Valuaion and Hedging of Correlaion Swaps Mas Draijer 4298829 Sepember 27, 2017
Absrac The aim of his hesis is o provide a formula for he value of a correlaion swap. To ge o his formula, a model from an aricle by Bossu is inspeced and is resuling expression for fair he fair value of a correlaion swap is simulaed. The Jacobi process will be defined and wo discreizaion schemes will be compared for i. Mehods are discussed o make simulaions of he Jacobi process as accurae as possible, making sure i crosses is boundaries -1 and 1 as lile as possible. I will be shown ha a correlaion swap can be hedged by dynamically rading variance dispersion rades. The main resul of he hesis is a parial differenial equaion for he fair value of a correlaion swap. I will be shown ha he expression for he value of a correlaion swap obained by Bossu s model saisfies his parial differenial equaion.
Conens 1 Inroducion 3 2 Financial background 4 2.1 Wha are correlaion swaps?..................... 4 2.1.1 Basic financial conceps................... 4 2.1.2 Volailiy........................... 5 2.1.3 Correlaion.......................... 6 2.1.4 Swaps............................. 7 2.2 Why correlaion swaps?....................... 8 2.2.1 Hedging............................ 8 3 Bossu s oy model 10 3.1 Saisical definiions used...................... 10 3.1.1 Sandard Brownian moion/wiener process........ 10 3.1.2 Adaped process o a filraion............... 10 3.2 Definiions in Bossu 2007)..................... 11 3.2.1 Realised volailiy and correlaion............. 11 3.2.2 Implied volailiy and correlaion.............. 13 3.2.3 Proxy formulas........................ 13 3.3 Comparison of he definiions in [1] o he regular definiions.. 14 3.3.1 Using he definiions of variance and correlaion..... 15 3.3.2 Jusifying he formulas in [1]................ 15 3.3.3 Implici assumpions..................... 17 3.3.4 The case ρ = 0........................ 18 3.4 The oy model............................ 19 3.4.1 One facor: variance swaps................. 19 3.4.2 Two facors: correlaion swaps............... 21 3.5 Wihou he assumpion σ I σ S.................. 26 4 The Jacobi Process 30 4.1 Wha is he Jacobi process?..................... 30 4.2 Discreizaion schemes........................ 31 4.2.1 Euler discreizaion...................... 32 4.2.2 Milsein discreizaion.................... 32 1
4.2.3 Comparison of he discreizaion schemes......... 33 4.3 Exceeding he boundaries...................... 35 4.4 Correlaion swaps........................... 38 5 Hedging 40 5.1 Inroducion.............................. 40 5.2 The mehod of variance dispersion rades............. 40 5.2.1 Wha are variance dispersion rades?............ 40 5.2.2 The hedging propery.................... 41 5.3 Fair srike of a correlaion swap................... 44 5.3.1 The vega-neural porfolio.................. 45 5.3.2 Comparison o he soluion in [1].............. 47 5.3.3 From no-arbirage price o fair srike............ 48 6 Conclusion 50 6.1 Furher research........................... 51 2
Chaper 1 Inroducion In recen years, volailiy has become an imporan facor on he financial marke. I is a raded asse wih grea liquidiy and furher derivaives on i are being widely raded as well. Volailiy swaps and variance swaps are popular financial producs, bu hey bring wih hem an exposure o correlaion. To hedge ha exposure, correlaion swaps can be used. In his hesis, a model by Bossu [1] for correlaion swaps will be inspeced. All is definiions and assumpions are being explored, resuling in a formula for he fair value of a correlaion swap. This resul will be simulaed and hese simulaions analyzed. Is drawbacks will be poined ou and explained. In addiion, he Jacobi process will be defined. This is a bounded process wih boundaries -1 and 1 and i is easy o simulae numerically. This makes i a fiing process for simulaing correlaion pahs. The Euler discreizaion scheme and he Milsein discreizaion scheme will be compared for he Jacobi process. Nex, we will look a why he simulaions cross he boundaries -1 and 1. We will look a several possible opions o reduce he amoun of pahs wihin a simulaion ha exceed he boundaries. The effec of hese opions will be observed and explained. Furhermore, he hedging of he correlaion swaps is looked ino. We define variance dispersion rades and look a how hey can be used o hedge correlaion swaps. A hedged porfolio consising of correlaion swaps and variance dispersion rades is hen inspeced o arrive a a parial differenial equaion for he fair value of a correlaion swap. Finally, his parial differenial equaion is used o confirm ha he formula given by [1] is he no-arbirage price of a correlaion swap. 3
Chaper 2 Financial background 2.1 Wha are correlaion swaps? 2.1.1 Basic financial conceps In finance, he an imporan aspec we deal wih is a financial marke. Mahemaically, a marke is represened as a probabiliy space. In his probabiliy space, we can use random processes o represen socks. Socks are basically small pieces of ownership of a company, bu hey are ofen bough and sold only for rading, raher han he buyer acually being ineresed in he company s policy. Someimes, socks are pu ogeher in a sock index. If you inves in a sock index, you immediaely inves in a lo of differen socks, and usually heir share in he index is weighed. The consiuen socks and he weighs are chosen and frequenly changed by he index managers. A sock index can be modeled by aking he appropriae linear combinaion of he socks in he index, he consiuen socks. On he financial marke, no only socks are raded, bu also many oher producs ha have o do wih socks. Traded producs ha are derived from socks are called financial) derivaives or simply financial producs. In his hesis, he erm financial producs will be used o preven confusion wih mahemaical derivaives of a funcion. An example of a financial produc is a fuure conrac. This is a deal beween wo paries saying ha one pary will buy a cerain amoun of a cerain sock from he oher pary a a predeermined dae expiry dae) and agains a predeermined price srike price). If he marke sock price a he expiry dae is higher han he srike price, he buying side of he fuure conrac makes a profi, because he can ge he sock cheaper han usual. If his is more likely o happen han he sock price being lower han he srike price, he seller is likely o lose. In his case, he seller may demand a paymen from he buyer when enering he conrac, as compensaion for he expeced loss. This paymen is called he price of he conrac. These definiions exend o many oher ypes of financial derivaives, including swaps, 4
Figure 2.1: High volailiy on he lef versus low volailiy on he righ. The sock wih high volailiy flucuaes around is expeced value much more han he sock wih low volailiy. [5] which will be explained in secion 2.1.4. Correlaion swaps have several hings in common wih fuure conracs, bu here are some large differences as well. To see his, we need several more definiions. 2.1.2 Volailiy The volailiy of a sock price is, simply pu, he square roo of is variance, so i is mahemaically equivalen o he sandard deviaion. Bu since we are dealing wih sock prices, and no jus any random variable, here is a bi more o i han ha. As can be seen in figure 2.1, he volailiy describes how much he sock price wiggles around is expeced value, no o be confused wih is oal difference from he previous value or previous average. This is in line wih our undersanding of variance and sandard deviaion: he higher he variance, he more observaions will differ from expecaions. This is he same wih volailiy. In mahemaical erms, he volailiy of a random variable X is defined as he square roo of he variance of X; σ X = VarX) = EX 2 ) EX)) 2. This is he exac definiion of volailiy, and here we view a sock price as jus a random variable. In realiy, sock prices are modeled in various ways, and he volailiy may be approximaed wih differen definiions o go along wih ha o make calculaions simpler. This will be eviden when he oy model from [1] is discussed. 5
