Changes of Numeraire for Pricing Futures, Forwards, and Options
|
|
- Brianne Johnson
- 5 years ago
- Views:
Transcription
1 Changes of Numeraire for Pricing Fuures, Forwards, and Opions Mark Schroder Michigan Sae Universiy A change of numeraire argumen is used o derive a general opion pariy, or equivalence, resul relaing American call and pu prices, and o obain new expressions for fuures and forward prices. The general pariy resul unifies and exends a number of exising resuls. The new fuures and forward pricing formulas are ofen simpler o compue in mulifacor models han exising alernaives. We also exend previous work by deriving a general formula relaing exchange opions o ordinary call opions. A number of applicaions o diffusion models, including sochasic volailiy, sochasic ineres rae, and sochasic dividend rae models, and jump-diffusion models are examined. A self-financing porfolio is called a numeraire if securiy prices, measured in unis of his porfolio, admi an equivalen maringale measure. The mos commonly used numeraire is he reinvesed shor-rae process; he corresponding equivalen maringale measure is he risk-neural measure. Geman, El Karoui, and Roche (1995 show ha oher numeraires can simplify many asse pricing problems. In his aricle, we build on heir resuls and, using he reinvesed asse price as he numeraire, unify and exend he lieraure on opion pariy, or equivalence, resuls relaing American call and pu prices for asse and fuures opions. The same numeraire change is used o obain new pricing formulas for fuures and forwards ha are ofen simpler o compue in mulifacor models. Finally, we use a numeraire change o simplify exchange opion pricing, exending a similar resul in Geman, El Karoui, and Roche o dividend-paying asses. The change of numeraire mehod is mos inuiive in he conex of foreign currency derivaive securiies. As discussed by Grabbe (1983, an American call opion o buy 1 DM, wih dollar price process S, for K dollars is equivalen o an American pu opion o sell K dollars, wih DM price process K/S, for a srike price of 1 DM. The dollar price of he call mus herefore equal he produc of he curren exchange rae, S, and he DM price of he pu. The call price is compued using he dollar value of a U.S. I am graeful o Kerry Back (he edior and Cosis Skiadas for heir many helpful suggesions. Thanks also o Peer DeMarzo, Ma Jackson, Naveen Khanna, David Marshall, Rober McDonald, Jim Moser, Phyllis Payee, and an anonymous referee for heir commens. This aricle subsumes Schroder (1992. Address correspondence o Mark Schroder, The Eli Broad Graduae School of Managemen, Deparmen of Finance, Michigan Sae Universiy, 323 Eppley Cener, Eas Lansing, MI , or schrode7@pilo.msu.edu. The Review of Financial Sudies Winer 1999 Vol. 12, No. 5, pp c 1999 The Sociey for Financial Sudies
2 The Review of Financial Sudies/v12n51999 money marke accoun as he numeraire, while he pu price is compued using he dollar value of a German money marke accoun as he numeraire. Corresponding o he change of numeraire is a change in probabiliy measure, from he risk-neural measure for dollar-denominaed asses o he risk-neural measure for DM-denominaed asses. As suggesed in Grabbe (1983, and developed in laer aricles, an analogous relaion applies o any asse opion. A call opion o buy one uni of an asse, wih dollar price process S, for K dollars is he same as a pu opion o sell K dollars, worh K/S unis of asse, for one uni of asse. Muliplying he asse denominaed pu price by he curren asse price convers he price ino dollars. The same numeraire change can be used o obain he ineres pariy heorem which expresses he ime zero dollar forward price, G (T, for ime T delivery of one DM as he spo currency rae imes he raio of wo discoun bond prices: G (T = S B (T /B (T, where B (T is he ime zero dollar price of a discoun bond paying $1 a T, and B (T is he ime zero DM price of a discoun bond paying 1 DM a T. This resul can be exended o forward conracs on any asse. A key issue