,000 ( 10;000  10) a year to have thisposition on.This is just a simplified example of the mechanics ofrepo. In reality, there is a very sophisticated repo marketto borrow and lend stocks. This means that the repo isnever fixed and will depend on demand from borrowersand supply of lenders.
13.2
REPO IN CASE OFA TAKEOVER
A takeover usually has a positive impact on the shareprice of the takeover target and a negative impact on theshare price of the acquiring company. The reason for thisis that the acquiring company expects to get synergiesand cost savings from this takeover and is therefore willing to pay a premium to the current share price of thetakeover target. Because there are usually a few companies looking at the same takeover target, the premiumpaid over the current share price is usually so much thatit only pays back when all the theoretical synergies andcost savings work out. This means that if somethingunexpected happens the acquiring company is likely tolose on the deal.4 For this reason a takeover tends to4This does not mean that every takeover is a bad deal. Very often, companieshave to make takeovers in order to cut costs, which is necessary to surviveamid fierce competition.
140
AN INTRODUCTION TO OPTIONS TRADING
attract short players of the stock of the acquiring company, which pushes up the repo of this stock. In takeoversituations one can see repo rates as high as 10% a year.
13.3
REPO AND ITS EFFECTON OPTIONS
Dividends and repo rates have the same effect on theforward. In both cases a higher dividend or repo bringsthe forward down. This also means that repo rates canhave a substantial impact on option prices.If a stock’s repo increases to particularly high levels, putoptions will become more expensive and call options lessexpensive. The reason becomes apparent when one looksat the different ways calls and puts are being hedged. If atrader buys a call option he will have to sell shares tohedge himself. In case of a high repo on a stock it wouldcost him a lot to put his hedge on because, if he wants toshort the stock, he would have to borrow it at a very highrate. Since he only wants to pay a certain implied volatility he needs to lower the price of the call comparedwith a situation where there is a much lower repo. For aput option the reverse holds. If a trader buys a put he willbuy the underlying stock to hedge himself. If this stockhas a very high repo the investor can lend these sharesand make additional money by doing so. Therefore, hecan pay a higher premium and still get the same impliedvolatility compared with a situation where the underlying is treated as general collateral.
REPO
13.4
141
TAKEOVER IN CASHAND ITS EFFECT ONTHE FORWARD
When a company is taken over in cash, this has a bigimpact on the prices of the options on this company. Thereason is that, when a cash offer is submitted and accepted by shareholders, one can reasonably expect thatthe share price of the takeover target will no longermove and, more importantly, the forward changes significantly. As an example, consider the cash bid byTelefónica on O2 for 200 pence per share. Suppose thatthe shares of O2 will continue trading for another3 months after Telefónica’s offer has been made public,after which the shares will be delisted and the shareholders will receive 200 pence per share in cash. The question is obviously: What effect does this cash offer have onthe price of a 2-year call option on O2 ? To answer thisquestion one first has to realize that the volatility of O2 isnext to 0. The next step is to think about what happens tothe forward of O2 . Since O2 shares will be delisted in 3months a 2-year forward is equal to a 3-month forward.Since any forward longer than 3 months is equal to the 3month forward, one can easily misprice call and putoptions with maturities over 3 months, especially ifthere is a big difference between the dividend yield andthe risk-free interest rate. Knowing this, it is relativelyeasy to price a 2-year call option on O2 with a strike priceof 180 pence. Assuming O2 will not pay any dividends,this call can be priced using equation (3.8), which expresses the price of a call in terms of its volatility and
142
AN INTRODUCTION TO OPTIONS TRADING
the forward. Since the stock will be delisted after 3months the maturity (T  t) of the call should be takento be 3 months. Since the share price is higher than thestrike price of 180 and the volatility is next to 0, d1 inequation (3.8) is 1 and, therefore, Nðd1 Þ and Nðd2 Þ areequal to 1. Plugging in all these variables into equation(3.8) gives a call price of St  erðTtÞ K,5 where T  t is 3months. This price can be made intuitive by going backto how an option on an underlying with zero volatilityand a maturity of 3 months is hedged. Since the shareprice is 200 pence, the strike price is 180 and the volatilityis 0, the delta of the option is 1. So, if an investor buys thecall he will hedge himself by selling the shares at 200pence, St . The call will give the investor the right to buythe shares at 180 in 3 months’ time. For this reason thevalue of the investor’s portfolio will be St erðTtÞ  K in3 months’ time. Since the price of the call option is equalto the present value of the portfolio’s value in 3 months’time, the call is worthSt erðTtÞ  K erðTtÞ ¼St  erðTtÞ K. By the same argument the value of anout-of-the-money call on O2 should theoretically be 0.6
5This price holds if the options are closed out at intrinsic. In some cases theoption exchange will compensate for the time value.6In practice, an out-of-the-money call on a share on which a cash offer hasbeen submitted will not be 0 because people will speculate on a counter offer.