I is imporan o noe he difference beween implied and realized volailiy. Volailiy is a variable ha will change over ime. This is parly because wihin he company selling he socks, here are periods of sabiliy low volailiy, sock price moves very lile), and periods of uncerainy high volailiy). This means fuure volailiy is uncerain, and needs o be prediced. If a model yields such a predicion for he fuure volailiy, his is called implied volailiy. The realized volailiy, however, represens he exac value he volailiy will have in he fuure. Implied volailiy is ofen aken as he expeced value of he realized volailiy, using a disribuion or pah and a probabiliy space given by he model. The use of he wo differen erms implied and realized volailiy helps o disinguish when predicions for fuure volailiy are discussed implied volailiy), or when here is alk of a fuure siuaion in which he volailiy a ha poin in ime will be known realized volailiy). 2.1.3 Correlaion Volailiy is an essenial variable when regarding sock prices, bu i is no enough when several sock prices or a sock index are involved. Differen sock prices may no move compleely independenly from each oher, e.g. when one bank collapses, oher banks may sar doing bad as well. This means he sock prices of hose banks are correlaed/here is a correlaion. Two random variables may be represened by a correlaion coefficien, a number o describe how much influence one random variable has on anoher one. I is similar o he saisical idea of covariance, bu he correlaion coefficien is scaled o be beween -1 and 1. The correlaion coefficien for wo random variables is defined if and only if heir covariance is defined. Since he economic siuaion is very imporan for he sock marke, i is logical ha every sock is influenced by a good or bad economy. This gives every sock a basic bu very low) level of correlaion wih he oher socks, which is furher increased if he companies are in he same secor, e.g. echnology or banking. Many sock indices pick heir consiuen socks from he same secor, and in addiion, here is a logical correlaion beween he index and he consiuen socks. All of his means ha correlaion is an imporan concep in he world of finance. The mahemaical definiion of he correlaion coefficien ρ X,Y beween wo socks X and Y is: ρ X,Y = covx, Y ) EXY ) EX)EY ) = σ X σ Y [EX2 ) EX)) 2 ][EY 2 ) EY )) 2 ], where covx,y) is he covariance beween X and Y, and σ X and σ Y are he volailiies of X and Y respecively. As wih volailiy, differen definiions of correlaion or correlaion coefficiens may be used o simplify models. Several differen examples of correlaion are shown in figure 2.2. 6
Figure 2.2: Scaerplos of oucomes from wo random variables. The op row shows ha differen correlaion coefficiens mean differen levels of noisiness, while he slope remains he same. The middle row shows ha changing he slope can only change he sign of he correlaion coefficien, bu no is absolue value. In he middle plo on he middle row, he random variable on he Y-axis has variance 0, so he correlaion coefficien is no defined. The boom row shows ha he correlaion coefficien does no say anyhing abou non-linear relaionships beween he wo random variables. [6] Like volailiy, correlaion is a variable which can change over ime. Therefore, we also have implied correlaion and realized correlaion, jus like we have implied and realized volailiy. Their definiions are analogue: implied correlaion is wha we currenly expec he correlaion o be a a fuure poin in ime an expeced value using a model), while realized correlaion is he rue value of he correlaion a ha poin in ime which is currenly unknown). 2.1.4 Swaps A swap can be seen as a be on a marke variable, such as he correlaion coefficien beween wo socks. Basically, a swap is a conrac saing ha one pary will pay he oher pary a cerain amoun of money, and ha amoun of money depends on he changes in ime of he underlying marke variable. Of course, he formal definiion of a swap is quie differen. A swap is a financial insrumen raded beween wo companies. The swap has a price, a fixed predeermined paymen from one side o he oher. I also has a payou, based on a changing variable such as correlaion, and on he predeermined srike price. This srike price is always uned so ha he up-fron price of he swap is 0. The swaps we will be looking a will also have a predeermined expiry dae. Anoher propery of swaps is ha hey are raded over he couner, meaning ha companies make he rade among hemselves, wihou 7
he inervenion of a sock exchange and is rules. Wih his, we can define correlaion swaps. Correlaion can be described as a number using he correlaion coefficien, which can change over ime, so a swap can be applied o i. An amoun of money is assigned for each poin he correlaion coefficien goes up or down; his is known as he noional amoun. For example, i migh be a good idea o inves in a correlaion swap beween Volkswagen and McDonalds if Volkswagen announces i will only sell burgers insead of cars from now on, which probably means he correlaion beween heir wo socks will rise. Conversely, he correlaion beween Volkswagen and Mercedes-Benz will hen likely drop. In he same way, volailiy and variance swaps can be defined; jus ake he correlaion swap and replace he correlaion coefficien wih volailiy and variance raes respecively. These will laer be used as a sepping sone o correlaion swaps, because heir complexiy makes i hard o give a mahemaical expression for heir payoff sraigh away. 2.2 Why correlaion swaps? As wih all financial producs, correlaion swaps can be raded for wo purposes: speculaing and hedging. Speculaing can be seen as gambling on he fuure correlaion: if you hink correlaion will go up, you can inves in a correlaion swap. You can win money, or you can lose, similar o a casino. Hedging, however, is a bi more complicaed. 2.2.1 Hedging If you inves o speculae, you bear a risk. The purpose of hedging, however, is o reduce or even eliminae risks. Hedging always goes ogeher wih oher invesmens, oherwise here would be no risks o reduce. For example, a bank migh have ons of invesmens, and a ne resul of losing AC200 per poin he correlaion beween Apple and Google increases and gaining AC200 for every poin i decreases). To ge rid of his correlaion-relaed risk, a correlaion swap could be enered which makes he bank gain AC200 per poin he correlaion beween Apple and Google increases. This makes heir invesmen safe for any changes in correlaion; heir exposure o correlaion is zero. This mehod is mos ofen used by invesmen banks, who ry o have no exposure and make marginal profis by selling financial producs for slighly higher prices han wha hey are worh. Anoher use of hedging is o ge rid of unwaned exposure, meaning e.g. ha a financial produc migh come wih exposure o boh volailiy and correlaion when you only wan o inves in 8
volailiy. In his case, a correlaion swap could cancel ou profis or losses because of correlaion changes, and you are lef wih only he desired invesmen in volailiy. 9
Chaper 3 Bossu s oy model In his chaper, we will look a he oy model described in [1] and he resuls derived from i. The firs secion will give some sandard definiions of known saisical erms used in he aricle. Using hese saisical erms, [1] defines is own approximaions of) variables. These are covered in secion 3.2, and in secion 3.3, hese definiions are compared o he acual definiions. All of his is combined in secion 3.4 o define he acual oy model, which is hen used o calculae prices of variance swaps and correlaion swaps. Finally, in secion 3.5 i is explored if an assumpion which in realiy migh no be rue can be lef ou. 3.1 Saisical definiions used 3.1.1 Sandard Brownian moion/wiener process A Wiener process W, also called Sandard Brownian moion, is a random process characerized by he following properies: 1. W 0 = 0 2. W has independen incremens: W +u W is independen of W s : s for u 0 3. W has Gaussian incremens: W +u W is normally disribued wih mean 0 and variance u: W +u W N0, u) 4. W has coninuous pahs: wih probabiliy 1, W is coninuous in. 3.1.2 Adaped process o a filraion A filraion F i )...n of he se of evens F in a probabiliy space is an indexed se of subses of F. This index se I is subjec o he condiion ha if i j in I, hen F i F j. The index is usually a ime parameer, in his case 10