examined in his aricle is he change of measure ha corresponds o a change of numeraire. Under he risk-neural measure, he drif rae of he reurns of he asse price S is he shor rae minus he dividend rae. In Secion 1 we show ha he drif rae of he reurns of S 1 (he price of dollars in unis of asse under he new measure is he dividend rae minus he shor rae. The reversal of he roles of he shor rae and dividend rae is inuiive because under he new numeraire he asse is riskless while dollars are risky. Example 1 shows ha he change of measure can resul in more suble modificaions and can change boh he inensiy and disribuion of jumps in jump-diffusion models. In Secion 2 we show ha he measure change also alers he drif erms of nonprice sae variables, such as in sochasic volailiy and sochasic ineres rae models. Example 1. Assume ha he shor rae and dividend rae are boh zero, and he asse price follows a Poisson jump process wih inensiy λ under he risk-neural probabiliy measure, Q. A jump ime τ i,i = 1, 2,...,he sock price raio has he Bernoulli disribuion { us(τi, wih Q-probabiliy p S(τ i = ds(τ i, wih Q-probabiliy 1 p, and beween jumps, ds /S = (1 µλ d, τ i < <τ i+1, i =, 1,..., where µ pu + (1 pd is he expeced price raio a jumps, and τ. 1144
3 Changes of Numeraire for Pricing Fuures, Forwards, and Opions A change of numeraire o he underlying asse price is associaed wih he new measure Q, where d Q/dQ = S T /S. A jumps, he value of a dollar measured in unis of he asse saisfies 1 { S(τ i 1 = u 1 S(τ i 1, wih Q-probabiliy puµ 1 d 1 S(τ i 1, wih Q-probabiliy (1 pdµ 1, and beween jumps ds( 1 /S( 1 = (µ 1λ d, τ i < <τ i+1, i =, 1,... The inensiy of he jump process under Q is µλ, which can be obained using he maringale propery of S under Q: Q(τ 1 > = E Q (1 {τ1 >}S T /S = E Q (1 {τ1 >}S /S = e (1 µλ Q(τ 1 >. The disribuions of he reurns of S under Q and S 1 under Q are idenical only in he special case when u = d 1 and µ = 1. We show ha subjec o some common echnical resricions (Assumpions 1 and 2 below, any American call price formula is he same, afer a change of numeraire, o an American pu price formula. This resul is useful for obaining prices, derivaives of prices wih respec o model parameers, and early exercise boundaries for pu opion formulas from he properies of he corresponding call opion formula. Previous aricles derive he correc pu-call equivalence formulas only for some special cases. The geomeric Brownian moion case (see Example 2 below is derived in McDonald and Schroder (199, Bjerksund and Sensland (1993, and, for fuures opions, in Byun and Kim ( Chesney and Gibson (1993 use a change of numeraire o obain a closed-form European formula for sock-index opions when he shor rae is sochasic from Jamshidian and Fein s (199 closedform European formula for opions on asses wih a sochasic payou rae. However, he change of measure is incorrec, in par because i neglecs o make he appropriae modificaion o he drif erm of he sae variable. 1 The jump probabiliies under Q can be verified using he general resuls in he appendix, or from Q ({S(τ 1 = us(τ 1 } {τ 1 } = pue Q (1 { τ1 }S(τ 1 /S(, for any T, and S(τ 1 /S( = ue (1 µλτ 1 on {S(τ 1 = us(τ 1 }. 2 Bjerksund and Sensland (1993 apply a resul in Olsen and Sensland (1991 which demonsraes ha he curren asse price can be facored ou in cerain conrol heory problems where he fuure reward is muliplicaive in he price of an asse. Their resul could be used o derive he pariy resul in a diffusion seing when he reurn volailiy is any funcion of he price, subjec o he price process being sricly posiive (such as he CEV model below. The Olsen and Sensland (1991 resuls can be generalized by allowing he payoff in Proposiion 1 below o depend on a vecor of conrols. See also Kholodnyi and Price (1998, who derive equivalence resuls for geomeric Brownian moion and he binomial model. They use no-arbirage argumens o derive general equivalence resuls in a seing where each opion price is a deerminisic funcion of he curren underlying asse price (for example, Markovian S and deerminisic r and δ. In he foreign currency conex, he equivalence resuls are in erms of he generaors of he domesic and foreign evoluion (or presen value operaors. 1145
