A Conditional Valuation Approach for Path-Dependent Instruments

Transcription

1 A Coditioal Valuatio Approach for Path-Depedet Istrumets August 5 Date Lomibao ad Steve Zhu Capital Marets Ris Maagemet Ba of America, N.A., New Yor. Itroductio I a effort to improve credit ris maagemet, fiacial istitutios have developed various measures to maage their exposure to couterparty ris. Oe importat measure of couterparty ris is potetial future exposure (PFE, which is a percetile (typically 95 or 99 percet of the distributio of exposures at ay particular future date. Credit exposure is the amout a ba ca potetially lose i the evet that oe of its couterparties defaults. The measuremet of exposure for derivative products is very importat because it is used ot oly to set up tradig limits but also as a essetial iput to ecoomic ad regulatory capital. The iteral ecoomic capital models used by most techologically advaced bas require the calculatio of the distributio of the exposure at specified future times. For bas itedig to use the iteral model method i the ew Basel II revised framewor o tradig activities [], specific exposure measures such as the expected exposure (EE ad expected positive exposure (EPE are required i the calculatio of the regulatory capital. This paper focuses o the methodology for calculatig the potetial future exposure of path-depedet derivative istrumets. Ulie loa products, the value of derivatives ad other maret-drive cotracts ca chage sigificatly over time as a result of maret movemets. This may lead to a potetial credit exposure with the tradig couterparty should it default i the future ad its trasactios has a positive maret value to the ba. Most bas use a variety of methods to maage such ris at the couterparty level, which may iclude limits o potetial exposures, ettig, collateral agreemets, ad early termiatio agreemets. Sice most credit limits are based o potetial exposure, it is importat for a ba to have robust ad accurate ris models, as well as systems ifrastructure, to quatify the potetial exposures of its derivatives positios. There are two mai compoets i calculatig credit exposures o a trasactio: sceario geeratio ad istrumet valuatio. The sceario geeratio is a simulatio process that geerates the future scearios of various maret ris factors at differet future times (or simulatio dates as termed i this paper. Similar to frot office trasactio pricig, exposure calculatio also requires a valuatio model i order to calculate the value of a trasactio over differet times i the future. The similarity stops here, however, as the calculatio of credit exposure requires modelig that may ot be cosistet with the frot office valuatio model, particularly the sceario geeratio process. For credit exposure, our cocer is o the potetial future maret value of a trasactio. Future maret scearios are usually geerated usig evolutio models of the uderlyig ris factors uder the real measure istead of the ris-eutral pricig measure used i the calibratio of the frot office models. Furthermore, the calculatio of credit exposure requires a valuatio model to value the trade ot oly at The authors would lie to tha Sergey Myagchilov for his cotributio i developig may of the techiques used i this article. We are also grateful to our colleague Michael Pyhti for providig may helpful commets. The opiios expressed i this article are those of the authors ad do ot ecessarily reflect the views or policies of Ba of America, N.A. For questios ad commets, please steve.zhu@bofasecurities.com. The Expected Exposure (EE, as defied i [], is the mea of the distributio of exposures at ay particular future date before the logest-maturity trasactio i the ettig set matures. The Expected Positive Exposure (EPE is the time-weighted average of the idividual expected exposures estimated for give forest horizos.

2 the curret time, but also to Value-at-Future (VaF 3 or price the trade cosistetly across differet times i the future (see illustratio below. Figure Mar-to-Maret ad Value-at-Future i Exposure Modelig Value 5 Mar-to Maret Future Sceario Distributio of Trasactio Values 5 Value at Future Simulatio The calculatio of credit exposure relies heavily o simulatio 4, especially whe couterparty s portfolio is depedet o multiple ris factors. Because of the computatioal itesity required to calculate couterparty exposures, especially for a ba with a large portfolio, compromises are usually made with regards to the umber of simulatio times or the umber of scearios. For example, the simulatio times (also called time bucets used by most bas to calculate credit exposure usually have daily or weely itervals up to a moth, the mothly up to a year ad yearly up to five years, etc. We geerate maret scearios across these simulatio times. The basic problem i valuig path-depedet istrumets i this framewor is that we simulate future scearios oly at discrete set of dates, while the value of the istrumet may deped o the full cotiuous path prior to the simulatio date (or a discrete set of dates differet from the give set of simulatio dates. Therefore, the valuatio models used to calculate exposure could be very differet from the frot-office pricig models. For credit exposure calculatios, pricig is ot a ed i itself. What is importat is the distributio of istrumet values (uder the real measure at differet times i the future. The valuatio models eed to be optimized i order to perform sufficietly large umber of calculatios required to obtai such distributio. Term structure models 5 such as HW, HJM ad BGM are ot adequate for exposure calculatio because these models require either Mote-Carlo or lattice-based modelig, which is computatioally itesive. Furthermore, the stadard valuatio models used to price the istrumets for mar-to-maret are ot applicable for calculatig exposures o may path-depedet products whose value at the future time may deped either some evet at a earlier time (such as exercise of a optio or i some cases o the etire path leadig to the future date (such as the case of oc-i ad oc-out barriers. For such path-depedet istrumets, we propose i this paper the otio of coditioal valuatio, which is based o probabilistic coditioal expectatio techiques ad develop value-at-future models for calculatig the exposure of several path-depedet derivative istrumets.. Sceario Geeratio The first step i calculatig credit exposure is to geerate potetial maret scearios (e.g., FX rates, equity prices, iterest rates, etc. at differet times i the future. Oe obvious choice is to use the same model for istrumet pricig to geerate the scearios, but the evolutio dyamics of this type of models are ofte 3 The otio ad cocept of VaF is similar to Mar-to-Future (MtF, the termiology origially itroduced by Ro Dembo, et al, i their paper [] for Algorithmics ris maagemet framewor. 4 Duffie ad Caabarro [3] provided a descriptio of Mote-Carlo simulatio approach o modelig the derivatives exposure. 5 Reboato [7] provided a excellet overview of iterest rate term structure models.