he filraion can be seen as including all hisorical daa and informaion bu no fuure daa. Then, a process ha is adaped o a filraion can for he purposes of his hesis be inerpreed as a process ha canno see ino he fuure. This means ha a process X is adaped if and only if for every realizaion and every n, X n is known a ime n; i can be evaluaed using he available informaion. In shor, X only depends on pas daa and informaion and no on fuure daa. 3.2 Definiions in Bossu 2007) In his secion, he definiions used in Bossu s aricle [1] will be presened. These definiions are used o derive he oy model for correlaion swaps, so o undersand he oy model, we mus undersand he definiions. These definiions will be looked a more in he nex secion o furher improve our undersanding of hem. 3.2.1 Realised volailiy and correlaion To define any sor of volailiy and correlaion, we mus firs define a sock marke. We consider a universe of N socks S = S i )...N. We ake a probabiliy space Ω, E, P ) wih P -filraion F, and we assume ha he vecor S is an F-adaped, posiive Io process. 1 We denoe S i ) he price of sock S i a ime. To define he index, we firs define he weighs: a vecor of posiive real numbers w = w i )...N ha sum up o 1. We define he index as I) N S i )) wi. This is a simplified model for he calculaion of sock index I wih consiuen socks S and weighs w; in pracice, however, mos sock indices are defined as an arihmeic weighed average. Given a ime period τ and a posiive Io process S, furher definiions are given. For he lengh of a ime period, a definiion is given ha also covers ime periods consising of muliple inervals. Le τ consis of n inervals, where he i-h inerval ranges from a i o b i. The lengh of he ime period τ is hen given 1 An Iô process is a random process X such ha X = X 0 + σ sdw s + µ sds. 0 0 The firs inegral is a so-called Iô inegral, which is defined in he same way as a Riemanninegral, only wih a Brownian moion W replacing he inegraion variable. See also foonoe 2. 11
so ha i can be wrien as he sum of he lenghs of he inervals i consiss of: τ τ ds = n b i a i ). The coninuously sampled realized volailiy of a consiuen sock or index is defined using a sochasic inegral 2 : σ S τ) τ 1 d ln S x ) 2. 3.1) Because only variance has a liquid marke, he volailiies of he individual socks are squared before we ake heir weighed average. Finally he square roo is aken o ge o he coninuously sampled average realized volailiy of he consiuen socks, or consiuen volailiy: σ S τ) N w i σ Si τ)) 2. In he same way, we find he realized residual: ɛτ) N wi 2 τ)) 2. σsi Using he above definiions, we define he coninuously sampled average realized dispersion dτ) beween consiuen socks, and heir coninuously sampled average realized correlaion ρτ), which will laer be referred o as canonical realized correlaion: τ σ dτ) S τ) ) 2 σi τ)) 2 3.2) σ I τ) ) 2 2 ɛτ)) ρτ) σ S τ) ) 2 3.3) 2 ɛτ)) 2 Suppose ha W is a Wiener process, H is a righ-coninuous, adaped and locally bounded process, and ha {π n} is a sequence of pariions of [0, ] wih mesh going o zero. Then he Iô inegral of H wih respec o W up o ime is a random variable HdW = lim H i 1 W i W i 1 ). 0 n [ i 1, i ] π n For an Iô process X wih sub-processes σ and µ see also foonoe 1), he following sochasic inegral is defined: HdX = H sσ sdw s + H sµ sds. 0 0 0 12
3.2.2 Implied volailiy and correlaion Using condiional expecaions, we can exend he above definiions o implied values. Firs we define implied volailiy: [ ] σ S := E σ S τ)) 2 σ S [0, ]), 3.4) where E[ σ X [0, ])] denoes condiional expecaion. P is a P -equivalen, F-adaped measure, meaning ha P X) = 0 P X) = 0. In he same way, we define he implied consiuen volailiy and he implied residual as: σ S τ) N ɛ τ) N ) 2 [ σ w i σ Si τ) = E S τ) ) ] 2 σ S [0, ]), 3.5) w 2 i σ Si ) 2 [ τ) = E ɛτ)) 2 ] ɛ[0, ]). 3.6) We consider a variance marke on S, w, I), where agens can buy fuure realized variance agains paymen a mauriy of a pre-agreed price. In his case, implied variance he square of implied volailiy) of S as defined above is he no-arbirage price of fuure realized variance of S. In he definiions, denoes ime a which he variance is bough or sold, and τ = [0, T ] denoes he period over which he variance is raded. The squares of implied consiuen volailiy and implied residual are he no-arbirage prices of a porfolio of he N fuure realized variances of he consiuen socks wih weighs w and w 2 i )...N, respecively. We can now define implied dispersion and canonical implied correlaion: σ d S τ) τ) ) 2 σ I σ ρ I τ) ) 2 ɛ τ) τ)) 2 σ S τ) ) 2 ɛ τ)) τ) ) 2 = E [ dτ)) 2 ] d[0, ]), 3.7) 2 1. 3.8) Here, d is he no-arbirage price of realized dispersion as defined earlier, bu in general, ρ τ) Eρτ) ρ[0, ]), so canonical implied correlaion is no he no-arbirage price for realized correlaion. 3.2.3 Proxy formulas If he amoun of consiuen socks N goes o infiniy, he residual erms ɛτ) and ɛ τ) vanish, as will be shown laer in his secion. This yields he following 13
proxy formulas: ρτ) N + ρ τ) N + σ I ) 2 τ) σ S ˆρτ) 3.9) τ) σ I τ) σ S τ) ) 2 ˆρ τ) 3.10) ˆρτ) is called realized correlaion, and ˆρ τ) is called implied correlaion. For mos indices, his is economically correc: fify socks is already enough o almos nullify he difference beween canonical realized correlaion and realized correlaion, as shown in [1]. We can archieve a beer undersanding of he reason why he ɛ erms are lef ou by looking a he definiions of all erms appearing in he original formula for ρ: σ I ) 2 = τ 1 d ln I) 2 σ S ) 2 = ɛ 2 = = τ 1 τ τ d w i ln S i ) 2 3.11) w i σ Si ) 2 3.12) wi 2 σ Si ) 2 3.13) We can see ha in he definiions of σ I ) 2 and σ S ) 2, he weigh erms w i appear wihou a power. In he definiion for σ I ) 2, he weigh erm is squared, bu only afer some oher operaions. In he definiion for ɛ 2, he weigh erm is immediaely squared. This means ha if he weighs would go o zero, ɛ 2 would vanish in comparison o σ I ) 2 and σ S ) 2. However, he weighs do no necessarily go o zero if N : if here are 1000 consiuen socks, i is possible for one sock o have a weigh of 0.9 while he oher 999 socks have a oal weigh of 0.1. So [1] implicily makes he assumpion ha his does no happen, and ha lim w i = 0 for all i. Since i is numerically shown in [1] ha he canonical N values and he proxy values are very close, his may be a reasonable assumpion. 3.3 Comparison of he definiions in [1] o he regular definiions The definiions for volailiy and correlaion in [1] are differen from he common definiions. Tha is why we compare he definiions used in [1] wih he sandard 14
definiions in his secion. Firs, some resuls will be derived from he exbook definiions of variance and correlaion. These resuls are rewrien o be as close as possible o he definiions in [1]. I will urn ou ha here are dispariies beween he rue values derived from he regular definiions and he formulas used in he aricle. The consequences of assuming hese formulas o be close o he rue values are discussed as well. Lasly, he case ρ = 0 is explored. 3.3.1 Using he definiions of variance and correlaion Take wo random variables S 1 and S 2. According o he sandard definiions, he variance of S 1 is: vars 1 ) = ES 2 1) ES 1 ) 2. The correlaion beween S 1 and S 2 is: ρ S1,S 2 = covs 1, S 2 ) vars1 ) vars 2 ). vars1 ) may also be subsiued by he volailiy σ X = vars 1 ). This gives rise o he insigh ha he price of a correlaion swap migh also be a value divided by he price of he corresponding volailiy swaps. This will laer urn ou o be rue, where he price of he volailiy swaps is represened by he price of consiuen volailiy. If we ake muliple S i, i = 1, 2,..., n, we can ake he average of he pairwise correlaions as he correlaion among all X i : ρ Si = 1 n 2 n n j=1 ES i S j ) ES i )ES j ) σ Si σ Sj. This formula is no very informaive. Anoher approach for finding a relaion is looking a vars 1 + S 2 ). Expanding his gives: vars 1 + S 2 ) = vars 1 ) + vars 2 ) + 2covS 1, S 2 ), σ S1+S 2 = σ 2 S 1 + σ 2 S 2 + 2σ S1 σ S2 ρ S1,S 2. Bringing he correlaion coefficien o he oher side gives us an expression for he correlaion in erms of he volailiies: ρ S1,S 2 = 1 σs1+s 2 σ S 1 σ ) S 2. 3.14) 2 σ S1 σ S2 σ S2 σ S1 3.3.2 Jusifying he formulas in [1] Going furher ino 3.14), we migh be able o jusify formula 3.3) for correlaion given in [1], repeaed here: σ I τ) ) 2 2 ɛτ)) ρτ) = σ S τ) ) 2, 3.15) 2 ɛτ)) 15