4 The Review of Financial Sudies/v12n51999 Example 5 below shows he correc measure change in ha model. Carr and Chesney (1996 derive a formula relaing call and pu prices in a one-facor model in which he volailiy of he underlying price obeys a symmery condiion (see Example 3 below. Baes (1991 derives equivalence formulas for American pu and call opions on fuures for some special cases o es classes of opion pricing models. Example 8 builds on his idea and derives general condiions under which he equivalence formula akes a paricularly simple form: swiching he roles of he curren fuures price and he srike price in he American call opion formula gives he price of an oherwise idenical American pu opion. Secion 1 presens he numeraire change mehod and he general resuls using he reinvesed asse price as he numeraire. Secion 2 presens examples of hese resuls. The Appendix derives he numeraire change for a general jump-diffusion model ha includes all he Secion 2 examples as special cases. 1. The Reinvesed Asse Price as he Numeraire We presen he general change of numeraire argumen before giving he main resuls. Fix a finie ime horizon [, T ]. 3 Le Y denoe some reinvesed asse price process. Tha is, Y is he ime balance of an invesmen sraegy of buying an asse and reinvesing all dividends ino new shares. Le R represen he reinvesed shor rae wih uni iniial invesmen: R = exp( r sds, where r is he shor rae process. If π is he sae price densiy process, hen πy and π R are P-maringales. I follows ha Y/R is a Q-maringale, where dq/dp π T R T. Tha is, when measured in unis of he numeraire R, Y is a maringale wih respec o he risk-neural probabiliy measure Q. Geman, El Karoui, and Roche (1995 show ha we ge he same resul when we replace R wih anoher self-financing porfolio V wih V = 1 (and V/R a Q-maringale. Then Y/V is a Q-maringale, where d Q/dP π T V T (or, equivalenly, d Q/dQ V T /R T. Tha is, when measured in unis of he numeraire V, Y is a Q-maringale. This simple change of numeraire is he basis for all he pricing resuls below. The resuls are very general in ha we allow for incomplee markes and price and sae variable dynamics which are neiher coninuous nor Markovian. The main assumpion is he exisence of a risk-neural measure. Assumpion 1. There exiss a risk-neural measure, Q, such ha every reinvesed price process relaive o he reinvesed shor-rae process is a Q-maringale. 3 We assume hroughou he exisence of a complee probabiliy space (, F, F, P and a filraion F = {F ; } which saisfies he usual condiions. See Proer (1992 for he required condiions and for all he resuls on semimaringale heory needed in his aricle. All processes are assumed o be adaped. 1146
5 Changes of Numeraire for Pricing Fuures, Forwards, and Opions The self-financing porfolio ha serves as he numeraire for our main resuls is he reinvesed asse price process wih uni iniial balance. Le S be a semimaringale represening he price process of an asse wih a proporional dividend payou rae δ. 4 We assume hroughou ha S is sricly posiive. 5 The value of he numeraire porfolio a any ime is S exp( δ sds/s. The probabiliy measure Q ha corresponds o he new numeraire is Q(A = E Q (1 { A } Z T, A F. (1 where Z is defined as he raio of he new and old numeraires: Z e (δ s r s ds S /S, [, T ]. (2 In oher words, he Radon-Nikodym derivaive is d Q/dQ = Z T. All he resuls of his secion hold when S is replaced by a fuures price process F (wih delivery dae D T ifweseδequal or. To jusify his, we consruc a numeraire porfolio wih value F exp( r sds/f a any ime by mainaining a long posiion of exp( r sds/f fuures conracs a and adding or subracing mark-o-marke gains and losses from a money marke accoun, whose ime-zero balance is se o $1 [his sraegy is described in Duffie (1992: chap. 