Financial (and Commodity) Derivatives

Transcription

1 Financial (and Commodity) Derivatives RNDr. Jiří Witzany, Ph.D. NB 178, Tuesday 10-12) 1

2 Literature Requirement ISBN Title Author Year of Publication Required Options, Futures, and Other Derivatives, 789 p. Hull, John C. 2006, 6 th edition Optional International Financial Witzany, J Markets, Oeconomia VŠE, 180 p. Optional Deriváty, 297 s. Dvořák, Petr Optional Finanční a komoditní Jílek, Josef Optional deriváty v praxi, 630 s. Financial derivatives in theory and practice, 393 p. P.J.Hunt, J.E. Kenedy ) The course should cover Chapters 1-15 from John Hull 2

3 Content Introduction principles of financial derivatives Future and forward markets Determination of forward and futures prices Interest rate and currency swaps Mechanics of options markets 3

4 Content - continued Modeling of market rates and valuation of options Options on stock indices, currencies, and futures (Greek letters, Delta-hedging etc., volatility smiles etc.) (Credit risk and credit derivatives, exotic, weather, energy, and insurance derivatives if time allows) 4

5 Part 1 Introduction to Derivatives Futures Markets 5

6 Introduction Principles of Derivatives A derivative contract can be defined as a financial instrument whose value depends (is derived from) the values of other, more basic underlying variables (financial assets - FX rates, interest rates, stock prices, bond prices, commodities, weather,..) Derivatives are settled at a certain future time Derivative markets trading with pure risk physical settlement of underlying assets often eliminated Derivative markets are often more liquid than the spot markets (e.g. commodity futures) spot prices derived from derivative prices 6

7 Interest Rate Derivatives Market Development Derivative markets have become increasingly important during the last 25 years 7

8 Credit Derivatives Market Development Some derivative markets grow exponentially 8

9 Exchange-Traded Markets versus OTC Markets Standardized derivative contracts defined by the exchange The contracts are always between the exchange and a participant OTC derivatives generally between any two subjects Derivative Exchanges exist since 19th century (Chicago Board of Trade futures like contracts, ) 9

10 OTC versus Exchange Market Development Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 10

11 Global Capital Markets 11

12 Market Value Versus Outstanding Amount Source: BIS 12

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14 Futures 14

15 Basic Derivative Types Forward Contracts Forward contract an agreement to buy or sell an asset at a certain future time for a certain price Payoff=S T -K Payoff=K-S T Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 15

16 Forward contracts Arbitrage-less market any possible arbitrage opportunity quickly disappears in an efficient market FX forward transaction can be equivalently achieved as a combination of a reverse FX spot and two deposits relatively straightforward pricing Generally for FX direct quotations FC/DC: 1 r FX forward FXspot 1 r DC FC d 360 d ( r DC r 1 r FC FC d ) 360 d ( r DC r FC d )

17 FX Forward Arbitrage Argument T+2 T+2+d Domestic Currency 1. Borrow at r DC 5. Repay the loan 2. Buy FC at S 4. Sell FC at F Foreign Currency 3. Deposit at r FC 17

18 FX Forward pricing Example: Assume that the spot rate EUR/PLN equals to 3,76. Estimate the 6 M (months) forward exchange rate if the 6 M interest rates in EUR and PLN are 4,35% and 4,85%. Solution: 1 r F S 1 r PLN EUR d 360 d ,0485 3,76 * , ,0244 3,76 * 1,0219 3,769 18

19 Other Basic Derivative Types Futures contracts similar to forwards but traded on an exchange (Chicago Board of Trade, ), standardized, margins to cover daily P/L Options right to buy/sell certain asset (stocks, currencies, ) by (at) a certain date for a certain price call/put options, strike/exercise price, expiration/maturity, European/American options, traded OTC (FX,..) or on an exchange (Chicago Board Options Exchange, ), binary, barrier, 19

