Numerical Mehods for European Opion Pricing wih BSDEs by Ming Min A Thesis Submied o he Faculy of he WORCESTER POLYTECHNIC INSTITUTE In parial fulfillmen of he requiremens for he Degree of Maser of Science in Financial Mahemaics May 2018 APPROVED: Professor Sephan Surm, Major Advisor Professor Luca Capogna, Head of Deparmen
Absrac This paper aims o calculae he all-inclusive European opion price based on XVA model numerically. For European ype opions, he XVA can be calculaed as soluion of a BSDE wih a specific driver funcion. We use he FT scheme o find a linear approximaion of he nonlinear BSDE and hen use linear regression Mone Carlo mehod o calculae he opion price.
Acknowledgemens I would like o express my graiude o my advisor, Professor Sephan Surm, who has given me a lo of help boh on my maser projec and my PhD applicaion. I need o appreciae my parens for supporing me hrough my educaion in WPI, boh financially and menally. I wan o hank my bes friends, Shan Jiang, Hanzhao Wang and Zhenyu Qiu. Your friendship is really imporan o me. i
Conens 1 Inroducion 1 2 Models 3 2.1 Marke seup and noaions....................... 3 2.1.1 Socks securiy.......................... 3 2.1.2 Risky bonds securiies...................... 5 2.1.3 Funding accoun......................... 5 2.1.4 Collaeral process and collaeral accoun............ 6 2.2 Replicaion of opions........................... 7 2.2.1 Risk neural measure....................... 7 2.2.2 Replicaion of opions and collaeral specificaion....... 8 2.3 XVA model and driver funcions..................... 10 2.3.1 XVA models............................ 10 2.3.2 Drivers............................... 13 3 Numerical mehods 17 3.1 FT scheme................................. 18 3.2 Linear regression Mone Carlo mehod................. 20 3.3 Pricing algorihm............................. 21 ii
4 Example 24 4.1 Resuls................................... 25 4.2 Check sabiliy.............................. 27 5 Conclusion 30 References 31 iii
Lis of Figures 3.1 Regression-Based Mone Carlo Algorihm................ 22 4.1 XVA adjusmens wih α = 0.7, 0.8, 0.9................. 26 4.2 Relaive XVA adjusmens wih α = 0.7, 0.8, 0.9............ 27 4.3 XVA adjusmens wih r f = 0.07, 0.08, 0.09............... 28 4.4 Relaive XVA adjusmens (%) wih r f = 0.07, 0.08, 0.09....... 28 4.5 All XVA adjusmens, he verical line is our resul, he hisogram shows all resuls from boosrapping.................. 29 iv
Chaper 1 Inroducion The Black-Scholes model for opion pricing assumes ha no paricipan will defaul. Bu defauls do happen in he real world. They maybe forced o close ou heir posiions as hey defaul, hence he rader should consider hose probabiliies of addiional coss when developing his porfolio. The Black-Scholes model assumes he same shor rae r for he borrowing and lending raes, bu hese raes are differen in realiy. Anoher limiaion a Black-Scholes model is ha we canno shor socks or oher asses as freely in realiy as he Black-Scholes model suggesed. This hesis follows he (Bichuch, Capponi, & Surm, 2016) marke seup, bu we use a much simpler version. The rader ges his funding from his reasury desk and mus pay back he money. The borrowing rae depends on his own credi level and curren marke condiions; i is usually higher han his lending rae. The difference beween hese wo raes is called funding spread, which is he funding cos needed o be considered in our model. As in (Bichuch e al., 2016) and (Burgard & Kjaer, 2011), wo corporae bonds are inroduced in order o hedge he credi defaul risk from boh rader and his counerpary. Using he repo marke mechanism, we can shor socks in his marke. Usually here is a difference beween borrowing and 1