where σ S τ) = N w i σ Si τ)) 2, ɛτ) = N wi 2 σsi τ)) 2, τ is a ime period and w i are he weighs of he consiuen socks in he sock index. σ Si and σ I also have oher definiions han usual using Iô inegrals, see secion 3.2), bu we will work wih he sandard definiions here, since he difference beween he definiions is small. Looking a a porfolio I consising of socks S 1, S 2 wih corresponding weighs w S1, w S2 and using he same reasoning as for 3.14), we obain for he correlaion: vari) = w 2 S 1 vars 1 ) + w 2 S 2 vars 2 ) + 2w S1 w S2 covs 1, S 2 ) σ 2 I = w S1 σ S1 ) 2 + w S2 σ S2 ) 2 + 2w S1 σ S1 w S2 σ S2 ρ S1,S 2 ρ S1,S 2 = σ2 I w S 1 σ S1 ) 2 w S2 σ S2 ) 2 2w S1 σ S1 w S2 σ S2 This resul can be exended for an index I consising of N socks S i wih weighs w i and volailiies σ Si. Define a correlaion value ha can be seen as he weighed average correlaion in he index: ρ = w i w j σ Si σ Sj ρ i,j i j. w i w j σ Si σ Sj i j Using his definiion and he same reasoning as for he case wih 2 socks, we lose he dependence on pairwise correlaions and we gain a dependence on he index volailiy: σ I ) 2 N wi 2σSi ) 2 ρ = w i w j σ Si σ Sj i j We can see ha he numeraor of his expression is exacly he same as he numeraor in equaion 3.3), bu he denominaor is differen. We can rewrie he denominaor o a erm closer o he denominaor in 3.3). Going back o he case wih only wo consiuen socks, we can subsiue 2w S1 w S2 σ S1 σ S2 wih 16
w S1 σ S1 + w S2 σ S2 ) 2 ɛ 2, jusified below: w S1 σ S1 + w S2 σ S2 ) 2 w 2 S 1 σ 2 S 1 w 2 S 2 σ 2 S 2 = w 2 S 1 σ 2 S 1 + w 2 S 2 σ 2 S 2 + 2w S1 w S2 σ S1 σ S2 w 2 S 1 σ 2 S 1 w 2 S 2 σ 2 S 2 = 2w S1 w S2 σ S1 σ S2. Using his subsiuion in he case wih N consiuen socks, he formula for ρ becomes: σ I ) 2 ɛ 2 ρ = ) 2 3.16) w i σ Si ɛ 2 By his, he erm ɛ 2 is incorporaed, bu we sill have ) 2 w i σ Si σ S ) 2 = N ) w i σ S i 2. This inequaliy is disregarded in [1]; i is assumed ha he difference beween hese wo erms is small. This becomes clear from he saemen ha consiuen volailiy is more frequenly defined as he weighed arihmeic average of volailiies of consiuen socks, w i σ Si. By aking a differen definiion for consiuen volailiy, i is implicily assumed in [1] ha he difference beween hese wo definiions remains small: N ) 2 w i σ Si σ S ) 2. 3.17) The difference beween hese wo erms is assumed no only o be close o zero, bu also o always have he same sign. The following inequaliy is saed in [1]: which implies ρ = σi ) 2 ɛ 2 σ S ) 2 ɛ σ I ) 2 ɛ 2 2 ) 2, w i σ Si ɛ 2 N ) 2 w i σ Si σ S ) 2. 3.18) 3.3.3 Implici assumpions In his subsecion, assumpions 3.17) and 3.18) are looked ino. The meaning of hese assumpions on economical level is discussed, as well as how reasonable he assumpions and heir economical equivalens are. 17
Consider he following: N ) 2 w i σ Si, = σ S ) 2 = w i σ Si w j σ Sj j=1 w i σ Si σ Si). This enables us o see ha he assumpion made a 3.17) is equivalen o w i σ Si 1 N σ Si, or in words: he weighed average of he volailiies of he consiuen socks is close o he non-weighed average. This means ha he weighs are more or less evenly disribued for consiuen socks wih large and small volailiies, and ha he consiuen socks wih large volailiies generally do no have significanly higher or lower weighs han hose wih small volailiies. This is a reasonable assumpion, especially for a large number N of consiuen socks. However, inequaliy 3.18) is no explained by his assumpion. I is equivalen o: 1 σ Si w i σ Si 0, N while his value migh jus as easily be posiive. This means ha for he inequaliy o hold, he consiuen socks wih low volailiies mus have larger weighs han hose wih high volailiies. When combined wih he reasoning above, his means he following assumpion is done in [1]: Consiuen socks of he index wih low volailiies have slighly larger weighs han hose wih high volailiies. Economically, his assumpion means ha he relaively sable socks low volailiy) are preferred when creaing or updaing he sock index. For many indices, his assumpion is correc: he risk of high-volailiy socks is only acceped if in reurn here s a higher expeced profi. Even when he index doesn explicily selec socks based on volailiy levels, many indices give weighs o consiuen socks corresponding o heir marke caps. Generally, socks wih high marke caps, i.e. socks of large companies, are less volaile han socks of small companies. This means ha in such indices, socks wih low volailiies have higher weighs han socks wih high volailiies, in accordance wih he assumpion. 3.3.4 The case ρ = 0 Wheher we use he formula in [1] or he rue formula for ρ, he resul for aking ρ = 0 remains he same, since he numeraor of he fracion needs o be 0 in 18
ha case, and he numeraors are he same in boh formulas. So boh formulas yield he following: σ I ) 2 ɛ 2 = 0; σ I ) 2 = wi 2.σ Si ) 2. This means ha he index variance equals he weighed squared weighs, so hey do no add o 1) arihmeic average of he variances of consiuen socks if and only if he average correlaion equals 0. This conclusion agrees wih our inuiion: if here is a correlaion, i would aler he index variance wih regard o he average consiuen variance. Since we are only considering averages, here may be some pairwise correlaions, bu if hey cancel each oher ou, heir effecs on he index variance apparenly cancel ou as well. The squared weighs make sense as well, since a scalar aken ou of a variance mus be squared. 3.4 The oy model 3.4.1 One facor: variance swaps Definiion Now he oy model in [1] for derivaives on realized variance will be defined using he definiions discussed in he previous secions. These derivaives can be seen as variance swaps. We sar simple, by aking a marke on a single asse S, where agens can rade he asse s realized variance σ S τ) ) 2 over a fixed ime period τ = [0, T ]. For [0, T ], we ake he variance price v o be he bes esimae of he variance we can give a ime : we ake he realized variance unil ime, and from ime unil ime T, we subsiue i wih implied variance. This is defined mahemaically as v = T σ S [0, ]) ) 2 + T T σ S [, T ]) ) 2, 3.19) where σ S and σ S are volailiy and implied volailiy of S respecively, defined according o 3.1) and 3.4). The forward-neural dynamics of v - he way in which v changes over ime - are no deermined by 3.19). In [1] i is assumed ha hese forward dynamics have he following srucure: dv = 2ω T T v dw, 3.20) where ω is a posiive volailiy of volailiy parameer, and W is a sandard Brownian moion under P*. The flexibiliy of his srucure lies in he parameer ω, which can be chosen using pas daa and a maximum likelihood 19
esimae. The erm T T ges closer o 0 as T, meaning he price of he variance swap becomes less volaile as he ime approaches mauriy, which agrees wih our inuiion. As expeced, he volailiy of he swap price is proporional o he swap price v iself. The Brownian moion W gives he variance process is randomness. This model is used o define he fair srike of a volailiy swap, which we will call fair volailiy. The payoff is V T vt, quie logically he square roo of he price of realized variance. The fair value of he swap a any ime is calculaed in [1] using 3.20) and some properies of Iô inegrals: V = [ v exp 1 ) ] 3 T 6 ω2 T. T Seing = 0, we find for he fair volailiy V 0 : V 0 = v0 [ exp 1 ] 6 ω2 T v0 is known as he fair srike of a variance swap, or fair variance in his hesis. Noice ha he erm srike is used insead of he erm price: his is he fair price for a variance swap wih srike 0, bu as is he rule wih swaps, he srike is adjused so ha he price becomes 0. In our case, his means he srike mus be v0. The erm exp [ 1 6 ω2 T ] is known as he convexiy adjusmen; a erm ha compensaes for he fac ha variance is convex in volailiy. Simulaion This oy model can be simulaed in MATLAB using he Euler discreizaion scheme on is forward-neural dynamics. For deails abou he Euler discreizaion scheme, see chaper 4 of his hesis or [4]. An example of several variance pahs simulaed using his model can be seen in figure 3.1. ω is aken as 0.61: his is a reasonable value for a mauriy of 1 year according o an esimaion done in [1] using real-world daa. v 0 has been arbirarily picked, because here are many differen socks wih many differen variances. I can clearly be seen from his plo ha he volailiy of a pah is lower when v is lower because of he erm v in he forward dynamics) and when is higher because of he erm T T T ). Boh for T 0 and for v 0 he volailiy should approach 0, since boh hese erms appear as a scaling facor in 3.20). The siuaion T T 0 occurs in he simulaion, a he righ side of figure 3.1, and we see ha he volailiy does indeed approach 0 in ha par. To illusrae wha happens for v 0, he blue pah has been modeled wih v 0 = 0.02. We can see ha he volailiy for his pah is indeed very low, as expeced, and he value of v barely changes. 20