20 Options Example: profit/loss on call/put options Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 20

21 Other Derivative types Swaps: two parties agree to a periodic exchange of certain cash flows interest rates, equity returns, nominals in different currencies etc., always OTC FRA Forward Rate Agreements payoff is defined as the difference between agreed and future interest rates - OTC Credit derivatives: the payoff on the creditwothiness of one or more companies or countries credit default swaps, CDO, - OTC Other underlyings: weather derivatives (daily temperature), energy derivatives - crude oil, natural gas, electricity, 21

22 Types of Traders Hedgers fundamental need to reduce (insure) risks Speculators use derivatives as an easy way to take a position/speculate on the market for example the hedge funds Arbitrageurs combine different products in different markets to lock in a risk-less profit see e.g. the relationship between spot and forward prices only small arbitrage opportunities exist 22

23 Speculation example $2000 to speculate with: The Amazon.com stock $20, 2-month call option with strike $22,50 is sold for $1 Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 23

24 Derivatives Risk Management Derivatives can be used to take huge risk with minimal initial investment Future derivatives settlement may allow to hide transactions for some time Nick Leeson Barrings Bank Singapore office 1 billion dollar loss Jerome Kerviel Societte General Equity Index Derivatives 5 bln EUR loss UBS, Merill Lynch, Morgan Stanley, AIG CDOs 100s bln USD losses Derivatives can be compared to electricity very useful but dangerous Need for high quality risk management risk limits on exposures, products, counterparties etc. 24

25 Futures Markets 25

26 Futures Markets Financially similar to forwards but exchange traded Futures exchange/clearinghouse stands between the market participants (compare to forwards) Majority of future contracts do not lead to physical delivery Futures Exchange +N -N Counterparties with a long position Counterparties with a short position 26

27 Newspaper quotes 27

28 Specification of a Futures Contract The asset: commodities must be exactly specified The contract size: depend on average user and delivery costs Delivery arrangements: cash or physical, important for commodities (cattle, lumber, cotton ). The party with short position usually files a notice of intention to deliver selection of location etc. Delivery time: usually end of month, or a longer delivery period Price quotes conventions: e.g. $/barrel with two decimal places; price and position limits 28

29 Daily Settlement and Margins Settlement/counterparty risk: counterparty does not deliver Margin mechanism daily financial settlement Initial Margin set so that daily losses are covered with high probability (e.g.99,5%) Daily P/L (variation in futures price) is debited/credited on the margin account Maintenance margin (around 75% of the initial margin) must be always maintained If the balance falls bellow the maintenance margin there is a margin call, the account must be topped to the initial margin level The account balance is release when the position is closed out or settled 29

30 Daily Settlement and Margins Example: +2 (100 ounces) gold futures contracts, price in $ per troy oz. (Initial Margin $2000, Maintenance Margin $1500 per contract) Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 30

31 Forwards versus Futures Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 31

32 Hedging Using Futures If future P/L depends linearly on the future price of an asset then futures contracts can be used for a perfect hedge P/L = N x P, P future price of a unit of the asset, N positive or negative / the sensitivity Take futures position equivalent to -N units of the asset (short for N positive, long otherwise) Then P/L = N x P - N x P = 0 for the total position after the futures contract settlement There are arguments for and against hedging 32

33 Hedging Using Futures Example: A farmer plans to sell his cattle on a local market one year from now, let us say in August The market prices of live cattle are quite volatile so the farmer decides to use ten live cattle futures contracts to fix his selling price. The table on the next slide shows quoted live cattle futures (trade unit is pounds and the price is in cents per pound). Propose an effective hedging strategy for the farmer. 33

34 Source: 34

35 Basis Risk Basis = spot price of the asset futures price of the contract = S F = b due to: time, not exactly identical assets Source: John Hull, Options, Futures, and Other Derivatives, 6th edition The effective hedging price = F 1 +b 2 = F 1 +(S 2 -F 2 ) 35

36 Cross Hedging The hedged asset and the futures asset are not same, but the prices are correlated For example hedging the price of jet fuel using heating oil futures, or hedging FX P/L from on equity investment using FX futures Find the OLS regression coefficient: S= h* F +, i.e. h= * S / F The hedge ratio h minimizes the variance of P/L caused by S 36