lending raes, bu in his paper we assume hey are he same for simplificaion. (Bichuch e al., 2016) hen generae a Backward Sochasic Differenial Equaion, BSDE in shor, for he opion pricing via XVA model. Once we ge he BSDEs, we can solve i numerically. This hesis uses nonlinear Mone Carlo mehods o solve i. (Crépey & Nguyen, 2016) used a perurbaion mehod, following (Fujii & Takahashi, 2012a, 2012b), o find a linear approximaion of he soluion, and solve he BSDE by leing he perurbaion parameer equal o 1. We expanded his mehod, which is called FT scheme, a lile bi o solve our problem. Since our driver is pah dependen, linear regression Mone Carlo mehod is also used. (Glasserman, 2013) uses linear regression Mone Carlo mehod for American opion pricing problems. Bu in our problem, insead of looking one sep forward, we need o remember everyhing in all he fuure ime since we will fuure value o define driver funcion a each ime. In his hesis, chaper 2 focuses on mahemaical BSDE models of European call and pu opions, and derives he drivers for boh opions. Chaper 3 presens he numerical mehod used o solve previous BSDEs. Chaper 4 uses he numerical algorihm developed in Chaper 3, and XVAs under differen collaeral levels are compared. Chaper 5 concludes. The codes are included in he Appendix. 2
Chaper 2 Models A probabiliy space, (Ω, G, P), is used o describe he physical world. We refer o he invesor or rader as I, and counerpary o invesor as C. The background filraion F := (F ) 0, augmened by (G, P)-nullses, includes all he informaion of he marke excep for defauls. The filraion H := (H ) 0 has all he informaion abou defaul evens. The filraion G := (G ) 0 is given by leing G := F H, augmened by (G, P)-nullses. 2.1 Marke seup and noaions 2.1.1 Socks securiy Le F := (F ) 0 be he filraion generaed by Brownian moion W P, where P is he physical measure. Then he dynamics of sock price is given by ds S = µd + σdw P, (2.1) 3
where he µ, σ are appreciaion rae and volailiy as common respecively, assumed o be consan in our model. In realiy, we canno shor sock freely. Shoring is conduced hrough he securiy repo marke. In he (Bichuch e al., 2016) and (Adrian, Begalle, Copeland, & Marin, 2013) marke seup, wo ypes of repo ransacion are considered. The firs one is called securiy driven ransacion. This ransacion is used o circumven he prohibiion of he rader from selling a sock which he or she doesn have, also called naked shor sales of socks. I works as follows: he rader signs a repo conrac wih some paricipan in he repo marke. The rader lends some money o he paricipan, which is used o buy socks and pos hem as collaeral o he rader. Thus he rader can sell socks and mus reurn socks o paricipans in exchange of a pre-specified amoun of money, which is usually higher han he lending amoun. So implicily, here is a reurn rae on rader, called r r +. The second ype of repo ransacion is called cash driven ransacion, which is exacly he oher side in his repo marke. When he rader wans o have a long posiion in socks, he borrows money from he reasury desk and uses hem o buy socks which are posed as collaeral for a loan a he repo marke. The rader agrees o purchase hose collaeral back a a pre-specified price, which is usually slighly higher han he original price of collaeral. So here is a cos rae, named as rr. In his paper, we assume r r + and rr are he same, denoe as r r. The relaion beween repo marke accoun and he socks is given by ψ r B rr = ξ S, (2.2) where B rr is he repo marke accoun, ξ is he number of shares in securiy accoun. This ideniy sems from he fac ha sock is only bough and sold via repo marke. 4
