profit equation over the range ST > X. For each share of stock, Profit = ST – (S0 + P) 0 = ST – ( ) ST = 48 Therefore, breakeven price is $48/share.

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Portfolio Insurance Long 1 stock (portfolio) and Long a Put Option on the stock (portfolio), S=20, X=20, P=5. Blue line, stock profit Red line, option payoff Black, portfolio profit.

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Covered call Writing a call option on an asset together with buying the asset. Written option is “covered” because the potential obligation to deliver the stock is covered by the stock held. Strategy produces immediate cash flows through the sale of the call options. The payoff to a covered call equals the stock value minus the payoff of the call. The call payoff is subtracted because the covered call position involves issuing a call to another investor who can choose to exercise it to profit at your expense. Here you are the call option writer. So remember the payoff profile of the call writer.

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**Covered call: Payoff & profit at expiration**

S0 = initial asset price, and C = call option premium. Cost of the position = asset price - call premium = S0 - C ST ≤ X ST > X Payoff of stock ST - Payoff of call – (ST – X) Total payoff X Profit ST – (S0 – C) X – (S0 – C) Remember that you earn the call premium. So that helps to REDUCE the overall cost of setting up the position. The cost = stock price minus call premium.

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**Payoff & profit of covered call position at expiration**

Figure C: total position is worth ST when the stock price at time T is below X and rises to a maximum of X when ST exceeds X. In essence, the sale of the call option means that call writer has sold the claim to any stock value above X in return for the initial premium (the call price). Therefore, at expiration the position is worth at most X. The dashed line is the net profit to the covered call position. Profit = payoff – cost of establishing position = Payoff – (S0 – C) , where C is call option premium Writing covered call options has been a popular investment strategy among institutional investors. Consider the managers of a fund invested largely in stocks. They might find it appealing to write calls on some or all of the stock in order to boost income by the premiums collected. Although they forfeit potential capital gains should the stock price rise above the exercise price, if they view X as the price at which they plan to sell the stock anyway, then the call may be viewed as enforcing a kind of “sell discipline”. The written call guarantees the stock sale will occur as planned.

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Covered call problem You establish a covered call position on XYZ stock today by buying 100 shares of XYZ stock at $16 per share and writing a 3-month call option contract on the same stock. Each call option has a strike price of $17 and a premium of $0.25. At the end of 3 months, compute the profit from the position if XYZ’s stock price is $14. What is the breakeven stock price for this strategy? Q1) The stock price < exercise price, so look at the profit equation when ST <= X. Profit = (ST x 100) – ( (16 x 100) – (0.25 x 100) ) = 1400 – (1600 – 25) = 1425 – 1600 = -$175 Q2) Look at a single share. From the payoff diagram you know that the breakeven stock price < X. Therefore, look at ST <= X Profit = ST – (S0 – C) 0 = ST – (16 – 0.25) 0 = ST – 15.75 ST = 15.75 The break even stock price is $15.75/ share

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Straddle A combination of a call and put, each with the same exercise price (X) and expiration date (T). Rationale You believe a stock will move a lot in price but are uncertain about the direction of the move. Straddle allows you to benefit from a price move in either direction. Example of how straddle is used: Suppose you believe an important court case that will make or break a company is about to be settled, and the market is not yet aware of the situation. The stock will either double in value if the case is settled favorably or will drop by half if the settlement goes against the company. The straddle position will do well regardless of the outcome because its value is highest when the stock price makes extreme upward or downward moves from X.

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**Straddle: Payoff & profit at expiration**

C = call option premium P = put option premium Cost of the position = call premium + put premium = C + P ST < X ST ≥ X Payoff of call ST – X + Payoff of put X – ST Total payoff Profit X – ST – (C + P) ST – X – (C + P) The worst case scenario for a straddle is no movement in the stock price. If ST equals X, both the call and the put expire worthless, and the investor’s outlay for the purchase of both options is lost. Straddle positions are basically bets on volatility. An investor who establishes a straddle must view the stock as more volatile than the market does. Conversely, investors who write straddle must believe the market is less volatile. They accept the option premiums now, hoping the stock price will not change must before option expiration.

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**Payoff & profit of straddle position at expiration**

Fig C. Notice that the portfolio payoff is always positive, except at the one point where the portfolio has zero value, ST = X. you might wonder why all investors don’t pursue such a no-lose strategy. To see why, remember that the straddle requires that both the put and call be purchased. The value of the portfolio at expiration, still must exceed the initial cash outlay for a straddle investor to clear a profit. The dashed line of figure C is the profit to the straddle. The profit line lies below the payoff line by the cost of purchasing the straddle, P + C. The straddle position generates a loss unless the stock price deviates substantially from X. The stock price must depart from X by the total amount expended to purchase the call and the put in order for the purchaser of the straddle to clear a profit.