37 Cross Hedging Example Source: John Hull, Options, Futures, and Other Derivatives, 6th edition h = 0,78 using elementary statistics formulas 37

38 Stock Index Futures Cash settlement based on actual index value 38

39 S&P 500 Futures on CME 39

40 Hedging an Equity Portfolio Example Value of S&P 500 index = 1000 Value of the hedged portfolio = $5 mil. Risk-free interest rate = 10% p.a. Index dividend yield = 4% p.a. Beta of the hedged portfolio = 1,5 Use 4-month S&P 500 futures (currently valued at 1020,20) to hedge the value of the portfolio. Simulate the outcome using CAPM given different values of the index in 3 month (800, 900, 1100 etc.). 40

41 Hedging an Equity Portfolio Example Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 41

42 Hedging an Equity Portfolio Index futures can be used in case the portfolio manager assumes a temporary decline of the market, or wants to speculate against the benchmark (index), or wants to adjust the beta of the portfolio, or wants to take a temporary position without investing into the stocks etc. Index futures can be also used to hedge a single stock against the market volatility 42

43 Rolling the Hedge Forward Example of April 2004 June 2005 hedge using three 6 month futures rolled forward (short positions) Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 43

44 Rolling the FX Hedge Forward Example: You run a CZK denominated money market fund for US investors. Show how to use 6 months forwards for a continuous FX hedging of the portfolio. The initial value of the portfolio is 1.7 bln CZK, and the expected growth is 4% annually. Give an example if the exchange rate development is e.g , 17.50, 18.00, 18.70, CZK/USD in 6 months periods. 44

45 Part 2 Forward and Futures Pricing Interest rates 45

46 Interest Rates Present value cashflow discounting key concept in derivatives valuation Risk free rate x credit margin Treasury rates and LIBOR rates Different interest rate conventions Different compounding frequency Continuous compounding FV=Ae Rt Bond pricing and zero rates - bootstrapping 46

47 Bootstrapping Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 47

48 Bootstrapping Example Calculate EUR 6M, 1Y, 2Y, and 3Y zero coupon discount rates in continuous compounding given the following bid rates: 48

49 Forward interest rates Forward rates can be derived from the zero rates e R(T)T = e R(t)t e R(t,T)(T-t) solve for R(t,T), the forward interest rate from t to T FRA forward rate agreement cash settlement of the difference between agreed rate R K and the actually observed rate R M (Libor) usually at the beginning of the interest period Valuation of FRA: V=L(R K -R F )(T 2 -T 1 )e -R2.T2, if R K is the contracted interest earned, R F is the forward rate from T 1 to T 2 at the time of valuation 49

50 Example Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 50

51 FRA Quotes 51

52 Forward and Futures Pricing Forwards and Futures have different settlement, however the price are very close Investment x consumption assets arbitrage argument can be used only for investment assets Short selling possible if the asset can be borrowed in case of stock dividends must be paid to the owner A forward contract can be replaced by shortselling/investing into the asset and depositing/borrowing the corresponding amount 52

53 Arbitrage Example Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 53

54 Forward Pricing If the current price of an investment asset with no income is S 0 then the forward price F 0 =S 0 e rt If the investment asset provides income with a present value I, then F 0 =(S 0 I)e rt If the investment asset provides known yield q with continuous compounding then F 0 =S 0 e (r-q)t, r-q is also called the cost of carry 54

55 General Forward Arbitrage Argument Spot Market Domestic Currency Financing Cost Forward Market Buy the asset at S Sell the asset at F Underlying Asset Storage Cost - Income 55

56 Arbitrage Example Source: John Hull, Options, Futures, and Other Derivatives, 6th edition 56

57 Market Value of a Forward Initially the value of a forward contract is zero, later it may be positive or negative f=(f 0 -K)e -rt, where K is the contracted price, F 0 the actual forward price, and T time to delivery from today. Notice that futures contracts settle F 0 -K, not (F 0 - K)e -rt It can be shown however that forward and futures prices are theoretically same if the margin account yields the market rate 57