2.1.2 Risky bonds securiies Two risky bonds wrien by he rader and he counerpary are inroduced. Denoe heir defauling imes as τ i, where i {I, C}, as rader and counerpary defauling ime respecively. We suppose he τ i s are following an exponenial disribuion wih inensiy h P i, i {I, C}, and are independen of F and each oher. H i () = 1, 0, is he defaul indicaor process. So he defaul evens filraion is given as H = (H ) 0, H = σ(h I (u), H C (u); u ). In paricular, his implies F Brownian moion W P is also a G Brownian moion. Assume hese wo bonds are zero recovery, and boh expires a ime T. Denoe he bond price wrien by rader as P I, denoe he bond price wrien by counerpary as P C. Accordingly, heir prices are given as dp i = µ i d P i dhi, P i 0 = e µ it (2.3) wih µ i as heir reurn raes. Le τ = τ I τ C T denoe he earlies sopping ime of mauriy ime T, rader defaul ime τ I and counerpary defaul ime τ C. 2.1.3 Funding accoun As menioned before, he rader receives or provides funding o his reasury desk wih differen raes. Usually he borrowing rae is higher han lending rae. We denoe r + f as he lending rae, r f as he borrowing rae. So he money marke accoun has he dynamics db r± f = r ± f Br± f d, (2.4) 5
where B r± f denoes he funding accoun. Le ξ f be he number of shares in funding accoun, and define B r f := B r f (ξ f ) = e 0 r f (ξs f )ds, (2.5) where r f := r f (y) = r f 1 {y<0} + r + f 1 {y>0}. (2.6) 2.1.4 Collaeral process and collaeral accoun Collaeral is used o reduce one s loss if he oher pary defaul before expiry. We denoe he collaeral process as C := (C ) 0, which is an F adaped process. If C > 0, we regard he rader as collaeral provider. In his case, he rader measures a posiive risk oward he counerpary, and poss collaeral o he counerpary o reduce counerpary s loss if defaul happens. On he oher hand, if C < 0, he rader is he collaeral aker, who measures a posiive risk oward he counerpary, and akes collaeral o miigae loss if he counerpary defauls. According o (ISDA, 2014), he mos popular ype of collaeral is cash collaeral. When he rader is he collaeral provider, le r + c be he rae on he collaeral amoun he will receive from he counerpary. If he rader is collaeral aker, we le r c be he rae on he collaeral amoun he will pay o his counerpary. In his hesis, we assume r + c = r c = r c. Le B rc be he collaeral accoun, so he dynamics of collaeral cash accoun is given by db rc = r c B rc d. (2.7) Furhermore more, if we le ψ c hen we have be he shares of B rc held by he rader a ime, ψ c B rc = C. (2.8) 6
The inuiion here is ha C is he amoun posed o he oher par by he rader, he collaeral accoun is he cash amoun will be received by rader if no defaul happens before T. So hey have he same amoun bu differen sign. 2.2 Replicaion of opions 2.2.1 Risk neural measure In order o replicae he derivaives, we need o define a risk neural measure. As (Bichuch e al., 2016), we firs inroduce he defaul inensiy model. Given he physical measure P, defaul imes of rader or counerpary are defined as independen exponenially disribued random variables wih consan inensiy h P i, i {I, C}. I holds hen ha for each i {I, C}, ϖ i,p := H i 0 (1 H i u)h P i du (2.9) is a (G, P)-maringale. We defined he discouned rae as r D, which is he discoun rae of valuaion pary used for collaeral and closeou. The risk neural measure Q is given by he Radon-Nikodm densiy dq dp Gτ = e rd µ σ W P τ (r D µ)2 2σ 2 τ ( µ I r D h P I ) H I τ ( e (r D µ I +h P I )τ µc r ) H C D τ e (r D µ C +h P h P C )τ. C (2.10) Under measure Q, he dynamics of our hree risky asses are given by ds = r D S d + σs dw Q, (2.11) dp I = r D P I d P I dϖ I,Q, (2.12) 7
dp C = r D P C d P dϖ C C,Q. (2.13) The W Q := (W Q, 0 τ) is (G, Q)-Brownian moion, and ϖ I,Q := ϖ I,Q, 0 τ as well as ϖ C,Q := ϖ C,Q, 0 τ are (G, Q)-maringales. These hree dynamics can be derived by Io s formula direcly hough (2.1), (2.2) and (2.7), and h Q i = µ i r D, i {I, C}. 2.2.2 Replicaion of opions and collaeral specificaion We focus on European call and pu opion. The Black-Scholes price given by he valuaion agen is used o calculae he closeou value and collaeral. Under he risk neural measure Q, we have ˆV := e r D(T ) E[Φ(S T ) F ], (2.14) where ˆV is he Black-Scholes opion price a ime as calculaed by he valuaion pary. Φ(S T ) is he payoff of European opions, which is given by { (ST K) + Φ(S T ) = European call opion, (K S T ) + European pu opion. When he rader is he pu or call opion seller, he needs o replicae his payoff Φ(S T ). Thus he could build a porfolio o hedge his posiion and use i o pay his counerpary. On he oher hand, when he rader buy one opion, he need o replicae he payoff of Φ(S T ) in order o hedge opion value flucuaion. In addiion, we need o consider collaeral for his opion conrac. We define he collaeral level as α, so under he assumpion ha neiher he rader nor coun- 8