Figure 3.1: 5 pahs of he oy model using Euler discreizaion. ω = 0.61, = 0.001, T = 1, v 0 = 0.02 for he boom blue) pah, v 0 = 0.2 for he oher pahs. I can be seen ha he volailiy of hese pahs is lower for low values of v and for approaching mauriy T. 3.4.2 Two facors: correlaion swaps In his subsecion, he fair value of a correlaion claim is modeled by exending he oy model o wo facors, being index variance and consiuen variance. This correlaion claim is inspired by he proxy formulas discussed in 3.2.3. Is payoff is σ I ) 2 τ) c T ˆρτ) = σ S. τ) This is differen from a sandard correlaion swap wih srike 0, which has payoff w i w j ρ SiSj τ) i<j ρτ) =, w i w j i<j where ρ SiSj is he pairwise correlaion beween socks S i and S j. The difference in payoff beween his correlaion claim and he sandard correlaion swap is almos zero shown in [1] using real-world daa), so heir prices will also be close o each oher. In realiy, all swaps including correlaion swaps) have a srike such ha he up-fron price of he swap becomes zero. This is no considered in [1], bu in chaper 5 we use he price of a correlaion swap wih srike 0 o find he value he srike should have o make he price equal o 0. 21
Definiion The oy model is exended o wo facors; hose facors are he index variance and he consiuen variance. We denoe wih v I he marke price of realized index variance a ime, wih v Si he marke price of he realized variance of sock S i a ime, and we define he no-arbirage price of realized consiuen variance a ime as: w i v Si. 3.21) v S In he same way as for he one-facor oy model, we define a srucure for he forward-neural dynamics of boh pahs: dv I dv S = 2ω I T T = 2ω S T T v I dw I, 3.22) v S dw S, 3.23) where ω I and ω S are consans volailiy of volailiy parameers and W I and W S are sandard Brownian moions under P*. We assume ha W I and W S have a consan correlaion χ, i.e. ) ) dw I dw S = χd. Noe ha his value denoes he correlaion beween changes in index and consiuen volailiies, raher han he correlaion beween he absolue levels. Equaion 3.23) is only an approximaion: i is assumed ha each v Si follows his ype of dynamics, bu heir arihmeic average v S does no necessarily have o follow he same dynamics. Assuming ha dv S v S i+1 v S i 3.24) for some poins in ime i+1 and i close o each oher, we can jusify formula 3.23), using ha each v Si follows his ype of dynamics. Firs we subsiue definiion 3.21) in 3.24): dv S = j=1 j=1 j=1 w j v Sj i+1 j=1 w j v Sj i ) w j v Sj i+1 v Sj i w j dv Sj, using approximaion 3.24) wih dv Sj insead of dv S ) in he las sep. 22
Now we can use ha each dv Sj follows he dynamics in 3.23): dv S j=1 = 2 T T 2w j ω Sj T T j=1 v Sj dz Sj 3.25) w j ω Sj v Sj dz Sj. 3.26) Each volailiy of volailiy parameer ω Sj of he individual consiuen socks is much higher han he volailiy of consiuen volailiy parameer ω S, so he weighed) mean of he ω Sj will also be much higher. On he oher hand, he weighed) mean of he Wiener process incremens dz Sj is much lower han dz S in absolue value, which is a Wiener process as well: because of he law of large numbers, he weighed mean of Wiener process incremens converges o zero. I is a reasonable assumpion ha hese wo dispariies cancel each oher ou, i.e. j=1 w j ω Sj dz Sj ω S dz S. for each j. Mak- This means ha ω Sj dz Sj can be approximaed by ω S dz S ing his subsiuion in 3.26), we ge dv S 2 T T ω S = 2 T T j=1 ω Sv S dz S, which is exacly how i is defined in 3.23). w j v Sj dz S Now he analyically derived formula for he fair value of he correlaion claim will be inroduced. The payoff of he correlaion claim is c T v I T v S T Using he definiions for index volailiy and consiuen volailiy given earlier in his chaper and some properies of Iô inegrals, he following fair price for he correlaion claim a ime is obained in [1]: c Ec T v I, v S ) = v I v S exp [ 4 3. ) ] ) 3 ω 2 T S χω S ω I T. T In words: he fair value of he correlaion claim is equal o he raio of fair index variance o fair consiuen variance, muliplied by an adjusmen facor which depends on he volailiy of index volailiy, he volailiy of consiuen 23
Figure 3.2: 5 pahs of he fair value of he correlaion claim. ω i = 0.61, ω s = 0.54, χ = 0.9, v0 I = 0.5, v S 0 = 0.6, = 0.001, T = 1. The op pahs purple and red) do no say below heir boundary of 1 because here is no mahemaical condiion enforcing he boundary. volailiy, and he correlaion beween index volailiy and consiuen volailiy. Taking = 0 and rewriing he equaion, we obain for he fair correlaion adjusmen, he raio beween implied correlaion and he rading price of he correlaion claim: ˆρ ) 0 4 = exp χωs ω I ω 2 ) S T. c 0 3 Hisorical daa shows ha he fair correlaion adjusmen is close o one for mos mauriies, meaning implied correlaion and fair correlaion are very close. [1] Simulaion The wo-facor oy model can be simulaed as a simple exension of he simulaion of he one-facor oy model. Boh v I and v S are modeled in he same way as v in he one-facor oy model. Using hese values, he formula for he fair price of a correlaion claim can be filled in. In figure 3.2, 5 of hese pahs are shown. ω i, ω s and χ are aken o be equal o he esimaes in [1] using real-world daa. v0 I and v S 0 are arbirarily picked, bu sill saisfying v0 I v S 0. This inequaliy canno be mainained for he res of he pahs, as can be seen in figure 3.3, since here is no condiion buil ino he model ha prevens v I from geing higher han v S. There is only he fair correlaion adjusmen, which is lower han 1 for mos mauriies when using he parameer esimaes from [1], since ω S χω I. However, his adjusmen is 24
Figure 3.3: The corresponding variance pahs for he correlaion pahs in figure 3.2. Each color represens he variance pahs corresponding o he correlaion pah of he same color. The solid lines are index variance pahs v I, he doed lines are consiuen variance pahs v S. ω i = 0.61, ω s = 0.54, χ = 0.9, v0 I = 0.5, v S 0 = 0.6, = 0.001, T = 1. I can be seen ha whenever he correlaion in figure 3.2 is above 1, he index variance pah is above he consiuen variance pah, as expeced. quie small, especially for larger. This is wha causes he problem ha he fair correlaion becomes higher han 1. When comparing figure 3.2 wih figure 3.3, we see ha he correlaion being higher han 1 corresponds o he index variance being higher han he consiuen variance. This problem does no appear if v0 I is much lower han v S 0 and χ says high. Tha way, he high correlaion prevens v I from growing oo fas wihou v S growing as well. This problem is also illusraed in appendix G in [1], where he probabiliy of he payoff being higher han 1 is shown as a funcion of χ for differen values of ρ 0. The resul of his appendix can be seen in figure 3.4. In our example in figures 3.2 and 3.3, we used correlaion beween volailiies χ = 0.9 and ˆρ 0 = 0.5/0.6 0.83. If we look a he plo in figure 3.4, following he line for ρ = 0.8, we find a probabiliy of around 0.25 for c T > 1. This agrees quie well wih our simulaions in figure 3.2, where a mauriy T, he op pah purple) clearly exceeds 1 and he second pah from he op red) is close o 1. To make sure he correlaion does no exceed 1, we apparenly need high χ and low ˆρ 0. Since he χ we used was already quie high 0.9), we simulae correlaion pahs for lower ˆρ 0. According o figure 3.4, he probabiliy for c T > 1 is almos 0 for ˆρ 0 = 0.5. To archieve his, we keep v S 0 = 0.6, meaning we have o se v I = 0.3. The resuls of his simulaion can be seen in figure 3.5. We 25
