A Two-Asse Jump Diffusion Model wih Correlaion Mahew Sephen Marin Exeer College Universiy of Oxford A hesis submied for he degree of MSc Mahemaical Modelling and Scienific Compuing Michaelmas 007
Acknowledgemens I would like o exend my graiude o my supervisor Sam Howison for his guidance and advice hroughou he preparaion and wriing of his disseraion. I would also like o hank Chrisoph Reisinger for his advice whils Dr Howison was away. Finally and mos imporanly I would like o hank my parens for heir coninued emoional and financial suppor, wihou which I would no be in his posiion oday.
Conens 1 Inroducion 1 1.1 Modelling Jumps in he Underlying Sock Price............ 1. The Meron Model............................ 6 1.3 Opion Pricing under he Meron Model................ 9 1.4 The Srucure of his Projec...................... 10 Correlaed Jumps 1.1 Covariance of wo asses......................... 14. Characerisic Funcions......................... 15.3 Momens.................................. 19 3 Mone Carlo Pricing 3.1 Equivalen Maringale Measures and Marke Incompleeness..... 3. Implemening Mone Carlo Mehods.................. 3 3.3 Pricing Opions on One Asse...................... 6 3.4 Implied Volailiy............................. 9 4 Exoic Opions 31 4.1 Exchange Opions............................. 31 4. Pricing Formula for an Exchange Opion................ 3 4.3 Evoluion of log(ξ)............................ 33 4.4 Change of Measure............................ 34 4.5 Pricing an Exchange Opion using Mone Carlo............ 41 4.6 Max-Call Opions............................. 4 4.7 Implied Correlaion............................ 45 5 Conclusion 49 A Expecaion of a compound Poisson process 51 Bibliography 53 i
Chaper 1 Inroducion Here we concern ourselves wih he pricing of opions on financial socks. The marke for opions is huge, for example in January 007 on average 10 million opions conracs were raded per day in he Unied Saes. Therefore need for an accurae opion pricing model is obvious as, bearing in mind he volumes being raded, any sligh inaccuracies in he price could cause grea losses o an invesor. In pracice he models presened by Fischer Black and Myron Scholes [] in heir seminal paper published in 1973 are mos widely used by praciioners in he markes and in recen years mos research ino derivaive pricing has concenraed on refining hese models. The populariy of he models is mainly due o he fac ha he majoriy of he required inpus are observable variables; he only unobservable variable required is he volailiy of he underlying sock. The model assumes ha he price of he underlying asse follows a geomeric Brownian moion (GBM), ds S = µd + σdw. (1.1) Under his assumpion he sock price follows a log-normal disribuion beween any wo poins in ime. Black and Scholes hen used he no arbirage principle (NAP) o derive a unique price for he opion. On op of he assumpion of a GBM modelling he evoluion of he underlying sock price, he Black-Scholes model also ress on he following assumpions abou he marke: 1. here are no ransacion coss or axes;. rading akes place coninuously in ime, 3. borrowing and shor-selling are allowed, 4. borrowing and lending raes are equal, 5. he shor-erm ineres rae is known and is consan, 6. he sock does no pay dividends, 1
Furher research has refined he original Black-Scholes model and shown ha he mehod holds when he ineres rae is assumed o be non-consan or even sochasic; when a sock pays dividends; when resricions are made on he use of shor sales and when he opion is of he American ype, i.e. i can be exercised any ime before or a he expiry dae. As menioned above, mispricing of an opion can lead o huge financial losses and empirical sudies of he prices admied by he Black-Scholes model show ha using his model his is he case, see Black [3]. The criical assumpions in he Black-Scholes derivaion is ha rading akes place coninuously in ime and ha he sock price has a coninuous sample pah wih probabiliy one. Realisically, coninuous rading is no possible, however rendering he Black- Scholes model invalid because of his would be an over-reacion as he coninuous rading soluion is a valid asympoic approximaion o he discree rading soluion, provided he sock price dynamics have coninuous sample pahs (which of course hey are no, bu are close o). The Black-Scholes arbirage porfolio under coninuous rading has zero-risk due o coninuous hedging. However under discree rading condiions we inroduce some risk since he marke moves beween rades. The porfolio risk has he order of he rading inerval lengh and hus he risk will end o zero as he inerval beween rades ends o zero. Therefore provided he ime inerval beween rades is no oo large, he error beween he Black-Scholes price and he realisic discree rading price will no differ by much. The Black-Scholes model is no valid hough, even if we rade in he coninuous limi, if he sock price dynamics do no have a coninuous sample pah. The Black- Scholes formula is valid if he sock price can only change by a small amoun over a small inerval of ime. Again empirical sudies show ha his is no he case and a more sophisicaed model of he underlying sock price is required. Marke reurns are generally lepokuric meaning he marke disribuion has heavier ails han a normal disribuion. The model should permi large random flucuaions such as crashes or upsurges. The marke disribuion is generally negaively skewed since downward ouliers are usually larger han upward ouliers. Rober Meron [16] proposed a soluion o his problem. He suggesed adding anoher erm o he evoluion of he underlying sock price given by equaion (1.1), his new erm would model jumps in he asse price. The evoluion for he underlying sock becomes, ds S = µd + σdw + dj, (1.) where dj represens a jump erm. 1.1 Modelling Jumps in he Underlying Sock Price Using equaion (1.), we will have wo ypes of changes affecing he overall change in he sock price. As wih he original Black-Scholes model we have flucuaions in he price due o general economic facors such as supply and demand, changes