erpary have defauled by ime, he collaeral process is given by C := α ˆV 1 {τ>}, wih 0 α 1. (2.15) The collaeral is allowed o be rehypohecaed by he collaeral aker. This means ha he collaeral aker can use cash collaeral o inves in oher invesmen opporuniies. We define our sraegy process as ϕ := (ξ, ξ f, ξ I, ξ C ; 0), where ξ denoes he shares in securiy accoun, which is he underlying in our case. ξ f denoes he number of shares in funding accoun. ξ I, ξ C denoe he number of shares in rader and counerpary bonds respecively. Combining wih (2.2) and (2.8), he porfolio process is given by V (ϕ) := ξ S + ξ f B r f + ξ I P I + ξ C P C + ψ r B rr ψ c B rc. (2.16) In his paper, we follow he risk-free closeou convenion. I means ha he surviving pary liquidaes all his posiions once someone defauls. We denoe θ as he closeou value a ime τ, where τ is specified is secion (2.1.2). This θ is given by θ := θ(τ, ˆV ) = ˆV τ + 1 {τc <τ I }L C Y 1 {τi <τ C }L I Y + = 1 {τc <τ I }θ I ( ˆV τ ) + 1 {τi <τ C }θ C ( ˆV τ ), (2.17) where Y := ˆV τ C τ = (1 α) ˆV τ is he value of he opion a defaul ime, need wih he collaeral and θ I (v) = v L I ((1 α)v) +, θ C (v) = v + L C ((1 α)v). The L i saisfy 0 L i 1, i {I, C} and he loss raes agains rader and counerpary. This θ is exacly he erminal amoun we wan o replicae, more deails are in (Bichuch e al., 2016) Remark 3.3. 9
2.3 XVA model and driver funcions In his par, we are using he assumpion ha r D = r ± r = r ± c = r + f r f. By (Bichuch e al., 2016) secion 4, his assumpion saisfies rader s non-arbirage condiion. For simpliciy, we use r D o represen r ± r and r ± c, and we sill use r + f in order o make difference wih r f. Bu finally we will change r+ f o r D in drivers funcion. 2.3.1 XVA models According o (Bichuch e al., 2016) secion 4, we can derive he following BSDEs by considering he dynamics of equaion (2.16), and using (2.2) & (2.15), dv + = f + (, V +, Z +, Z I,+, Z C,+ ; ˆV )d Z + dw Q Z I,+ dϖ I,Q Z C,+ dϖ C,Q, (2.18) V + τ = θ I ( ˆV τ )1 {τi <τ C T } + θ C ( ˆV τ )1 {τc <τ I T } + Φ(S T )1 {τ=t }, (2.19) and dv = f (, V, Z, Z I,, Z C, ; ˆV )d Z dw Q Z I, dϖ I,Q Z C, dϖ C,Q, (2.20) V + τ = θ I ( ˆV τ )1 {τi <τ C T } + θ C ( ˆV τ )1 {τc <τ I T } + Φ(S T )1 {τ=t }. (2.21) 10
Noice ha here we only replicae one share of claim. The drivers are given by f + (, v, z, z I, z C ; ˆV ) := (r + f (v + zi + z C α ˆV ) + r f (v + zi + z C α ˆV ) r D z I r D z C + r D α ˆV ), (2.22) and f (, v, z, z I, z C ; ˆV ) := f + (, v, z, z I, z C ; ˆV ). (2.23) V + is he value process of he porfolio which hedges 1 share of opion, V is he value process of porfolio which hedges 1 share of opion. We le Z = ξ σs, Z I = ξ I P I, Z C = ξ C P C. From (2.18-2.21), if we can solve hese BSDEs, hen we have he all-inclusive price of opions. Since here is no Z in wo drivers above, we will omi his parameer in following drivers, and his is because of our assumpion of r D = r r. Le ˆV be he Black Scholes opion price. We can define XVA in our model from (Bichuch e al., 2016) Definiion 4.6. Definiion 1. The seller s XVA is a G-adaped process, which is given by XV A + := V + ˆV, (2.24) and he buyer s XVA is given by XV A := V ˆV. (2.25) By Black-Scholes pricing heorem, he dynamics of ˆV is given by d ˆV = r D ˆV d ẐdW Q (2.26) 11