Appendix
APROBABILITY THATAN OPTION EXPIRESIN THE MONEY
144
APPENDIX A
W
henever the stock price movement is only knownup to time t and the price at time t is St , the Black–Scholes formula gives that:1 2
ST ¼ St eðr2 ÞðTtÞþy
ðA:1Þ
provided it is a non-dividend paying stock. In the aboveformula y is normally distributed with mean 0 and variance 1. Now it is easy to calculate the probability that anoption expires in the money. Take a call option. A calloption expires in the money when ST > K, where K is thestrike price. This means:1 2
ST ¼ St eðr2 ÞðTtÞþy > Km Stlnþ r  12 2 ðT  tÞKpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃy> Tt¼ d2Since y has a standard normal distribution:Pðcall expires in the moneyÞ ¼ PðST > K Þ ¼ Nðd2 ÞðA:2ÞIn the same way one can derive:Pðput expires in the moneyÞ ¼ PðST < K Þ ¼ Nðd2 Þ:ðA:3Þ
Appendix
BVARIANCE OF ACOMPOSITE OPTION
146
I
APPENDIX B
t is very easy to prove that the variance of a compositeoption is equal to:2compo ¼ 2S þ 2S FX þ 2FX
ðB:2Þ
One just has to compare the movement of the underlyingin the composite currency with the movement of a geometric basket option composed of the underlying in thelocal currency and the FX rate. The formula for thisbasket option is as follows:Ft ¼ St;local St;FX
ðB:2Þ
dSt;local¼ rlocal dt þ S dW1;tSt;local
ðB:2Þ
where
and1dSt;FX¼ rcompo  rlocal dt þ FX dW2;tSt;FX
ðB:3Þ
SinceVar½lnðFt Þ ¼ Var ln St;local St;FX¼ Var ln St;local Þ þ ln St;FX¼ 2S þ 2S FX þ 2FX t
ðB:4Þ
1To get an intuition for equation (B.3), imagine an exchange rate with zerovolatility, and therefore the value of the exchange rate only changes becauseof differences in risk-free interest rates between the currencies. In this case
1 þ rcompo dt. Taylor expansion gives that this is equal to1 þ rlocal dt St;FX 1 þ rcompo  rlocal dt .Stþdt;FX ¼ St;FX 
APPENDIX B
and
147
E½lnðFt Þ ¼ E ln St;local St;FX

ðB:5Þ
¼ rlocal t þ rcompo  rlocal t
ðB:6Þ
¼ rcompo t
ðB:7Þ
the movement of the stock in the composite currency canbe modelled according to equation (12.4).