3 costraied by arbitrage argumets. I cotrast, the dyamics for ris measuremet are usually built o a real measure based o historical data ad ot ecessarily costraied to a ris eutral framewor. For example, i a frot-office pricig model, the iterest rate scearios are usually geerated by costructio 6 of zero rates or discout factors usig the maret prices of cash, euros ad swap rates. However, such costructio is ofte computatioally expesive, as it requires the search algorithm for the busiess day cout library. Furthermore, the forward rates implied from these scearios ca be osesical as a result of arbitrage-free costraits. I this paper, we assume without loss of geerality the simple logormal model for uderlyig prices (. ( St ( = S exp µ σ t+ σwt ( where Wt ( is the Browia motio, µ is the drift ad σ is the volatility. However, distictio must be made betwee sceario geeratio ad istrumet valuatio for these parameters. Whe the model is used for pricig or istrumet valuatio, we ow that the volatility σ = σ iv, i.e., the implied volatility for the optio o uderlyig price, ad the drift is set uder the ris-eutral measure to µ = r d for stoc or idices with r = iterest rate ad d = divided yield. O the other had, whe the model is used for geeratig future scearios for ris maagemet, we ormally use the drift µ = µ h ad volatility σ = σ h, which are usually estimated from the historical data as follows: (. T σ St ( h l h T t S( t µ = =, T St ( µ h = l T t = S( t ad we the adjust the drift µ = µ h + σ to compesate σ term i the model (.. There are two ways that we ca geerate the possible future values of the maret factors. The first is to geerate a path of the maret factors through time, i.e., each simulatio describes a possible trajectory from time t = to the logest simulatio time t = T. The other method is to simulate directly from time t = to the relevat simulatio date t. We will refer to the first method as Path-Depedet Simulatio (PDS ad the secod method as Direct Jump to Simulatio Date (DJS. Path-Depedet Simulatio PDS is a form of discrete-evet simulatio where a maret factor is evolved through discrete time itervals. Therefore, the value of a maret factor at ay give simulatio date is depedet o its values at previous simulatio dates. A PDS sceario is a path that a maret factor taes through time. With PDS, the maret factor dyamics is expressed i term of the previous simulated values X ( t i ad the differece i simulated times: i+ i i+ i For example, the logormal evolutio fuctio taes the form (.3 X ( t = ψ X ( t,( t t (.4 ( ( exp ( ( X ti+ = X ti µ σ t i+ ti + zσ ti+ t i where z is a ormal variat ad X ( t i + represets the shoced maret factor at time t i + that coects from the particular sceario X ( t i + at previous time t i. Figure A illustrates a sample path for X ( t i while Figure 3A shows a 3-sceario radom simulatio of a stoc price usig PDS. 6 The costructio of zero curves is commoly referred as curve costructio, which is a ecessary step i pricig most iterest rate istrumets. 3

4 Direct-Jump to Simulatio Date With DJS, the evolutio process depeds o the iitial value, X ( t, ad the distace, t i +, from the origial time poit t ( i (.5 ( ψ ( Whe fuctio taes the form: X t = ' X t, t. i+ + ( t =, this ca be simply writte as ( ψ ( (.6 ( ( exp ( X t = X µ σ t+ zσ t X t = ' X, t. For example, the logormal evolutio where z is a ormal variate ad X ( t represets the shoced maret factor at time t. Figure A illustrates a path-depedet simulatio ad Figure B illustrates a direct-jump to a simulatio date. Figure Two ways of geeratig maret scearios A. Path-Depedet Simulatio (PDS B. Direct-Jump to Simulatio Date (DJS X 4 X X 4 X X X X 3 X 5 X X X 3 X 5 t t t t 3 t 4 t 5 t t t t 3 t 4 t 5 Figure 3 Path-Depedet Simulatio ( scearios - Oe-Factor Logormal Model 35 A. Path-Depedet Simulatio 35 B. Direct-Jump to Simulatio Date 3 3 Sto c Pri ce (U SD 5 5 Sto c Pri ce (US D Simulatio Time (Yrs Simulatio Time (Yrs Meawhile, Figure 3A shows a 3-sceario path-depedet simulatio ad Figure 3B the DJS radom simulatio of a stoc price usig a logormal evolutio model. The results of simulatio test o a logormal evolutio model will show that the price factor distributio is almost idistiguishable betwee PDS ad DJS. 4

