U.U.D.M. Projec Repor 2012:9 Pricing opions on defaulable socks Khayyam Tayibov Examensarbee i maemaik, 30 hp Handledare och examinaor: Johan Tysk Juni 2012 Deparmen of Mahemaics Uppsala Universiy
Pricing opions on defaulable socks Khayyam Tayibov Maser hesis in mahemaics Uppsala Universiy June 2012
Absrac This work deals wih imporan issue of pricing of opions wih defaul risk. We use resuls obained by Carr and Linesky in pricing European and American-ype opions. The key noion in his work is he risk-neural survival probabiliy which makes i possible for us o accoun for possibiliy of defaul. To price American-ype opions, we will use Leas Square Mone Carlo mehod, because i is easier o apply survival probabiliy in repeaed simulaions.
Conens 1 Inroducion... 6 1.1 Mone Carlo simulaions... 6 1.2 Opion ypes... 6 2 Jump o defaul CEV model... 7 2.1 Bessel processes... 7 2.2 Applicaion of Bessel processes o a jump o defaul CEV model.... 7 3 Compuaions... 13 3.1 Numerical resuls... 13 3.1 Valuaion of European opions... 13 3.1.1 Parameer sensiiviy... 16 3.2 Valuaion of American opions... 21 3.2.1 Parameer sensiiviy... 23 3.2.2 Early exercise curve... 26 4 Conclusions... 27 4.1 Framework... 27 4.2 Fuure research... 27 Bibliography... 28
Inroducion 1.1 Mone Carlo simulaions In his work I use Mone Carlo simulaions o calculae he price of opions on underlying asses which can defaul. Mone Carlo simulaions are based on repeaed sampling in order o calculae he value of he opion. In his work, implemenaion is based on simulaion of sock pahs in he new jump o defaul consan elasiciy of variance model (CEV). Then we incorporae survival probabiliy ino he scheme. 1.2 Opion ypes The following ype of opions are considered: European opions - which give he holder he righ bu no he obligaion o exercise he opion a fixed ime, called mauriy. American opions - are he ype of opions which can be exercised a any ime up o he ime of mauriy.
Jump o defaul CEV model 2.1 Bessel processes Much of he heoreical resuls in his work are based on he properies of Bessel processes, so we give definiion of hese processes and some of heir properies. Before ha noe ha for every ε 0, x 0 he soluion of he equaion R = x + ε + 2 0 R s db s is unique. Definiion: For every ε 0, x 0 he unique srong soluion o he equaion above is called he square of ε dimensional Bessel process wih saring poin a x and denoed as BES ε (x). 2.2 Applicaion of Bessel processes o a jump o defaul CEV model. We sar by inroducing some empirical findings ha our model is based upon. One of hese is "implied volailiy skew". I is a name for he inverse relaionship beween implied volailiy and an opion's srike price. This relaionship has been observed for boh single name and index opions since 1987 in (Dennis). Anoher ineresing fac is he empirical evidence ha sugges ha realized sock volailiy is negaively relaed o sock price. This is called a leverage effec and has been suppored by empirical sudies in (Chrisie). When we model a diffusion process we design i so ha i encompasses he above menioned empirical findings. To fulfill he leverage effec and he implied volailiy skew we assume a consan elasiciy of variance (CEV). Also, in his model i is possible ha he process will jump o zero. We assume ha he insananeous risk-neural hazard rae of defaul is an increasing affine funcion of he insananeous variance of he underlying sock. This new model is called he jump o defaul exended CEV or JDCEV process. We assume marke wih no arbirage and an equivalen maringale measure. The following is he sochasic differenial equaion (SDE) for he pre-defaul sock process {S, 0}. ds = r q + λ S S d + σ S, S db, S 0 > 0, (1)
