CQG Integrated Client Options User Guide. November 14, 2012 Version 13.5

Transcription

1 CQG Integrated Client Otions User Guide November 4, 0 Version 3.5

2 0 CQG Inc. CQG, DOMrader, Snarader, Flow, and FOBV are registered trademarks of CQG.

3 able of Contents About this Document... Related Documents... Customer Suort... Otions in CQG... 3 Entering Otions Symbols... 4 Oening Otions Alications... 5 CQG API and Otions... 6 Greeks and Volatility Definitions... 7 Standard Otions Pricing Models... 9 Exotic Otion Models... 3 Interest-Rate Otion Models... 3 Sread Otions Model... 4 Cumulative Normal Distribution Function Aroximation Numerical Methods for Solving Equations Numerical Differentiation rading Otions Setting Otions Preferences... 5 Setting Otions Window View Preferences Setting Otions Calculator View Preferences Setting Volatility Worksho View Preferences Setting Strategy Analysis View Preferences Setting Volatility Preferences Setting Interest Rate Preferences Setting Price Filter Preferences... 6 Setting Greeks Scale Preferences... 6 Setting Advanced Preferences Setting Model Preferences Setting Udate Frequency Preferences Udating the Refresh Rate Otions Window Otions Window oolbar... 7 Customizing Columns Changing the Order of Columns... 77

4 Marking At-the-Money Changing the Dislay ye Oening Another Alication from an Otions Window Setting What If Otions Parameters... 8 Coying Data to Excel... 8 Placing Orders from the Otions Window Otions Calculator Otions Calculator Comonents Otions Calculator oolbar Using the Otions Calculator Inutting What Ifs... 9 Viewing Summary Statistics Using the Otions Calculator Grah Using Cursors with an Otions Calculator Grah... 0 Information Dislayed in an FX OC View... 0 Selecting the Proerties for the Otions Calculator Grah Lines Otions Grah Otions Grah oolbar Define Otions Grah Curves... 3 Volatility Worksho... 3 Volatility Worksho Comonents... 4 Volatility Worksho oolbar... 7 Saving the Volatility Curve Oening a Saved Volatility Curve Adjusting the shae of the curve Removing Corrections Selecting the Colors for the Volatility Worksho Grah Lines Designating the Aroximation Characteristics Modifying the Volatility Curve Resetting the Volatilities Using 3D Strategy Analysis Window... 5 Strategy Analysis Window Comonents... 5 Strategy Analysis oolbar Selecting a Strategy Selecting an Underlying Model for Strategy Dislays... 6

5 able abs Using the Dislay abs Setting Proerties for 3D Strategy Grah Underlying Information Dislay Proerties Creating and Editing Strategies Saving an Otions Strategy... Loading a Saved Strategy... 4 Using the Strategy Worksace Manager Window... 5 Weights... 7 Using Advanced Strategy Features... Otions Strategy Color Windows... 7

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7 Page About this Document his document is one of several user guides for CQG Integrated Client (CQG IC. his guide details otions-secific tools in CQG. You can navigate the document in several ways: Click a bookmark listed on the left of the age. Click an item in the able of Contents. Click a blue, underlined link that takes you to another section of the document. o go back, use Adobe Reader Page Navigation items (View menu. If you are looking for a articular term, it may be easier for you to search the document for it. here are two ways to do that: Right-click the age, and then click Find. Press Ctrl+F on your keyboard. his document is intended to be rinted double-sided, so it includes blank ages before new chaters. Please note that images are examles only and are meant to demonstrate and exose system behavior. hey do not reresent actual situations. o ensure that you have the most recent coy of this guide, lease go to the user guide age on CQG s website. Related Documents CQG IC user guides: CQG Basics Charting and Studies Advanced Analytics rading and CQG Sreader Otions User Guide

8 Page Customer Suort CQG Customer Suort can be reached by hone from Sunday, :30.m. C through Friday, 5:00.m. C. hese hours also aly to Live Chat. United States United Kingdom +44 ( France +33 ( Germany +49 ( Jaan +8 ( Russia Singaore Sydney +6 ( websut@cqg.com 4 hours a day, 7 days a week. If you have questions about CQG documentation, lease contact the hel author. About this Document

9 Page 3 Otions in CQG CQG IC includes five otions alications: Otions Window Otions Calculator Otions Grah Volatility Worksho Strategy Analysis All CQG IC users have access to the Otions Window and the Otions Grah. If you would like to learn more about our advanced otions offering, which includes Otions Calculator, Otions Strategy, and Volatility Worksho, lease contact CQG. CQG offers seven basic otion models that serve as the framework for valuing otions: Black, Black-Scholes, Bourtov, Cox-Ross-Rubinstein, Garman-Kohlhagen, Merton, and Whaley. Otions User Guide

10 Page 4 Entering Otions Symbols he format for otions on futures is: C.<symbol><month code><year><strike rice> for calls and or P.<symbol><month code><year><strike rice> for uts. he strike rice is -5 digits. Examle: C.SPZ08500 = December call on the S&P 500 futures contract. An alternate format is C.<symbol>_<month code><year>.<strike rice> for calls and with P. for uts. C.SP_U8.500 = Setember call on the S&P 500 futures contract. On Otions windows, you can enter the symbol only. For at the money for the nearby month, tye C. or P., the symbol, and?. For at the money for some other month, tye C. or P., the symbol, the month, the year, and? and then ress CRL+ENER. For strikes for the most active month, tye C. or P. and the symbol and? and then ress CRL+ENER. Otions in CQG

11 Page 5 Oening Otions Alications Click the Otions button on the main toolbar, and then click the name of the otions window you want to oen: his button rovides access to all otions windows without having to dislay the button for each window. If the Otions button is not dislayed, click the More button, and then click Otions. You can also add individual otions windows to the toolbar: Otions User Guide

12 Page 6 CQG API and Otions CQGs API suorts efficient access to otions strike roerties through the use of the CQGInstrumentsGrou interface. With one request to CQG servers, your alication can subscribe to all strikes in any given contract month or a range of months. Data subscrition levels can also be configured to otimize instrument resolution for strike roerties and market data, allowing for the delivery of critical information without unnecessary overhead. CQG also offers access to common real-time values for all subscribed otions strikes: Greeks, theoretical values and imlied volatilities. hrough the API, CQG offers in-deth ortfolio analysis caabilities. Otions in CQG