5 Sice cotiuous-time models are used for the maret factor evolutio (rather tha simple discretizatio, the maret factor distributio at a give simulatio date usig either PDS or DJS will be idistiguishable i the limit of large umber of samples. Sice we will be usig coditioal valuatio methods i this paper, what matters is the distributio of the maret factor scearios at a simulatio date rather tha the path it too to get there. 3. The Browia Bridge The cocept of coditioal expectatio ca be illustrated by usig the example of a Browia bridge. The Browia bridge is a set of Browia paths W(t that start from oe poit (i.e., the origi at time ad ed at aother poit WT ( = wpre-specified at a future time T > as illustrated below: Figure 4 Graphical illustratio of a Browia bridge 8 6 W(T = w t < T T The questio of iterest is to determie the distributio of W(t at ay time t (, T coditioal o owig the ed poit W(T = w! Mathematically, the desity of a Browia bridge 7 ca be explicitly derived such that ( = ( = = exp ( π t( t/ T (3. f ( W t x W T w ( x wt/ T t( t/ T where W(t is a stadard Browia motio 8. Compared to stadard Browia motio, the Browia Bridge possesses some uique properties. I particular, the coditioal mea ad variace of the Browia Bridge are give by (3. Wt WT = ( ad E ( ( ttwt / ( ( Var Wt ( WT ( = t tt / (3.3 E[ exp{ Wt ( } WT ( ] = exp { t( tt / + ( ttwt / ( } 7 Browia Bridge ad its desity ca be easily exteded to the case of multi-dimesioal Browia process. 8 We will use the otatio Wt ( to deote Browia motio uder the real (historical measure ad Wt ( for riseutral measure. 5

6 { } { Wt} WT t( tt ( ttwt ( Var exp ( ( = exp / + / for the geometric Browia bridge. For exposure calculatio, the coditioal valuatio is a probabilistic techique to adjust the valuatio models for istrumets whose value at the future date may deped o the sceario before that time. Such approach fits aturally i the value-at-future exposure framewor, where oe ca separate istrumet valuatio completely from the maret ris factor sceario geeratio. Furthermore, this approach provides a cosistet treatmet across differet types of istrumets, which the eable us to aggregate the exposures betwee the istrumets of differet types, such as swaps ad swaptios. I the followig sectios, we will demostrate the powerful techiques of coditioal valuatio approach i the exposure calculatio for several well-ow istrumet types with the path-depedet features. 4. Formulatio of Coditioal Valuatio We agai emphasize that the eed for coditioal valuatio stems from the fact that we ca oly simulate future scearios at discrete time itervals because of limited computer resources. However, the value of a derivative product at ay of these dates may deped o the full path over the cotiuum of dates prior to the simulatio date. We provide i this sectio the formulatio of the coditioal valuatio approach for calculatig credit exposures that are cosistet across all derivative products, path-depedet or ot. For simplicity of expositio, we will assume i this paper that the direct-jump to simulatio date (DJS approach as explaied i sectio 3 is used to geerate maret scearios. The exposure calculatio is performed o a discrete set of the future times, which are called i this paper as the simulatio dates. As described i sectio, there are two steps i the calculatio of future exposure of a derivative istrumet. First, the future scearios of maret factors must be geerated uder the real measure (as opposed to the ris-eutral measure o the simulatio dates. Secod, oe eeds to adjust the real measure to a ris-eutral measure whe applyig the valuatio fuctios to calculate the value of derivative cotract for each of the maret scearios o the simulatio date. For this purpose, we first itroduce the otatios used to describe the evolutio of maret ris factor ad exposure calculatio: Discrete simulatio dates: { t = t, t, K, t } Maret ris factor scearios: X ( t = X ( t, X ( t, K, X ( t N { N } { N } The future values of the trasactio: V ( t = V ( t, V( t, K, V( t Sice the credit exposure of a derivative cotract at a future time depeds o the expected value of the cotract give the sceario of uderlyig maret ris factors at that time, we eed a valuatio methodology that ca calculate the future value of derivative cotract which may be cotiget o the scearios of uderlyig maret ris factors betwee today ad the future time.. The scearios ca be geerated from the Mote-Carlo simulatio of a ris factor evolutio model i two differet ways: path-depedet scearios ad direct-jump scearios, as described i sectio. There are may situatios where the future value of a give trasactio is ot uiquely determied by the state of uderlyig ris factors at the simulatio date. For example, we cosider a swap-settled swaptio (or a Bermuda i a geeral case where the future value of such trasactio ca be ambiguous o the simulatio date past the expiry date of optio, because we could either have a swap as the result of optio exercised or othig if the swaptio expires worthless. Other examples iclude barrier (i.e., oc-i or oc-out ad average optios, where the payoff is truly path-depedet i the sese that the future values of such optios at the simulatio date deped o the etire history of uderlyig maret factor. For path-depedet istrumets, their values at the future simulatio date may thus deped o either the evet occurrig at a time before the simulatio date or the etire sceario path leadig to the simulatio date. Hece, the valuatio at the future date ca be formulated as the coditioal expectatio ( < t t (4. ( { } V t, x = E f t, X( t X( t = x 6