where r 0 he risk-free ineres rae, q() 0 is he dividend yield, σ(s, ) 0 is he insananeous sock volailiy and λ S, 0 is he defaul inensiy. To be consisen wih he leverage effec and he implied volailiy skew, we define he insananeous volailiy as follows σ S, = a S β, where β < 0 is he volailiy elasiciy parameer and a > 0 is he volailiy scale parameer. Addiionally, we assume ha he defaul inensiy is an affine funcion of he underlying sock: λ S, = b + cσ 2 S, = b + ca 2 S 2β, where b 0 and c > 0. Before proceeding wih he soluion of SDE we make some definiions. The jump-o-defaul hazard process is defined in (Peer Carr) as follows: Λ = λ S u, u du, < T 0 0, T 0 where T 0 = inf 0: S = 0. A random ime of jump o defaul is he firs ime when Λ is greaer or equal o he random level e~ exp 1 : ξ = inf{ 0: Λ e}. And, finally he ime of defaul is composed of wo pars: ξ = T 0 ξ. In he above menioned work, he auhors consider hree "building block claims" as a basis of more complex securiies: 1) A European-syle coningen claim wih payoff Ω S T a ime T, given no defaul by T, and no recovery in case of defaul. 2) A recovery paymen of one dollar a he mauriy dae T if defaul occurs by T; 3)A recovery paymen of one dollar paid a he defaul ime if defaul occurs by T. They show ha he valuaion of hese hree "building blocks" reduces o compuing riskneural expecaion of he form
E e T λ S u,u du Ω S T 1 S m,t >0 S = S. (2) This is calculaed, and as a resul an analyical soluion is obained, by using he heory of Bessel processes. We will denoe a Bessel process of index µ saring a x (Bes µ (x)) by R µ. According o Proposiion 5.2, for any 0 < T he following holds: T E exp exp c a 2 ( = E x (μ ) u) S 2β u du Ω S T 1 S m,t >0 S = S exp{ c β 2 0 τ du T 1 α s ds }Ω(e ( β R τ ) β )1 {T0 R >τ}, (3) R u 2 where T τ, T = a 2 (u) e 2 β u α s ds du. To ge rid of he erm exp{ c τ du β 2 0 R2 u } he following proposiion from (Piman) is used: Proposiion: Le P µ be he law of he Bessel process R µ sared a x > 0 and le R be is canonical filraion. Then, for ν 0 and μ 0 he following absolue coninuiy relaion holds P ν x R = ( R x )ν μ exp ν2 μ 2 du (μ P ) 2 R2 x R. (4) 0 u And for ν < 0 and μ 0 he following absolue coninuiy relaion holds before he firs hiing ime of zero, T R 0 : P ν x R T R 0 = ( R x )ν μ exp ν2 μ 2 2 du R2 0 u P x (μ ) R. (5) Consider he following Bessel process indexes ν + = ν + 1 β = c + 1 2 β > 0 and ν = c 1 2 β. According o (5), we have P ν x R T R 0 = ( R x )v μ exp ν2 μ 2 2 du R2 0 u (μ P ) x R = P ν x R T R 0 ( R x ) ν+μ exp ν2 μ 2 2 2 μ 2 P x (μ ) R = P x ν + R ( R x ) ν ++μ exp ν + 2 P x (μ ) R du R2 0 u 0 du R u 2
P x ν R T 0 R exp ν2 μ 2 ν + 2 + μ 2 2 du R2 0 u = ( R x ) ν ++μ +ν μ P x ν + R. (6) Noe ha Plugging i in (6) we ge ν ν + = c 1 2 c 1 2 β = 1 β ν 2 ν + 2 = c 2 c + 1 4 c2 c 1 4 = 2c β 2. P x ν R T 0 R exp c β 2 du R2 0 u = ( R τ x ) 1 β P x ν + R. (7) Applying his o (3) we obain he following equaliy: T E exp exp c a 2 ( = E x (ν + ) R τ x 1 β T Ω(e α s ds u) S 2β u du Ω S T 1 S m,t >0 S = S 1 ( β R τ ) β ). (8) Thus, he problem has been reduced o compuaion of he expeced value of a funcion of he sandard Bessel process. From (Göing) and we have he following expression for he densiy of R τ : P x τ R τ dy = p ν τ; x; y dy = y τ y x ν exp x2 + y 2 2τ I τ xy τ dy, where I ν z = n=0 1 n! Г n + ν + 1 z 2 ν+2n is he Bessel funcion of he hird kind of index ν. We use he fac ha Bessel process densiy can be expressed in erms of he non-cenral chisquares densiy. For his, noe ha he non-cenral chi-squares disribuion wih δ degrees of freedom and non-cenraliy parameer α > 0 has he densiy f X 2 x; δ, α = 1 α+x x e 2 2 α v 2 Iv αx 1 x>0, and we arrive o he following expression for he Bessel process disribuion funcion:
p v τ; x; y dy = 2y τ f y2 x2 X 2 ; δ, τ τ. Finally, o calculae (8) we will need an expression for momens of chi-square disribuion. Lemma 5.1 (from (Peer Carr)): Le X be a X 2 δ, α random variable, v = δ 1, p > v 1 2 and k > 0. The p-h momens and runcaed p-h momens are given by M p; δ, α = E X2 δ,α X p = 2 p e α 2 Г p + v + 1 Г v + 1 F 1 p + v + 1, v + 1, α 2, 9 Φ + p, k; δ, α = E X2 δ,α X p 1 X>k = 2 p e α 2 n=0 α 2 n Г p + v + n + 1, k 2 n! Г v + 1 + n, (10) Φ p, k; δ, α = E X 2 δ,α X p 1 X k = 2 p e α 2 n=0 α 2 n γ p + v + n + 1, k 2 n! Г v + 1 + n, (11) The formulas (9) - (11) enables us o calculae he expeced value of (8) hus enabling us o derive formulas for more complex securiies.
