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Lecture 7 The Fundamentals of Options Primary Text Edwards and Ma: Chapter 18

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Options An option is a contract that gives its holder a right but not an obligation to purchase or sell a specific asset (e.g., commodity futures or security) at a specific price on or before a specified date in the future. To acquire this right, the buyer of the right (i.e., the option buyer or holder) pays a premium to the seller of the right (i.e., the option seller or writer). If the option holder chooses to exercise her right to buy or sell the asset at the specified price, the option writer has an obligation to deliver or take delivery of the underlying asset. The potential loss to an option writer is unlimited (?). In contrast, if the option holder chooses not to exercise her right, but to allow the option to expire, her loss is limited to the premium paid.

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Options Terminology Option Holder (Buyer) – An individual (or firm) who pays the premium to acquire the right. Option Writer (Seller) – An individual (or firm) who sells the right in exchange for a premium. Premium – the market value of the option, in effect the price of the insurance. Strike Price – The fixed price specified in an option contract is called the option’s strike price or exercise price. Expiration Date – The date after which an option can no longer be exercised is called its expiration date or maturity date.

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Options Terminology Call Option – An option (a right but not an obligation) to buy a specified asset at a set price on or before a specified date in the future. Put Option – An option (a right but not an obligation) to sell a specified asset at a set price on or before a specified date in the future. American-type Option – An American-type option can be exercised at any time prior to the contract’s expiration date, at the holder’s discretion. European-type Option – A European-type option can only be exercised on the contract’s expiration date.

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Options Exchange-Traded Options Exchange-traded options contracts are standardized and traded on organized (and government designated) exchanges. An exchange-traded option specifies a uniform underlying asset, one of a limited number of strike prices, and one of a limited number of expiration dates. Strike price intervals and expiration dates are determined by the exchange. Performance on options contracts is guaranteed by a clearing corporation that interposes itself as a third party to all option contracts. Thus, contract standardization and a clearing corporation guarantee provide the fundamental structure for exchange-traded options.

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Options Exchange-Traded Options Once an exchange-traded option contract is purchased, contract obligation may be fulfilled in one of these three ways: The option holder exercises her right on or before the expiration date – the option writer must then adhere to the terms of the option contract, and accept the other side of the position. The option writer keeps the premium. The option holder allows the option to expire unexercised – the premium is retained by the option writer, and the writer’s obligation is discharged. Either or both the option holder and/or writer executes an offsetting transaction in the option market, eliminating all future obligations. In this case, the rights or obligations under the original contract are transferred to a new option holder or writer.

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Options How Options Work In addition to its type (call or put) and the name of the underlying asset/security, an option is identified by its strike price and expiration date. For exchange traded options, the strike price and expiration date are determined by the rules of the exchange. On the Chicago Board Options Exchange (CBOE), a single call option contract gives its holder the right to buy 100 shares of the underlying stock and is of the American-type. Original maturities of CBOE options vary from three months to three years, and they all expire on the third Friday of the month in which they mature.

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CallPut StrikeExp.Last saleChangeOpen Int.Last SaleChangeOpen Int April , , , , , , May , , , , , , Options Listing of Home Depot Option Prices: CBOE, 24 March 2009

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The HD April 09 call option with a strike price of $20.00 per share was last traded at a price (premium) of $3.60 per share. A buyer of this option, therefore, would have to make an immediate payment of $3.60 per share (or $360 per contract) to the writer of the option. The buyer of the call option would have the right (but not the obligation) to purchase 100 shares of HD at $20 until April 17 (the third Friday in April). If exercised, the holder’s net cost per share of the HD stock would be $23.60 per share. If the holder let the option expire without exercising the right, her net loss would be $360. Options Listing of Home Depot Option Prices: CBOE, 24 March 2009

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The HD April 09 put option with a strike price of $20.00 per share was last traded at a price (premium) of $0.35 per share. A holder of this option, therefore, would have to make an immediate payment of $0.35 per share (or $35 per contract) to the writer of the option. The holder of the put option would have the right (but not the obligation) to sell 100 shares of HD at $20 until April 17 (the third Friday in April). The holder’s net revenue per share of the HD stock would be $19.65 per share. If the holder let the option expire without exercising the right, her net loss would be $35. Options Listing of Home Depot Option Prices: CBOE, 24 March 2009

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The level of the strike price and the value of option: Call options with lower strike prices are more valuable and the put options with higher strike prices are more valuable to the holders. Call:Strike Price ↑ => Premium ↓ Put:Strike Price ↑ => Premium ↑ Intrinsic value versus time value: Option premiums have two components Intrinsic value Time value Options Properties of Option Pricing

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Intrinsic value: If the current stock price is above the strike price of a call (or below the strike price of a put), the option has intrinsic value. An option with intrinsic value is said to be in-the-money. If the current stock price is equal to or below the strike price of a call (or equal to or above the strike price of a put), the option has no intrinsic value. An option with no intrinsic value is said to be at-the-money if current market price of the stock is equal to the strike price An option with no intrinsic value is said to be out-of-the-money if current market price of the stock is below the strike price of a call and above the strike price of a put Options Properties of Option Pricing