7 where the coditioig is o the state of maret ris factor X ( t = x, for a particular sceario at < deotes the etire path of ris factor evolutio. Whe a istrumet is ot path-depedet, { X ( t: t t } t ad such as i the case of swap, forward ad cash-settled optio, the coditioal expectatio i (4. above simply degeerates to (4. V ( t, x = f ( t, X( t = x which is just a simple MtM valuatio at the simulatio date t. I this paper, we refer to this type of valuatio as the coditioal value-at-future or simply value-at-future (VaF as described i sectio. However, VaF is ot the MtM valuatio at the future simulatio date. To illustrate the differece, we cosider two special cases i the formulatio (4. where the valuatio fuctio is separable i the followig sese: (4.3 { } (, ( = (, ( ({ ( } < t t < t t f t Xt gt Xt h Xt (4.4 f ( t,{ Xt ( } < t t = gt (, Xt ( + h( { Xt ( } < t t I each of two cases, we ca respectively rewrite the coditioal expectatio explicitly { } { } (4.5 ( ( ( < (4.6 ( ( ( < V t, x = g t, x E h { X( t} X( t = x t t V t, x = g t, x + E h { X( t} X( t = x t t where g( t, x is the mar-to-maret valuatio of such trasactio at the simulatio date. As show later i this paper, the barrier optio is a example of the case (4.3 where (4.7 h( { X( t} < t t = I{ X ( t < H : t< t } is the idicator fuctio of breachig the up barrier. The average optio is a example of the case (4.4 where (4.8 h ({ X ( t t } < t t = X ( t t i t t is the average of the sceario history leadig to t. Fially, the formulatio of coditioal valuatio i (4. provides the cosistecy for trasactios with ad without path-depedece, ad thus maes it possible for ettig ad aggregatio across multiple ris factors. Such approach is relatively easy to implemet, as it is feasible i may cases to explicitly compute the coditioal expectatio for may istrumets such as the barrier optio, average optio, swaptio ad variace swap. 5. Barrier Optio The barrier optios are typical examples of path-depedet optios where their payout at maturity is determied by the etire history of the uderlyig asset prices. They are ofte embedded i the iterest rate products such as the ocout swap, ocout cap ad floor, where the swaps, caps or floors will cease to exist if the forward rate rises or falls below a pre-specified level (i.e., the barrier. The pricig of barrier optios is well ow, ad their (ris-eutral values are give by (5. MtM barrier E t[max{, ST ( K} I{ S (, }], for up-out Call Max t T < H ( t = E t[max{, K S( T} I{ S (, }], for dow-out Put Mi t T > L 7

8 where SMax (, t T = max{ S( τ, t < τ T} ad SMi (, t T = mi{ S( τ, t < τ T}, t E deotes the expectatio uder the riseutral measure, H = the up barrier ad L = the dow barrier. If the maret ris factor S(t is assumed to follow a (ris-eutral logormal process, (5. St ( = S exp σ t+ σwt ( the the aalytic solutios such as Blac-Scholes type formula to (5. ca be foud i Haug [4] ad Hull [5]. However, our iterest here is to calculate the exposures of these istrumets at some future simulatio date. For exposure calculatio, we first specify the evolutio of ris factor i the actual measure (5.3 ( St ( = S exp µ σ t+ σwt ( which cotais a drift ad the volatility that are differet from the ris-eutral process i (5.. The drift, which represets the ris premium for the future ucertaity, ca be calibrated to the historical time series. For a fixed sceario St ( = x, we illustrate four types of paths i the figure show below: Path has ot passed through the fixed sceario at the simulatio date Path ever breached the barrier before the maturity, Path 3 breached the barrier before the simulatio date, Path 4 breached the barrier before the maturity but after the simulatio date. Figure 5 Differet sceario paths for crossig the barrier Stoc Price Barrier = H S (t = x Path Path Path 3 S Path 4 t= (today t (simulatio date T (maturity date Thus, the third path will affect the calculatio of exposure at the simulatio date sice the crossig of barrier before the simulatio date will determie the existece or extictio of uderlyig optio. Sice the digital optios are simple case of barrier optios, we describe oly the coditioal valuatio of barrier optios, particularly the up-out ad dow-out call optios. The value-at-future calculatio of such barrier optios are give by the coditioal expectatio (5.4 VaF uo ( t ; x = E max{, ST ( K} I{ S (, } (, ( Max t T < H ISMax t < H St = x E max{, ST ( K} I E I St ( = x = { SMax ( t, T < H} { SMax (, t < H} = ( ; Prob (, ( MtM uo t x SMax t < H S t = x 8