Then we can derive he BSDEs for XVA, by combining he BSDE for V wih Black Scholes BSDE of ˆV : dxv A ± = f ± (, XV A ±, Z ± dw Q I,± C,± Z, Z ; ˆV ) I,± Z dϖ I,Q C,± Z dϖ C,Q, (2.27) XV A ± τ = θ C ( ˆV τ )1 {τc <τ I T } + θ I ( ˆV τ )1 {τi <τ C T }, (2.28) where Z ± := Z ± Ẑ, ZI,± θ I (v) := L I ((1 α)v) +. = Z I,±, ZC,± = Z C,± and θ C (v) := L C ((1 α)v), The drivers f are given by f + (, xva, z I, z C ; ˆV ) := (r + f (xva + zi + z C α ˆV ) + r f (xva + zi + z C α ˆV ) r D z I r D z C + r D α ˆV ) + r D ˆV, (2.29) f (, xva, z I, z C ; ˆV ) = f + (, xva, z I, z C ; ˆV ). (2.30) Nex, as (Bichuch e al., 2016), we can move one sep forward by using reducion echnique developed by (Crépey & Song, 2015) o generae anoher BSDE, which sops a expiry ime T wih zero erminae value. This is he exacly BSDE and drivers we are gonna use in chaper 3. Theorem 1. The BSDEs dǔ ± = ǧ ± (, Ǔ ±, Ž± ; ˆV )d Ž± dw Q (2.31) Ǔ ± T = 0 12
in he filraion F wih ǧ + (, ǔ, ž ; ˆV ) := h Q I ( θ I ( ˆV ) ǔ)+h Q C ( θ C ( ˆV ) ǔ)+ f + (, ǔ, ž, θ I ( ˆV ) ǔ, θ C ( ˆV ) ǔ; ˆV ), (2.32) ǧ (, ǔ, ž ; ˆV ) := ǧ + (, ǔ, ž ; ˆV ), (2.33) admis unique soluions Ǔ ± such ha Ǔ ± = XV A ± τ. (2.34) On he oher hand, we can ge he XV A soluion from Ǔ by XV A ± := Ǔ ± 1 {<τ} + ( θc ( ˆV τc )1 {τ C <τ I T } + θ I ( ˆV τi )1 {τ I <τ C T } ) 1 { τ} 2.3.2 Drivers We are using he BSDE given by Theorem 1. The drivers are given by (2.32) and (2.33). Firs le s focus on selling one single opion, hus he rader wan o hedge payoff Φ(S T ). The driver we are using is ǧ +. For simpliciy, we define ǧ ± = ǧ ± (, ǔ, ž ; ˆV ). (2.35) When selling an European opion, he opion value will always be posiive. Thus 13
θ C ( ˆV ) = 0 according o our definiion. ǧ + = h Q I ( θ I ( ˆV ) ǔ) h Q Cǔ [r+ f ( ǔ + θ I ( ˆV ) + (1 α) ˆV ) + r f ( ǔ + θ I ( ˆV ) + (1 α) ˆV ) r D ( θ I ( ˆV ) ǔ) + r D ǔ + r D α ˆV ] + r D ˆV = h Q I ( θ I ( ˆV ) ǔ) h Q Cǔ r+ f ( ǔ + θ I ( ˆV ) + (1 α) ˆV ) + + r f ( ǔ + θ I ( ˆV ) + (1 α) ˆV ) + r D ( θ I ( ˆV ) ǔ) r D ǔ + r D (1 α) ˆV. (2.36) Since we wan o ge a simpler version of he driver funcion, we can discuss differen cases for posiive and negaive ( ǔ + θ I ( ˆV ) + (1 α) ˆV ). Then we may cancel some erms and collec erms having ǔ. I s shown as follows, i. If ǔ + θ I ( ˆV ) + (1 α) ˆV 0, hen ǧ + (ǔ) = h Q I ( θ I ( ˆV ) ǔ) h Q Cǔ r Dǔ = h Q I θ I ( ˆV ) (h Q I + hq C + r D)ǔ, (2.37) ii. If ǔ + θ I ( ˆV ) + (1 α) ˆV < 0, hen ǧ + (ǔ) = h Q I ( θ I ( ˆV ) ǔ) h Q Cǔ + r f ( ǔ + θ I ( ˆV ) + (1 α) ˆV ) + r D ( θ I ( ˆV ) ǔ) r D ǔ + r D (1 α) ˆV = (h Q I + r f + r D) θ I ( ˆV ) + (r f + r D)(1 α) ˆV (h Q I + hq C + r f + 2r D)ǔ. (2.38) Before we use (2.37) and (2.38) as formula for drivers, we need o check condiions (i) & (ii). As hese condiions are pah dependen, which means he resuls varies a differen ime, we need o check hem sep by sep when we race back he XVA. 14
The idea is similar wih wha we do for American opions. When we wan o hedge he payoff Φ(S T ), which is he case of buying an European opion, we need o use ǧ as our drivers. Also, compare o selling one opion, θ I ( V ) = 0 in his case. ǧ = h Q I ( θ I ( ˆV ) + ǔ) h Q C ( θ C ( ˆV ) + ǔ) + [r + f ( ǔ + θ I ( ˆV ) + ǔ + θ C ( ˆV ) + ǔ (1 α) ˆV ) + r f ( ǔ + θ I ( ˆV ) + ǔ + θ C ( ˆV ) + ǔ (1 α) ˆV ) r D ( θ I ( ˆV ) + ǔ) r D ( θ C ( ˆV ) + ǔ) r D α ˆV ] + r D ˆV = h Q I ǔ hq C ( θ C ( ˆV ) + ǔ) + [r + f (ǔ + θ C ( ˆV ) (1 α) ˆV ) + r f (ǔ + θ C ( ˆV ) (1 α) ˆV ) r D ǔ r D ( θ C ( ˆV ) + ǔ) + (1 α)r D ˆV ] (2.39) Similarly, we need o discuss he sign of (ǔ + θ C ( ˆV ) (1 α) ˆV ). iii. if ǔ + θ C ( ˆV ) (1 α) ˆV 0, hen ǧ = h Q I ǔ hq C ( θ C ( ˆV ) + ǔ) + [r + f (ǔ + θ C ( ˆV ) (1 α) ˆV ) r D ǔ r D ( θ C ( ˆV ) + ǔ) + (1 α)r D ˆV ] = h Q I ǔ hq C ( θ C ( ˆV ) + ǔ) r D ǔ (2.40) = h Q C θ C ( ˆV ) (h Q I + hq C + r D)ǔ. And we add ime ino ǧ, so ǧ (ǔ) = h Q C θ C ( ˆV ) (h Q I + hq C + r D)ǔ (2.41) 15
iv. If ǔ + θ C ( ˆV ) (1 α) ˆV < 0, hen ǧ = h Q I ǔ hq C ( θ C ( ˆV ) + ǔ) + [ r f (ǔ + θ C ( ˆV ) (1 α) ˆV ) r D ǔ r D ( θ C ( ˆV ) + ǔ) + (1 α)r D ˆV ] = (h Q C + r D r f ) θ C ( ˆV ) (r f r D)(1 α) ˆV (2.42) (h Q C + hq I r f + 2r D)ǔ and we plug in ime, ǧ (ǔ) = (h Q C +r D r f ) θ C ( ˆV ) (r f r D)(1 α) ˆV (h Q C +hq I r f +2r D)ǔ. (2.43) Wih drivers above, we can use he FT scheme o approximaely calculae he XVA by he linear regression Mone Carlo algorihm. 16