58 Futures on Stock Indices The underlying index is assumed to provide an expected yield q (which should represent the average annualized dividend yield during the life of the contract) Then F 0 =S 0 e (r-q)t The average dividend yield q is usually lowerthanthenr (F 0 >S 0 ), but may be also higher in some periods (F 0 <S 0 ) 58

59 Forwards and Futures on Currencies F 0 =S 0 e (r1-r2)t, if the exchange rate is quoted as currency1 for one unit of currency2, i.e. currency2 plays the role of an investible asset with the yield r 2. Example: 59

60 Futures on currencies 60

61 Futures on Commodities Commodities generally have storage costs and do not provide income, except gold and silver. Storage costs U (present value) can be treated as negative income, hence F 0 =(S 0 +U)e rt Some assets like crude oil may provide so called convenience yield y, then F 0 =(S 0 +U)e (r-y)t = S 0 e (r+u-y)t Futures prices of oil tend to decrease, i.e. r+u<y, conevenience yield possible shortages 61

62 Futures price versus expected spot price Long futures position is equivalent to investment into the asset An investor generally require extra return for systematic risk If k is the required rate of return on the asset then it follows F 0 =E(S T )e (r-k)t, hence F 0 <E(S T ) if k>r This is the case in particular for futures on stocks or indexes positive beta systematic risk If the asset has negative systematic risk, then F 0 > E(S T ) 62

63 Normal and Inverted Futures Prices 63

64 Contango and Normal Backwardation 64

65 Interest Rate futures Eurodollar futures Eurodollars are dollars deposited outside of the United States Three month Eurodollar futures (CME) contracts 3-month interest rate. Delivery March-June- September-December up to 10 years in the future The quote = 100 r, where r is the annualized 3- month rate in the Act/360 convention. One basis point is equivalent to $25 settlement amount. Example: Use futures contract to lock 3-month interest rate earned on $1 mil. one year later 65

66 Eurodollar futures Wall Street Journal, Feb 5,

67 Eurodollar Fututres Prices on 67

68 Long Term Interest Rate Futures Day count conventions: Act/Act (Treasuries), 30/360 (Corporate bonds), Act/360 (Money market) Bond prices: cash price = clean price + accrued interest Treasury bond futures are quoted in the same way as the bonds ( means 110 and 3/32 of the face value $ ) Any Treasury bond with at least 15 years to maturity can be delivered using so called conversion factors (6% YTM convention) 68

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72 Hedging Example We hold 100 T-Bonds currently priced at 95% (Nominal=$ ), with 20 years to maturity and duration 15 years. We need to sell the bonds in 2 months. Propose an appropriate number of 3 months futures contracts on 30Y T-Bonds to hedge the price provided the quoted futures price is 90%, the current conversion factor for the CTD is 1.3, and its duration is

73 Are forward and futures prices equal? Yes, if we assume that the interest rates are constant, or at least sufficiently independent on underlying asset prices This assumption does not hold for interest rate futures and forwards IR futures rates are higher that IR forward (FRA) rates due to the discounting difference and the convexity adjustment Forward rate = Futures rate - 2 T 1 T 2 /2 73

74 Most of the volume Interest Rate Swaps 74

75 Swaps Exchange of cash flows in the future plain vanilla interest rate swaps Currency swaps Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 75

76 Interest Rate Swap Example Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 76

77 Interest Rate Swap Example Intel transforms fixed 5,2% payments to Libor + 0,2% Microsoft transforms Libor + 0,1% payments to 5,1% fixed payments Assets can be transformed as well 77

78 Role of Financial Intermediary Banks play the role of intermediaries The banks moreover exchange the positions mutually, some play the role of market-makers Banks use so called ISDA master agreements 78

79 IRS Quotes Source: 79

80 Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 80

81 Interest Rate Swap Example Valuation of IRS Can be regarded as an exchange of a fixed-coupon bond (FCB) for a floating rate note, MV = Value of the FRN Value of the FCB Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 81