13 Page 7 Greeks and Volatility Definitions As you work with otions in CQG IC, it s helful to understand how imlied and average volatility are calculated and how the Greeks are defined. Delta Delta shows the change in the rice of a derivative to the change in the rice of the underlying assets. Sometimes delta is known as the hedge ratio, as delta indicates how much of the underlying asset needs to be bought or sold to hedge the otion. raders take advantage of delta by creating delta hedging, delta sreads, and delta neutral. Delta values are ositive numbers less than or equal to 00. hey reresent the ratio of the change in the theoretical value over the change in the underlying rice. Values: Out of the money = close to 0 At the money = close to +0.5 In the money = close to + Calls = ositive Puts = negative Delta values for the out-of-the-money series move closer to 0 as exiration nears. Likewise, more in-the-money otions have deltas close to as exiration aroaches. For examle: If the underlying S&P 500 contract stands at 3400, with a delta of 5.73, and a theoretical value of 600.5, and the underlying rice increases to 340, while the delta rises to 54.0, the theoretical value increases to 707. he calculations are: = 00 ( / = * = he deltas from one underlying rice to the next are interolated = new theoretical value Gamma Gamma is the amount the delta changes when the underlying rice changes by one tick. Gamma is greatest for at-the-money otions. Gamma increases as the otion moves closer to exiration. raders try to limit gamma risk because short gamma ositions create a otential for losses. For examle: If the delta of an S&P future was 9.80, the gamma was.0 and the rice of an S&P future increased from to i.e., a one-tick increase, the delta would increase to 9.8. Otions User Guide

14 Page 8 heta heta reresents the loss in theoretical value in one day, if all other factors are constant. In other words, it attemts to isolate the time decay factor. For examle: Assume the amount showing the Value column was 75., with 5 days until exiration and a theta value of You would exect to see the amount in the Value column decrease aroximately 9 dollars the following day. A more recise definition of the amount of the time value lost is an average of the hetas on the dates under consideration. So, if the theta on the following day was 95.0, the decrease in theoretical value would be: ( / = 93.6 Vega Vega is the amount that the theoretical value changes when the volatility changes by oint. For examle: Assume a June Corn contract had a vega of.4, a volatility of 5.90, and a theoretical value of If the volatility were to increase to 6.90, the vega says that the theoretical value would increase by.4 dollars to 46.8, rovided the other factors affecting otions rices remained constant. he dislay also indicates the days until exiration, as well as the volatility and interest rate assumtions underlying the data. Rho Rho is the change in otion rice to a unit change in interest rates. When the interest rate increases, the call otion rice increases also and ut otion rice falls. For examle: Assume the starting call value is 4.0, the interest rate r is 5% and zerocouon rate b is %. Rho(r(er %= 0.43, and Rho(b(er %=0.38, If r rises to 6% and b stays at 5%, the call value is If r stays at 5% and b rises to 3%, the call value is Imlied Volatility he imlied volatility calculated from an otions dislay reresents the volatility that, if entered into a theoretical ricing model, would roduce a theoretical value equal to the market rice of the otion. Unlike the Historical Volatility study, the Imlied Volatility calculation deends on the model selected, the calculation method chosen and the arameters inut in the What if? column. Average Volatility he average volatility is calculated using the following formula: IV AvgV = L ( SPH UP + IVH ( UP SPL ( SP SP H L Otions in CQG

15 Page 9 Standard Otions Pricing Models Otions ricing models describe mathematically how a set of inut arameters tyically underlying rice, strike rice, time to exiration, interest rate, and volatility combine to determine a theoretical value of an otion. CQG offers seven basic otion models that serve as the framework for valuing otions: Black, Black-Scholes, Bourtov, Cox-Ross-Rubinstein, Garman-Kohlhagen, Merton, and Whaley. erm heov sigma, ImV Greeks Delta, Gamma, Γ Vega heta, Θ Rho, ρ N(x Definition otion theoretic value volatility of the relative rice change of the underlying stock rice imlied volatility Partial derivatives of the otion rice to a small movement in the underlying variables. Main greeks are delta, gamma, theta, vega, rho. delta is the first derivative of the otion rice by underlying rice gamma is the second derivative of the otion rice by underlying rice vega is the first derivative of the otion rice by volatility theta is the first derivative of the otion rice by time to exiration rho is the first derivative of the otion rice by interest rate the cumulative normal distribution function x z N x = e ( π dz n(x normal distribution function n( x = e π x x, n ( x = x e π S X r underlying rice strike rice of otion risk-free interest rate otion time to exiration in years Otions User Guide

16 Page 0 erm b Definition volatility of the relative rice change of the underlying instrument the cost-of-carry rate of holding the underlying security For further reading, we suggest: he Comlete Guide to Otion Pricing Formulas. ISBN Otions, Futures, and Other Derivatives. ISBN Otion Volatility and Pricing Strategies. ISBN X. Otions in CQG

17 Page Black Model In 976, Fisher Black develoed a modification to the Black-Scholes model designed to rice otions on futures more recisely. he model assumes that futures can be treated the same way as securities, roviding a continuous dividend yield equal to the risk-free interest rate. he model rovides a good correction to the original model concerning otions on futures. However, it still carries the restrictions of the Black-Scholes evaluation. Notation C P U E r t ν N(x heoretical value of a call heoretical value of a ut Underlying rice Strike rice Interest rate ime to exiration in years Volatility Cumulative normal density function he theoretical values for calls and uts are: C = Ue P Where: rt Ue rt N( h Ee N( h ν t rt = rt N ( h + Ee N ( ν t h ln( U / E ν t h = + ν t Note: Although similar, this definition of model. An alternative form for h is: ln( U / E + ν t h = ν t h is different from the one used in the Black-Scholes Otions User Guide