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Straddle problems You establish a straddle position on ABC Inc.’s stock by buying a three-month call option and a three-month put option. Both options have an exercise price of $50. ABC’s current stock price is $50 per share. The call option premium is $5 and the put option premium is $3. Compute the straddle’s profit on the expiration date if ABC’s stock price on expiration date is $60. How far would ABC’s stock price have to fall for you to make a profit on your initial investment? Q1) Use the payoff/profit structure of the straddle. Look at the profit equation where ST >= X. Call payoff = 60 – 50 =10 Put payoff = 0 Total payoff = = 10 Profit = payoff – cost of straddle = 10 – (5 + 3) = 2. Q2) This is a question about the breakeven point for this position. You need to solve for ST such that your profit is 0 over the price range of ST < X. (see payoff diagram) For ST < X, Profit = X – ST – (C + P) Breakeven means: 0 = ST – (5 + 3) 0 = 50 – 8 – ST 42 = ST Difference between current price and breakeven price = 50 – 42 = 8. Therefore, price must fall by more than $8 for you to make a profit. (Falling by exactly $8 only lets you breakeven, so 0 profit).

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Which strategy? (1) You are the portfolio manager of Nohope Equity Fund. One of your stock holdings is Refin Corp. Your equity analyst tells you that Refin’s stock price is not expected to rise substantially within the foreseeable future. At the same time, you need to raise cash right now to meet fund redemptions. What would be a simple options strategy to exploit your conviction about the stock price’s future movements and allow you to earn immediate income? Long call. Long put. Protective put. Covered call. Long straddle. Answer: D. Only this strategy produces income immediately and benefit from the stock price not rising. Covered call does not allow you to gain from substantial price gain. Since the equity analyst does not expect substantial price gain, you should adopt this strategy.

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Which strategy? (2) PUTT Corporation’s common stock has been trading in a narrow price range for the past month, and you are convinced it is going to break far out of that range in the next three months. You don’t know whether it will go up or down, however. What would be a simple options strategy to exploit your conviction about the stock price’s future movements? Long call. Long put. Protective put. Covered call. Long straddle. Based on BKM chp14 Q6 Answer: E.

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Practice 10 Chapter 15: 7,8,11,14 (b),15,20,21, 22, 25 Hints: to graph the payoff diagram of strategies, you need to firstly draw the payoff tables as demonstrated in slide 54, 59, 63. Based on the tables, you can draw the diagram. See the following example, which is the solution to 14 (a).

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Position S < X1 X1 < S < X2 X2 < S < X3 X3 < S Long call (X1) S – X1 Short 2 calls (X2) –2(S – X2) Long call (X3) S – X3 Total 2X2 – X1 – S (X2–X1 ) – (X3–X2) = 0

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**Determinants of option value**

The values of call and put options are affected by: Underlying asset price Exercise price Volatility of the asset price Option’s time to expiration Riskfree interest rate Cash payouts from underlying asset, e.g., dividend.

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**Determinants of call option value**

If this variable increases Value of call option Asset price, S Increases Exercise price, X Decreases Volatility, Time to expiration, T Interest rate, rf Cash payouts e.g., dividend Asset price: If S increases, intrinsic value increases which in turn increases the call’s value. Exercise price: If X increases, intrinsic value decreases since the difference (S – X) becomes smaller. This reduces the call’s value. Volatility: As volatility increases, the chance of a large price rise or a large price drop increases. The owner of a call benefits from price increases (because call is more likely to be in the money at expiration) but his loss is limited to the call premium in the event of price decreases. Since the call holder’s loss is limited while his gains are unlimited, an increase in volatility will increase the call’s value. Thus, extremely good stock outcomes can improve the option payoff without limit, but extremely poor outcomes cannot worsen the payoff below zero. This asymmetry means volatility in the underlying asset price increases the expected payoff to the option, thereby enhancing its value. Time to expiration: As T increases, call value increases. For more distant expiration dates, there is more time for unpredictable future events to affect prices, and the range of likely stock prices increases. Thus as T increases, the call is more likely to end up in the money and become more valuable at expiration. Interest rate, rf: Holding the stock price constant, as interest rate rises, the PV of the exercise price decreases. Consider the adjusted intrinsic value of the call = S0 – PV(X). As the PV(X) decreases, the adjusted intrinsic value increases which in turn increases call value. Cash payout, e.g., dividends. Take the case of dividends paid on a stock. A high dividend payout policy puts a drag on the rate of growth of the stock price. For any expected total rate of return on the stock, a higher dividend yield must imply a lower expected rate of capital gain. This drag on stock appreciation decreases the potential payoff from the call option, thereby lowering the call’s value.