Chaper 3 Numerical mehods We define ǧ (u) = ǧ ± (, u; ˆV ) (3.1) for wriing simpliciy. Noice we omi Ž± here since Ž± doesn appear in our final drivers according o (2.37), (2.38), (2.41) and (2.43). And le E [ ] = E[ G ] (3.2) o be he condiional expecaion given G. Before digging ino he Mone Carlo mehod, we are changing BSDE (2.31) ino he expecaion form, and hen ake condiional expecaion of boh sides given G. Thus Ǔ = E [ T ] ǧ(ǔs)ds, (0, T ). (3.3) 17
3.1 FT scheme By (Fujii & Takahashi, 2012a, 2012b), a perurbaion parameer ɛ and he following perurbaion form of BSDE (3.3) are inroduced: Ǔ ɛ = E [ T ] ɛǧ s (Ǔ s)ds ɛ. (3.4) I s exacly he same as (3.3) when ɛ = 1. Suppose he soluion of (3.4) can be represened as a power series of ɛ: Ǔ ɛ = Ǔ (0) + ɛǔ (1) + ɛ 2 Ǔ (2) + ɛ 3 Ǔ (3) +. (3.5) Then consider he Taylor expansion of ǧ a Ǔ (0), ǧ (Ǔ ɛ ) = ǧ (Ǔ (0) )+(ɛǔ (1) +ɛ 2 Ǔ (2) + ) u ǧ (Ǔ (0) )+ 1 (1) (ɛǔ +ɛ 2 Ǔ (2) + ) 2 2 2 uǧ (Ǔ (0) )+. (3.6) By collecing he erms wih same order wih respec o ɛ in (3.6), and comparing hem wih (3.5), we have he following relaionships: Ǔ (0) = 0, (3.7) [ T ] = E ǧ s (Ǔ s (0) )ds, (3.8) Ǔ (1) Ǔ (2) Ǔ (3) = E [ T = E [ T Ǔ (1) u ǧ s (Ǔ (0) Ǔ (2) s u ǧ s (Ǔ (0) s ] )ds, (3.9) ] )ds, (3.10) where he hird order erm should have a second order parial derivaive erm. Bu all of our drivers are linear funcion wih respec o Ǔ, so he second order derivaive is 0 and we can omi i. By leing ɛ = 1, we can generae a approximaion of he 18
BSDE soluion, Ǔ Ǔ (1) + Ǔ (2) + Ǔ (3). (3.11) To calculae he inegral inside condiional expecaions, (Fujii & Takahashi, 2012b) inroduce a random variable o randomize he inegral. Thus he problem becomes figuring ou he expecaion which could be done numerically by Mone Carlo mehod. This is called he FT scheme. Assume η 1 is a ime random variable wih densiy as φ(s, ) = 1 {s } µe µ(s ), (3.12) hus we have Ǔ (1) = E [ T = E [ T = E [ T = E [1 {η1 T } ] ǧ s (Ǔ s (0) )ds 1 {s } ǧ s (Ǔ s (0) )ds] φ(s, ) eµ(s ) µ e µ(η 1 ) µ ] (0) ǧs(ǔ s )ds ] ǧ η1 (Ǔ η (0) 1 ). (3.13) Similarly, we can derive [ Ǔ (2) = E 1 {η1 T }Ǔ (1) η 1 e µ(η 1 ) µ ] ǧ η1 (Ǔ η (0) 1 ), (3.14) plug he resul from (3.13) ino (3.14) and use ower propery, we ge Ǔ (2) = E [1 {η2 T } e µ(η 2 η 1 ) µ ǧ η2 (Ǔ (0) η 2 ) eµ(η 1 ) ] u ǧ η1 (Ǔ η (0) µ 1 ), (3.15) 19
where η 2 is a random variable wih densiy φ(s, η 1 ) = 1 {s η1 }µe µ(s η 1). Similarly, Ǔ (3) = E [1 {η3 T } e µ(η 3 η 2 ) µ ǧ η3 (Ǔ (0) η 3 ) eµ(η 2 η 1 ) µ u ǧ η2 (Ǔ (0) η 2 ) eµ(η 1 ) ] u ǧ η1 (Ǔ η (0) µ 1 ), (3.16) where η 3 has densiy of φ(s, η 2 ) = 1 {s η2 }µe µ(s η 2). One imporan hing is ha for all [0, T ], we have Ǔ (0) = 0 from (3.7). Once we calculae hese hree condiional expecaions, he approximaed resul is jus he sum of hem. 3.2 Linear regression Mone Carlo mehod An inuiive idea is o use a ime grid and sample N random vecors (η 1, η 2, η 3 ) o calculae he Ǔ from = T o = 0 backwards sep by sep. Bu in every sep (say a ime n ) we need o calculae many Ǔ n in order o use he Mone Carlo mehod for ime n 1, so he complexiy is exponenially increasing in ime. A more efficien way is o use he linear regression Mone Carlo mehod o do his, similar o is use for calculaing American opion prices. A firs, we have o specify some model seups. We define our ime grid i = i, where i = (0, 1,, n) and = T. Thus, according o (Glasserman, 2013) chaper n 8.6, E i [f i+1 (X i+1 )] = β T i ψ i (x), (3.17) where f( ) is a pre-specified funcion, β i is our coefficiens vecor of lengh m, ψ i (x) is he vecor of basis funcion values of lengh m and x is he parameers given a ime i. We need o simulae b independen pahs of (X ) 0 for he calculaion. The fied β i is given by ˆβ i = ˆB 1 ψ ˆB ψv, (3.18) 20