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85 Currency Swaps (CCS) Exchange of interest payments (usually fix-fix) in two currencies Exchange of principals at the beginnig and at the end of the contract Valuation similar to IRS Credit risk of CCS is higher than that of IRS 85

86 Part 3 Options Markets 86

87 Underlying assets Call/Put, long/short position, European/American, in/at/out of the money, intrinsic/time value Stock options mostly exchange traded CBOE, PHLX, AMEX, PACIFEX, EUREX Options on indices exchange traded cash settlement Currency options - exchange traded and OTC Options on futures contracts exchange traded options to acquire long or short position in a futures contract Expiration date and strike price is defined options are traded at a premium 87

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90 EUR/USD options volatility quotes Source: Reuters, 27/11/07 90

91 Margins Investors writing an option (short position) must maintain a margin Naked option is an option written without the offsetting position in the underlying stock the margin than must be at least 10%, resp. 20% of the underlying price Writing covered calls the underllying is already owned An Option Clearing Corporation usually guarantees the settlement 91

92 Factors affecting option prices The current stock price, S 0 The strike price, K The time to expiration, T The volatility of the stock price, The risk-free interest rate, r The dividends expected during the life of the option 92

93 Factors affecting option prices 93

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96 Factors affecting option prices Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 96

97 Assumptions and notation No transaction cost, equal taxes, borrowing and lending at the risk-free rate, no arbitrage opprotunities, no bid-ask spreads Notation: 97

98 Upper and lower bounds for option prices A call option can never be worth more than the stock S 0 A European put option can never be worth more than the discounted strike price Ke -rt A European call option on non-dividend stock is worth at least S 0 -Ke -rt A European put option on non-dividend stock is worth at least Ke -rt -S 0 98

99 Put-Call Parity Portfolio A: one European call option plus cash Ke -rt Portfolio B: one European put option plus one stock The value of both portfolios at T is max(k,s T ) Hence c+ke -rt = p+s 0, otherwise there is an arbitrage opportunity 99

100 Put-Call Parity Example Strategy: short B, buy A Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 100

101 Trading strategies involving options Different strategies combining a long/short position in a stock and a long/short position in a put/call option More complex strategies involve two different options Bull/Bear/Butterfly/Calendar/Diagonal spreads, straddles, strips, straps, strangles 101

102 Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 102

103 Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 103

104 Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 104

105 Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 105

106 Part 4 Pricing of options 106

107 Pricing of options Binomial trees Example: European call option to buy a stock in three months for $21 Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 107

108 One step binomial tree example It is possible to set up a riskless portfolio combining a position in the stock and in the option The portfolio is riskless if the profit/loss is the same in all (both) scenarios Long: 0,25 shares Short: 1 option The value of the portfolio is $4,5 in three months in both cases Hence the present value of the option is calculated as $5 minus discounted $4,5 108

109 General one-step tree The riskless portfolio can be set up in general 109

110 Risk-neutral Valuation Note, that the result does not depend on probabilities of the two scenarios However if we set up p, probability of the movement up, so that E(S T )=S 0 e RT (the stock return equls to the risk-free rate) then it turns out that the price of the option equals to the discounted expected pay-off Risk neutral valuation principle: we can assume that the world is risk neutral when pricing an option. The result is valid also in the real world which is not risk-neutral! 110

111 Two-step binomial trees Time steps are three months, r=12% 111

112 Two-step binomial trees The option price equals to the expected pay-off in a risk neutral world 112

113 European x American options 113

114 Delta Riskless portfolio: -1 call option, + stock The change in the option price is offset by * the change of the stock price Delta equals to 25% in the one-step binomial tree example Delta changes in a two-step binomial tree 114

115 Volatility Volatility is measured as the standard deviation of the return (with expected value ) normalized by the square root of the time u and d are usually derived from those parameters N-step binomial tree can be then used for a Monte-Carlo simulation When t is small 115