Compuaions 3.1 Numerical resuls Formulas (10) and (11) pose cerain problems for calculaions. To avoid hese problems we will employ Mone Carlo mehod for compuaion of opion prices (in his case, for simpliciy, call opion). In his case, o facor in he probabiliy of defaul by jump o zero as opposed o simple diffusion process, we use he risk neural survival probabiliy. This probabiliy has he following form Q S, ; T = E e T λ S u,u du 1 S m,t >0 S = S (12) and afer applying Lemma 5.1 we arrive a he following formula Q S, ; T = e T b u du x 2 τ 1 2 β 1 M ; δ 2 β +, x 2 τ, (13) where δ + = 2(v + + 1) and τ is defined as in he equaion (3). 3.1 Valuaion of European opions The survival probabiliy is applied as follows: In each of he simulaions we compare he expression above wih a uniformly disribued random level. If he survival probabiliy is larger han his random level, hen we accep he curren pah S (i.e., defaul does no occur), oherwise we consider he curren pah equal o zero (i.e., defaul occurred). For convenience we will wrie he SDE (1) in a more compac way: ds = a(, S )d + b(, S )db Wih his in mind, and using Euler's discreizaion we can formulae he Mone-Carlo mehod for CEV and JDCEV models as follows:
Algorihm 1 = T M for j=1,...,n for i=1,...m i = i 1 + w = u n, u n N(0,1) S i = S i 1 + a i 1, S i 1 + b i 1, S i 1 w end Compue he value V T j = ( S T K) +. end Compue E V T = 1 N N j=1 V T j. Compue he discouned value V 0 = e rt E V T.
Algorihm 2 (when defaul is possible) = T M for j=1,...,n for i=1,...m i = i 1 + w = u n, u n N(0,1) S i = S i 1 + a i 1, S i 1 + b i 1, S i 1 w Calculae risk-neural survival probabiliy in (13): Q i = Q S j, ; T end Generae normally disribued sample e n E 0,1, n=1,..., N. if(q j > e n ) Compue he value V T j = ( S T K) +. end Compue E V T = 1 N N j=1 V T j. Compue he discouned value V 0 = e rt E V T.
3.1.1 Parameer sensiiviy The following parameers are considered in his Mone Carlo simulaion: σ = 0.2, β = 0.2, r = 0.1, T = 0.5, K = 16, S 0 = 12. Also, for convenience, we use he following values for he funcions which form he basis for JDCEV: q = 0, a = 1, c = 1, r = r. We simulae he price process using Euler discreizaion. Fig.1 In Fig.1 we model he price process as we vary he iniial sock price. As one can noe, here is a considerable difference in opion prices beween hese wo cases. This is caused by he fac ha in he second case we ake ino accoun he possibiliy of defaul and so, some of he pahs in our simulaion we accep as zeros. Figures 2, 3 and 4 show us how he price process behave wih respec o variaions in sock process wih differen volailiy scale parameer and wih respec o srike price. I is imporan o remember ha he volailiy in his model is no a consan bu a power funcion of he form σ S, = as β. I is easy o show ha he picure roughly corresponds o he sandard CEV process 1 wih consan coefficiens, when we adjus he parameers of our curren model accordingly. 1 The same Mone Carlo mehod can be used wih minor changes; The main, and he mos imporan difference would be he absence of he risk-neural probabiliy of survival, because in sandard CEV model defaul occurs hrough diffusion o zero.