BIBLIOGRAPHYAarts, P. (1999). Pricing the basket option and exploring its volatility structure. Mathematics thesis, Utrecht University,Utrecht, The Netherlands.Durrett, R. (1996). Probability: Theory and Examples, 2nd edn.Duxbury Press, Belmont, California.Grimmett, G.R. and D.R. Stirzaker (1992). Probability andRandom Processes, 2nd edn. Clarendon Press, Oxford, UK.Hull, J.C. (1993). Options, Futures, and Other Derivative Securities, 2nd edn. Prentice Hall, Englewood Cliffs, New Jersey.Hull, J.C. (1997). Options, Futures and Other Derivatives, PrenticeHall, Englewood Cliffs, New Jersey.Lamberton, D. and B. Lapeyre (1996). Introduction to StochasticCalculus Applied to Finance, Chapman & Hall, London.Roelfsema, M.R. (2000). Exploiting the Dependencies betweenIndex and Stock Options. Delft University Press, Delft,The Netherlands.Weert, F.J. (2002). Basket en skew arbitrage. Mathematics thesis,Utrecht University, Utrecht, The Netherlands.Wilmott, P. (1998). Derivatives: The Theory and Practice ofFinancial Engineering, university edn. John Wiley & Sons,Chichester, UK.
INDEXAhold 99 100, 138 9American optionsconcepts 3, 7, 13 14, 31, 48 9,57 60dividend-paying stocks 57 60early exercise 13 14, 48 9, 57 60European options 7, 57 9pricing 31, 57 60appendices 143 7arbitrage, concepts 16 19at-the-money optionsconcepts 9 12, 23 7, 65, 92,97 100, 106 9, 111 16, 131,135 6gamma 65, 92, 94 5, 98prices 65, 92, 97 8, 111 16quantos 131straddles 111 13, 114 16strangles 114 16AXA 122banksexotics 130pre-paid forwards 123 5ratings 125types 125basis points (bps) 138 9basket options 146 7bearish strategies 107 8, 138 9binomial tree modelBlack Scholes pricing formula 31concepts 31Black Scholes pricing formulabinomial tree model 31concepts 2, 21 31, 35 40, 42 5,55 6, 133 5, 144
dividends 22 31, 35 40, 57 60,133, 144forwards 36 8important dependencies 26 7interest rates 22 3, 29 31, 42 3,133 5parameters 2, 22 31, 35 40, 42,44 5, 133riskless portfolios 55 6volatility 22 5, 28 9, 42 5, 133 5BMW 106 7, 110 11BP 121, 130 2bps see basis pointsBrownian motion 133 5bullish strategies 106 7, 109 11,130 1buy-backs 126 7buyers, options 8 12‘buying the call spread’ 106call optionscollars 109 11concepts 2 6, 8 16, 23 9, 37 9,48, 49 53, 54, 74 86, 106 7, 144delta values 48, 49 54, 63 5, 74 80dynamic hedging 74 86early exercise 13 14, 48 9, 57 60employee incentive schemes 126examples 3 6, 13 14, 23 9, 35 6,37 9, 44 5, 49 54, 74 86, 144gamma 63 5, 75 86, 98hedging 52 7, 74 86interest rates 30 1, 86 92long positions 8 12, 15 16, 54 7,74 7, 80 6, 89 92, 106 7pre-paid forwards 123 5
152prices 23 7, 29, 37 9, 49 52put call parity 16 19, 39 40repos (repurchase agreements) 140short positions 8 12, 15 16, 54,77 86, 98 9, 106 7straddles 111 12synthetics 39 40theta 66 9, 75 80vega 69 71call spread, concepts 106 7cash takeovers, forwards 141 2Coca-Cola 123collarsconcepts 109 11, 121 2delta 111insurance companies 122commodities, underlying assets 2composite optionsconcepts 135 6, 145 7variance 145 7corporate investors 118, 119 27correlationFX factors 130 5types 131 2cost of carry 35cross-holdings, corporate investors119 20currencies see foreign exchangeDaimler Chrysler (DCX) 35 6,118 20delta 47 60, 74 92, 94 8, 103, 111,142collars 111concepts 48 60, 62 5, 74 87,94 8, 103, 111, 142definition 48dividend-sensitive options 57examples 48, 74 86gamma 62 5, 75 86, 94 6, 98hedging 52 7, 59 60, 62 5, 74 86,94 8, 111negative values 48neutral positions 56 7, 63, 74 86,91 2discounted expected payoffs, delta49 52discounting, concepts 11 12diversification 119 20dividend yields 37 8, 122dividends