in economic oulook ec. These facors cause small movemens in he price and are modelled by a geomeric Brownian moion wih a consan drif erm. The jump erm models he arrival of imporan informaion ino he marke ha will have an abnormal effec on he price. This informaion could be indusry specific or even firm specific. By is very naure imporan informaion only arrives a discree poins in ime and will be modelled by a jump. The causes of he jumps in he sock price can be pu in hree separae caegories. 1. Firm specific jumps - These jumps only affec individual firms. They may be caused by news enering he marke abou an individual firm s profi repor or managemen news ec.. Indusry/secor specific jumps - These jumps are caused by news enering he marke ha may only affec a specific indusry, for example a naional holiday in which weaher is paricularly bad may affec he socks relian on he Briish ourism indusry. 3. Marke specific jumps - These jumps affec every company in he marke. They may be caused by news affecing he general marke such as ineres raes, credi spreads or oil prices. No all firms are affeced in he same way however, for example some firms may jumps up, ohers down and he jumps may be of varying magniude. When modelling an opion dependen on he one individual sock, differeniaion beween causes of jumps is no necessary since we only need o know when he sock jumps and no wha caused he jump or wheher oher socks were affeced by he same informaion. However when modelling wo socks he differen causes of jumps is of ineres o us and jumps may occur independenly in each sock or hey may be common o boh. Wha properies should hese jumps have? Here we assume ha each jump in he sock price is independen of he ohers. This assumpion is no necessarily realisic bu we make he assumpion for modelling purposes. Our jump erm will ake he form of a random variable J ha can ake posiive or negaive values and deermines he jump magniude, muliplied by an ineger valued process N ha iniiaes a jump. N mus have he following properies, N N s is independen of N s. N 0 = 0, N N +, N s N if s <. We also require ha in a given shor space of ime δ, he likelihood of a jump is roughly proporional o he lengh of δ. The proporionaliy consan is denoed by λ and is called he jump inensiy. If δ is aken o be small, he probabiliy of wo jumps in he inerval is negligible. Therefore anoher propery our couning process 3
mus saisfy is, N N +δ = N + 1 N + k wih probabiliy 1 λδ o(δ) wih probabiliy λδ + o(δ). wih probabiliy o(δ). A process saisfying he four properies above is called a Poisson process wih inensiy λ. I has he Poisson probabiliy densiy funcion, P(N = j) = (λ)j e λ j = 0, 1,,... j! Proof. Define p n () P(N = n). Consider p n ( + δ), i.e. P(N +δ = n). There are n + 1 ways in which N +δ can equal n: 1. N = n and no jumps occur in δ,. N = n k and k jumps occur in δ, k = 1,,..., n. These evens are disjoin herefore p n ( + δ) = P(N = m)p(n +δ = n N = m) 0 m n = 0 m n P(N = m)p(n m arrivals in δ) = P(N = n)(1 λδ) + P(N = n 1)(λδ) + o(δ) = p n ()(1 λδ) + p n 1 ()(λδ) + o(δ). This final equaion holds for n 0. For n = 0 we find p 0 ( + δ) = p 0 ()(1 λδ) + o(δ). We expand p n ( + δ ) abou using Taylor series, p n ( + δ) = p n () + δ dp n() d Therefore our wo equaions for p n () and p 0 () become and p n () + δ dp n() d + (δ) d p n () d +... + = p n ()(1 λδ) + p n 1 ()(λδ) + o(δ) p 0 () + δ dp 0() + = p 0 ()(1 λδ) + o(δ). d Therefore upon cancelling he p n () and p 0 () from boh sides respecively, dividing hrough by δ and leing δ 0 we ge dp n () d = λp n 1 () λp n () n = 1,,... (1.3) 4
and dp 0 () d = λp 0 (). (1.4) We now have a sysem of equaions wih boundary condiions p n (0) = δ n0 where δ ij is he Kronecker Dela. We solve his sysem by inducion. Equaion (1.4) wih p 0 () = 1 is solved o give p 0 () = e λ. (1.5) We subsiue equaion (1.5) ino equaion (1.3) wih n = 1 and p 1 () = 0 and solve o ge p 1 () = λe λ. Now suppose ha and look o solve equaion (1.3). We have p n 1 () = (λ)n 1 (n 1)! e λ dp n () d = λ n n 1 e λ λp n () wih p n (0) = 0. This is solved by as required. p n () = (λ)n e λ n! Now we mus consider how he jump variable J will be disribued. Suppose ha informaion eners he marke causing an insananeous jump in he asse price afer which he price has moved from S o J S. Here J is he absolue magniude of he jump. Therefore he relaive price change will be given by ds S = J S S S = J 1. The simples opion would be o choose all jumps o be equal o some consan J = J c for all, however his would be highly unrealisic. I would no be ridiculous o assume ha he larger he magniude of he jump, he less probable i would be. Meron [16] suggess modelling he jump variables as non-negaive log-normal random variables in order o provide a more realisic jump erm. A random variable X has a log-normal disribuion wih mean µ and variance σ if ln(x) is normally disribued wih mean µ and variance σ. Equivalenly X has a log-normal disribuion if X = exp(y ) where Y has a normal disribuion wih mean µ and variance σ. If X N(α, δ) hen X log-normal(e α+1 δ, e α+δ (e δ 1)). 5
The jump variable J is disribued as follows, j = ln(j ) IID N(α, δ ). (1.6) The relaive jump size is (J 1) where J is log-normally disribued wih mean α and variance δ, herefore E[(J 1)] = e α+1 δ 1 K (1.7) E [( (J 1) E[J 1] ) ] = e α+δ (e δ 1). 1. The Meron Model Using he las secion, he evoluion of an asse under a jump diffusion model is ds S = µd + σdw + (J 1)dN. Here µ and σ are consans. W is a Wiener process wih respec o he marke probabiliy measure P. N is a Poisson process (wih consan inensiy λ) wih respec o he marke probabiliy measure also, he Poisson process is assumed o be independen of he Wiener process. J is he random jump variable ha is assumed o follow he log-normal disribuion. We also assume ha J is independen of he oher wo random processes in he evoluion, namely he Wiener process and Poisson process. The variable J is also independen hrough ime, Cov(J s, J ) = 0 for s. The model evolves as follows: { ds µd + σdw if no Poisson even occurs = S µd + σdw + (J 1) if a Poisson even occurs. Therefore if a jump is iniiaed and he log-normal random variable akes he value 1.1, he sock price will jump up by 10%. Likewise a drawing of 0.9 for J will cause a 10% fall in he sock price. We now look o solve equaion (1.) and find an expression for he log prices ln(s ). For his we need o use Io s formula for jump-diffusion processes as given by Con and Tankov [7], we will inroduce his shorly. 1..1 Noaion Due o he insananeous naure of jumps in he price of an asse, we will have a disconinuiy in he price when a jump occurs. To disinguish beween he prices eiher side of he disconinuiy we inroduce he following noaion. If a ime a jump occurs, denoed by S hen S = lim r S r and S = S + S. 6