18 Page Generalized Black-Scholes (Black-Scholes extended Model he generalized Black-Scholes model can be used to rice Euroean otions on stocks without dividends [Black and Scholes (973 model], stocks aying a continuous dividend yield [Merton (973 model], otions on futures [Black (976 model], and currency otions [Garman and Kohlhagen (983 model]. heov Call ( br c CGBS S e N( d X e r = = N( d Put ( PGBS X e r br = = N( d S e N( d where d d ln( S / X + = = d ( b + / N(x the cumulative normal distribution function; S underlying rice; X strike rice of otion; r risk-free interest rate; time to exiration in years; volatility of the relative rice change of the underlying stock rice. b the cost-of-carry rate of holding the underlying security. b = r gives the Black and Scholes (973 stock otion model. b = r q gives the Merton (973 stock otion model with continuous dividend yield q. b = 0 gives the Black (976 futures otion model. b = r r f gives the Garman and Kohlhangen (983 currency otion model (r f - risk-free rate of the foreign currency. Delta Call = e Put = e ( br ( br N( d [ N( d ] Otions in CQG

19 Page 3 Otions User Guide Gamma Gamma is identical for ut and call otions. S e d n r b Γ = ( ( where ( x e x n = π - normal distribution function. Vega Vega is identical for ut and call otions. d n e S Vega r b = ( ( heta Call ( ( ( ( ( ( d N X r d N e S r b d n e S r r b r b + + Θ = Put ( ( ( ( ( ( d N X r d N e S r b d n e S r r b r b Θ = Rho Call = <> = 0 0, ( b when c b when d N e X r ρ where c call heov Put = <> = 0 0 (, b when b when d N e X r ρ where ut heov

20 Page 4 Otions in CQG Imlied volatility o find imlied volatility the following equations should be solved for the value of sigma: Call ( ( ( d N e X d N e S c r r b = Put ( ( ( d N e S d N e X r b r = where ( b X S d + = / / ln( d d = his equation has no closed form solution, which means the equation must be numerically solved to find. Bourtov s Model Bourtov s model is based on the Black-Scholes model. It defines a secial method to calculate volatility, which is an inut arameter of the Pricing Model Calculator.

21 Page 5 Cox-Ross-Rubinstein Model he Cox-Ross-Rubinstein binomial model can be used to rice Euroean and American otions on stocks without dividends, stocks and stock indexes aying a continuous dividend yield, futures, and currency otions. heov he main binomial model assumtion is the underlying rice can either increase by a fixed amount u with robability, or decrease by a fixed amount d with robability -. So the underlying rice at each node is set equal to i S u d where ji, i = 0,,..., j S underlying rice; u, d u and down jum sizes that underlying rice can take at each time ste. Otion ricing is done by working backwards, starting at the terminal date. Here we know all the ossible values of the underlying rice. For each of these, we calculate the ayoffs from the derivative, and find what the set of ossible derivative rices is one eriod before. Given these, we can find the otion one eriod before this again, and so on. Working ones way down to the root of the tree, the otion rice is found as the derivative rice in the first node. Call At exiration date: i ni fi n = max( S u d X, 0, i 0,,..., n, = where n number of time stes. At each revious ste: Euroean exercise f e [ f + f ] r t i, j = i+, j+ ( i, j+ American exercise f i ji r t ( S u d X, e [ f + ( f ] i, j = max i+, j+ i, j+ where u = e t rice u movement size; d = e t t = / n = / u b t e d = u d rice down movement size; size of each time ste; u movement robability; b the cost-of-carry, defined as: b = r to rice Euroean and American otions on stocks; Otions User Guide

22 Page 6 b = r q to rice Euroean and American otions on stocks and stock indexes aying a continuous dividend yield q; b = 0 to rice Euroean and American otions on futures; b = r r f to rice Euroean and American currency otions (r f risk-free rate of the foreign currency. Put At exiration date: i ni i, n = max( X S u d,0, i = 0,,..., n At each revious ste: Euroean exercise f [ f + f ] r t i, j = e i+, j+ ( i, j+ American exercise f i ji r t ( X S u d, e [ f + ( f ] i, j = max i+, j+ i, j+ Delta Given the f, values calculated for the rice, Delta aroximation is i j f = S f, f,0 = S u S d Gamma Gamma aroximation is γ = f = S heta [( f, f, ( S u S u d ] [( f, f,0 ( S u d S d ] 0.5( S u S d heta can be aroximated as f,0 f θ = t Vega, Rho 0,0 System uses the numerical differentiation to calculate the Greeks. Imlied volatility System numerically finds imlied volatility. Otions in CQG

23 Page 7 Garman-Kohlhagen Model his model, develoed to evaluate currency otions, considers foreign currencies analogous to a stock roviding a known dividend yield. he owner of foreign currency receives a dividend yield equal to the risk-free interest rate available in that foreign currency. he model assumes rice follows the same stochastic rocess resumed in the Black-Scholes model. his model is used to evaluate otions written on currencies. he interest rate of the native currency is used as the default, but you can set the foreign interest rate in Model references. his model corrects the difference between native and foreign interest rates. However, as a modification of Black-Scholes model, it ossesses all its limitations. Notation C P U E r r f t ν heoretical value of a call heoretical value of a ut Underlying rice Strike rice Interest rate Interest rate in the foreign country ime to exiration in years Volatility he Euroean call rice is given by: C r f t = Ue rt N ( h Ee N ( h ν t Where: ln( U / E + ( r r h = ν t + ν / t he Euroean ut rice is given by: P = Ue r f t f rt N( h + Ee N( ν t h Otions User Guide

24 Page 8 Merton Model In 973, Merton roduced a model with a non-constant interest rate. He assumed that interest rates follow a secial tye of random rocess. By taking into consideration the dynamic rocess of interest rate determination, and the correlation between the underlying rice and the otions rice, this model rovides an imrovement over the Black-Scholes model. his model is generally used to value Euroean otions written on stocks. Notation C P U E t N(x ν heoretical value of a call heoretical value of a ut Underlying rice Strike rice ime to exiration in years Cumulative normal density function Volatility ν R(t ρ Volatility of an interest rate contract Interest rate Correlation between the underlying and interest rate contracts he theoretical values for Euroean calls and uts are: C = UN( h B( t EN( h ϑ t P = UN ( h + B( t EN ( ϑ t h Where: ln( U / X ln B( t + ϑ( t / h = ϑ t t ϑ( t = ( ν + ν ρνν dt 0 B( t = e R( t t Otions in CQG