Figure 3.4: The resul of appendix G in [1], comparing he probabiliy of he correlaion being higher han 1 a mauriy T for T = 1 year) as a funcion of he correlaion beween volailiies χ wih differen values of he saring value denoed wih ρ in his image). This image shows ha he probabiliy for c T > 1 goes down for higher χ, bu for higher saring correlaions ρ he effec is diminished, only having a real impac for χ 100%. of correlaion ˆρ 0 = v0 I /v S can see from he lef plo ha he correlaion pahs say well below zero. The righ plo shows ha he condiion v I < v S is well saisfied. We can conclude ha his model may give unrealisic values of c T > 1, bu if he parameers are righ, i.e. high χ and low ˆρ 0, he model does work. Furhermore, he diminishing volailiy of he fair correlaion over ime is visible again, as in figure 3.1. This is because of he erms T T in he pahs for he implied and consiuen volailiies. The erm T T )3 in he fair correlaion adjusmen makes his erm go o 1 for T, as i should, since he payoff of he correlaion swap is c T = v I, wihou any adjusmen facor. T /v S T 3.5 Wihou he assumpion σ I σ S From formula 3.16) for he rue, unapproximaed correlaion value, i is clear ha σ I ) N ) 2 2 w i σ Si, because he correlaion value needs o say below 1. By making approximaion 3.17) and subsiuing i in he formula for he correlaion, [1] implies ha 26
Figure 3.5: 5 simulaions of fair correlaion pahs lef) and in he same color heir corresponding variance pahs righ). Doed lines are consiuen variance pahs v S, solid lines are index variance pahs v I. ω i = 0.61, ω s = 0.54, χ = 0.9, v0 I = 0.3, v S 0 = 0.6, = 0.001, T = 1. The parameers are chosen such ha χ is high and ˆρ 0 = v0 I /v S 0 is low, o avoid any correlaion pahs going over 1. This was succesful, since all correlaion pahs say well below 1. he correlaion value remains below 1, so: σ I σ S. To simplify noaion, we use he following definiion for he weighed arihmeic consiuen volailiy for his secion: σ A := w i σ Si. This gives a shorer formula for he rue correlaion: and for he proxy formula: ρ = σi ) 2 ɛ 2 σ A ) 2 ɛ 2, ) σ I 2 ˆρ =. σ A This all works fine wihou approximaion 3.17). The par where he assumpion simplifies he calculaions is in he wo-facor oy model. Definiion 3.21) is he logical definiion for a variance process corresponding o σ S) 2. Below he definiion, i is explained how his process can be approximaed by 27
he forward dynamics in 3.23). If we look a he variance process corresponding o σ A : N ) 2 v A = w i v Si, 3.27) we can expec ha his process is furher away from he ype of dynamics in 3.23) han v S. To find ou wha kind of process approximaes 3.27), we sar rewriing he process from is definiion. Again, we use he approximaion dv A dv A v A k+1 v A k 3.28) for some poins in ime k+1 and k close o each oher, o allow us o use definiion 3.27). Subsiuing his: N ) 2 N ) 2 w i v Si k+1 w i v Si k 3.29) = = + w i v Si k+1 ) 2 + ) 2 w i v Si k w 2 i j i j i ) v Si k+1 v Si k j i w i w j v Si k+1 v Sj k+1 w i v Si k+1 w j v Sj k+1 3.30) w i v Si k w j v Sj k 3.31) v Si k 3.32) ) v Sj k. 3.33) The firs sep we ook here was wriing ou he square in he firs erm in N ) 3.29) o w i v Si N k+1 j=1 w j v Sj k+1 ). For i = j, his is equal o he firs erm in 3.30) and for i j, i is equal o he second erm. We do he same for he second erm in 3.29), which becomes he wo erms in 3.31). Nex, we subrac he firs erm in 3.31) from he firs erm in 3.30) o ge o 3.32) and we subrac he second erm in 3.31) from he second erm in 3.30) o ge o 3.33). Term 3.32) can be approximaed by N wi 2dv Si, bu erm 3.33) is a bi more involved. I can be a subsanial par of he full value of dv A, so we canno neglec i. We can ake v Si k+1 and v Sj k+1 ogeher, as well as v Si k and v Sj k, bu here is no way o express his par in erms of dv Si. So o be able o model 3.33), we need o define a model for v Si k+1 v Sj k+1 28 v Si k v Sj k. 3.34)
We expec ha if we have a good model for 3.34), we can subsiue ha in he oy model and use v A insead of v S. The model would mos likely become more accurae from his, and depending on he accuracy of he model for 3.34), he modeled value of he correlaion claim migh no exceed 1 anymore. Of course his is a good hing, bu i is hard o find a saisfacory approximae for 3.34). The model would have o conain an expression for he correlaion in he porfolio I, bu we are rying o model he correlaion ourselves, so if we had had an expression for he correlaion, we would no be doing all hese calculaions. This also explains why [1] suggesed approximaion 3.17); even hough he accuracy of he model suffers, he model becomes so much more complex when no making he assumpion ha he addiional accuracy is no worh i. 29
Chaper 4 The Jacobi Process To see if our calculaions acually work, i is useful o perform simulaions. Simulaions can generae plos o help us explain cerain phenomena or hey can provide esimaes for cerain parameers by doing a large number of simulaions and looking a he properies of his se of simulaions, such as heir mean. The Jacobi process is a very useful process, since i can be easily simulaed and is parameers can be chosen such ha i approximaes a correlaion process. 4.1 Wha is he Jacobi process? The bounded Jacobi process, which we will call simply he Jacobi process in his hesis, has several properies ha make i ideal o model correlaion pahs. The dynamics of he process are given by dρ) = κ ρ µ ρ ρ)) d + γ ρ 1 ρ2 )dw ). 4.1) The erm µ ρ ρ)) makes he process mean-revering. If ρ) < µ ρ his erm makes he process go up over ime and if ρ) > µ ρ i makes he process go down over ime, so he process will always be inclined o move in he direcion of µ ρ. How srong his effec is can be conrolled wih κ ρ. For κ ρ, he condiion 0 κ ρ < 1 mus hold, since for negaive κ ρ he process would move away from is mean and for κ ρ > 1 he process would move pas is mean insead of owards i. The mean µ ρ mus be beween -1 and 1, since a correlaion value canno exceed hese values. The mean-revering propery can also be seen in realiy on he sock marke, wih he correlaion beween socks usually being around he same level for a period of ime. The erm γ ρ 1 ρ2 )dw ) in 4.1) is he random par of he process, wih W denoing a Wiener process. The erm 1 ρ 2 ) makes sure he process does no cross is boundaries 1 and -1, because 1 ρ 2 ) 0 for boh ρ 1 and ρ 1. This means ha whenever ρ approaches one of is boundaries, he random par of he process will sar conribuing less and less o he 30
oal shif in ρ and he mean-revering par will ake he process back away from he boundary. The square roo makes sure ha his erm does no conribue oo much o he process when ρ is far from is boundaries. The parameer γ ρ can be seen as a volailiy parameer of he enire process, since i scales he erm wih he Wiener process in i. Since volailiy mus always be posiive, we have γ ρ > 0. To ensure ha he process will say wihin is boundaries of -1 and 1, he following parameer consrain is given in [4]: ) γ 2 ρ γρ 2 κ ρ > max,. 4.2) 1 µ ρ 1 + µ ρ If we subsiue 4.2) ino 4.1), we find lim dρ) < ρ 1 γ2 ρd, lim dρ) > ρ 1 γ2 ρd. Since γ 2 ρ is always posiive, his makes sure he pah will go down if i approaches 1 and up if i approaches -1. All parameers i.e. κ ρ, µ ρ and γ ρ ) have ρ in heir subscrips o denoe ha hey can be funcions of ρ. This gives even more versailiy o he Jacobi process, bu for simpliciy reasons hey will be modeled as consan values in his hesis. 4.2 Discreizaion schemes The Jacobi process defined in 4.1) is an example of a so-called sochasic differenial equaion: an equaion where a pah is specified implicily by specifying he shif in is value using he value of he pah iself. The general formula for a sochasic differenial equaion for a pah X is as follows: dx) = α, X))d + σ, X))dW ). 4.3) Every sochasic differenial equaion has a erm dependen on he shif in ime d, and a random erm dependen on he incremen dw ) of a Wiener process W ). The soluion X of a sochasic differenial equaion is a coninuous pah. However, we canno simulae rue coninuiy, so we have o use a discreizaion scheme o divide he ime period ino small inervals [ i 1, i ] and change he erms appearing in 4.3) accordingly. Two poenially useful discreizaion schemes for he Jacobi process are he Euler discreizaion and he Milsein discreizaion. In his secion, we check how well boh discreizaion schemes perform wih he Jacobi process. A he end, we make a choice for which discreizaion scheme o use in he res of he hesis. 31