Theorem 1..1. (Io s Formula for Jump-Diffusions) Suppose X is a jump diffusion process wih evoluion given by X = X 0 + 0 a s ds + 0 N b s dw s + X i, where a is he drif erm, b is he volailiy erm and X i corresponds o jump i in he sock price. Then Con and Tankov [7] sae ha for a funcion f(x, ), df(x, ) = f(x, ) i=1 f(x, ) d + a d + b x f(x, ) d x f(x, ) + b dw + [f(x + X ) f(x )]. x (1.8) Using his heorem for f(.) = ln(.) and S described by he sochasic differenial equaion (1.) we ge d lns = ln S 1 = µs = This is solved o give ln S ln S d + µs d + σs dw + [ln j S ln S ] S ln S ( 1 S d + σ S S (µ σ ) d + σdw + ln J. ln S = ln S 0 + ) d + σs 1 S dw + [ln J + ln S ln S ] ) (µ σ + σw + N i=1 ln J i. (1.9) This is clearly he same as i would be in he Black Scholes geomeric Brownian moion case excep for he sum of log-normal jumps. We mus be careful when adding a erm of his ype o he price process. The erm N Q = ln J i i=1 is called a compound Poisson process and adding a erm of his form o a geomeric Brownian moion, as we have done above, will affec he drif of he asse. We see using a momen generaing funcion argumen (Appendix 1) ha E [ N ] ln J i = λk i=1 where K is defined in equaion (1.7). Therefore he addiion of a compound Poisson process will increase he mean of he asse by λk. We inroduce he compensaed Poisson process This process is a maringale. Q λk. 7
Proof. E[Q Kλ F s ] = E[Q Q s F s ] + Q s Kλ. Due o he memorylessness of exponenial random variables (see [18]) we have E[Q Q s F s ] = E[Q Q s ] = Kλ Kλs = Kλ( s). Therefore E[Q Kλ F s ] = Kλ( s) + Q s Kλ = Q s Kλs. Since he compensaed Poisson process is maringale we can added i o he price process wihou affecing he drif of he asse µ. Hence equaions (1.) and (1.9) mus become ds S = (µ λk)d + σdw + (J 1)dN (1.10) and ln S = ln S 0 + ) (µ λk σ + σw + N i=1 ln J i. (1.11) Upon aking exponenials of equaion (1.11) we ge he soluion for S, { ( ) } S = S 0 exp µ σ N + σw + j i. (1.1) where he variable j = ln J is normally disribued as saed in equaion (1.6). Equivalenly we can wrie S = S 0 exp i=1 ) } {(µ σ N + σw i=1 J i. (1.13) Noice ha if all he jumps ha occur in he inerval (0, ) are zero hen N i=1 j i = 0 or N i=1 J i = 1 as we would expec. We noe here ha an alernaive mehod for adjusing he Meron model o accoun for he predicable pars of he jumps involves incorporaing he expecaion of he jump erm ino he Wiener process. By definiion, a Wiener process saisfies W 0 = 0 and has normally disribued incremens wih zero mean, W W s N(0, s) = W N(0, ). In order o incorporae he predicable expecaion of he jump erm, he mean of our Wiener process would have o become Kλ σ. 8
1.3 Opion Pricing under he Meron Model Meron [16] derived a pricing formula for a European call opion on he asse S under he Meron jump diffusion model. He used a dela-hedging argumen similar o ha used by Black and Scholes in he derivaion of he call opion pricing formula under geomeric Brownian moion. If no jump occurs in he asse price hen he only risk in he asse evoluion comes from he Wiener process W. The porfolio ha is long one call opion and shor unis of he asse S is perfecly hedged of his risk so he porfolio grows a he risk-free rae r. However if a jump does occur he porfolio is exposed o he jump risk as he hedge has no eliminaed he risk associaed wih he Poisson process N. Meron makes he assumpion ha he risk associaed wih he jumps in he asse price is diversifiable since he jumps in he individual asse price are uncorrelaed wih he marke as a whole, in oher words he risk is unsysemaic. If his is he case hen he Capial Asse Pricing Model (CAPM) says he jump erms offer no risk premium and he asse sill grows a he risk free rae. Meron derived a parial differenial equaion, which was solved by an infinie series. If we denoe he -price of he call on asse S wih srike K under he Meron jump diffusion model by V M (S, K, r, T) where r is he risk-free rae and T is he opion mauriy, hen V M (S, K, r, T) = λ(t e )(λ(t ))i V call (S, r i, σ i, T), (1.14) i! i=0 where λ = λe α+ δ r i = r λ(e α+ (α + δ δ) 1) + i (T ) σ i = δ σ + i T and he normally disribued jump variables j have mean α and variance δ. Here V call is he Black Scholes call price. This pricing formula is no a closed formula as i involves an infinie sum ha needs o be runcaed in order o calculae a price. However he sum converges rapidly so accurae prices can be achieved wih a relaively small number of erms. Figure 1.1 shows he prices admied by he Meron jump diffusion model for wo differen ses of jump parameers compared wih Black Scholes prices. The prices admied by he Meron model for hese ses of jump parameers are nearly always greaer han he Black Scholes prices and he difference beween he prices increases as he jump parameers increase in magniude. For opions deep in-he-money or deep ou-he-money he difference beween he prices is smaller han he opions ha are a-he-money. We will discuss his in furher deail in Secion 3.3. 9
60 50 European call opion prices for Black Scholes and Meron models Black Scholes λ = 0.5, α = 0.1, δ = 0.1 λ = 1, α = 0.5, δ = 0.5 40 Call price 30 0 10 0 50 100 150 00 Srike K Figure 1.1: Black Scholes prices and Meron prices for a European call opion. 1.4 The Srucure of his Projec Under he assumpion ha he underlying sock price evolves according o a geomeric Brownian moion we can compue explici pricing formulas for calls and pus and also a wide range of exoic opions oo, for example see Haug [11]. However under a jump diffusion model, explici pricing formulas are harder o come by. Meron [16] derived formulas for calls and pus under he jump diffusion model using a risk-hedging argumen and hese formulas have subsequenly also been derived using equivalen maringale measures. Few explici formulas for opions under jump diffusion exis due o he increased complexiy ha jumps cause. A furher complicaion occurs when he exoic opion o be priced has a payoff ha is a funcion of wo underlying asses raher han one. This is due o he correlaion ha may or may no exis beween he wo asses. In he geomeric Brownian moion case he only correlaion beween wo asses comes via he Wiener processes. In he jump diffusion environmen however, i is possible and more realisic o have correlaed jump erms. This is he main focus of his projec. Previous work by Ike Dike [8] focused on spread opions, which are exoic opions wih a payoff ha is a funcion of ypically wo energy commodiies. The underlying asses were modelled using jump diffusion processes in which he jump erms were perfecly correlaed and he focus was on he numerical soluion of he pricing equaions. In his projec we firs we propose a model ha allows us o realisically imiae correlaion beween he jumps in wo asses. We do no focus on a paricular indusry or commodiy as a model of his ype could be applied o any opion on wo correlaed asses ha exhibi disconinuous price processes. We hen derive a pricing formula for an exchange opion on wo asses when he asses are modelled using he jump diffusion model wih correlaion. We hen price exchange opions and anoher exoic opion called a max-call under 10