25 Page 9 Otions User Guide Whaley Model he quadratic aroximation method by Baron-Adesi and Whaley (987 can be used to rice American otions. heov Call < + = = * * * / (,,,,, ( (,,,,, ( S S when X S S S when S S A b r X S C stoks on otions r b when b r X S C c q GBS GBS where b the cost-of-carry rate; b = r to rice otions on stocks. b = r q to rice otions on stocks and stock indexes aying a continuous dividend yield q b = 0 to rice otions on futures. b = r r f to rice currency otions (r f risk-free rate of the foreign currency. C GBS the generalized Black-Scholes call heov exression; ( [ ] ( * ( * S d N e q S A r b = b X S S d + + = / ( / ln( ( / 4 ( ( K M N N q + + = / r M = / b N = r e K = S * the critical commodity rice for the call otion that satisfies ( [ ] (,,,,, ( * ( * * * S d N e q S b r X S C X S r b GBS + = he last equation should be numerically solved to find S *. Put > + = ** ** ** / (,,,,, ( S S when S X S S when S S A b r X S P q GBS

26 Page 0 where P GBS the generalized Black-Scholes ut heov exression; S A = q ** ( br ** [ e N( d ( ] S q ( N = ( N + 4 M / K S ** the critical commodity rice for the ut otion that satisfies X S ** = P GBS ( S ** S, X,, r, b, q ** ( br ** [ e N( d ( S ] he last equation should be numerically solved to find S **. Delta Call = GBS GBS ( S, X,, r, b, ( S, X,, r, b, + A q S q /( S * q when b = r ( otions on stoks when S < S when S S where GBS - the generalized Black-Scholes call exression. * * Put = S, X,, r, b, + A q ** q GBS ( q S /( S when S > S ** when S S ** where GBS - the generalized Black-Scholes ut exression. Otions in CQG

27 Page Gamma Call Γ Γ = Γ 0 Put Γ Γ = 0 GBS GBS GBS ( S, X,, r, b, ( S, X,, r, b, + A q ( q S q /( S q ** q ( S, X,, r, b, + A q ( q S /( S Vega Call VegaGBS ( S, X,, r, b, Vega = Numerical differentiation 0 * q when b = r ( otions on stoks when S < S when S S when S > S when S S * ** ** * when b = r ( otions on stoks when S < S when S S Put Numerical differentiation when S > S Vega = 0 when S S heta Call ΘGBS ( S, X,, r, b, Θ = Numerical differentiation 0 where Θ GBS - the generalized Black-Scholes call Θ exression. Put * * when b = r ( otions on stoks when S < S Numerical differentiation when S > S Θ = 0 when S S where Θ GBS - the generalized Black-Scholes ut Θ exression. * when S S ** ** * ** ** Otions User Guide

28 Page Rho Call ρ GBS ( S, X,, r, b, ρ = Numerical differentiation 0 where ρ GBS - the generalized Black-Scholes call ρ exression. Put when b = r ( otions on stoks when S < S * when S S Numerical differentiation when S > S ρ = 0 when S S where ρ GBS - the generalized Black-Scholes ut ρ exression. Imlied volatility System numerically finds imlied volatility. Imlied volatility can t be calculated for call otion if otion value is less than (underlying rice - strike. Imlied volatility can t be calculated for ut otion if otion value is less than (strike - underlying. * ** ** Otions in CQG

29 Page 3 Exotic Otion Models For further reading, we suggest: he Comlete Guide to Otion Pricing Formulas. ISBN Barrier Otions, Binary/Digital Otions, and Lookback Otions at Standard (Vanilla Barrier here are two kinds of the barrier otions: In = Paid for today but first come into existence if the underlying rice hits the barrier H before exiration. Out = Similar to standard otions excet that the otion is knocked out or becomes worthless if the underlying rice hits the barrier before exiration. heov In Barriers Down-and-in call c(x>=h = C + E η =, φ = c(x<h = A B + D + E η =, φ = U-and-in call c(x>=h = A + E η = -, φ = c(x<h = B C + D + E η = -, φ = Down-and-in ut (X>=H = B C + D + E η =, φ = - (X<H = A + E η =, φ = - U-and-in ut (X>=H = A B + D + E η = -, φ = - (X<H = C + E η = -, φ = - Out Barriers Down-and-out call c(x>=h = A C + F η =, φ = c(x<h = B D + F η =, φ = Otions User Guide

30 Page 4 Otions in CQG U-and-out call c(x>=h = F η = -, φ = c(x<h = A B + C D + F η = -, φ = Down-and-out ut (X>=H = A B + C D + F η =, φ = - (X<H = F η =, φ = - U-and-out ut (X>=H = B D + F η = -, φ = - (X<H = A C + F η = -, φ = - where ( ( x N e X x N e S A r r b = φ φ φ φ φ ( ( ( x N e X x N e S B r r b = φ φ φ φ φ ( ( ( y N S H e X y N S H e S C r r b = + η η φ η φ µ µ ( ( / ( / ( ( ( y N S H e X y N S H e S D r r b = + η η φ η φ µ µ ( ( / ( / ( ( ( [ ] y N S H x N e K E r = η η η η µ / ( ( ( [ ] z N S H z N S H K F + = + λ η η η λ µ λ µ / ( / ( X S x + + = µ ( / ln( H S x + + = µ ( / ln( ( ( X S H y + + = µ ( / ln ( S H y + + = µ ( / ln S H z + = λ / ln( / µ = b

31 Page 5 r λ = µ + K ossible cash rebate, b the cost-of-carry. b = r to rice otions on stocks. b = r q to rice otions on stocks and stock indexes aying a continuous dividend yield q b = 0 to rice otions on futures. b = r rf to rice currency otions (rf risk-free rate of the foreign currency. Delta, Gamma, Vega, heta, Rho he system uses the numerical differentiation to calculate the Greeks. Imlied volatility he software shall numerically find imlied volatility. Otions User Guide