INDEXBlack Scholes pricing formula22 31, 35 40, 57 60, 133, 144concepts 22, 25 6, 33 40, 51 2,57 60, 122, 140 2effects 22 31, 33 40, 51 2, 57 60ex-dividend stocks 58 9options pricing 22 31, 34 40,51 2, 57 60risk 38 9dynamic hedgingconcepts 56 7, 74 92, 94 6downward stock movement 82 6long call options 74 7, 80 6, 89 92profit formula 80 6profits 74 92, 94 6short call options 77 86theta 75 80, 86 92upward stock movement 81 6early exercise, American options13 14, 48 9, 57 60employee incentive schemes 126European optionsAmerican options 7, 57 9concepts 3 7, 9 10, 15 19, 22 31,51 2, 63 5examples 3 6, 9 12, 13 14, 23 9,35 6, 37 9, 44 5, 51 2gamma 63 5interest rates 30 1pricing 22 31, 51 2theta 66 9event risk 99 100ex-dividend stocks 58 9exchange-traded options, concepts2 3, 7exercised rightsearly exercise 13 14, 48 9, 57 60options 2 7, 13 14, 17 19exotic optionscomposite options 135 6, 145 7concepts 130 6quanto options 130 5expiration, concepts 2 12, 143 4foreign exchange (FX)concepts 2, 130 6, 146 7quantos 130 5volatility 132 5forwards
INDEXBlack Scholes pricing formula36 8concepts 34 5, 36 8, 109 11, 122,123 5, 140 2definition 34options pricing 36 8pre-paid forwards 123 5pricing 34 5, 36 8repos (repurchase agreements) 140,141 2synthetics 39 40takeovers 141 2FTSE indices 122FX see foreign exchangegammaconcepts 62 5, 75 86, 94 6, 98definition 62delta 62 5, 75 86, 98European options 63 5expiring at-the-money options 65,92, 94 5, 98vega interaction 94 6gc see general collateralgearing 96 7, 109 11, 118collars 109 11spreads 96 7general collateral (gc) 138 40geometric basket options 146 7GlaxoSmithKline (GSK) 135 6the Greeks 29, 31, 48 60, 62 71,74 87, 94 100see also delta; gamma; rho; theta;vegaconcepts 29, 48 60, 62 5, 69 71,74 87, 94 100direction of the underlying 81 6,97 8in practice 94 100GSK see GlaxoSmithKlinehedging 34 5, 52 7, 59 60, 62 5,74 86call options 52 7, 74 86concepts 34 5, 52 7, 74 80definition 52delta 52 7, 59 60, 62 5, 74 86,94 8, 111put options 54 6repos (repurchase agreements) 140
153historic volatility 42 3, 99 100Hull, J.C. 44implied volatility 41 6, 69, 88, 91 2,95 103, 127see also skew; volatilityconcepts 42 6, 69, 88, 91 2,95 103, 127definition 44examples 44 5, 99 100premiums 99 100, 103 4strategies 45 6, 69, 91 2, 95 6strike price 102 3in-the-money optionsconcepts 9 12, 25 7, 49 52, 54 9,66 7, 143 4theta 66 7insurance companies, collars 122interest rates 2, 12, 16 19, 22 31,42 3, 66 9, 86 92, 124 5Black Scholes pricing formula22 3, 29 31, 42 3, 133 5concepts 22 3, 29 31, 42 3, 66 9,86 92, 124 5dynamic hedging 86 92estimates 43per annum 29 31per trading day 29 31rho 31risk-free rates 56, 110 11, 133 6theta 66 9, 86 92intrinsic value, options 11, 13, 48 52,58 60investment banksexotics 130ratings 125investorscorporates 118, 119 22panics 104portfolio managers 118 27strategies 106 16, 117 27takeovers 120 2, 141 2types 118Libor 124 5loanspre-paid forwards 123 5repos (repurchase agreements)138 42long correlation, concepts 131 2, 136