he jump diffusion wih correlaion model using Mone Carlo mehods. The price of an exchange opion found using Mone Carlo mehods can hen be compared wih he price admied by he pricing formula already derived. We also invesigae implied volailiy and implied correlaion of jump diffusion prices. 11
Chaper Correlaed Jumps Suppose we choose o model wo correlaed asses S (1) and S () using a jump diffusion model. Boh asses are modelled by equaion (1.10), ds (i) S (i) = (µ i λ i K i )d + σ i dw (i) + (J (i) 1)dN (i) i = 1,. (.1) Here µ i is he consan mean of he asse i, σ i is he consan volailiy, W (1) and are Wiener processes wih correlaion ρ 1 and volailiy marix {σ ij } i,j=1. The W () variables J (i) are he log-normally disribued jump variables. The Poisson processes N (1) and N () have consan inensiies denoed by λ 1 and λ respecively. The wo jump erms are o be parially correlaed meaning ha if one asse jumps hen he oher will jump wih a probabiliy p. We consruc N (1) and N () in such a manner o achieve his. We consruc hem using hree independen Poisson processes denoed by n (1), n () and n (3). These independen Poisson processes n (i) have inensiy denoed by λ i and he discree probabiliy densiy funcion P(n (i) = j) = (λ i ) j j! e λ i derived in Secion 1.1. We hen consruc N (1) and N () as follows, (.) N (1) = n (1) + n (3) (.3) N () = n () + n (3). (.4) Changing he consrucion of he Poisson process N (1) in his way does no change i s disribuion. This is seen using characerisic funcions. For a random variable Y he characerisic funcion of Y is defined as ϕ Y (ω) = E(e iωy ). If Y has probabiliy densiy funcion f Y, he characerisic funcion is given by { E(e iωy ) = eiωx f Y (x)dx eiωx f Y (x) 1 if Y is coninuous if Y is discree. (.5)
We know n (i) has a Poisson disribuion wih inensiy λ i and discree probabiliy densiy funcion given by equaion (.), herefore i has a characerisic funcion equal o φ n (i)(θ) = i=0 e iθx(λ i ) x e λ i x! = e λ i (λ i e iθ ) x i=0 x! = e λ i (e iθ 1). We define N (i) = n (i) + n (3) and using he fac ha he characersic funcion of he sum of wo independen random variables is simply he produc of he characerisic funcions of he individual random variables we ge, φ N (i)(θ) = φ n (i)(θ) φ n (3)(θ) I follows ha N (i) Poiss(λ i + λ 3 ). The Poisson processes N (1) and N () hrough n (1) and n () = e λ i (e iθ 1) e λ 3 (e iθ 1) = e (λ i +λ 3 )(e iθ 1) [17], a change of variables and inegraing ou n (3) are capable of producing independen jumps and simulaneous jumps hrough n (3). Using work by M Kendrick resuls in he join probabiliy densiy funcion for N (1) and N (), min(i,j) P(N (1) = i, N () = j) = Boh asses are modelled by k=0 e (λ 1 +λ +λ 3 ) λ i k 1 λj k λk 3 (i k)!(j k)!k!. (.6) ds (i) S (i) = (µ i λ i K i λ 3 K 3 )d + σ i dw (i) + (J (i) 1)dn (i) + (J (3) 1)dn (3) (.7) for i = 1,. Here K i = exp ( ) α i + 1 δ i for i = 1,, 3. Using he same mehods as in Secion 1. his is solved for S (i) o give, n (i) n (3) S (i) = S (i) 0 exp (µ i λ i K i λ 3 K 3 ) + σ i W (i) + j (i) k + j (3) l. (.8) Here we have chosen ha here be hree jump variables, one for each independen Poisson process. Figure.1 shows an example of a simulaion of his model. We see ha asse 1 exhibis an independen jump near ime = 0.51 whereas boh asses exhibi common jumps near ime = 0. and = 0.65. Alernaively we could have chosen no o differeniae beween jumps iniiaed by eiher Poisson process in each asse. This is a simple case of he model oulined in equaion (.7) in which J (i) = J (3). We will use his model when we derive a pricing formula for an exchange opion in Secion 4. 13 k=0 l=0
14 Price simulaions of correlaed asses, λ 1 =λ =1, λ 3 =3 1 = 1, S() 0 = 1 Sock Price, S (1) 0 10 8 6 4 0 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure.1: Simulaion of asse prices; λ 1 = λ = 1, λ 3 = 3..1 Covariance of wo asses We have ha he wo asses are modelled by equaion (.1) and herefore [ ] ds (i) E = µ S (i) i d. The variance of asse i is found by ( (1)) ds Var S (1) [( (1) ds = E S (1) ( (1))) ] ds E S (1) [ ( = E (λi K i + λ 3 K 3 )d + σ 1 dw (i) Using he following rules (d) = 0, ] + (J (i) 1)dn (1) + (J (3) 1)dn (3) ) ddw (i) = 0 for i = 1,, (dw (i) ) = σi d for i = 1,, dw (1) dw () = ρ 1 d, his simplifies o ( (1)) ds Var = (σ1 + λ iki + λ 3K3 )d. S (1) 14
The covariance is found as follows. ( (1) ) [( ds Cov, ds() (1) ( (1) ds ds = E E S (1) S () S (1) [ ( = E (λ1 K 3 + λ 3 K 3 )d + σ 1 dw (1) S (1) ))( ds () S () ( ()))] ds E S () + (J (1) 1)dn (1) + (J (3) 1)dn (3) ( (λ K + λ 3 K 3 )d + σ dw () + (J () 1)dn () + (J (3) 1)dn (3) ) ] Again using he rules oulined above, his simplifies o ( (1) ) ds Cov, ds() = E[σ S (1) S () 1 σ ρ 1 + (J (3) 1)(J (3) 1)dn (3) ] = (σ 1 σ ρ 1 + K3 λ 3)d The correlaion of he wo asses is defined as ( ) ds ( (1) ) Cov (1), ds() ds Corr, ds() S (1) S () = S (1) S () ( ) Var Var( ds (1) S (1) ) ds () S () ) = σ 1 σ ρ 1 + K3 λ 3 σ 1 + K1 λ 1 + K3 λ 3 σ + K λ + K3 λ. (.9) 3 I is clear o see ha increasing he correlaion ρ 1, beween he wo Wiener processes will increase he correlaion beween he wo asses. Increasing he size of λ 3 compared wih λ 1 and λ, and increasing K 3 compared wih K 3 and K will also increase correlaion beween he asses. Figure. shows how he correlaion beween he asses varies when K 1 and K are kep fixed and K 3 is varied beween 0 and. For each plo we se σ 1 = σ = 0.5, ρ 1 = 0.5 and λ i = 1 for i = 1,, 3. Figure.3 shows how he correlaion varies when λ 1 and λ are kep fixed and λ 3 varies beween 0 and 10. σ 1, σ and ρ are as above and we se K i = 1 for i = 1,, 3.. Characerisic Funcions We find he log-price process for asse S (i) by dividing equaion (1.1) by S (i) 0 and aking he naural logarihm. We denoe he log-price process of asse S (i) by X (i), X (i) ( ) S (i) ln = S (i) 0 ( ) n (i) µ i σ i λ ik i λ 3 K 3 + σ i W + j=0 n (3) j (i) j + k=0 j (3) k, where j (i) and j (3) are normally disribued as in equaion (1.6). If he compound Poisson processes were removed he sock would follow a geomeric Brownian moion 15