32 Page 6 Asset-or-Nothing Binary At exiry, the asset-or-nothing call otion ays 0 if S <= X and S if S > X. Similarly, a ut otion ays 0 if S >=X and S if S < X. heov Call c = S e Put = S e where ( br ( br N( d N( d ln( S/ X + b+ d = b the cost-of-carry. b = r b = r q b = 0 b = r rf Delta to rice otions on stocks. to rice otions on stocks and stock indexes aying a continuous dividend yield q to rice otions on futures. ( br nd ( call = e ( Nd ( + ( br n( d ut = e ( N( d Gamma e Γ = call Γ = ut ( br ( br e n d to rice currency otions (rf risk-free rate of the foreign currency. d nd ( ( S d ( ( S Otions in CQG

33 Page 7 Vega ln( S / X + b = ( ( br Vcall Se n d ln( S / X + b = ( ( br Vut Se n d heta Θ Θ call ut Rho ρ ρ ρ ρ call ut call ut = S e = S e n( d ln ( b r N( d + ( S / X ( br n( d ln ( b r N( d ( S / X + b + + b + ( br ( br Se n( d = b 0 ( br Se n( d = b 0 r = Se N( d b = 0 r =Se N( d b = 0 Imlied volatility o find imlied volatility the following equations should be solved for the value of sigma: Call c = S e Put = S e ( br ( br N( d N( d System numerically solves these equations. Otions User Guide

34 Page 8 Otions in CQG Floating Strike Lookback he Lookback models are used to rice Euroean lookback otions on stocks without dividends, stocks and stock indexes aying a continuous dividend yield and currency otions. A floating strike lookback call gives the holder of the otion the right to buy the underlying security at the lowest rice observed, Smin, in the life of the otion. Similarly, a floating strike lookback ut gives the otion holder the right to sell the underlying security at the highest rice observed, Smax, in the otion s lifetime. heov Call + + = ( ( ( min min ( a N e b a N S S b e S a N e S a N e S c b b r r r b where b the cost-of-carry; b = r to rice otions on stocks; b = r q to rice otions on stocks and stock indexes aying a continuous dividend yield q; b = r r f to rice currency otions (r f risk-free rate of the foreign currency; b S S a + + = / ( / ln( min a a = Put + + = ( ( ( max ( max b N e b b N S S b e S b N e S b N e S b b r r b r where b S S b + + = / ( / ln( max b b =.

35 Page 9 Delta, Gamma, Vega, heta, Rho he system uses the numerical differentiation to calculate the Greeks. Imlied volatility he system uses numerically find imlied volatility. Otions User Guide

36 Page 30 Otions in CQG Fixed Strike Lookback In a fixed strike lookback call, the strike is fixed in advance, and at exiry the otion ays out the maximum of the difference between the highest observed rice, Smax, in the otion lifetime and the strike X, and 0. Similarly, a ut at exiry ays out the maximum observed rice, Smin, and 0. heov Call when X > S max + + = ( ( ( ( d N e b d N X S b e S d N e X d N e S c b b r r r b where b the cost-of-carry; b = r to rice otions on stocks; b = r q to rice otions on stocks and stock indexes aying a continuous dividend yield q; b = r r f to rice currency otions (r f risk-free rate of the foreign currency; b X S d + + = / ( / ln( d d = when X <= S max = ( ( ( ( max max ( max e N e b e N S S b e S e N e S e N e S X S e c b b r r r b r where b S S e + + = / ( / ln( max e e = Put when X < S min + + = ( ( ( ( d N e b d N X S b e S d N e S d N e X b b r r b r when X>=S min

37 Page 3 Otions User Guide = ( ( ( ( max min ( min f N e b f N S S b e S f N e S f N e S S X e b b r r r b r where b S S f + + = / ( / ln( min f f = By defining the following variables all four formulas can be combined into one: z - otion tye adjustment, = otion ut otion, call z rice extreme observed, S = ut otion; calculating a if, otion, call calculating a if min max, S S S rice limit, S L < > = otherwise;, for uts, for calls or if, X X S X S S S L Now the formulas transform into: ( + = = = b d N(z S/S d N(z e b S d N(z S X S e z d N(z e b d z N S S b e S z d N(z e S z d N(z e S z X (S e z heov b L b L L r b b r r r (b L r L max Delta, Gamma, Vega, heta, Rho he system uses the numerical differentiation to calculate the Greeks. Imlied volatility he systems finds imlied volatility numerically.

38 Page 3 Otions in CQG Interest-Rate Otion Models For further reading, we suggest he Comlete Guide to Otion Pricing Formulas. ISBN he Vasicek Model he Vasicek (977 model is a yield-based one-factor equilibrium model. he model allows closed-form solutions for Euroean otions on zero-couon bonds. heov Call ( ( h N P X h N P L c τ = Put ( ( h N P L h N P X + = τ where L bond rincial (i.e. face value, τ bond time to maturity, P = P(, τ P(τ P =, P(-the rice at time zero of a zero-couon bond that ays $ at time, ln X P P L h τ + = d = ( ( a e e a d a a ( = τ r B e A P = ( ( ( where r the initial risk-free rate a e B a = ( ( ( = a B a b a B A 4 ( / ( ex (

39 Page 33 Otions User Guide a the seed of the mean reversion, b the mean reversion level. Delta Since, Delta is the otion value sensitivity to small movements in the underlying rice then Call ( ( ( P h n P L h N X h n X P c = = τ Put ( ( ( P h n P L h N X h n X P + + = = τ Gamma Gamma is identical for ut and call otions. ( ( P h h n X h P h n P L P τ + = Γ = Vega System uses the numerical differentiation to calculate Vega. heta System uses the numerical differentiation to calculate heta. Rho Since the rice at time zero of a zero-couon bond that ays $ at time t is r t B e t A t P = ( ( ( then P B P = τ τ τ P B P = B B h τ = where τ B(τ B =, B = B( Call

40 Page 34 Otions in CQG ( ( ( ( B B h n P X h N B P X h N B P L B B h n P L r c ρ τ τ τ τ τ + = = Put ( ( ( ( B B h n P X h N B P X B B h n P L h N B P L r ρ τ τ τ τ τ + + = = Imlied volatility System numerically finds imlied volatility.