Fig.2 Fig. 3
There is a visible posiive relaionship beween he Call prices and volailiy scale parameer a. This is due o he fac ha here is a posiive relaionship beween he defaul inensiy and survival probabiliy as one can see in he formulas (12) and (13). We see he similar picure in he case of pu opion. Fig. 4 Fig. 5
Fig. 6 Table 1 (β = 0.8) Table 2 (β = 0.5) K Pu (defaulable) Pu K Pu (defaulable) Pu 25 3,26735117 3,37148058 30 7,82562315 8,14186813 35 12,4480029 12,8884104 40 17,1506247 17,6395995 45 21,8090426 22,4007288 50 26,5075384 27,1578761 55 31,0806412 31,9098313 60 35,6210944 36,6947347 65 40,2683308 41,4184517 70 44,3155414 46,1953553 25 0,01781364 0,02553169 30 3,04649288 3,35879985 35 7,50912999 8,08391345 40 11,8624989 12,880693 45 15,924856 17,6390138 50 20,2246887 22,3972671 55 24,3074233 27,1182497 60 29,3135234 31,8994619 65 33,6505195 36,6437196 70 38,3330697 41,4137091
K Call (defaulable) Call K Call (defaulable) Call 5 15,1560141 15,6707037 6 14,2510955 14,6853808 7 13,3162802 13,7484261 8 12,4260429 12,7959076 9 11,5313328 11,8642969 10 10,5697552 10,9030618 11 9,61016665 9,94096613 12 8,68677413 9,00332434 13 7,75157435 8,05452331 14 6,83653674 7,06132437 15 5,93273754 6,15594622 16 5,06006772 5,19226843 17 4,11322989 4,23818106 5 14,3739878 15,7327635 6 13,3010966 14,7362367 7 12,7703187 13,7639406 8 11,8110208 12,7975732 9 11,1017502 11,8675297 10 9,98200418 10,8944848 11 9,06502021 10,0442589 12 8,42090692 9,01952865 13 7,40877195 8,0690498 14 6,68541555 7,09601792 15 5,56055892 6,16713181 16 4,78794227 5,17958325 17 3,88482316 4,2420216 Parameers: S=20, q=0, r=0.05, q=0, T=1 year. As eviden from Tables 1 and 2, he smaller he β he closer he value of he opion on defaulable sock process is o he opion on non-defaulable sock. This is caused by he behavior of he survival probabiliy (12). For example, for he above parameers we have he following picure of he risk-neural survival probabiliy: Fig.7
I is clear ha here is posiive relaionship beween β and defaul probabiliy, which explains why he difference beween opion value on defaulable and non-defaulable sock is bigger wih smaller volailiy elasiciy parameer β. 3.2 Valuaion of American opions American ype opions differ from European ypes in ha hey can be exercised a any ime during is life. Consequenly, opion price depends on exercise sraegy. This makes Mone Carlo simulaions, which are very simple o implemen in a European ype opions case, a lile less rivial. There are several mehods for pricing American opions. By doing ime discreizaion of Black-Scholes formula we ge a linear complimenary problem. One of he mehods of soluion is projeced successive over relaxaion (PSOR) mehod. As PSOR is inefficien for some cases of space discreizaion, oher mehod called operaor spliing(os) is applied o solve he problem. The main idea of his mehod is spliing of operaor ino separae fracional ime seps in he discreizaion (Toivanen). We won' go ino his mehod, as i can be problemaic o inegrae he risk-neural survival probabiliy in his scheme. For numerical calculaions we will use leas squares Mone Carlo simulaions wih Laguerre polynomials as se of basis funcions. In his regression mehod American-ype opion is approximaed by using a Bermuda -ype opion. This is Leas Square Mone Carlo mehod by Longsaff and Schwarz (Schwarz). We sar by simulaing sock pahs and hen, do backward ieraions. A each sep of ieraions we perform a leas square approximaion of he coninuaion funcion. Longsaff and Schwarz mehod is based on he assumpion ha he coninuaion value can be expressed as a linear combinaion of basis funcions: C i s = E V i+1 S i+1 S i = s = m j =1 β ij L j (s), where β ij are consans, and L j s is he Rodriguez represenaions of Laguerre polynomials: L n x = e x n! d n dx n (e x x n ). The Longsaff and Schwarz mehod algorihm as presened in (Korn):
Algorihm 3 1. Choose k basic funcions L 1,, L k. 2. Generae he N underlying geomeric Brownian pahs of he sock prices a each ime sep: {S( 1 ) j,, S( m ) j }. 3. Generae normally disribued sample e n E 0,1, n=1,..., N. 4. Se he erminal value for each pah: V m,j = e r f S T j, j = 1,, N. 5. Coninue backward in ime and a each ime i, i = m 1,..., 1: - Using linear leas square mehod compue β i. - Compue C i using β i. - Compue he risk-neural survival probabiliy Q i = Q S j, ; T. - Decide o exercise or hold he opion. Se for j = 1,, N V ij = e r if S i j, V i+1,j, else if e r if S i j C i S i j, i and Q i > e n 6. Compue V 0 = 1 N (V 11 + + V 1N ).