1 Correlaion beween he asses for varying values of K i 0.9 0.8 Correlaion 0.7 0.6 0.5 0.4 0.3 0. K 1 = K = 0 0.1 K 1 = K = 1 K 1 = K = 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 1.8 The value of K 3 Figure.: Correlaion beween he asses for varying values of K 3 0.9 Correlaion beween he asses for varying values of λ i 0.8 0.7 0.6 Correlaion 0.5 0.4 0.3 0. 0.1 λ 1 = λ = 1 λ 1 = λ = 5 λ 1 = λ = 10 0 0 1 3 4 5 6 7 8 9 The value of λ 3 Figure.3: Correlaion beween he asses for varying values of λ 3 16
and as in he Black Scholes case, X (i) would be normally disribued. However he compound Poisson processes make he log-reurn process non-normal. We will use he law of oal probabiliy o derive an expression for he probabiliy densiy funcion of he variable X (i). We will hen calculae he characerisic funcion and deduce he momens of X (i). A special case of he law of oal probabiliy for discree random variables says ha given n muually exclusive evens, B 1,...,B n, whose probabiliies sum o one, we have n P(A) = P(A B i ). i=0 Using he law of oal probabiliy and denoing he probabiliy densiy funcion on X (i) by f (i) X we have = P(n (i) = j)p(n (3) = k)p(x (i) n (i) = j, n (3) = k), f X (i) j=0 k=0 where P(X (i) n (i) = j, n (3) = k) is he probabiliy densiy funcion of X (i) condiional on he compound Poisson processes being equal o j and k respecively. The probabiliy densiy funcions for he Poisson processes were inroduced in Secion 1.1. As menioned above he probabiliy densiy funcion of X (i) is no normal. However when we condiion on he Poisson processes being equal j and k respecively, he probabiliy densiy funcion of X (i) is normal. If he jump erms above are removed, X (i) becomes he familiar Black-Scholes log-reurn which is normally disribued. The jump variables j are also normally disribued. Therefore when condiioned on he Poisson processes X (i) is simply a sum of independen normally disribued variables, which iself is normal. We have ha j (i) N(α i, δi ) for i = 1,, 3. (.10) I follows ha if we condiion on n (i) = j and n (3) = k, (( X (i) N µ i σ i λ ik i λ 3 K 3 ) + jα i + kα 3, σ i + jδ i + kδ 3 If we le µ = (µ i σ i λ i K i λ 3 K 3 ) + jα i + kα 3 and σ = σi + jδ i + kδ 3 hen P(X n i = j, n3 = k) = 1 { } (x µ) σ π exp. σ Using he definiion of a characerisic funcion in equaion (.5) and he definiions of µ and σ from above he characerisic funcion of X (i) is { } ϕ (i) X (ω) = e iωx e λi (λ i ) j e λ3 (λ 3 ) k 1 (x µ) j! k! σ π exp σ j=0 k=0 { } e λi (λ i ) j e λ3 (λ 3 ) k 1 (x µ) = j! k! σ π exp + iωx σ j=0 k=0 ). 17
We now complee he square wihin he exponenial brackes o ge ϕ (i) X (ω) = j=0 k=0 1 σ π exp e λi (λ i ) j e λ3 (λ 3 ) k exp {i µω ω σ } j! k! { x ( µ + iω σ } ) σ dx We see ha he erms on he second line are he only erms conaining x, herefore hey are he only erms ha remain under he inegral. They consiue he probabiliy densiy funcion of a normally disribued variable wih mean µ + iω σ and variance σ. Thus when inegraed over he whole real line his erm is equal o one. We now subsiue µ and σ back in and rearrange, (ω) = e λi (λ i ) j e λ3 (λ 3 ) k ϕ X (i) j=0 k=0 j=0 k=0 j=0 k=0 j! k! e λi (λ i ) j e λ3 (λ 3 ) k j! k! { [( ) ] exp iω µ i σ i λ ik i λ 3 K 3 + jα i + kα 3 { ( exp iω µ i σ i λ ik i λ 3 K 3 } ω σi + jδi + kδ3 = ) } ω σ i exp{iωα i ω δi } j exp{iωα 3 ω δ3} k = e λi (λ i e {iωα i ω δi } ) j e λ3 (λ 3 e {iωα 3 ω δ3 } ) k j! k! { ( ) } exp iω µ i σ i λ ik i λ 3 K 3 ω σ i. We separae he double series ino he produc of wo single infinie sums and herefore we ge { } { } ϕ (i) X (ω) = exp λ i (e {iωα i ω δi } 1) exp λ 3 (e {iωα 3 ω δ3 } 1) { ( ) } exp iω µ i σ i λ ik i λ 3 K 3 ω σ i. is (.11) The characerisic funcion of a Poisson disribued random variable P Poiss(λ) ϕ P (θ) = exp{λ(e iθ 1)} and he characerisic funcion of a normally disribued random variable Q N(µ, σ ) is } ϕ Q (θ) = exp {µiθ σ θ. s 18
I is clear ha he characersic funcion of X (i) given in equaion (.11) is he produc of he characerisic funcions of wo Poisson variables and he characerisic funcion of a normal variable, his resul makes sense inuiively..3 Momens Now we have he characerisic funcion for he random variable X (i) we are ineresed in using i o analyse cerain properies of he disribuion of X (i). If we ake he naural logarihm of ϕ (i) X and expand as a Taylor series we ge, [19], (iω) (iω) r g(ω) = ln ϕ (i) X (ω) = κ 1 (iω) + κ + + κ r r! +... where κ i is he ih cumulan of he disribuion of X (i). We can hen find he mean, variance, skewness and excess kurosis of X (i) using, E[X (i) ] = κ 1, Var[X (i) ] = κ, Skewness[X (i) ] = κ 3, (κ ) 3 Excess Kurosis[X (i) ] = κ 4. The nh cumulan is found using he following: κ κ n = i n d(n) g dω (n)(0) (.1) where he superscrip n denoes he nh derivaive wih respec o ω. Using equaion (.1) we find ( ) E[X (i) ] = µ i σ i λ i K i λ 3 K 3 + λ i α i + λ 3 α 3, Var[X (i) ] = σ i + λ i (α i + δ i ) + λ 3 (α 3 + δ 3) Skewness[X (i) ] = λ i (3δ i α i + α i) + λ 3 (3δ 3α 3 + α 3) Excess Kurosis[X (i) ] = λ i (3δ i + 6δ i α i + α i) + λ 3 (3δ 3 + 6δ 3α 3 + α 3). The evoluion of an asse under a jump diffusion model is equal o a Black Scholes geomeric Brownian moion when boh λ i and λ 3 are zero and/or when he expeced jumps are exacly zero for all, i.e when µ i, µ 3, δ i and δ 3 are zero. Trivially, i is clear from he expression of he variance of X (i) under a jump diffusion ha if he jumps erms are exacly zero for all, he variance of he asse under modelled using a jump diffusion is equal o ha of a geomeric Brownian moion. As he mean and variance of he jump variables increase, so does he variance of he asse, as one would expec. 19