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**Determinants of put option value**

If this variable increases Value of put option Asset price, S Decreases Exercise price, X Increases Volatility, Time to expiration, T Increases/ Uncertain Interest rate, rf Cash payouts e.g., dividend Asset price: If S increases, intrinsic value decreases which in turn decreases the put’s value. Exercise price: If X increases, intrinsic value increases since the difference (X – S) becomes bigger. This increases the put’s value. Volatility: As volatility increases, the chance of a large price rise or a large price drop increases. The owner of a put benefits from price decreases (because put is more likely to be in the money at expiration) but his loss is limited to the put premium in the event of price increases. Since the put holder’s loss is limited while his gains are unlimited, an increase in volatility will increase the put’s value. Thus, extremely bad stock outcomes can improve the option payoff without limit, but extremely good outcomes cannot worsen the payoff below zero. This asymmetry means volatility in the underlying asset price increases the expected payoff to the option, thereby enhancing its value. Time to expiration, T: For American puts, increase in time to expiration must increase value. One can always choose to exercise early if this is optimal: the longer expiration date simply expands the range of alternatives open to the option holder, thereby making the option more valuable. For a European put, where early exercise is not allowed, longer time to expiration can have an indeterminate effect. Longer maturity increases volatility value since the final stock price is more uncertain, but it reduces the present value of the exercise price that will be received if the put is exercised. The net effect on put value is ambiguous. Interest rate, rf: Holding the stock price constant, as interest rate rises, the PV of the exercise price decreases. Consider the adjusted intrinsic value of the put = PV(X) - S0 . As the PV(X) decreases, the adjusted intrinsic value decreases which in turn decreases put value. Cash payout, e.g., dividends. Take the case of dividends paid on a stock. A high dividend payout policy puts a drag on the rate of growth of the stock price. For any expected total rate of return on the stock, a higher dividend yield must imply a lower expected rate of capital gain. This drag on stock appreciation increases the potential payoff from the put option, thereby raising the put’s value.

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**Binomial option pricing model (16.2)**

Proposed by Cox, Ross and Rubinstein “Option Pricing: A Simplified Approach”, Journal of Financial Economics, 1979, 7, Replication principle: Two portfolios producing the exact same future payoffs must have the same value. Otherwise, there will be opportunities for riskless arbitrage. Use this model to price European call options. We study the pricing of European options. Such options are simpler and easier to value since exercise is only possible at expiration. Unless otherwise stated, all options are assumed to be European. We use the binomial option pricing model to find the European call option price. Then, we use the Put-Call parity relationship to find the European put option price. The pricing model relies on the notion of replication. This principle says that if two portfolios give you the same payoffs in the future must have the same value right now. If this does not hold, investors will exploit the mispricing through arbitrage. Such activities will ensure that the option price returns to fundamental value. Riskless arbitrage: able to make a sure profit without any risk. such an strategy either entails (i) zero cost and positive profits/cash flows or (ii) positive cash flows upfront and zero cash flows in the future.

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**Single-Period Binomial Model (1)**

S=100, it will move to either 110 or 90 in one year X=100 If the investors borrow money, the interest rate=6% for one year. What is the price of the European call option?

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**Single-Period Binomial Model (2)**

Try to find a synthetic portfolio including stocks and bonds, which will replicate the payoff of a Call option. Solve:

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**Single-Period Binomial Model (3)**

S=100, it will move to either 110 or 90 in one year X=100, r=6% Form a synthetic portfolio: short position in a bond (sell a bond to borrow money) at $ and long position in ½ share of stock after 1 year

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**Single-Period Binomial Model (4)**

Since the payoff (value) for the synthetic portfolio is exactly the same as that for the Call option in all circumstances, the price (initial value) of the portfolio must be the same as that of the Call.

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**Another way to Price a call option (1)**

Compute the price of a call option written on the stock of ABC Inc. The stock is currently selling for S0 = $100. The stock price will either increase by a factor of u = 2 to $200 or fall by a factor of d = 0.5 to $50 by year end. The call option has a strike price of $125 and a time to expiration of one year. The risk-free interest rate is 8% p.a. Assume you only exercise at end of the year. Assume the interest rate is effective annual yield. This is to reconcile with the use of continuously compounding in the put-call parity relationship later on.

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**Another way to Price a call option (2)**

Terms: Su = u x S0 = year end stock price if price rises to $200 Sd = d x S0 = year end stock price if price falls to $50 C0 = call option price Cu = call option payoff if stock price is Su Cd = call option payoff if stock price is Sd 1. Call option payoffs Cu = 200 – 125 = 75 Cd = 0 (not optimal to exercise)

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**Another way to Price a call option (3)**

2) Hedge ratio, H H = (75 – 0)/(200 – 50) = 75 /150 = 0.5/1 3) Form a portfolio that is short 1 call and long 0.5 shares of ABC stock.