3.2.1 Parameer sensiiviy As eviden from Tables 3 and 4, lower β resuls in bigger difference beween he opions on defaulable and non-defaulable opion. This is consisen wih he European ype opion cases. The reason is, as a par of numerical mehod, we add he ime poins when he riskneural survival probabiliy is smaller han a random level, o he se of ime poins in which we accep he sock price as equal o zero. Table 3 (β = 0.5) K Pu(defaul) Pu 25 0 0 30 0,11153476 0,12710312 35 4,97005468 4,97061257 40 9,96857552 9,96151396 45 14,9636761 14,9632046 50 19,9544929 19,9548011 55 24,9452399 24,9506628 60 29,9385832 29,9459733 65 34,9375425 34,9447761 70 39,9391133 39,9394238 Table 4(β = 0.2) K Pu(defaul) Pu 25 4,92638962 4,95357142 30 9,93429036 9,95427846 35 14,9170587 14,9454879 40 19,9193856 19,9396028 45 24,919995 24,9331697 50 29,9245166 29,9215411 55 34,8940074 34,9265439 60 39,8803758 39,9177479 65 44,8960086 44,9192177 70 49,8789401 49,9058121 Again we use he above menioned values for parameers: S=20, q=0, r=0.05, q=0, T=1 year, a=0.2, c=1.
Fig. 8 These plos show American pu opion price as a funcion of volailiy scale parameer. Predicably, his relaionship is posiive. Fig. 9
Fig. 10 The parameers in above simulaions: K=50, r=0.1, c=1, b=0, T=0.5, a=0.2 (volailiy scale parameer). We can see a clear paern in Fig. 9-10. Decrease in β leads o a shrinking of he coninuaion region. Remember ha β < 0 is a volailiy elasiciy parameer, and as such, his relaionship is qui predicable. This is because of he inverse relaionship beween he price and he volailiy, which has been shown in empirical sudies (Chrisie).
3.2.2 Early exercise curve We define he value of American pu opion in our case as follows: P S,, T = sup τ T E,S e τ r s ds K S + τ, where τ is he exercise ime, and he coninuaion region is defined as Ω = (s, ) P(s,, T) > K s + }. Because he price funcion is coninuous, i is obvious ha a some poin i his he sraigh line of pay-off funcion a some S f. This is called an early exercise curve. K=60, r=0.1, c=1, b=0, T=1, a=0.2, β = 0.5 Noe ha, he early exercise curve is no convex, as opposed o he geomeric Brownian moion case. We did no have ime o verify his, so i migh be an ineresing opic of fuure research.
Conclusions 4.1 Framework We can conclude ha he framework proposed by P.Carr and Linesky offers a relaively easy way o price differen ypes of financial insrumens wih a posiive probabiliy of defaul. In his hesis, we focus on American and European-ype opions, bu as described in (Peer Carr), his can be exended o all kinds of differen securiies. 4.2 Fuure research I can be ineresing o find ou how his new model affecs he hedging, as porfolio may consiss of risky asses. In his case, using mehods oher han Mone Carlo would be more appropriae considering he problems associaed wih muliple simulaions o calculae he greeks. Anoher ineresing nuance is, because American-ype opions can be exercised a any ime before he mauriy, he holder migh choose, in he face of probable defaul o alleviae he hrea of losing more money, o exercise he opion before defaul happens, and losing less. I would be ineresing o conduc furher research ino his scenario.
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