The real ineres hough is in he skewness and kurosis. The whole aim of his model is o incorporae skewness and lepokuriciy of reurns, characerisics of marke reurns ha he Black Scholes model fails o reproduce. Provided ha α j 0 and δ j α j 3 for j = i, 3, hen he reurns will be posiively or negaively skewed. Wheher he reurns are posiively or negaively skewed will depend on he relaive sizes of he jump parameers. The reurns will display excess kurosis provided he jump parameers are no exacly zero. This means he log-reurns should display heavy ails ha ge hicker as he jump parameers increase. Firs we verify he skewness of he log-reurns by ploing he probabiliy densiies of he log-reurns for differen choices of α i and α 3. We ake µ i = 0.05, σ i = 0. and plo he log-reurns over he inerval (0, 0.5) for he asse wih saring price S (i) 0 = 50. We fix λ i = λ 3 = 0.5 and δ i = δ 3 = 0.1 and observe he cases when α i and α 3 are boh equal o -0.5, 0 and 0.5. Figure.4 plos he log-reurn densiies for he hree choices of jump means. We see ha when he jump means are negaive he 4 3.5 Log reurn densiy of Meron jump diffusion model α i = α 3 = 0.5 α i = α 3 = 0 α = α = 0.5 i 3 3.5 Densiy 1.5 1 0.5 0 1 0.5 0 0.5 1 1.5 Log reurn Figure.4: Log-reurn densiies for various choices of α i and α 3. log-reurn densiies are negaively skewed, when he he jump means are posiive he log-reurns are posiively skewed and when he jump means are zero he log-reurn densiy is no skewed. This agrees fully wih he analysis above and is confirmed by he summary saisics in Table.1. Also for non-negaive jump means he log-reurns are bi-model. We now se he jump means o zero so ha he skewness remains zero and keep he oher parameers as in Figure.4 apar from he jump inensiies λ i and λ 3. If we increase hese jump inensiies we should see heavier ails in he log-reurn disribuion. This is eviden in Figure.5. As he jump inensiies increase, he ails in he log-reurn densiy clearly ge hicker. This is because an increase in jump inensiy causes a greaer number of jumps in he asse price. I is hese jumps ha cause he oulier reurns. Exra ouliers give he disribuion heavier ails. This is 0
Jump means mean(x ) var(x ) skewness(x ) excess kurosis(x ) α i = α 3 = -0.5-0.037 0.0758-1.737 3.5601 α i = α 3 = 0 6.0747e-04 0.015-0.0055 0.5080 α i = α 3 = 0.5-0.04 0.0747 1.7086 3.547 Table.1: Saisics of log-reurn densiies for differen jump means. 3.5 3 Log reurn densiy of Meron jump diffusion model λ i = λ 3 = 1 λ i = λ 3 = 10 λ = λ = 50 i 3.5 Densiy 1.5 1 0.5 0.5 1.5 1 0.5 0 0.5 1 1.5.5 Log reurn Figure.5: Log-reurn densiies for various choices of λ i and λ 3. furher raified by a QQ-plo of he log reurns when he jump means are se o zero in Figure.6. If he log-reurns were normal he QQ-plo would be linear along he red line. Normal Probabiliy Plo Probabiliy 0.999 0.997 0.99 0.98 0.95 0.90 0.75 0.50 0.5 0.10 0.05 0.0 0.01 0.003 0.001 6 4 0 4 6 Daa Figure.6: QQ-plo of log-reurns when jump means are zero. 1
Chaper 3 Mone Carlo Pricing 3.1 Equivalen Maringale Measures and Marke Incompleeness The only probabiliy measure we have menioned so far is he marke measure or physical measure denoed by P. This is he measure acually observed in he marke under which asses wih more risk generally display a greaer expeced reurn. We are now ineresed in he equivalen maringale measure. Two probabiliy measures P 1 and P on a sample space Ω wih even spaces F 1 and F respecively are equivalen if F 1 = F and for all evens ω in F 1 we have ha P 1 (ω) = 0 P (ω) = 0. Under a maringale measure all asses have an expeced reurn equal o he risk-free rae irrespecive of he amoun of risk associaed wih he asse under he marke measure P. Therefore for an asse S we have E MM [S ] = S 0 e r if ineres raes are consan where E MM is he expecaion under he maringale measure. In he Black Scholes model he asse evoluion follows a geomeric Brownian moion, which consiss of a drif erm and Brownian moion erm as in equaion (1.1). The only source of risk in his model comes from he Brownian moion erm. We define W = W + µ r σ where r is he risk-free rae and he erm µ r σ is he marke price of risk. By Girsanov s heorem here exiss a measure Q under which W is a Brownian moion and under his measure he asse price evoluion becomes ds S = rd + σd W. (3.1)
I is clear ha he expeced reurn of he asse is now equal o he risk-free rae and Q is he unique equivalen maringale measure. When we inroduce jump erms ino he evoluion of he asse as we have done in he previous secion, he ideas above no longer hold. We can now also change he probabiliy measure by changing he jump inensiies λ i and λ 3. Therefore for a space of pahs of he price of he asse S (i), λ i and λ 3 can be whaever we choose and we can hen simply adjus he drif of he Brownian moion o make he new measure a maringale measure. We have an infinie number of choices for he jump inensiies herefore we have an infinie number of equivalen maringale measures. This is known as marke incompleeness. In order o comba his problem we follow he example se by Meron and inroduce he idea of sysemaic and unsysemaic risk. Sysemaic risk is he risk inheren o he enire marke. In Secion 1.1 we discussed he possible ypes of jumps ha could occur in an asse price. Jumps ha are firm-specific or indusry-specific are associaed wih unsysemaic risk. We assume ha he only cause of jumps in he asse price is firm-specific or indusry-specific informaion enering he marke. If he Capial Asse Pricing Model (CAPM) holds hen his means he jump erm is zero-bea and hus offers no risk premium. I follows ha if we hen make a measure change by adjusing he drif of he Brownian moion and apply Girsanov s heorem, he jump diffusion evoluion is a maringale under he risk-neural measure Q. Shreve [18] discusses a jump diffusion model under which a change of measure for he Wiener process and he jump inensiies in he compound Poisson process resuls in a complee marke model, however his model requires ha he jump variables J be finiely discree disribued. In our model we have assumed ha he jump variables are log-normally disribued and are herefore clearly no finiely disribued. 3. Implemening Mone Carlo Mehods Wih many exoic derivaives, closed pricing formulas are pracically impossible o derive. As he number of socks on which he payoff is dependen increases, he complexiy increases. Therefore we mus use alernaive mehods for pricing. The basic idea behind Mone Carlo simulaion is ha running many simulaions of he payoff under he risk-neural measure Q and aking an average will give a good indicaion of he derivaive price once discouned by he risk-free rae. Suppose an opion on wo asses has a payoff funcion H(S (1) T, S() T ). If ineres raes are consan, he price a ime under he equivalen maringale measure Q is given by V (S (1), S (), ) = e r(t ) E Q [H(S (1) T, S() T )], (3.) where E Q is he expecaion under his measure. For example a call opion on asse S wih srike K has payoff funcion Ψ = max(s K, 0), herefore he -price of he call opion can be found by [ ( V call (S, K, ) = e r(t ) EˆQ Ψ S exp {(r 1 ) })] σ + σw ˆQ, (3.3) 3