154long positionsdynamic hedging 74 7, 80 6,89 92, 95 6options 8 12, 15 16, 34 5, 54 7,71, 74 7, 80 6, 89 92, 94 6,106 7, 118 19, 127lossesoptions 8 12, 25, 76 86, 88 92,100theta 89 91volatility 25, 100
INDEX
Newton Raphson method 44normal distribution 27 9, 44, 133 5,144
54 7, 71, 74 7, 80 6, 89 92,94 6, 106 7, 118 19, 127losses 8 12, 25, 76 86, 88 9, 100obligations 2 3parties 8prices 4 8, 11 12, 16 19, 21 31,42 60, 102 4, 131 5profits 4 6, 8 12, 15, 25, 46, 74 92,94 6, 114 16, 118, 126 7repos (repurchase agreements)140 2short positions 8 12, 15 16, 54,77 86, 94 6, 98 9states 9 12strategies 45 6, 69, 91 2, 94 100,105 16, 117 27takeovers 120 2, 141 2terminology 2 3, 8 12types 2 3out-of-the-money options, concepts9 12, 25 7, 49 52, 54 5, 142
O2 141 2obligations, options 2 3option part to the price of an optionsee time value . . .option tradersportfolio managers 118problems 97profits 74 92, 94 6, 118, 126 7vision 91 2, 118, 126 7volatility views 118, 126 7optionssee also American . . .; call . . .;European . . .; prices; put . . .background 1 19composite options 135 6, 145 7concepts 1 19definitions 2 3, 8dividend risk 38 9dividend-sensitive options 57dividends 22 31, 33 40, 51 2,57 60employee incentive schemes 126examples 3 6, 9 12exercised rights 2 7, 13 14, 17 19exotic options 130 6historical background 1intrinsic value 11, 13, 48 52, 58 60long positions 8 12, 15 16, 34 5,
panics, investors 104parameters, Black Scholes pricingformula 2, 22 31, 35 40, 42,44 5, 133parties, options 8payoffsconcepts 15 16, 48 9, 106 16,119 20delta 48 52examples 3 14, 39 40, 106 16intrinsic value 48 9, 58 60pension funds 118 19pin risk, concepts 99plain vanilla optionssee also optionsdefinition 130portfolio managersconcepts 118 27option traders 118portfoliosput call parity 17 19riskless portfolios 56 7, 63, 74 86,91 2pre-paid forwards 123 5premiums 99 100, 103 4, 107 9,112, 121, 127, 139 40present value (PV) 36pricesAmerican options 31, 57 60
maturity see expirationMitsubishi 120movements in underlying prices81 6, 97 8, 104
INDEXbinomial tree model 31Black Scholes pricing formula 2,21 31, 35 40, 42 5, 133 5, 144call options 23 7, 29, 37 9, 49 52concepts 21 31, 34 40, 42 5delta 47 60, 62 5, 74 92, 94 6,103, 142dividends 22 31, 34 40, 51 2,57 60European options 22 31, 51 2expiring at-the-money options 65,92, 97 8, 111 16forwards 34 5, 36 8intrinsic value 11, 13, 48 52, 58 60options 4 8, 11 12, 16 19, 21 31,42 60, 102 4, 131 5premiums 99 100, 103 4, 107 9,112, 121, 127, 139 40put options 22 3, 29, 37 9, 50 2put call parity 16 19quantos 131 5rho 31skew 102 4supply/demand factors 43takeovers 139, 141 2theta 62, 65 9, 75 80, 86 92time value of an option (option part)49 52, 58 60, 88 92underlying assets 22 31, 35 40,42 6, 48 60, 62 5, 74 92, 94 8,104, 139 42vega 29, 62, 69 71, 94 8probability, expiring in-the-moneyoptions 143 4profit formula, dynamic hedging 80 6profitsdynamic hedging 74 92, 94 6long call options hedging 74 7,80 6, 89 92, 95 6option traders 74 92, 94 6, 118,126 7options 4 6, 8 12, 15, 25, 46,74 92, 94 6, 114 16, 118, 126 7short call options hedging 77 86volatility 25, 46, 