116 Behaviour of Stock Prices Stochastic process variable whose value depends on time and changes in an uncertain way Discrete/continuous time, discrete/continuous variable Markov process only the present value of a variable is relevant for the future Market rates (stock prices, exchange rates, interest rates) are assumed to follow the Markov process (x technical analysis) 116

117 Wiener process Continuous time process: any time period can be divided into arbitrary number of steps Wiener process (Brownian motion) Markov process where z(1)-z(0) has distribution N(0,1) and the distributions are uniform for smaller time steps The square root of time rule: z(t 0 +t)-z(t 0 ) has the distribution N(0, t) Stochastic difference equation: dz= dt, where is randomply taken with the distribution N(0,1) Generalized Wiener process: dx=adt + b dt, i.e. x(t 0 +t)-x(t 0 ) has the distribution N(at,b t) 117

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120 Ito s process dx = a(x,t)dt + b(x,t)dz, where dz is the Wiener process The drift and the variance depend on x and t The process for Stock prices S: normally distributed annualized rate of return with expected value and standard deviation (observed for a small time periods, not one year) ds= Sdt + Sdz (geometric Brownian motion) It turns out that S(1)-S(0), or S(1)/S(0)-1, are not normally distributed lognormal distribution 120

121 Lognormal distribution Lognormal Distribution f(x)

122 Monte Carlo simulation 122

123 Ito s Lemma If G=G(x,t) where x follows an Ito s process dx = a(x,t)dt + b(x,t)dz, then G follows the Ito s process Where dz is the same as above 123

124 Application to forward contracts Forward price of a non-dividend paying stock F=Se r(t-t) Using the Ito s lemma it follows that the process for F is: 124

125 Lognormal property Let G=lnS, where S follows the geomentric Brownian motion, then applying the Ito s lemma we get: Consequently lns(t)-lns(0) has a normal distribution N(( - 2 /2)T, T) 125

126 The Black-Scholes Model Lognormal property of stock prices 126

127 Lognormal property of Stock It can be shown that prices Stockpricecanbemodelledas 127

128 Estimating volatility from historical data 128

129 Black-Scholes differential equation Principle: set up a risk-less combination of an option and the underlying stock the price of the option f depends on the underlying S and t use the derivative delta hedging The return on this portfolio in a short time period must be equal to the risk free interest rate 129

130 Black-Scholes differential equation Source: John Hull, Options, Futures, and Other Derivatives, 5th edition 130

131 Black-Scholes differential equation 131

132 Black-Scholes differential equation Ito s lemma: Discrete version: 132

133 Black-Scholes differential equation Risk less portfolio I.e. Use the equations above 133

134 Black-Scholes differential equation Hence the portfolio is risk less, as expected, and so And we get the Black-Scholes-Merton differential equation: 134

135 Risk-neutral valuation The Black-Scholes-Merton equation does not depend on the expected return, i.e. on investors risk preferences!!! We can assume that we are in a risk-neutral world. The result will be the same as in the real world with risk sensitive investors. Consequently we can simply discount the expected pay-off of an option using the risk free rate. The result will be the unique solution of the differential equation 135

136 The Black-Scholes Formula Consider a European call option, then in a risk neutral world The value of the the option at expiration is: And the price of the option: Using the model for the Stock price at time T: 136

137 The Black-Scholes Formula where and N(x) is the cummulative standardized normal distribution Similarly for a put option: 137

138 The Greeks Partial derivatives of an option (portfolio) market value measure sensitivity with respect to the relevant variables Delta, Gamma the 1st and the 2nd derivatives w.r.t. the underlying asset price Vega the derivative w.r.t. the volatility variable Rho the derivative w.r.t the interest rate Theta the derivative w.r.t. to time, ususally measured as the change of value per day The Greeks are used for hedging Delta-hedging, Vega-hedging, Gamma-hedging e.t.c 138

139 International Financial Markets Lecture Notes J.Witzany Includes valuation of forwards and a detailed chapter on derivatives more or less covering 1BP426 content Oeconomia, vailable in the VŠE bookstore 186 Kč, 180 pages, English 139