Rakesh Kumar, ARSD College



Chapter-10, MACHANICS OF OPTION MARKETIt confers the right to do something, but not have to exercise it. It requires upfront payment.Types:Call option: Right to buy an asset by a certain date for a specified price.Put option: Right to sell an asset by a certain date for a specified price.Expiration Date or maturity Date: Price specified in the contractAmerican option- it can be exercised at any time up to expiration date. Most popular.European option: Exercised only on the expiration date. Easy to analyse.Call Option: A call option gives its owner the right to buy stock at a specified exercise or strike price on or before a specified maturity date. If the option can be exercised only at maturity, it is conventionally known as a European call; the option can be exercised on or at any time before maturity, and it is thenknown as an American call. Exercise price = $430If price at the end of expiration period is less than $430, then better in not exercising. If price at the end of expiration period is more than $430 (say $500), then it is worth buying the call option and exercising it. The value of call option at the expiration = $(500 – 430) = $70. This is not profit but pay-off. The circumstances in which the put turns out to be profitable are just the opposite of those in which the call is profitable. If the share is worth $340, Value of put option at expiration = exercise price - market price of the share = $430 - $340 = $90From seller’s/writer’s point of view: The buyer’s asset is the seller’s liability. it is the buyer who always has the option to exercise; option sellers simply do as they are told. If the share price is below the exercise price when the option matures, the buyer will not exercise the call and the seller’s liability will be zero. If it rises above the exercise price, the buyer will exercise and the seller must give up the shares. The seller of the put loses the difference between the share price and the exercise price received from the buyer. the seller will be safe as long as the share price remains above $430 but will lose money if the share price falls below this figure.Profit Diagram: Position diagrams show only the payoffs at option exercise; they do not account for the initial cost of buying the option or the initial proceeds from selling it. $54.35 cost of the Google callCall option:European call option: strike price=$100 per 100 shares; option price=$5; expiration date=4 moths; current stock price=$98; initial investment=$500If the stock price is less than $100 then do not exercise call option and hence losses=$500Suppose, stock price after 4 months = $115 then, gain=$1500 or $15 per share Net profit= $1000If stock price=$102 at the expiration, then losses=$3 per share; not exercising losses =$5Put option: Hoping stock price will fall.Initial investment=$700; stock price after 3 months=$55 per share then gain=$1500 or $15 per shareNet profit=$800If stock price after 3 months exceeds $70, then put option expires worthless and investment losses=$700Option Position: Long position implies option boughtShort position implies option sold or writtenWriter: Receives cash upfront but potential liabilities later; profit or losses reverse of purchase of option.Four types of option position:A long position in call option A long position on put optionA shot position on call optionA shot position on put optionPay-off with respect to purchaser of option.Profit from writing a European call option on one shareOption option=$5Strike price=$100Pay-off from long position in European call option = max (ST – K, 0)option exercised if ST > K; not if ST < K pay-off from short position of European call option=max (ST – K, 0) = min (K – ST,0)Profit from Writing a European put option: option price=$7strike price=$70payoff for holder for holding long position in European put option=-max (K – ST, 0)pay off from short position in European put option=max (K – ST, 0) = min (ST – K, 0)Underwriting asset:Stock option – right to buy or sell 100 shares at specified strike price i.e. ST. Often traded in a lot of 100 shares.Foreign currency option – European style contract – one contract is to buy/sell 10,000 units of a foreign currency for US dollars.Index option – S & P 500 index (SPX), S & P 100 index (OEX); Mostly contracts are European; one contract is usually to buy/sell 100 times the index at the specified strike price; Settlement is always in cash.Future option – Exchanges also trades in American options on that contract.When call option is exercised, the holder’s gain is excess of future price over strike price.When put option is exercised, the holder’s gain is excess of strike price over future price.CHAPTER – 11; PROPERTIES OF STOCK OPTION Factors affecting option prices:Current stock price (S0)Strike price (K)Time of expiration (T)Volatility of stock price (σ)Risk free rate of interest (rf)Dividend that are expected to be paidEuropean callEuropean putAmerican callAmerican putCurrent stock price+-+_Strike price-+-+Time to expiration??