4.2.1 Euler discreizaion The simples of he discreizaion schemes is he Euler discreizaion. I urns he ime variable ino a finie number of equidisan poins in ime i. The pah X) is made discree by aking heir values a ime i : X i := X i ), and he shif in is value becomes X i+1 X i. The funcions α and σ are evaluaed for i, X i and he shif in ime d is replaced by := i+1 i. Finally, since Wiener incremens W i+1 ) W i ) are normally disribued wih mean 0 and variance, dw ) is replaced wih Z, where Z is a draw from a sandard normal disribuion. Applying all his, he Euler discreizaion for a general sochasic differenial equaion 4.3) becomes X i+1 = X i + α i, X i ) + σ i, X i ) Z. 4.4) The Jacobi process can be wrien as a general sochasic differenial equaion by aking X = ρ, α, ρ)) = κµ ρ)) and σ, ρ)) = γ 1 ρ 2 ). Subsiuing his ino 4.4), we ge a discree version of he Jacobi process: ρ i+1 = ρ i + κµ ρ i ) + γ 1 ρ 2 i Z. 4.5) Since compuer programs like MATLAB can ake draws from a sandard normal disribuion, his is a formula ha can be used o numerically simulae a correlaion pah. An example of a simulaion of a Jacobi process is shown in figure 4.1. In his example, here were values of he Jacobi pah crossing he boundary of 1; hese have been runcaed, meaning any values higher han 1 were replaced by 1. For he coninuous Jacobi process, i should no be possible o cross he boundaries because of parameer consrain 4.2). This is because for he pah o become higher han 1, we mus firs have ρ) = 1. In his case, he parameer consrain makes sure he pah goes down immediaely aferwards. Afer discreizaion however, we can have ρ i < 1 and ρ i+1 > 1 wihou he pah ever being exacly equal o 1. Possible soluions o his problem are explored in secion 4.3. 4.2.2 Milsein discreizaion A slighly more complicaed way o urn he Jacobi process ino a discree pah is by using he Milsein discreizaion. This discreizaion akes one addiional erm from he Iô-Taylor expansion, he sochasic equivalen of he Taylor expansion. For a general sochasic differenial equaion as described in 4.3), he discreizaion looks like his: X i+1 = X i + α i, x i ) + σ i, x i ) Z + 1 2 σ i, x i ) Z 2 ) σ x i, x i ), 4.6) 32
Figure 4.1: 5 simulaions of Jacobi pahs wih values above 1 runcaed o 1. ρ 0 = 0, κ = 0.5, µ = 0.6, γ = 0.6, = 0.005. where Z is a draw from a sandard normal disribuion. process dρ) = κµ ρ))d + γ 1 ρ 2 )dw ), we have for he parial derivaive: σ x = γρ) 1 ρ2 ). For he Jacobi Subsiuing his and he oher necessary funcions ino he Milsein discreizaion, equaion 4.6): ρ i+1 = ρ i + κµ ρ i ) + Zγ 1 ρ 2 i + 1 2 γ2 ρ i Z 2 ). Noice ha in he las erm, he par 1 ρ 2 i in σ is canceled ou because i also appears in he denominaor of σ x i, ρ i ). Using his formula, we can simulae he Jacobi process in Malab. In figure 4.2, 5 pahs simulaed using he Milsein discreizaion are shown. The parameers are exacly he same as in figure 4.1. The figures obviously look very similar, since hey model he same process. An ineresing observaion is ha he amoun of exceedances of he boundary ρ = 1 does no appear o be much lower han for he Euler discreizaion. This will be looked ino in secion 4.2.3. 4.2.3 Comparison of he discreizaion schemes In his secion, we will compare he Euler discreizaion scheme wih he Milsein discreizaion scheme. A good way o do his is by comparing he amoun 33
Figure 4.2: 5 simulaions of Jacobi pahs simulaed using he Milsein discreizaion wih values above 1 runcaed o 1. ρ 0 = 0, κ = 0.5, µ = 0.6, γ = 0.6, = 0.005. N T = 1 T = 5 T = 10 T = 20 T = 50 100 0.019 0.01 0.482 0.05 0.804 0.1 0.971 0.2 1 0.5 1000 0.014 0.001 0.464 0.005 0.771 0.01 0.976 0.02 1 0.05 10000 0.009 0.0001 0.444 0.0005 0.792 0.001 0.958 0.002 1 0.005 100000 0.014 1 10 5 0.433 5 10 5 0.786 0.0001 0.961 0.0002 1 0.0005 Table 4.1: The proporion of exceedances for differen amouns of imeseps N and differen end imes T. represens he imesep size. The pahs are modeled using Euler discreizaion. ρ 0 = 0, κ = 0.5, µ = 0.6, γ = 0.6, M = 1000. of boundary exceedances for boh discreizaion schemes. These amouns will be calculaed and wih hem, a decision will be made abou which discreizaion scheme bes fis our needs and will be used for he res of he hesis. If we look a figures 4.1 and 4.2, our firs idea is ha he amoun of exceedances do no differ much beween he wo discreizaion schemes. This is no compleely unexpeced, since boh discreizaion schemes have he same order of weak convergence: he convergence of he simulaed process value o he rue value for 0. See [4] for deails). To invesigae his furher, a able comparing he imesep sizes o he proporion of pahs exceeding 1 or -1 is shown in able 4.1 for Jacobi discreizaion and in 4.2 for Milsein discreizaion. Tables 4.1 and 4.2 show he amouns of exceedances for differen imesep sizes and end imes T for Euler discreizaion and Milsein discreizaion. The reacion of he amoun of exceedances o he imesep size and he end ime will be discussed in secion 4.3. Here we noe ha he ables show almos exacly he same amouns of exceedances for he Euler discreizaion as for he 34
N T = 1 T = 5 T = 10 T = 20 T = 50 100 0.024 0.01 0.504 0.05 0.839 0.1 0.975 0.2 1 0.5 1000 0.014 0.001 0.494 0.005 0.813 0.01 0.977 0.02 1 0.05 10000 0.017 0.0001 0.440 0.0005 0.801 0.001 0.974 0.002 1 0.005 100000 0.016 1 10 5 0.434 5 10 5 0.786 0.0001 0.960 0.0002 1 0.0005 Table 4.2: The proporion of pahs modeled using Milsein discreizaion exceeding 1 or -1, for differen amouns of imeseps N and differen end imes T. represens he imesep size. ρ 0 = 0, κ = 0.5, µ = 0.6, γ = 0.6, M = 1000. Milsein discreizaion. Even hough he Milsein discreizaion is slighly more complicaed han he Euler discreizaion, adding an addiional erm wih a parial derivaive, his apparenly does no resul in a lower amoun of exceedances. This is because hey have he same order of weak convergence, as poined ou earlier. The increased complexiy of he Milsein discreizaion urns ou no o have any effecs for modeling he Jacobi process, so we use he Euler discreizaion in he res of his hesis. 4.3 Exceeding he boundaries Even hough he runcaed pah shown in figure 4.1 is heoreically a viable correlaion pah, he values have been changed. Tha means ha his is no longer a rue Jacobi pah, so i is no a good simulaion of he Jacobi process anymore. To preven his from happening, we need o make sure ha we do no have o runcae. For ha goal, we simulae 1000 pahs a once and look a he proporion of pahs for which a leas one value has been runcaed. Firs, we compare his proporion o a value derived from he parameer consrain 4.2) on he Jacobi process. The parameer consrain is repeaed here: γ 2 κ > max 1 µ, γ 2 ). 1 + µ Rewriing his consrain o be described by a single variable, we ge: ) A := max κ γ2 1 µ, κ γ2 > 0. 1 + µ In able 4.3, his value A is compared o he proporion of exceedances while moving around he parameers γ, κ and µ. I appears ha A is no represenaive for he proporion of exceedances. For example, for κ = 0.5, γ = 0.6, µ = 0.4 : A = 0.2429 and #exc = 0.2630, or 0.2470 in a differen se of simulaions wih he same parameers. However, for κ = 0.30, γ = 0.30, µ = 0.40 : A = 0.2357, bu #exc = 0.0000. In hese wo cases, A is almos he same, bu he number of exceedances vasly differs. This means ha he number of exceedances and he value of A reac differenly o changes in he underlying 35