where ˆQ is he risk-neural measure. In order o price an opion on wo asses under he jump diffusion model we mus firs simulae he evoluion of asses 1 and under he risk-neural measure. As discussed in he previous secion we assume he jump erms offer no risk premium and we make a change of measure so he adjused Brownian moion erm becomes a maringale under he risk-neural measure. S (i) = S (i) 0 exp (r λ ik i λ 3 K 3 ) + σ i d n (i) (i) W + j=1 n (3) j (i) j + k=1 j (3) k, (3.4) where W (i) ( is a) Wiener process under Q. We aim o simulae he evoluion of = log as we can hen ake exponenials and muliply by S (i) 0 o ge a X (i) S (i) S (i) 0 sample pah for he asse. In order o simulae equaion (3.4) over he inerval (0, T), we discreize ime ino inervals of lengh and over each inerval (, + ) follow he seps, 1. generae Z N(0, 1),. generae N i Poisson(λ i ) and N 3 Poisson(λ 3 ) ; if N i = N 3 = 0 se M i = M 3 = 0 and go o sep 5, 3. generae j i 1,...,j i N i and j 3 1,...,j 3 N 3 where j i k N(α i, δ i ) for k = 1,...,N i and j 3 l N(α 3, δ 3 ) for l = 1,...,N 3, 4. se M i = j i 1 + + ji N i and M 3 = j 3 1 + + j3 N 3, 5. se X i + = X i + (r 1 σ i λ i K i λ 3 K 3 ) + σ Z + M i + M 3. Figure.1 in Secion is a plo of he sample pahs of wo correlaed asses modelled under jump diffusions using his simulaion mehod. When simulaing he payoff of a European derivaive, we are only ineresed in he value of he underlying asses a ime T as his is he only ime a which he derivaive can be exercised. Therefore compuing a value for S (i) a many poins over he inerval (0, T) is no necessary. We ake o equal he whole inerval (0, T) and he seps oulined above can be repeaed. We compue n sample pahs and use he value of he asses a ime T o calculae he expeced derivaive payoff for each sample, P i = H(S (1) T, S() T The esimae of he expeced payoff P is found by P = 1 n 4 ) for 1,..., n. n P i. i=1
We hen discoun he expeced payoff by dividing by he price of zero-coupon bond wih price one a and mauriy a T. Thus H(S (1), S (), ) = e rt P. As one would expec, he sandard error of he esimaed derivaive price decreases as he number of sample pahs increases. In fac he Mone Carlo price is guaraneed o converge o he acual price a he number of simulaions ends o infiniy. The sandard error of he esimae is given by ( ) Error = 1 m ( (1 m) P P i ) This decreases proporionally wih he square roo of he number of simulaions. If we would like o halve he error in he esimae we mus run four imes as many simulaions. This is highlighed when we use Mone Carlo echniques o price a European call opion using a varying number of samples. Figure 3.1 plos he Mone Carlo European call price for an increasing number of simulaions agains he Black Scholes price. The error beween he Mone Carlo price and he Black Scholes price i=1 60 55 Comparison of Mone Carlo call price wih Black Scholes call price Mone Carlo Black Scholes 50 Call Price 45 40 35 30 5 0 15 10 1 3 4 5 6 7 Number of Simulaions, 10 x Figure 3.1: Mone Carlo and Black Scholes prices of a European call is exremely small around 10 6 simulaions. Therefore for he calculaions we will run here we will use 10 6 simulaions o ensure saisfacory accuracy. Obviously his number of simulaions is by no means small and compuing ime can become and issue. Since Malab is a vecor based program he shores compuaion ime is achieved when he simulaions are run simulaneously in vecor forma. However, his requires soring a huge amoun of daa as each simulaion requires a number of random variables o be sampled and sored. For sample sizes greaer han 10 7 we run ino memory problems. A way o comba his is o run he simulaions in 5
a loop wih each loop compuing one payoff and adding i o a sum ha can be divided by he number of samples in he sum for he esimaed payoff. This way we only need o sore he daa for one sample pah a a ime and we sidesep any memory problems. However his mehod is exremely ime consuming in Malab. Since Figure 3.1 suggess sample sizes of he order 10 6 suffice we will use he vecor echnique here. 3.3 Pricing Opions on One Asse Under he Black Scholes model he only source of volailiy in he underlying asse price comes from he driving Wiener process. The sandard deviaion of he asse under a Black Scholes GBM is sd BS (S ) = σ BS, (3.5) where σ BS is he sandard deviaion of he driving Wiener process. If we denoe he sandard deviaion of he Wiener process in he jump diffusion model by sd jd hen in ligh of Secion.3, he sandard deviaion of he asse under a jump diffusion model is sd jd (S ) = (σ jd + λ i(α i + δ i ) + λ 3(α 3 + δ 3 )). Since he jump inensiies λ i and λ 3 are always posiive i is clear ha if we se σ BS = σ jd hen he volailiy of jump diffusion model will always be greaer or equal o ha of he Black Scholes model. The price of a European call opion increases as volailiy increases herefore we would expec he jump diffusion model o reurn European call opion prices greaer han hose from he Black Scholes model. This is illusraed in Figures 3. and 3.3. We se S 0 = 100, σ jd = σ BS = 0. and vary he jump diffusion parameers o compare he prices admied by boh models over a range of srikes. The differen ses of jump diffusion parameers used are highlighed in Table 3.1. λ 1 λ α 1 α δ 1 δ Se 1 0.5 0.5-0.05-0.05 0.05 0.05 Se -0.1-0.1 0.1 0.1 Se 3 1 1-0.3-0.3 0.3 0.3 Se 4-0.3-0.3 0.3 0.3 Table 3.1: The jump parameers used in each se for Figure 3., 3.3 and 3.6 We see ha when he jump parameers are small as in se 1, here is very lile difference beween he Black Scholes price and he jump diffusion price for opions ha are deep in he money or deep ou of he money, however he difference increases as we ge closer o being a-he-money. When he jump parameers are large he difference beween he wo prices is an again larges when he opions are near ahe-money, however as he jump parameers increase in magniude he difference 6
90 80 70 European call opion prices for differen ses of jump diffusion parameers GBM Se 1 Se Se 3 Se 4 60 Call price 50 40 30 0 10 0 0 50 100 150 00 50 300 Srike K Figure 3.: Black Scholes price and jump diffusion price for differen ses of jump diffusion parameers for 30 K 70. 50 45 40 35 European call opion prices for differen ses of jump diffusion parameers GBM Se 1 Se Se 3 Se 4 Call price 30 5 0 15 10 5 0 70 80 90 100 110 10 130 Srike K Figure 3.3: Black Scholes price and jump diffusion price for differen ses of jump diffusion parameers for 70 K 130. 7