9 for the up-out barrier optios, ad similarly we have (5.5 VaF do ( t = E t max{, ST ( K} I{ S (, } { (, } ( Max t T > L I SMax t < L St = x E max{, ST ( K} I E I St ( = x = { SMax ( t, T > L} { SMax (, t > H} MtM do t x SMax t > L S t = x for the dow-out barrier optios. Thus, the VaF calculatio is simply the mar-to-maret value at the fixed sceario St ( = xmultiplied by the coditioal probability of crossig the barrier, which ca be computed usig the so-called reflectio priciple of Browia path as described i Karatzas ad Shreve [6] such that = ( ; Prob (, ( Max, (5.6 Prob S (, t < H S( t = x = exp h ( x h ad Max, (5.7 Prob S (, t > L S( t = x = exp l ( x l where x = l Sx ( / S / ( σ t, h = log ( H / S / ( σ t ad l log ( L/ S / ( σ t =, ote the above probabilities are well defied sice l x h. Fially, we cosider a example of a up-out barrier optio with oe-year maturity ad the barrier level at % above the ATM strie. Usig (5.4, we compute the exposure profiles at 95%, 5% ad 5% cofidece levels respectively o differet simulatio dates over a oe year period as show i the figure below. Note that the exposure profiles of the barrier optio are quite differet compared to that of a stadard optio (see figure below. The exposure profile of a up-out barrier optio exhibits a sharp covexity i cotrast to the cocave profile of a vailla optio. The peay shape of exposure profile o a up-out barrier optio is attributed to the egative covexity that the optio becomes more liely to be oced out as the uderlyig price reaches close to the barrier level. Figure 6 Exposure Profile for Up-Out Barrier at % Cofidece Vailla Optio (-Year At-the-Moey Barrier = % of Spot Price 3 6 Exposure (' USD 8 4 Exposure (' USD Simulatio Time (Years Simulatio Time (Years 95th Percetile 5th Pctile 5th Pctile 95th Percetile 5th Pctile 5th Pctile 9

10 6. Asia Optio A Asia optio is a optio o the average of uderlyig prices or iterest rates tae at certai frequecy (such daily or weely from the start date to the maturity date. There are geerally two types of averagig methods: arithmetic average ad geometric average. The arithmetic average is defied as (6. A = St ( / i= while the geometric average is defied as i (6. G i= / i = S( t where St ( i is the price of uderlyig at reset time t i. The price of uderlyig asset is assumed to follow a logormal process (6.3 ( λ St ( = S exp µ σ t+ σwt ( where σ is the volatility of uderlyig stoc, µ λ = r d + λσ is the ris-adjusted drift for a specified maret price of ris λ. The geometric average, as a product of uderlyig prices, is logormal sice the uderlyig price is logormal at each reset time. However, the arithmetic average as the sum of logormal prices will ot be logormal i geeral. Noetheless, to hadle the Asia feature, we approximate the arithmetic average with a logormal distributio with mea ad variace chose to match the actual mea ad variace. Whe average reset frequecy is high such as daily, the logormal volatility of arithmetic average ca be approximated by socalled the 3 -rule. This approximatio rule is derived i the followig: (6.4 Var[ A ] = Var [ St ( + ( St ( + L + St ( ] ( + Λ + + = ( T t σ = ( T t 3 σ ( ( Aualizig by ( T t ad taig the limit, we obtai Var A /( T t σ / 3. I geeral, the mea ad variace of arithmetic average ca be calculated i the followig (6.5 E [ A ] = F ( t, Var[ A ] E E[ ] = A A i = where F t S exp{ ( r d t} i ( = is the forward price at today for delivery at the future date t. Furthermore, we defie the variace of retur o arithmetic average (6.6 Var( A σ = + A T ( E[ A ] The optio o average price ca geerally be classified ito two differet types: Average Price Optio ad Average Strie Optio. The payoff o average price optio is defied as max, { φ ( A K } while the payoff o average-strie optio { φ ( ST A } max, ( 6

11 That is, average-price optio is a optio strie o a fixed price ad average-strie price is a optio strie o the average price. For valuatio of a Europea optio o arithmetic average, we usually approximate by a Blac-Scholes fuctio assumig a logormal average price: { } (6.7 φ ( where M E[ A ] MtM ( t = E max, A K = M N( d K N( d = ad A d σ as calculated i (6.6 l( M / K +.5 A T =, d = d σ A T σ T A σ At ay future simulatio date t > t, we deote (6.8 St ( j j= At (, t = t Stdt ( t t respectively as the discrete or cotiuous average betwee t ad t, similarly for A( t, T as the average betwee t ad T. The the value-at-future of average optio is give by the coditioal expectatio (6.9 ( = { φ ( } VaF t; S( t E max, A( t, T K S( t = Emax, φ A t, t + A t, T K W t ( ( ( where φ =+ for a Call, φ = for a Put, ad the expectatio is tae coditioal o the price of uderlyig stoc St ( at simulatio time. Sice A( t, T is idepedet of Wt (, the coditioal expectatio i (6.9 ca be rewritte as (6. VaF ( t ; S( t = E E max, φ A( t, T K A( t, t W ( t There is o aalytic solutio for the coditioal expectatio, i geeral, for arithmetic average optio. However, we ca fid a good semi-aalytic approximatio by assumig a logormal distributio for the averages: (, ; = exp { (, + (, } (, ; = exp { (, + (, } A t t z M t t V t t z A t T x M t T V t T x where M is the mea, V is the variace of the average, x ad z ~ N(, with (6. X, Z ( t Corr l A(, t,l A( t, T ρ = = The mea ad variace ca be computed oce their st ad d momets are ow (6. A= l( M l( M, V = l ( M l ( M For two momets over time iterval [ t, T ], we ca compute i straightforward fashio