#exc A κ γ µ 0,1240 0,4429 0,70 0,60 0,40 0,1830 0,3429 0,60 0,60 0,40 0,2630 0,2429 0,50 0,60 0,40 0,3380 0,1429 0,40 0,60 0,40 0,3580 0,0429 0,30 0,60 0,40 0,0000 0,2357 0,30 0,30 0,40 0,0010 0,1357 0,20 0,30 0,40 0,0170 0,0357 0,10 0,30 0,40 0,0000 0,6357 0,70 0,30 0,40 0,0000 0,4357 0,50 0,30 0,40 0,2470 0,2429 0,50 0,60 0,40 0,3180 0,2600 0,50 0,60 0,50 0,4860 0,2750 0,50 0,60 0,60 Table 4.3: The proporion of pahs exceeding 1 or -1 a leas once, wih 1000 simulaions for every combinaion of parameers. For every simulaion, ρ 0 = 0, T = 5, = 0.005. The Jacobi discreizaion is used. parameers. Noe ha, as menioned above for he case of κ = 0.5, γ = 0.6, µ = 0.4, here may be a difference in he number of exceedances per simulaion even when he underlying variables are kep he same. This means he amoun of simulaions 1000) is no enough o provide an accurae value for he proporion of exceedances. I is accurae enough, however, o show ha A is no represenaive for he proporion of exceedances: a difference of around 0.25 is oo big o be accouned for by he inaccuracy of he simulaions. To find a quaniy ha is represenaive of he amoun of exceedances, we ry looking a he process iself and he probabiliy ha a pah crosses 1 or -1. Unforunaely, his probabiliy canno be wrien in erms of he underlying parameers. This is firsly because we only have an implici formula for he pah ρ, where ρ i+1 depends on ρ i in several ways, and secondly because here is no explici cumulaive disribuion funcion for he normal disribuion, which Z follows. Inuiively, we expec he amoun of exceedances o be relaed o he size of he imeseps. To check his, he proporion of exceedances is compared wih he amoun of imeseps and he end imes in able 4.1. The amoun of exceedances does no appear o be conneced o he amoun of imeseps N. I personally hink his is very srange a firs sigh. Usually, lowering he size of he imeseps makes a numerical simulaion more accurae. However, a high amoun of exceedances means inaccuracy of he simulaion, because here 36
canno be any exceedances in he coninuous Jacobi process. Because his phenomenon seems so srange, i will be discussed furher below. The end ime T does maer for he amoun of exceedances in able 4.1. For T = 1, less han 2% of all pahs exceeds 1 or -1 for each N. This proporion is already much higher for T = 5, and coninues o rise for higher T, unil a T = 50 all pahs exceed 1 or -1. The unimporance of he imesep size In able 4.1, i appears as if decreasing for a cerain end ime T has lile o no effec on he proporion of pahs exceeding 1 or -1. However, i mus be noed ha since he number of imeseps N is increased as well, here are more opporuniies discree poins in ime) where ρ can cross hese values. Since lowering wihou increasing N, namely by decreasing T, does have effec, he mos logical explanaion would be his: he posiive effec in decreasing he amoun of exceedances gained from lowering is exacly canceled ou by he negaive effec gained from increasing N. In his par, i will be checked if his idea is in accordance wih he heory. Firs, we look a he formula used o simulae he Jacobi process, obained wih Euler discreizaion: ρ i+1 = ρ i + κµ ρ i ) + Zγ 1 ρ 2 i. The significance of here is a scaling facor for boh he mean revering par and, in a lesser manner, he random par. Now, le us look a a random value ρ i and he probabiliy ha i crosses 1. We will be comparing his probabiliy for differen and differen amouns of imeseps, so we can disregard he probabiliy ha i crosses -1 since i is calculaed in exacly he same way and hus has he exac same relaionship for differen and N) and we can assume wihou loss of generaliy ha ρ i 0. For ρ i+1 o be larger han 1, we need he following: ρ i+1 ρ i = κµ ρ i ) + Zγ 1 ρ 2 i > 1 ρ i. Since we keep ρ i and all parameers fixed for his argumenaion, we can subsiue κµ ρ i ) wih a consan c 1 and γ 1 ρ 2 wih anoher consan c 2. Then we ge c 1 + c 2 Z > 1 ρi. 4.7) If we know c 1, c 2, and ρ i, a probabiliy can be calculaed from his formula. For now, we will compare i o a case wih δ = 10 and look a he probabiliy ha ρ i+10 > 1. Looking from ρ i, we need o look 10 imeseps ahead o calculae his probabiliy: 10 n=1 ) c 1 δ + c 2 δzn > 1 ρ i, 37
M = 10 M = 100 M = 1 000 M = 10 000 M = 100 000 Mean 0.10624 0.11024 0.15608 0.15342 0.15275 Variance 0.046396 0.053783 0.045237 0.046241 0.046427 Table 4.4: The mean and variance of all simulaed values ρt ), for differen amouns M of simulaed pahs. There are 0 pahs exceeding 1 or -1 for each M. where each Z n is an independen draw from a sandard normal disribuion. 10 Defining Z := n=1 Z n, we ge 1 10 c 1 + c 2 Z > 1 ρi. The random variable 10 n=1 Z n is normally disribued wih mean 0 and variance 10, so Z follows he sandard normal disribuion. This means he above equaion is exacly equal o equaion 4.7). This shows ha he probabiliy for ρ crossing 1 or -1 in one imesep is exacly he same as wih en imeseps ha are en imes smaller. This explains why he proporion of exceedances says he same if we decrease he imesep size by only increasing he amoun of imeseps wih he same facor. 4.4 Correlaion swaps To model correlaion swaps using a Jacobi process, we will use he mean of many simulaed processes as he expeced value of he process. To be able o use his as an expeced value, we have wo requiremens: he amoun of exceedances should be as low as possible, and he mean mus converge o a single value for large numbers of simulaions M. How o keep he exceedances low has been discussed in he previous secion: we will ake T, γ and µ low, and κ high. I appears ha i does no maer wha we choose for N and wheher we use Euler or Milsein discreizaion schemes, so we will ake N = 100 and use he Euler discreizaion scheme. For T = 1, γ = 0.3, µ = 0.3 and κ = 0.7, some es runs of he simulaion all yield 0 exceedances. Firs, we will check if he mean of he resuls of he simulaion a = T converges. Looking a able 4.4, we can see ha for M higher han 1000, he mean is sable. The variance is quie low here, because we have T and γ low. Bu he mean does converge, so we can a leas use i as he expeced value of he correlaion wih hese parameers. If he variance is higher, he mean will sill converge as long as here are lile o no exceedances, alhough M may need o be higher.) In figure 4.3, i can be seen ha he disribuion followed by he simulaed correlaion values a ime T is a normal disribuion. The Jacobi process can be used o model he correlaion pah used in [1], if we ake he righ values or funcions) for γ, µ, κ and ρ 0. This is an iner- 38
Figure 4.3: A hisogram showing how he resuls a ime = T of he simulaions of he Jacobi process are disribued. M = 100000, T = 1, γ = 0.3, µ = 0.3 and κ = 0.7. The values are normally disribued around heir mean. esing opic for furher research. The forward dynamics of he correlaion pah ˆρ = v I /v S will firs have o be defined. This definiion of ˆρ will have o be rewrien in order o be able o use he forward dynamics of v I and v S, perhaps some approximaions will have o be made. When a formula for he forward dynamics of ˆρ is obained, i can be compared wih he forward dynamics of he Jacobi process, as defined in equaion 4.1). This will show wha he bes definiions for κ ρ, µ ρ and γ ρ would be; hey would probably have o be funcions of ρ and. Even wih he righ definiions, he Jacobi process would sill be differen from he correlaion pah in [1], because of he erm 1 ρ 2 ) in he forward dynamics of he Jacobi process. This is wha makes i ineresing, however. This erm aims o keep he value of he process below 1, so i may solve he problem of he correlaion pah defined in [1] geing higher han 1. Inspecing his difference may even lead o an improvemen of he model in [1]. Anoher use of he Jacobi process would be o model he correlaion swap in a differen way. I could be used o model he correlaion defined in [1] as described above, bu insead of inspecing he differences in he correlaion pahs hemselves, a swap on he correlaion pah could be inspeced. The fair value of his swap could be obained by running a large number of simulaions of he correlaion pah and aking he mean of he swap s payoff as he swap price discouning o compensae for he ineres rae as necessary). This price could hen be compared o he price given in [1], and an explanaion for any difference could be found. Alhough hese ideas are ineresing, here is no room o explore hem in his hesis. 39