where ˆB ψ is a m m marix wih qr enry as 1 b b ψ q (X ij )ψ r (X ij ), (3.19) j=1 and ˆB ψv is a m-vecor wih rh enry as 1 b b ψ r (X ik )f i+1 (X i+1,k ). (3.20) k=1 When pricing American opions, we usually use sock price pah as our X in he above model. However, our drivers ake (η 1, η 2, η 3 ) as he inpu parameers. So i s reasonable o se our (X ) 0 o be (η 1, η 2, η 3 ) 0, and hese hree process should have he following relaionships: (1) η 1 i is generaed wih densiy funcion φ(s, i ) = 1 {s i }µe µ(s i), (2) η 2 i is generaed wih densiy funcion φ(s, η 1 i ) = 1 {s η 1 i }µe µ(s η i 1 ), (3) η 3 i is generaed wih densiy funcion φ(s, η 2 i ) = 1 {s η 2 i }µe µ(s η2 i ). 3.3 Pricing algorihm Firs we need o decide he basis funcions, which are denoed as ψ( ). Second, generae processes η = (η 1, η 2, η 3 ) from he relaionships above and he Black-Scholes opion price process ˆV, which could be simulaed using sock price process. We also have ǧ( ) as our drivers. According o equaions (3.13), (3.14) and (3.15), we define Ǔ (y) i = E i [f (y) i (η i, ˆV i )] (3.21) Then le E i [f (y) i (η i )] = (β (y) i ) T ψ (y) i (η i, ˆV i ). (3.22) 21
And β i is given by ˆβ (y) i = ( ) ˆB(y) 1 ψ ˆB(y) ψv, (3.23) where ˆB (y) ψ is an m m marix wih qr enry as 1 b b j=1 ψ q (y) (η i,j, ˆV i,j)ψ r (y) (η i,j, ˆV i,j), (3.24) and ˆB (y) ψv is an m-vecor wih rh enry as 1 b b k=1 ψ r (y) (η i,k, ˆV i,k)f (y) i (η i,k, ˆV i,k), (3.25) where y {1, 2, 3}, m is he number of basis funcions. Noice in he above specificaion, f is no G -measurable. The algorihm is shown in figure 3.1. Regression-Based Mone Carlo algorihm (ǧ, T, η, ψ( ), f, ˆV ) (1) Generae b pahs of η as above, generae b pahs of ˇV (he clean BS price) (2) A erminal nodes, se Ǔ n,k = 0, k = 1, 2,, b (3) Apply backward inducion: for i = n 1,, 1 When = i, Ǔ for all > i are already known for k in 1, 2, 3,, b for y = 1, 2, 3 check condiions (i)&(ii) or (iii)&(iv) in secion 2.3.2 o decide driver ǧ decide funcion f (y) i (y) calculae ˆβ i = ( calculae Ǔ (y) i,k ˆB (y) ψ = ( ˆβ (y) i (η i,k, ˆV i,k) ) 1 Ǔ i,k = Ǔ (1) i,k + Ǔ (2) i,k + Ǔ (3) i,k (4) reurn Ǔ 0 = e r D 1 1 b b k=1 Ǔ 1,k ˆB (y) ψv ) T ψ (y) i by (3.24) and (3.25) (η i, ˆV i,k) Figure 3.1: Regression-Based Mone Carlo Algorihm In sep 3, we plug η ino funcion f, we need o calculae he drivers g η. Since 22
η is greaer han, Ǔη are already calculaed in previous loop. So everyhing is fine as long as we se Ǔη = Ǔ k, where k 1 < η k. We can simply sore he pah of Ǔ and search he Ǔη value. Anoher problem is how o choose basis funcion. We choose as basis funcion: ψ (y) = ψ( i ) = (1, S i, S 2 i ) T, y 1, 2, 3. (3.26) Polynomial funcions are smooh, which is a very good propery for he linear regression Mone Carlo mehod. Using i as basis funcion s variable insead of η i should be a reasonable guess, since all hese η i are generaed from i, hus heir mean should converge o some funcion of i. We will see how i performs in nex chaper. One may also be curious abou why we don apply backward inducion unil i = 0. The reason is ha a i = 0, he marix B ψ is no inverible because of he basis funcion we use as he iniial sock price is idenical. So using he XVA prices a ime 1 and hen discoun i o 0 should be a reasonable plan. 23
Chaper 4 Example We are using he following benchmarks: σ = 0.2, r D = r r = r c = r + f = 0.05, r f = 0.08, µ I = 0.21, µ C = 0.16, L I = L C = 0.5 and α = 0.9, h Q I and h Q C can be calculaed by h Q i = µ i r D, i {I, C}, which is also given previously. Assume he rader is selling one European call opion. The iniial price of he underlying sock is S 0 = 100, he srike price is K = 110 and he opion expires a T = 1. Since he rader has a shor posiion in opions, his corresponding driver is g + as specified in (2.37) and (2.38). The condiions needed o be checked are (i) & (ii). I s necessary o menion ha we only use b = 20, which is usually considered as oo small sample size, bu we will check how i works. We will use boosrapping as furher echnique in his chaper. Boosrapping is a resampling echnique which is used when he size of given sample is oo small. This echnique works as follows: firs we generae a new sample wih he same size as given sample by aking values from he given sample wih replacemen and calculae he XVA price wih his new sample; hen we repea he firs sep for many imes; finally we use all resuls generaed by he second sep o find a more sable resul (i.e. calculae he average) and check he sabiliy of resul (i.e. find he confidence 24