++Volatility++++Risk free rate+-+-Amt. of future dividend-+-+s+ → An increase in variable causes option price to increase or stay same- → An increase in variable causes option price to decrease or stay same? → UncertainStock price and strike price: If call option is exercised at some future time, then payoff = (ST – K); More valuable with an increase in the strike price or decrease in K; Opposite for put optionTime to expiration: call and put American option more valuable (at least not decrease in value) as expiration time increases. Long life option is always worth at least as much as short life option because the owner of long life option has all the exercise opportunities open to the owner of short life option.Euro call and put option usually becomes more valuable as time to expiration increases, not always. (i.e. two European call option with expiration date 1 month and 2 months; large dividend is expected in 6 weeks, dividend causes stock price to decrease and short life option could be worth more than long life option.Volatility: measure of uncertainty about future stock prices. Call owner benefits from increases in price but has limited downside risk in the event of decreases in price because the most the owner could lose is the price of option. Put owner benefits from a decrease in price. The value of both calls and puts increase as volatility increases.Risk Free Rate of Interest: with an increase in this rate –Expected return required by the investor from the stock increases; PV of future cash flows received by the option holder decreases; Value of call option increases and put option decreasesIf rate of interest increases due to decrease in price of stock then the value of call option decreases and put option increases.Amount of Future Dividends: Dividend decreases the stock price on the ex-dividend date which is bad news for the call option and good news for the put option. The value of dividend is negatively related to call option and positively to put option.Assumptions:No transaction costBorrowing and lending at risk free rate of interestMarket participants take advantage of the arbitrage opportunitiesNotations:S0=current stock priceK=option strike priceT=option time to expirationST=stock price on expiration dater=rf for maturing in time T (nominal rate and r>0)C=value of American call option to buy one shareP= value of American put option to buy one sharec= value of European call option to buy one sharep= value of European put option to buy one shareupper bounds and lower bounds for option prices: If an option price is above upper bound or below lower bound then there will be profitable opportunities for arbitrageurs.Upper Bound: option can never be worth more than the stock, so stock price is the upper bound for to the option price:c<S0 and C<S0; if not true then an arbitrageur could make riskless profit by buying the stock and selling the call option.American put option: Right to sell one share of stock for K, can never more be worth more than K:P<KEuropean option: At the maturity the option can’t be more than K. If can’t be worth more than the PV of K today:P<Ke-rTIf not true, then an arbitrageur can make a riskless profit by writing option and investing proceeds of the sale at rf.Lower Bound for Calls on non-dividend paying Stocks: Lower bound for Euro call option on non-dividend paying stockS0 – Ke-rTEx. See pg no 267Formal Argument: Portfolio A: one European call option + zero coupon bond that provides payoff of K at time TPortfolio B: one share of stockIn portfolio A, zero coupon bond is worth K at T. If ST>K, call option is exercised at maturity and portfolio is worth ST.If ST<K, call option expires worthless and portfolio is worth K. Hence at T, portfolio is worth max(ST, K)Lower Bound for European puts on non-dividend paying Stocks: For European put option on a non-dividend paying stock, a lower bound price isKe-rT – S0Formal Argument: portfolio C: one European put + one sharePortfolio D: one zero coupon bond paying off Kat time T.If ST <K, option exercised at option maturity in portfolio C and portfolio becomes worth K.If ST >k, the put expires worthless and the portfolio is worth ST at this time. Hence, portfolio C is worth Max(ST, K) in time T.Portfolio D is worth K in time T. Hence, Portfolio C is always worth as much as, and can be sometimes be worth more than, portfolio D in time T.In the absence of arbitrage opportunities portfolio C must be at least as much as portfolio D today. Sop + S0 > Ke-rTor p> Ke-rT – S0Because the worst that can happen to a put option is that it expires worthless, its