41 Page 35 Otions User Guide he Hull and White Model he Hull and White (990 model is a yield-based no-arbitrage model. his is extension of the Vasicek model. he model allows closed-form solutions for Euroean otions on zero-couon bonds. heov Call ( ( h N P X h N P L c τ = Put ( ( h N P L h N P X + = τ Where L bond rincial (i.e. face value, τ bond time maturity, P = P(, τ P(τ P =, P( - the rice at time zero of a zero-couon bond that ays $ at time, ( ( ln X P P L h τ + = ( ( a e e a a a ( = τ a the seed of the mean reversion. Unlike Vasicek model, P and P τ are inut arameters. Delta Call ( ( ( P h n P L h N X h n X P c = = τ Put ( ( ( P h n P L h N X h n X P + + = = τ

42 Page 36 Otions in CQG Gamma Gamma is identical for ut and call otions. ( ( P h h n X h P h n P L P τ + = Γ = Vega Because ( ( x n x n = ( ( τ h h n P X h h n P L c Vega + = = = heta Call = Θ = r g h n P X h N P r X r g h n P L c τ ' ( ( ' ( = X P P L g τ ln ( P r ln = Put = Θ = r g h n P L r g h n P X h N P r X τ ' ( ' ( ( where ( ( ( + = a a a a a e a e e a e e ( ( ' τ τ

43 Page 37 Otions User Guide Rho Since, the rice at time zero of a zero-couon bond that ays $ at time t is r t e t P = ( then P P = τ τ P τ P = h τ = Call ( ( ( ( h n P X h N P X h N P L h n P L r c τ τ τ ρ τ τ + = = Put ( ( ( ( h n P X h N P X h n P L h N P L r τ τ τ ρ τ τ + + = = Imlied volatility he system finds imlied volatility numerically.

44 Page 38 he Ho and Lee Model Ho and Lee (986 model is the no-arbitrage model. he model allows closed-form solutions for Euroean otions on zero-couon bonds. heov Call c = L Pτ N( h X P N( h Put X P N( h + L P N( h = τ Where L bond rincial (i.e. face value, τ bond time maturity, P = P(, P τ = P(τ, P( - the rice at time zero of a zero-couon bond that ays $ at time, h = L P( τ ln + P( X = τ ( he distinctions from Vasicek model are - P and P τ are inut arameters, - exression is different. Delta Call c = P = X n ( h X N( h L P n( h τ P Put h h = = LP n( h + X n( h P P P τ Otions in CQG

45 Page 39 Otions User Guide Gamma Gamma is identical for ut and call otions. ( ( + + = Γ = P h n X P h n P L P τ Vega Because ( ( x n x n = ( ( τ h h n P X h h n P L c Vega + = = = heta Call = Θ = r g h n P X h N P r X r g h n P L c τ ' ( ( ' ( = X P P L g τ ln ( P r ln = Put = Θ = r g h n P L r g h n P X h N P r X τ ' ( ' ( ( where [ ] 3 ' = τ,

46 Page 40 Otions in CQG Rho Since the rice at time zero of a zero-couon bond that ays $ at time t is r t e t P = ( then P P = τ τ P τ P = h τ = Call ( ( ( ( h n P X h N P X h N P L h n P L r c τ τ τ ρ τ τ + = = Put ( ( ( ( h n P X h N P X h n P L h N P L r τ τ τ ρ τ τ + + = = Imlied volatility he system finds imlied volatility numerically.

47 Page 4 Sread Otions Model For an overview of sread otions, we suggest: he Comlete Guide to Otion Pricing Formulas. ISBN Sread Otions at Kirk s Aroximation he aroximation method by Kirk (995 can be used to rice Euroean sread otions on futures. heov Call rice is c = r r ( F + X e ( F N( d N( d = e ( F N( d ( F + X N( d Put rice is = where d d r ( F + X e ( N( d F N( d ln( F + = = d ( /, F F = F + X,, and the volatility of F is aroximated by the combined value:,. = F F F X + κ + F + X F rice on futures contract, F rice on futures contract, volatility of futures, volatility of futures, κ - correlation between the two futures contracts. However, it should be noted that both formulas for c and above can be easily calculated using generalized Black-Scholes equation. Notice that if we take S = F K = F + X Otions User Guide

48 Page 4 Otions in CQG, and is calculated by the formula above than c may be exressed as ( ( d N Xe d N Se c c r r BS = =, which is exactly identical to BS equation. Similar is true for. hat also imlies that some of Greeks can be calculated by the corresonding BS formulas. Prior to giving formulas for Greeks lets introduce a few heler equations which may hel in imlementing the formulas found across the section. = β X F F +, thus simlifying β κ β + =. Put-call arity in Kirk s model is exressed as: ( F X F e c r + + =. Below are some artial derivatives used in equations ( = = F F d F d. he first derivative of sigma by the rice of the second futures is: ( X F X F + = κ β. he second derivative of sigma by the rice of the second futures is a bit more comlex and is: ( ( ( = 3 F X F X F X X F X F β κ. Partial derivatives of,d d by the rice of the second futures are also useful to have. hose are: ( X F d F F d + =, ( X F d F F d + =. Also, some artial derivatives of the combined volatility are as follow: βκ =, = κ β β,

49 Page 43 Otions User Guide β κ =. Finally it should be noted that 0 / ( ( = F d n d n, and hence: ( ( d n d n F =. Delta, Delta Each delta is calculated with resect to the corresonding asset rice movement. Sensitivity of call otion rice to rice change of the first futures is: ( d N e F c r c = =. Sensitivity of call otion rice to rice change of the second futures is: ( ( + + = = ( F d n X F d N e F c r c. By virtue of call-ut arity given above the following exressions are true for ut otion Deltas. Sensitivities of ut otion rice to rice change in rice of either the first or the second futures are, resectively: r c e F = =, r c e F + = =. Gamma, Gamma Each gamma is calculated similar to delta, with resect to the corresonding asset rice movement. he equation is identical for call and ut: ( [ ] ( [ ] ( F d n e d d n F d d n F e r r c = + + = = Γ Γ he gamma with resect to the second futures rice is identical for call and ut and is exressed as: ( ( = = Γ Γ F F d d F X F F F d d e r n c.