12 St ( (6.3 M ( t, T; x = exp[ µ j t] (6.4 M ( t T x j= ( µ σ j iv j µ ( j+ i σiv ( j+ 3i St ( +, ; = exp exp + + j= i= However, the calculatio of the two momets over the time iterval [ t, t ] is quite complex, as the repeated use of the variace formula (3.3 of the Browia Bridge is required (6.5 ( M t, t ; z E ( j ( E ( j ( S t S t = = j S t S t + = + j= ad S = µ t µ t σ h j t π exp z ψ z σh j= σh t (6.6 ( M t, t; z = E S( t ( j S t ( + j= j = E St ( ( E ( ( ( j St + Stj Sti St ( + j= i= S = ( + {( } h t z h t h t + π exp ( µ +.5 σ + σ /( σ ( µ +.5 σh t + zσh t σh j t ψ σ h t j= j µ t + zσh t σh ( j+ i t + exp{ i tσh } ψ i= σ h t Fially, we apply the Blac-Scholes optio formula to the ucoditioal expectatio i (6. (6.7 ad VaF ( t; S( t = ψ ( z dz BS At (, T; z, K( z, T t, ( ρ ( t V( t, T; x, φ + φ At (, t; z + At (, T; z K ψ( zdz K K (6.8 A( t, t; z + A( t, T; x K ψ ( z dz K { } KΦ( K + exp M( t, t; z + V( t, t ; z

13 = Φ( K + Vt (, t; z + Φ( K + ρ( t Vt (, t; z where ψ is the stadard ormal desity fuctio, Φ is the cumulative distributio fuctio ad BS [ X, K, T, σ, φ ] is the stadard Blac-Scholes optio pricig fuctio, [ ] K = l( K N/ M( t, t / V( t, t { } N Kt (, Z = K exp Mt (, t + Vt (, t Z N N { ( ρ ρ } A( t, TZ ; = exp Mt (, T + Vt (, T ( t + Vt (, T Z ( t The above valuatio eeds to perform a itegratio of the Blac-Scholes fuctio with a stadard ormal desity fuctio, which ca be calculated usig a simple umerical itegratio. As a example, we compute the exposure profiles o a.5-year average price optio with weely averagig frequecy ad a fixed strie set equal to the spot price. Compared with a stadard ATM optio, the average optio has a lower pea exposure ad the exposure profile of such optio exhibits a humped shape i the risig price scearios as show i the figure below. Figure 7 Exposure Profile for Asia Optio vs. Stadard Optio At-the-Moey Vailla Optio Average Rate Optio 6 8 Exposure (' USD 8 4 Exposure (' USD Simulatio Time (Years Simulatio Time (Years 95th Percetile 5th Pctile 5th Pctile 95th Percetile 5th Pctile 5th Pctile 7. Swap-Settled Swaptio A iterest rate swaptio is a optio to eter a swap. Depedig o the swap beig a payer or receiver, such optio is usually called right-to-pay (RTP or right-to-receive (RTR swaptio. I cotrast to regular swaptios where the optio will be cash-settled at the expiratio date, a swap-settled swaptio will settle ito its uderlyig swap if the optio is exercised at the expiratio of the optio. For credit exposure, this differece is very crucial as the future exposure o a cash-settled swaptio stops right before the expiry while the future exposure o a swap-settled swaptio ca potetially cotiue well beyod the expiry of the optio ito the remaiig life of uderlyig swap. The payout at optio expiratio date is defied as (7. MtM ( T = max{, Swap( T } where swpt e e 3