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Market ScenarioCallPut Market price > Strike PriceIn-the-MoneyOut-of-the-Money Market price = Strike PriceAt-the-Money Market price < Strike PriceOut-of-the-MoneyIn-the-Money Options Properties of Option Pricing Time value: Why, then, the options that are out-of-the-money have positive premiums? Because, it still has a time value. The difference between an option’s price (premium) and its intrinsic value is called the option’s time value. Time Value = Premium – Intrinsic Value

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Intrinsic value and Time Value: Call: Intrinsic Value = Market Price – Strike Price Time Value = Premium – Intrinsic Value Premium = Intrinsic Value + Time Value Put: Intrinsic Value = Strike Price – Market Price Time Value = Premium – Intrinsic Value Premium = Intrinsic Value + Time Value Options Properties of Option Pricing

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CallPut StrikeExp.PremiumInt. Val.Time Val.PremiumInt. Val.Time Val April May Options Components of Option Premiums HD Stock Closing price on 24 March 2009: $22.95 per share

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The relationship between time value and time to expiration: the longer the time remaining until an option’s expiration, the higher the premium tends to be, everything else being equal. The difference between the premiums of two options with the same strike price but different expiration date is the same as the difference between the time values of the two options. For example, the premium for the HD May 2009 call option with the strike price of is ($3.90) higher than the premium for the HD April 2009 call option ($3.60) with the same strike price. The difference between the time value of these two options is also $0.30 per share. This is because a longer time provides more opportunity for the price of the underlying asset to move to a level where the option is in-the-money, and where the purchase and sale of the asset at the specified strike price will be profitable. Options Properties of Option Pricing

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The relationship between time value and strike price: The magnitude of an option’s time value reflects the potential of the option to gain intrinsic value during its life. A deep out-of-the-money option has little potential to gain intrinsic value because to do so asset prices will have to change substantially. Therefore, it will have little time value. Similarly, a deep in-the-money option is as likely to lose intrinsic value as to gain it, as a consequence also has little time value. In general, time value is at the maximum when an option is at-the- money. Options Properties of Option Pricing

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Options make it possible for investors to modify their risk exposure to the underlying asset Denote the exercise (strike) price of an option on date t by SP t and the market price of the underlying stock by MP t At expiration the strike price of the call option and the market price of the underlying stock can be denoted by SP T, and MP T, respectively. At expiration the payoff from a call option is the larger number between its intrinsic value (time value is zero at expiration) and zero. Max (MP T −SP T, 0) At expiration the payoff from a put option is the larger number between its intrinsic value (time value is zero at expiration) and zero. Max (SP T −MP T, 0) Investing with Options

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Denote call and put premiums by C and P respectively. Profit/Loss of the Call Holder =Max (MP T −SP T, 0) − C Profit/Loss of the Call Writer = C − Max (MP T −SP T, 0) Profit/Loss of the Put Holder = Max (SP T −MP T, 0) − P Profit/Loss of the Put Writer = P − Max (SP T −MP T, 0)

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In addition to using options to modify risk exposure, buying or selling options is an alternative way to take a position in the market for a trader who does not own the underlying asset. For example, suppose that you have $100,000 to invest and you expect that the price of a stock is going to increase (bullish). Assume that the riskless interest rate is 5% per year and the stock pays no dividends. Compare your portfolio’s rate of return for three alternative investment strategies over a one-year holding period: Invest the entire $100,000 in the stock. Invest the entire $100,000 in call options of the stock. Invest 10% ($10,000) in call options of the stock and the rest ($90,000) in the risk-free asset. Investing with Options

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Assume that the current price of the stock is $100 per share, and the premium for the call option with $100 strike price is $10 per share. Under the three alternative strategies, the investment scenarios and corresponding rate of returns are Strategy 1: Buy 1,000 share of the stock Strategy 2: Buy calls (with strike price $100) on 10,000 share of the stock Strategy 3: Buy calls (with strike price $100) on 1,000 share of the stock and invest $90,000 in risk free asset (with 5% interest rate). Investing with Options

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Which of the three strategies is the best for you? It depends on your expectation about the stock price and your risk tolerance Suppose that you have scenario forecasts for three possible states of the economy. You believe that there is a probability of 0.2 (20% chance) that there will be a boom and stock price will rise by 50% during the year, a probability of 0.6 (60% chance) that the economy will be normal and market will rise by 10%, and a probability of 0.2 (20% chance) that there will be a recession and the stock will fall by 30%.

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Which of the three strategies is the best for you? Normal situation – Strategy 1 performs the best Boom – Strategy 2 performs the best Recession – Strategy 3 performs the best Thus, none of the strategies dominates the other. Depending on an investor’s risk tolerance, he or she might choose any one of them. Indeed, a very highly risk-averse investor might prefer the strategy of investing all 100,000 in the risk-free asset to earn 5% for sure.

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The Put-Call Parity Pricing Relationship: Common Stock We have just seen that a strategy of investing some of your money in the riskless asset and some in a call option can provide a portfolio with a guaranteed minimum value and an upside slope equal to that of investing in the underlying stock. There are at least two other ways of creating that same pattern of payoffs: Buy a share of stock and a European put option and buy a pure discount bond and a European call option. Consider a share of a stock with market price $100, and European call and put options with strike price $100 and premium $10.