50 Page 44 Vega he vega is chosen to reflect sensitivity of the sread rice with resect to movement of value of the combined volatility : Vega = F e heta n( d r. Call Θ = r c + g, Put Θ = r + g, where F + X g = e r F n ( d ( F ln + n ( d ( F ln + = e r F + X n ( d e r = n( d F Rho Call ρ = c Put ρ = Chi Chi χ (as defined in Carmona & Durrleman denotes the first derivative of otion rice by correlation coefficientκ. c = = e κ F F n ( χ r d. Otions in CQG

51 Page 45 Imlied Volatility & Correlation here is no definite way to calculate both, given a concrete sot rice. It is suosed to determine the value of the combined volatility by the standard aroach of solving the equation numerically as done in Black-Scholes model. However, it should be ossible to calculate imlied value of any of three,, κ variables rovided the other two are known. For that urose the artial derivatives of otion value by any of three variables may be required to aly Newton s equation solver, for instance. Let s denote a selected variable as ξ, which may be either of,, κ. he generic form of the artial derivative of otion value c = e ξ c is: ξ cbs F n( d = = Vega ξ ξ ξ r he exression demonstrates that values calculated with BS model can be used. Substituting ξ c c c with,, and κ the exressions for each of,, and can be obtained using the κ corresonding artial derivatives of given earlier.. Otions User Guide

52 Page 46 Cumulative Normal Distribution Function Aroximation For further reading, we suggest: he Comlete Guide to Otion Pricing Formulas. ISBN Handbook of Mathematical Functions. ISBN Abromowitz and Stegun aroximation he following aroximation of the cumulative normal distribution function N(x roduces values to within six decimal laces of the true value. When x >= 0 N(x = n(x(a * k + a * k^ + a3 * k^3 + a4 * k ^ 4 + a5 * k ^5 when x < 0 N(x = N(-x where n(x normal distribution; k = / ( * x; a = ; a = ; a3 = ; a4 = ; a5 = ; Hart s aroximation his algorithm uses high degree rational functions to obtain the aroximation. his function is accurate to double recision (5 digits throughout the real line. Otions in CQG

53 Page 47 Numerical Methods for Solving Equations he system offers several methods of the solving of the nonlinear equations. For further reading, we suggest Numerical Recies: he Art of Scientific Comuting, 3 rd ed. ISBN-0: Bisection Method he bisection method is a simle iterative root-finding algorithm. he method convergence is linear, which is quite slow. On the ositive side, the method is guaranteed to converge. Newton s Method Newton's method, also called the Newton-Rahson method, is an iterative root-finding algorithm. he method convergence is usually quadratic, however it can encounter roblems for function with local extremes. Newton's method requests that function is differentiable. Newton Safe Method Newton Safe method is an iterative root-finding algorithm, which combines the bisection and Newton s methods. he method, however if function has local extremes convergence can be linear. Like Newton's method, Newton safe method requests that function is differentiable. Brent s Method Brent s method is an iterative root-finding algorithm. his method is characterized by quadratic convergence in case of smooth functions and guaranteed linear convergence in case of non-smooth or sohisticated functions. Otions User Guide

54 Page 48 Numerical Differentiation he first derivative shall be calculated as df dx f f h i+ i x= x i he first derivative reresents instantaneous rate of change, which is limit of average rate of change where h is the small time interval, h=the time between oint t and oint t+=δt as (delta the second derivative shall be calculated d f dx x xi f i+ fi + h f i Otions in CQG

55 Page 49 rading Otions rading with CQG IC is exlained in detail in our trading user guide. As an otions trader, you may want to: Add a Greek column to DOMrader (rading Preferences > Dislay > Greek column for otions Highlight theoretical value on the DOMrader (rading Preferences > Dislay > Price Column Select on otions model (rading Preferences > Dislay > Otions Use theoretical value to calculate UPL/MVO (rading Preferences > Dislay > Status DOMrader and Order icket have otions-secific comonents. he current strike rice is dislayed, and you can change the model and Greeks directly on the trading alication. he Account Summary area of Orders and Positions also has otions-secific data. Note about otions rices on DOMrader You may wonder why rice calculations sometimes differ between the otions window and DOMrader. As exected, the rice on the otions window is calculated using market data and shows the current value of the Greek for a single rice. DOMrader offers an entire ladder of rices. Excet for the single cell where the last trade occurred, other rices are otential rices at which the otions contract may be traded later, if the market moves in that direction. Because we cannot calculate an actual rice for a future state, we use redictive mathematics to derive those otential rices. o calculate Delta for a otential rice of C.EP U3350 away from the current market (say, at 400, we use the rice of the underlying instrument F.EPU and other characteristics of the F.EP market movement that would result in market of C.EP U3350 moving to 400. hus, we are trying to redict what the value of Delta would be then if the otion rice achieves 400. CQG uses a comlex algorithm to make that rediction. Otions User Guide

56 Page 50 Because of this difference in calculation, the rices on the otions window may be different from the rices on DOMrader. Otions in CQG

57 Page 5 Setting Otions Preferences o set otions references, click the Setu button and then click Otions Preferences. You can also click the Prefs button on the Otions toolbar. o start, select the model and where to aly these references. If you select DDE & Oerator values, changes aly to other areas where otions are used, such as custom studies. Click the Summary button to view, rint, and save (.dat file the current settings. Click the Defaults button to return to default values. Other Otions references include (tabbed area at bottom of window: View settings allow you to show or hide Greek and imlied volatility scales, order columns, and set extended coloring arameters. Volatility settings allow you to set the imlied volatility tye, evaluation method for average volatility, and select a volatility calculation tye. Interest Rate settings allow you to set the interest rate for various currencies. Price Filter settings allow you to select which rice to use for underlying and otion. You can also choose to use most recent settlement rices. Otions User Guide