14 N e i i e i i=. (7. Swap( T = b d [ F( T, T K ] To price a swaptio (both cash-settled ad swap-settled, Blac s formula is usually applied to the forward swap rate of the uderlyig swap i practice, i.e., N (7.3 MtM swpt ( t = bi di ES { max[, S( Te K ]} i= N = bd i i BS ( S, K, Te t, σ S, Type i= where N is the umber of swap periods, bi is the day-cout fractio ad di is the discout factor. Here, σ S is the implied volatility ad E S deotes the expectatio uder the forward swap measure as explaied i Hull [5], such that E S[ ST ( e ] = S at expiry time T e ad the swap rate St ( follows a logormal process uder the forward measure (7.4 St ( = S exp σ ( St+ σswt. For exposure calculatio, we choose the swap rate as the ris factor ad model its evolutio uder the actual measure as (7.5 ( µ St ( = S exp σ t+ σwt ( for some drift µ ad volatility σ that are calibrated to the historical time series. For ay future time before the expiryt e, the exposure is simply give by the valuatio formula (7.3. For swap-settled swaptios, ote that whe the simulatio date t is past the expiry T e, we face a ambivalet situatio whether or ot to calculate the exposure o a uderlyig swap sice we are ot sure if the swaptio was exercised earlier. To illustrate this, cosider two paths leadig to the fixed sceario as i Figure 8. Path (solid lie implies a optio exercise ito the uderlyig swap sice the swap rate at expiry Te is above the strie rate. However, Path (dotted lie implies that the optio expires worthless. Therefore, the calculatio of the exposure at time t > T e should iclude the probability of optio exercise. Figure 8 Optio Expiries i a Swap-Settled Swaptio S (t Swap Rate Strie Rate=K Exercised Path S Path t (Today T e (Expiry Date t (Sim. Date T (Maturity Date 4

15 Thus, the value-at-future of such swaptio is give by the coditioal expectatio N i= (7.6 VaF swpt ( t = bd i i ES { Max[, S( Te K] S( t } N bd i i St ( K Prob ST ( e > K St ( i= = [ ] { } for the future time t > T e. The coditioal probability ca be computed by applyig the Browia Bridge to the swap rate evolutio up to the fixed sceario St (. The ris factor evolutio after the simulatio date t eeds to be adjusted bac to the ris-eutral process (7.4 for valuatio. For the fixed swap rate sceario, the swap rate at expiryt e ca be expressed i term of a Browia bridge such that (7.7 ( ( exp ( µ σ ( St = STe t Te + z σ t T e uder the actual measure. This eables us to compute the coditioal probability of optio exercise at the expiratio time Prob ST ( e K St ( N z ( Te, t * (7.8 { > } = where { [ S K] ( t Te [ S t S] } l / / l ( / * z ( Te, t = σ t ( t T e, usig the property of Browia bridge (3.3. Hece, the value-at-future after the expiratio date is the product of the MtM value of remaiig swap ad the coditioal probability of optio exercise at expiry. As a example, we cosider both the cash-settled ad swap-settled RTP swaptio stried at the moey with - year optio to eter ito a 5-year payer swap o USD m otioal. The figures below compare the exposure profiles betwee these two swaptios. The cash-settled swaptio reaches the pea exposure as expected at the optio expiry date ad the exposure does ot exted beyod the expiratio date, while the exposure of a swapsettled swaptio goes beyod the optio expiry ad exteds to the fial maturity date of uderlyig swap ad it reaches the pea exposure after the expiry date approximately at 3 rd of uderlyig swap life. Figure 9 Exposure Profiles of Cash-Settled ad Swap-Settled Swaptios, Cash-Settled Swaptio, Swap-Settled Swaptio 8, 8, Exposure (' USD 6, 4,, - (, Simulatio Time (Years Exposure (' USD 6, 4,, (, Simulatio Time (Years 95th Percetile 5th Pctile 5th Pctile 95th Percetile 5th Pctile 5th Pctile 8. Equity Variace Swaps Variace swaps are volatility cotract that pays off the realized variace o either the Equity idex or sigle stoc agaist a pre-specified strie. By defiitio, the payoff from variace swap at maturity is equal to 5

16 (8. ( σ Payoff = R K N where N is the otioal (i volatility uits, K is the strie, (8. Pi + 5 σ R = l i= Pi σ R is the realized variace, defied as ad where P i is the closig price for uderlyig idex or stoc o the i th day ito the swap, is the umber of days i the swap. Suppose that the swap started at time T S ( < with maturity T M ( >. The, the MtM value of the swap today is give by (8.3 5 Pi MtM ( + = l + + impl K N it : i Pi σ + + < + + where ti TS, TM is the ith -tradig day, is the umber of tradig days passed (history, + is the umber of tradig days left i the swap, icludig today, σ impl is the arithmetic average of implied volatilities for at-themoey calls ad puts with maturity T M. Without loss of geerality, we igore the effect of discoutig i (8.3 above. First we defie daily variace as (8.4 vt ( i ( P P l i / = t i i where ti = ti ti is expressed as the fractio of the year. I our model we assume cotiuous evolutio of vt (, the istataeous variace rate. Notably, it is the volatility of vt ( that the value of the swap deviates from at its iceptio ad therefore justifyig the existece of variace swaps. From historical data, the quatity vt ( i is mea revertig i the log-term with small set of outliers. Nevertheless, due to a short maturity of most variace swaps (teor about a year o average as well as the averagig ature of the payoff formula, we model vt ( as a logormal process with zero drift ad without jumps (8.5 dv( t σ v( t dw ( t vt = v Wt t = or ( ( exp σ ( σ where Wt ( is stadard Browia motio ad σ is volatility of vt (. For a give MTM-value of the trade v = K + MtM(/ N. today, oe ca bac out ( Next, we describe the value-at-future calculatio of the variace swap at future date t > ( (8.6 ( ( ad (8.7 v( t VaF t = v t K N t ( TS v( + vt ( ( TM t + E vtdt ( W( t = T T M S 6