beween he opions deep in-he-money and deep ou-of-he-money is also large, as we would expec. Now we choose a se of jump diffusion parameers and calculae he sandard deviaion of he asse under his model using equaion (3.5). We will hen se he Black Scholes sandard deviaion equal o his jump diffusion sandard deviaion, i.e. σ BS = (σ jd + λ i(α i + δ i ) + λ 3(α 3 + δ 3)). If we choose he sandard deviaion of he wo models o be he same hen we should be o see he effec of he skewness and lepokuriciy on he opion prices since he mean and variance of he asses under boh models should mach bu he skewness and excess kurosis will no. If we ake he jump parameers as in se oulined in Table 3.1 hen sd jd (S ) = (σjd + λ i(αi + δ i ) + λ 3(α3 + δ 3 )) = 0.34641. Figure 3.4 plos he jump diffusion prices for he parameers of se and he Black Scholes prices for sd BS (S ) = 0.34641 over a range of srikes. The prices mach closely 40 35 European call opion prices for Black Scholes and jump diffusion. GBM Jump Diffusion 30 Call price 5 0 15 10 5 70 80 90 100 110 10 130 Srike K Figure 3.4: Black Scholes price and jump diffusion price when sd BS (S ) = sd jd (S ). for a-he-money opions and deep ou-he-money opions bu as he srike passes he sock saring price of 100 he he difference beween he wo prices increases and he jump diffusion model slighly under-esimaes he price compared wih he Black Scholes price. One possible explanaion for his is ha under geomeric Brownian moion an opion ha is deep in-he-money has less risk of rereaing back o being ou-he money whereas adding he jump erms adds risk and he chance of he price falling is herefore greaer. 8
3.4 Implied Volailiy The Black Scholes pricing equaion for a European call on an asse S requires a number of inpu parameers o compue a price, V call (S 0, K, r, T, σ BS ) = price. These being he curren value of he sock, he srike of he opion, he risk-free ineres rae, he ime unil expiry of he opion and he sock volailiy. The firs four inpu parameers are eiher observable or are predeermined condiions of he conrac. The final inpu however, he sock volailiy, is unobservable. This means he value of he opion depends on an esimae of he consan volailiy of he underlying sock over he inerval (0, T). The opion price is a monoonically increasing funcion of he volailiy, as illusraed in Figure 3.5. Therefore he higher he volailiy of 35 European call price as a funcion of σ; S 0 =100, K=10 30 5 Call opion price 0 15 10 5 0 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 σ Figure 3.5: European call price as a funcion of σ. he underlying sock, he higher he opion price. I also means ha for every opion price here exiss a unique volailiy ha mus realise ha price when oher inpus are kep consan. This is he implied volailiy of ha price. In general, opions on he same asse bu wih differen srikes and expiraion daes have differen implied volailiies. For example, he plos of he implied volailiy of marke call opion prices agains he opion srike ofen resemble a smile meaning deep in-he-money or deep ou-of-he-money opions have higher implied volailiies han a-he-money opions. This conradics he assumpion made in he Black-Scholes model ha he underlier has consan volailiy. Calculaing he implied volailiy for he marke price of a call opion over a range or srikes or mauriy daes gives an insigh ino he marke expecaions of he asse volailiy and is herefore of ineres o an invesor. We can compue call prices under he jump diffusion model using Mone Carlo mehods for a range of srikes. We can hen use hese prices o find he Black Scholes 9
implied volailiy. The Black Scholes formula is difficul o inver in a closed form so we have o use a numerical mehod. If V call (.) is he Black Scholes price as funcion of volailiy and C jd is he call price from he jump diffusion model, we need o find σ imp such ha V call (σ imp ) C jd = 0 using a roo finding funcion. Malab acually has an inbuil funcion ha compues he implied volailiy given he required inpus. Figure 3.6 plos he implied volailiies of jump diffusion call prices for differen ses of jump diffusion parameers. The ses of jump diffusion parameers used are oulined in Table 3.. λ 1 λ λ 3 α 1 α α 3 δ 1 δ δ 3 Se 1 0.5 0.5 1-0.05-0.05-0.05 0.05 0.05 0.05 Se 1 1 1-0.1-0.1-0.1 0.1 0.1 0.1 Se 3 1 1 1-0. -0. -0.3 0.05 0.05 0.05 Se 4 0.5 0.5 1-0.05-0.05-0.05 0. 0. 0.3 Table 3.: The jump parameers used in each se for Figure 3.6 0.9 0.8 Black Scholes implied volailiy of jump diffusion call prices GBM Se 1 Se Se 3 Se 4 Black Scholes implied volaily 0.7 0.6 0.5 0.4 0.3 0. 0.1 70 80 90 100 110 10 130 Srike K Figure 3.6: Implied volailiies of jump diffusion call prices for differen ses of jump parameers; 70 K 130. As expeced he prices given by he model wih he larges jump parameers display he highes implied volailiy. The implied volailiy of he Black Scholes model is perfecly consan as we would hope bu he jump diffusion prices display slighly greaer implied volailiies for smaller srike prices. 30
Chaper 4 Exoic Opions 4.1 Exchange Opions Consider wo asses S (1) and S (), which have prices a ime denoed by S (1) and respecively. We assume no dividends are paid on he sock and all reurns are S () from capial gains only. An exchange opion gives he holder he righ bu no he obligaion o exchange a quaniy of asse S (1) for a quaniy of asse S (). Here we will discuss a European exchange opion, which can only be exercised a he expiry dae T in he fuure. Exchange opions can occur in many differen conexs. For example in a buyou or akeover, holders of shares in he arge company may be given he opion of exchanging heir holdings for a quaniy of shares in he acquiring company. Anoher example is in currency markes, an opion o swap a quaniy of one foreign currency for a predeermined volume of anoher is an exchange opion. Given he definiion above, he holder would only exercise he opion if a ime T he value of sock S () was greaer han S (1). Therefore he payoff is given by E(S (1) T, S() T ) = max(q S () T Q 1S (1), 0), (4.1) where Q i is he quaniy of asse i. If we assume ha fracions of socks can be raded, wihou loss of generaliy we can define he opion as he opion o swap one uni of asse S (1) for Q Q 1 unis of asse S (). Here we will assume ha Q Q 1 = 1. Upon exercising he opion he holder could hen insananeously sell he more expensive asse S () in he marke o receive a profi or hold i in his/her porfolio having acquired i for cheaper han he marke price. Noice ha he definiion of a European exchange opion on he asses S (1) and S () wih expiry T is also he definiion of a European call opion on asse S () wih srike price S (1) T and expiry T. Margrabe [14] derived a pricing formula for an exchange opion when he underlying asses are modelled using a geomeric Brownian moion. If we denoe he -price of a European exchange opion by V E (S (1), S (), ρ, ) hen V E (S (1), S (), ρ, ) = S (1) N(d 1 ) S () N(d ) (4.) T 31