58 Page 5 Greeks Scale settings allow you to set the rice scale, time direction, and time scale and to choose ercent or fractions for imlied volatility and delta and gamma. Advanced settings allow you to select the underlying contract tye and increase days to exiration. Model settings allow you to define arameters for each model. Udate Frequency settings allow you to set the refresh eriod for average volatility, interest rate, and new/removed contract and to set udate delays for theoretical value and the Greeks. Setting Otions Preferences

59 Page 53 Setting Otions Window View Preferences View settings allow you to show or hide Greek and imlied volatility scales, order columns, and set extended coloring arameters for old and stale. Aearance Select this check box to dislay the scale setting (ercent or fraction in the header. Column order Click the Months check box to arrange the columns by month. Click the Puts/Calls check box to arrange the columns so that all calls columns come before uts columns. Extended Coloring: Mark as Set the threshold for old rices and stale movement. Otions User Guide

60 Page 54 Setting Otions Calculator View Preferences Degree of Polynomial Enter a value u to 8. he higher value, the slower the drawing of the grah but the better the curve fits the Volatility Skew grah. Points to Plot Enter a value u to 0. he higher the number, the slower the drawing of the grah but the higher the definition. Setting Otions Preferences

61 Page 55 Setting Volatility Worksho View Preferences Show Choose the elements to add to the Volatility Worksho dislay: Yesterday curve, Yest. IVs, Call/Put curve, or Net Change. Each of these becomes an additional row in the table about the grah and are dislayed on the grah. he curves are added to the grah. Yesterdays IV (each otion s settlement IV is reresented as circles on the grah. Net change is reresented as a vertical line between the current IV and yesterday's settlement IV. Strikes Range Exand the curves on the left and right side by a designated ercentage. his facilitates estimating the IVs of otions that have not yet been listed. For examle, if the range rior to the exansion was from 000 to 3000 and the range was exanded on the right side by 0 ercent, the new range would be from 000 to 300 [(.*( ]. Otions User Guide

62 Page 56 X-Axis tye Select the variable reresented by the X-axis: Strike Price or Delta. Mark as Set the threshold for old rices, in hours, and stale movement, in ercent. Setting Otions Preferences

63 Page 57 Setting Strategy Analysis View Preferences Dislay tye Select whether to dislay the P&L grah using rofit/loss as a function of the underlying rice or value of the ortfolio as a function of rice. Otions User Guide

64 Page 58 Setting Volatility Preferences Volatility settings allow you to set the imlied volatility tye, evaluation method for average volatility, and select a volatility calculation tye. Volatility for calculation Select one of: Aly vol surface = 3-D value from the Volatility Worksho Aly vol curve = -D value from the Volatility Worksho Use IV for Greeks&heoV = Used in conjunction with the Imlied Volatility ye, raded or Momentary. Use IV for Greeks = Used in conjunction with the Imlied Volatility ye, raded or Momentary. Use Average Vol = Used in conjunction with the Average Volatility evaluation method. he average volatility using Put-Call Searate and Put-Call Combined is calculated by taking a weighted average of the imlied volatilities for the strikes encomassing the at-the-money-strike. For examle, with the underlying at and the imlied volatility of the calls at 6.0 and the imlied volatility of the calls at 5.4 the average call volatility would be:.6( (5.4 = his volatility would be used to value all the calls. he average ut volatility would be calculated the same way and that value would be used to value the uts. If the Put-Call Combined choice were selected, the call volatility and ut volatility would be averaged and that volatility would be used for all the otions series of that articular underlying. Please note that theoretical value cannot be calculated using imlied volatility. If you select the Use imlied volatility checkbox, CQG uses imlied volatility to calculate all the values excet theoretical value, where it uses one of the selections from the drodown list: Put-Call Searate, Put-Call Combined or Historical. However, if the Use imlied volatility box is not selected, all the values are calculated using one of the three methods. Setting Otions Preferences

65 Page 59 Imlied Volatility ye Select one of: raded = Matches the otions rice with the underlying rice, based on a time when the two rices were in sync, that is, the otions rice haened no later than 3 hours after the underlying rice. his could lead to a value that is in sync but not current. Using this value involves taking the synced underlying rice (also referred to as the coherent underlying rice, which is the close of the underlying instrument during the minute rior to the last otion tick. However, if the underlying has not traded during this minute, the system uses the underlying tick closest to the time of the otion trade, as long as it haened during the current trading day. If the otions rice is a closing value, the settlement rice for the underlying is used as the coherent underlying rice. Momentary = Matches the otions tick with the nearest tick in the underlying, even if the underlying trade haened after the otions rice. Volatilities calculated this way may be off by a large amount if the underlying trade took lace substantially before or after the otions trade. If you select this value, the calculation uses the most current underlying rice and the most current otions rice. Volatilities calculated this way may be off by a large amount, if the underlying rice has changed significantly since the last otions tick. In other words, momentary imlied volatility takes the most current underlying tick without matching it to the time of the otions. his may or may not result in the same volatility as the traded imlied volatility. hese selections are global, which means they aly to all models. (Imlied volatility selections made on the Model tab are only relevant to the selected model. Average volatility Select one of: Put-Call Searate = wo values, one for the calls and one for the uts, are calculated and given searately. hese values are then used as the volatility inut for the selected otions model. Put-Call Combined = he searate call and ut volatilities are averaged together and one value is given. his value is then used as the volatility inut for the selected otions model. Historical Volatility = Reresents the standard deviation of a series of rice changes measured at regular intervals. You define the Historical Volatility using either Percent or Logarithmic rice changes. Percent changes assume that rices change at fixed intervals. Logarithmic changes assume that rices are continuously changing. Historical Volatility requires a eriod value. Constant value requires a ercentage value. Constant Volatility = If selected, you must also select a ercentage for the volatility. For examle, if the selected contract was trading at 300 and the volatility value selected was 0%, you would be imlying an underlying rice of 300+ or - 0%, i.e., over the next year. Otions User Guide