17 which cosists of the cumulative variace over three time itervals: the period before today [ T S,], the period from simulatio date to maturity [ t, TM] ad the period from today to simulatio date [, t ]. The last oe is a coditioal expectatio that depeds o the Browia Bridge scearios from today leadig up to the realizatio of Wt (. We ca compute the coditioal expectatio by chagig the order of itegratio ad expectatio (8.8 t t E vtdtw ( ( t = E vt ( W( t dt ad applyig the variace formula of a Browia Bridge (3.3 (8.9 E vt ( W( t E v(exp[ σwt ( σ t] Wt ( = Taig the itegrals leads to t π t (8. E vt ( Z dt= v( exp( Z N( Z N( Z t σ where N ( σ x u / x = e du, ad Z π ( W t = ~N(,. t σwt ( t σ t = v( exp t t Substitutig bac ito the formula for v( t, we obtai the MtF value of cumulative variace (8. v( t ( T exp( σz t σ t ( T t S + M + v( = T M T π t S exp( Z { N ( Z N ( Z σ t } σ The graph below shows the exposure profiles o a oe-year variace swap with 3, variace uits of the NASDAQ idex at three scearios: risig, fallig, ad uchaged volatility. Compared to exposure profiles o iterest rate swaps or optios, the exposure of a variace swap does t amortize to zero at maturity, which is typical for a iterest rate swap, rather it teds to exhibit a humped profile (relative to forward or optio due to the averagig ature of variace over the time to maturity. Figure Exposure Profile for Variace Swap. Exposure (USD,, 8, 4, - (4, (8, Simulatio Time (Years Fallig Vol (5th Percetile Risig Vol (95th Percetile Media (5th Percetile 7

18 9. Coclusio The accurate calculatio of exposure is a essetial compoet i maagig credit ris with the tradig couterparties. I this paper, we have preseted a techique usig coditioal expectatio valuatio to improve the accuracy of simulated exposures for path-depedet products. Sice a typical couterparty portfolio geerally cosists of may types of istrumets as well as a variety of maret factors affectig the values of these istrumets, simple formula approaches (such as add-o factors used to calculate couterparty exposure are mostly iadequate. Thus the simulatio approach as described i Duffie ad Caabarro [3] is usually employed to calculate exposure across the future time horizos, usually up to the logest maturity i the portfolio. I the simulatio approach, each trasactio is revalued at future times usig simulated future scearios of maret factors. However, simulatio ad revaluatio cosumes so much computer resources that certai simplificatios are ecessary i order for a ba to have its daily exposure report delivered to its users i a reasoable amout of time. Oe simplificatio that is metioed i this paper is to have discrete sets of simulatio dates, usually with icreasig time itervals for the loger maturities. However, this simplificatio creates some difficulties i valuig a path-depedet istrumet at a future simulatio date sice we do ot have the cotiuum of the evolutio of maret factors up to that date. Examples are give i this paper of pathdepedet istrumets that deped o the history of a maret factor leadig up to its value at a give simulatio date. Frot-office pricig models are clearly iadequate to calculate the value-at-future of a path-depedet istrumet sice these models assume that o previous cotiget evet has tae place prior to the valuatio date. I this paper we have preseted a methodology to accout for the possibilities of particular prior evets that may affect the exposure, coditioal o the simulated value of the relevat maret factor at a give simulatio date. Furthermore, usig the properties of the Browia Bridge, we have derived aalytic expressios to calculate the exposure or value-at-future o a umber of path-depedet istrumets such barrier optios, average optios, variace swaps, ad swap-settled swaptios.. Refereces [] Basel Committee, The Applicatio of Basel II to Tradig Activities ad the Treatmet of Double Default Effects, Ba for Iteratioal Settlemets, April 5. [] Dembo, Ro, A. Aziz, D. Rose ad M. Zerbs, Mar-to-Future: A Framewor for Measurig Ris ad Reward, Algorithmics Publicatio,. [3] Duffie, Darrell, E. Caabarro, Measurig ad Marig Couterparty Ris, Chapter 9 i ALM of Fiacial Istitutios, edited by Leo Tilma, Istitutioal Ivestor Boos, 4. [4] Haug, Espe. G. The Complete Guide to Optio Pricig Formula, McGraw-Hill, New Yor, 997. [5] Hull, Joh, Optios, Futures ad Other Derivative Securities, Fifth Editio, Pretice-Hall, 3. [6] Karatzas, Ioais, S.E. Shreve, Browia Motio ad Stochastic Calculus, Spriger-Verlag, 99. [7] Reboato, Riccardo, Review of Term Structure Models, RBS QUARC ad Oxford Uiversity, February 3. 8