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Payoff Structure for Stock plus Put Strategy

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Payoff Structure for Bond plus Call Strategy

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Thus, a portfolio consisting of a stock plus a European put option (with strike price SP) is equivalent to a pure discount default-free bond (with face value SP) plus a European call option (with strike price SP). So, by the Law of One Price, they must have the same price. The following equation expresses the pricing relation: The equation is known as the put-call parity relationship. The relationship allows one to determine the price of any one of the four securities from the values of the other three. The Put-Call Parity Pricing Relationship: Common Stock

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The put-call parity relationship can also be used as a recipe for synthesizing any one of the four from the other three. Use the sign of each variable to determine long or short A plus sign indicates cash inflow – thus, short A minus sign indicates cash outflow – thus, long Rearranging the put-call parity relationship we have: That is the characteristics of a short call option can be broken into three components: Short the put option with the same strike price Short the underlying stock at the prevailing market price Long the bond with the face value equal to the SP of the options The Put-Call Parity Pricing Relationship: Short Call Synthetic

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Rearranging the put-call parity relationship we have: That is the characteristics of a long call option can be broken into three components: Long a put option with the same strike price of the call long the stock at the prevailing market price Short the bond with face value equal to the SP of the options The Put-Call Parity Pricing Relationship: Long Call Synthetic

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The put-call parity relationship can also be rearranged as : That is the characteristics of a short put option can be broken into three components: Short a call option with the same strike price Long the underlying stock at the prevailing market price Short the bond with the FV equal to the SP of the options The Put-Call Parity Pricing Relationship: Short Put Synthetic

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The put-call parity relationship can also be rearranged as : That is the characteristics of a long put option can be broken into three components: Long the call option with the same strike price Short the underlying stock at the prevailing market price Long the bond with the FV equal to the SP of the options Similarly, long or short stock or bond can also be synthesized using the put-call parity relationship – following the same principle. The Put-Call Parity Pricing Relationship: Long Put Synthetic

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Call-Put Arbitrage The equations for call and put premiums can also be regarded as formulas for converting a put into call and vice versa. For example, suppose thatMP = $100, SP = $100, P = $10, T = 1 year, and r = 0.08 Then the price of the call option, C, would have to be C = – 100/1.08 = If the price of the call is too high or too low relative to the price of the put, and there are no barriers to arbitrage, arbitragers can make a certain profit. For example, if C is $18 and there are no barriers to arbitrage, an arbitrager can lock in a riskless profit by selling a call with strike price $100 and simultaneously buying a put with the same strike price and expiration date, borrowing the capital at the risk-free interest rate, and buying the underlying stock.

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Call-Put Arbitrage

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Put-Call Arbitrage If the price of the put option is high relative to call prices, an arbitrager can lock in a riskless profit by selling the overpriced put and buying a call, selling the underlying stock, and lending the proceed at the risk-free interest rate (8%).

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The Put-Call Parity Pricing Relationship We can gain some additional insight into the nature of the relationship among puts, calls, stocks, and bonds by rearranging the terms in the put-call parity relationship. If MP = SP/(1+r) T => C = P If MP > SP/(1+r) T => C > P If MP C < P

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Option Pricing: The Black-Scholes Model Black and Scholes derive the following equations for pricing European call options on non-dividend-paying stocks: C = price (premium) of the call MP = current market price of the stock SP = current strike price of the call r = riskless interest rate T = time to maturity of the option in years σ= standard deviation of the annualized rate of return on the stock N(d)= probability that a random draw from a standard normal distribution will be less than d

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The formulation of the model is the construction of a hypothetical risk-free portfolio, consisting of long call options and short positions in the underlying stock, on which an investor earns the riskless rate of interest. We can derive the formula from the price of a put option by substituting C in the put-call parity condition: Option Pricing: The Black-Scholes Model

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Estimation of Price Volatility Price volatility refers to the degree of volatility of price change – the percentage changes in prices. A commonly used measure of this volatility is the standard deviation of previous daily, weekly, or even monthly percentage changes in prices. The percentage price change is often calculated as the difference between the natural logarithms of the current and previous prices. N = Number of observations X t = ln(MP t /MP t-1 ), i.e., the percentage price change MP t = stock price at the end of period t X-bar = mean of X t

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Estimation of Price Volatility Depending on which data interval is used to calculate the standard deviation of prices, annualized price volatility (σ) is obtained by multiplying the calculated standard deviation by the square root of the number of periods: For monthly data: σ = × monthly standard deviation For weekly data:σ = × weekly standard deviation For daily data:σ = × daily standard deviation

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Fundamental Determinants of Option Prices In general, prior to expiration the value of an option depends upon six variables: the current value of the underlying asset or stock (MP t ) the options strike price (SP t ) the time remaining until the option expires (T − t) the current level of the risk-free interest (r) the anticipated volatility of the price of the underlying asset or stock (σ) Cash dividend yield (d)

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Fundamental Determinants of Option Prices

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