inerval). 4.1 Resuls Under he assumpions above, he Black-Scholes price of his European call opion can be calculaed by BS formula as follows, ˇV 0 = S 0 Φ(d 1 ) Ke rt Φ(d 2 ), (4.1) d 1 = log S 0 + (r + 1 K 2 σ2 ) T σ, d 2 = d 1 σ T, T where Φ( ) is cumulaive densiy funcion of sandard normal disribuion. The resuls of our XVA adjusmen price and Black Scholes price are given in he below able, B-S price XVA adjusmen ˇV 0 = 39.2 U 0 = -1.443 I migh be a lile srange ha we have a negaive XV A which leads o a lower all-inclusive price wih such a high collaeral level α = 0.9. The reason could be ha we have r c = r D = 0.05, which is higher ha (Bichuch e al., 2016) example wih r D = 0.05 bu r c = 0.01. In our assumpion, he rader would ge more reurn from his posed collaeral accoun and hus have a lower cos. By modifying he driver funcion o include r c = 0.01, we ge a posiive resul wih XV A = 5.08, which verifies our argumen. We also calculaed he resul under differen collaeral level α. As shown in Figure 4.1, we noice ha when he collaeral level α decreases, he value of XVA is decreasing, which is consisen wih (Bichuch e al., 2016) resul. An inuiive explanaion is wih a lower α, he rader has less limiaions since he has o pos 25
less collaeral money o his counerpary and hus his funding cos is reduced, which leads o lower selling price. The relaive XVA adjusmens are also showed in Figure 4.2. Figure 4.1: XVA adjusmens wih α = 0.7, 0.8, 0.9 Furher more, we compare he XVA under differen r f. As shown in Figure 4.3 & Figure 4.4, XVA value decreases when he r f increases under he assumpion of α = 0.9. 26
Figure 4.2: Relaive XVA adjusmens wih α = 0.7, 0.8, 0.9 4.2 Check sabiliy We use boosrapping o check he variance of our resul under he assumpion of α = 0.9 and r f = 0.08. Figure 4.5 shows he resul of all of our XVA adjusmens, and he variance is 0.028, 95% confidence inerval is [ 1.80, 1.147]. Even hough we only use 20 sample pahs, he error is jus abou ±0.32, which is only 0.83% of he agen price or Black Scholes price. We consider his as an accepable resul. By doing a furher sep, we can use boosrapping easily wih almos no cos o ge a much more converged resul as wha has been done he secion 4.1. 27
Figure 4.3: XVA adjusmens wih r f = 0.07, 0.08, 0.09 Figure 4.4: Relaive XVA adjusmens (%) wih r f = 0.07, 0.08, 0.09 28
Figure 4.5: All XVA adjusmens, he verical line is our resul, he hisogram shows all resuls from boosrapping 29
Chaper 5 Conclusion Following he (Bichuch e al., 2016) marke seup and XVA model, we derive a numerical mehod o price European Opions via BSDEs. Under he specific assumpion of r r ± = r c ± = r + f = r D < r f, which saisfies non-arbirage condiion, we generae driver funcions for boh selling and buying posiions. Then he FT scheme is used by leing perurbaion parameers equal o 1 and we derive a linear approximaion. Since he funcions inside condiional expecaion are pah dependen, we use he Linear Regression Mone Carlo mehod which is used o price American opions. An example is given in Chaper 4. The resuls generae by he numerical mehod are quie sable and reasonable for only using 20 sample pahs, which is always considered as a small sample size. Anoher very powerful daa science ool boosrapping is also used wih very low cos bu increase he sabiliy of our resul significanly. 30
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