value can’t be negative. Hencep> max (Ke-rT- S0, 0)Put call Parity: Relationship between prices of European put and call that have the same strike price k and time to maturity T. ConsiderPortfolio A: one European call and one zero coupon bond that provides a payoff of K at T.Portfolio C: one European put and one share of a stockValues of portfolio A and portfolio C at time TST >KST < KPortfolio ACall optionST - K0Zero coupon BondKKTotalSTKPortfolio CPut option0K - STShareSTSTTotalSTK If ST > K, both portfolios are worth ST at. If ST < K, both portfolios are worth K at T. i.e. both are worthmax (ST, K), when option expires at TSince, portfolios have identical values at time T so they must have same values today, otherwise an arbitrageur will buy cheaper portfolio and sell expensive portfolio. Since, portfolios cancel each other at T, this strategy would lock arbitrage K = difference between two portfolio values.c + Ke-rT = p + S0LHS = Portfolio A today; RHS = Portfolio B todayIt shows that the value of a European call with certain exercise price and exercise date can be deducted from the value of European put with the same exercise date and exercise price, and vice-versa. Call on Non-dividend Paying Stock: It is never optimal to exercise US call option on a non-dividend paying stock before the expiration date. Then, the strike price has to be paid later on (on which the investor earns interest for the full month with no chance on losing due to dividend). Moreover there is some chance that the stock price may move below the strike price. Formally, c > S0 – Ke-rT Given, r > 0, c > S0 – K when T > 0 then c is always greater than option’s intrinsic value prior to maturity. If it were optimal to exercise prior to maturity, C would equal the option’s intrinsic value at that time. Call option insures option holder against stock price falling below strike price. For option holder the later the strike price is paid out the better.Bounds: Since, US call options are never exercised early when there are no dividends, they are equivalent to European call option, so that C = c. Hence, the lower and upper bounds are given bymax(S0 – Ke-rT, 0) and S0 respectively.Puts on Non-Dividend Paying Stock: Put should be exercised as early if it sufficiently deep in money.Let ST =$10 & S0 =0; if exercised immediately then gain is $10; if waited, gains falls below but it can never be more than $10 because negative stock prices are impossible. Further, receiving $10 now is preferable to receiving $10 in future. A put option plus a stock is an insurance against the stock price falling below a certain level. Early put exercise is attractive if S0 falls, as r rises, and the volatility decreases.Bounds: Lower and upper bound of a European put, when there is no dividends, are given bymax(Ke-rT –S0, 0) < p < Ke-rTFor American put on a non-dividend paying stockP > max(K – S0, 0)Lower and upper bound for an American put on a non-dividend paying stock aremax( K – S0, 0) < P < KVariations of the price of American put option with stock prices: It is optimal to exercise American put option early, the value of the option is K – S0, hence the value of put merges into the put’s intrinsic value, K – S0. The curve representing the relationship between American put price and stock price moves outwards if T increases or stock price volatility increases or r decreases. An American put option is always worth more than a European put option. Because, a American put is sometimes be worth its intrinsic value but a European put option must sometimes be worth less than its intrinsic value.Variation of price of a European put option with the stock price: The point B, at which the price of the put option is equal to its intrinsic value, must represent a higher value of the stock price than point A. Point E in the fig is where S0 = 0 and the European put price is Ke-rT.Effects of Dividend: Assumption: The dividends paid during the life of the option are known. A dividend is assumed to occur at the time of its ex-dividend date.D = PV of dividends during the life of the option.Lower Bounds and upper bounds of calls and puts: Portfolio A: European call and cash = D + ke-rTPortfolio B: One shareWe havec > max(S0 – D – Ke-rT, 0)Portfolio C: One European put and one sharePortfolio D: Amount of cash = D + e-rTWe have p > max(D + Ke-rT – S0, 0)Early Exercise: When dividend id expected, an American call option will not be exercised early. Sometimes it is optimal to exercised American call immediately prior to an ex-dividend date. It is never optimal to exercise a call at other times.Put – Call Parity: With dividend, call-put parity becomes:c + D + Ke-rT = p + S0With dividend, American call-put parity becomes:S0 + D + K < C – P < S0 – Ke-rT ................
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