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**Another way to Price a call option (4)**

4) Compute the end-of-period payoff. Payoff in 1 year for each possible stock price Sd = $50 Su = $200 Write 1 call -(200 – 125) = -75 Buy 0.5 shares 0.5 x 50 = 25 0.5 x 200 = 100 Total 25 Year end payoff is certain!

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**Another way to Price a call option (5)**

5) Compute present value of $25 with a one-year risk-free interest rate of 8%. PV = 25/1.08 = $ 6) Set value of hedged position equal to present value of the certain payoff and solve for call option’s value. 0.5S0 – C0 = 50 – C = C = = 26.85

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**Binomial option pricing model**

Basic idea: You can form a portfolio consisting of stock and call options that produces a certain (no-risk) payoff in the future. This is also called the hedged position. Discounting this payoff at the risk-free rate gives the portfolio value today. Using this present value and the current stock price, you solve for the call option price.

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**Another pricing problem**

You are attempting to value a call option with an exercise price of $100 and one year to expiration. The underlying stock pays no dividends, its current price is $100, and you believe it has a 50% chance of increasing to $120 and a 50% chance of decreasing to $80. The riskfree interest rate is 10% p.a. What is the hedge ratio? Calculate the call option’s value using the Binomial pricing model. Verify that the call option price is $13.64. BKM Chp15Q29 List the relevant parameters: X=100, T=1, S0=100, Su=120, Sd=80, u=1.2, d=0.8, r=0.1 1) Compute hedge ratio: Compute call option payoffs at year end Cu = =20, Cd = 0 H = (20 – 0)/(120 – 80) = 20/40 = 0.5. The hedge ratio is 0.5. 2) Form portfolio that is long 0.5 shares of stock and short 1 call. Calculate year-end payoff If ST = 80, Payoff = (0.5 x 80) = 40 If ST = 100, Payoff = (0.5 x 120) – (120 – 100) = 60 – 20 = 40 3) Discount the certain payoff at the riskfree rate of 10% and set that equal to the cost of the hedged portfolio and solve for call option price. 40/1.1 = 0.5*100 – C C = /1.1 = 50 – = = (to 2 d.p.) The call option price is $13.64.

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**Price a call option using hedge ratio**

Implication: form a portfolio that is long A shares of stock and short B calls

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**Price a put option using hedge ratio**

Implication: form a portfolio that is long A shares of stock and long B puts Example: Chapter 16, 8

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**Multi-period Binomial Tree**

88
**Put-Call Parity Relationship (1)**

Links the prices of European call and put options. Given: European call option price underlying asset price Riskfree rate The Put-Call Parity Relationship produces the put price. Once you know the call price, put pricing is easy. We make use of the put-call parity relationship to find the European put price.

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**Put-Call Parity Relationship (2)**

Assumptions: Options are European options. Both call and put options are written on the same underlying asset. Underlying asset does not pay any cash flow (e.g., dividends) before option expiration. Continuous compounding. If the parity relationship is violated, an arbitrage opportunity arises. Later on, we extend the basis put-call parity relationship to include dividend-paying stocks.

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**Put-Call Parity Relationship (3)**

The relationship says that: A portfolio that is long 1 call and short 1 put has the same payoffs at expiration as… A portfolio made up of the underlying asset plus a borrowing position. The riskfree interest rate is r.

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**Put-Call Parity Relationship (4)**

Suppose you buy a call option and write a put option, each with the same exercise price, X, and the same expiration date, T. At expiration, the portfolio payoff: ST ≤ X ST > X Long call ST – X Short put -( X – ST) Total

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**Put-Call Parity Relationship (5)**

Compare this payoff to that of a portfolio made up of: 1 share of stock borrowing equal to the present value of the exercise price, Xe-rT. At maturity, repay X. At expiration, the portfolio payoff: ST ≤ X ST > X Long stock ST Loan repayment - X Total ST – X This portfolio is a levered equity position.

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**Put-Call Parity Relationship (3)**

Call option price Put option price Current stock price Present value of exercise price Using continuous compounding, we have Since the two portfolios have the same payoff at expiration, the costs of establishing the two portfolios must be equal. The net cash outlay necessary to establish the option position is C – P. The call is purchased for C, while the written put generates income of P. Likewise, the levered equity position requires a net cash outlay of S0 – Xe-rT, the cost of the stock less the proceeds from borrowing. Equating these costs, we have the put-call parity relationship. The term, Xe-rT is simply the PV of a riskfree bond with a face value of X. The bond matures at T. Re-arranging this equation, we derive the formula for the put option price. See next page. Time to expiration in years r = Riskfree rate (per annum basis)

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Pricing a put option (1) Put option price, P

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Practice 11 Chapter 16: 5,8,9,29,30

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The basics for simulations

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