7]. Alernaively, we can use he fac ha F is a Q- maringale and direcly define Z o be F /F. The following proposiion provides a general pricing formula under a change of numeraire o he reinvesed asse price. The consan K will serve as he srike price in he opion pricing applicaions below. The process S represens he price of KS dollars measured in unis of he asse S. Proposiion 1. Define S = KS /S and Q by (1. Then he ime-zero price of an asse wih he F τ -measurable payoff P τ a he sopping ime τ [, T ] is ( E Q e τ rsds P τ = E Q (e τ δsds P τ S τ /K. Furhermore ds = S (r δ d + dm d S = S (δ r d + d M, S = K, 4 I is easy o exend he resuls o discree dividends. In addiion o he proporional dividend rae δ, suppose he asse pays discree cash dividends C i a sopping imes T i, i = 1, 2,...All he resuls are hen generalized by adding he erm log[1 + C i /S(T i ]o δ sds hroughou. The exponenial T i [,] of his addiional erm represens he addiional shares of asse accumulaed by reinvesing he discree dividends ino new shares purchased a he ex-dividend price S. 5 In many applicaions he resuls exend o he case where S has an absorbing boundary a zero. In a diffusion model, for example, consruc a modified sock price process whose diffusion erm is killed he firs ime he price his a small posiive consan. The dominaed convergence heorem can be used o evaluae he limi of he expecaion (in Corollary 1, for example as his consan goes o zero. 1147
6 The Review of Financial Sudies/v12n51999 where M and M and local maringales under Q and Q, respecively. 6 The quadraic variaions of M and M saisfy (dm 2 /S 2 = (d M 2 / S 2 beween jumps. 7 Proof. Assumpion 1 implies ha he price is given by he firs expecaion. Applying he numeraire change, ( E Q e τ ( rsds P τ = E Q Z τ e τ δsds P τ S τ /K = E Q (e τ δsds P τ S τ /K, where he las equaliy is obained using ieraed expecaions and he maringale propery of Z. The equaion for he reurns of S follows because he raio of he reinvesed price process o he reinvesed shor rae process is a Q-maringale. The equaion for he reurns of S follows because he raio of he shor rae price process o he reinvesed price process is a Qmaringale. The equaliy, beween jumps, of he insananeous volailiies of reurns follows from Iô s lemma and from he Girsanov Meyer heorem (a generalizaion of Girsanov s heorem o a non-brownian seing, which implies ha M M is absoluely coninuous in beween jumps. The proposiion shows ha he insananeous reurn variances of S and S are idenical beween jumps. Example 1 illusraes ha a jumps, he squared reurns will generally be differen. The firs applicaion of Proposiion 1 relaes call prices o pu prices under a change of numeraire. Corollary 1. Define S = KS /S and Q by Equaion (1. Then he value of a call opion on S is he same, afer a change of numeraire, as he value of a pu opion on S: ( E Q e τ rsds max[s τ K, ] = E Q (e τ δsds max[s S τ, ], for any sopping ime τ T. 6 A sufficien condiion for M and M o be maringales under Q and Q, respecively, is ha r and δ are bounded processes. 7 The quadraic variaion of any semimaringale Y is denoed by [Y, Y ], and can be decomposed ino is coninuous and jump componens: [Y, Y ] = [Y, Y ] c + s ( Y s 2, where [Y, Y ] c = [Y c, Y c ] and Y c is he pah-by-pah coninuous par of Y. For coninuous Y (or for beween jumps, i is common o wrie (dy 2 insead of d[y, Y ]. The quadraic variaion is invarian o changes in measure. See Proer (1992: chap. II for he formal definiions. 1148
7 Changes of Numeraire for Pricing Fuures, Forwards, and Opions The lef-hand side represens he value of a European call opion expiring a τ wih srike price K and underlying price process S. The righ-hand side represens he value of a European pu opion, also expiring a τ, bu wih a srike price S and underlying price process S. The roles of he shor rae and asse payou rae are reversed in he call and pu price expressions. Corollary 1 also holds for American opions under Assumpion 2 below. Corollary 2. Define S = KS /S and Q by Equaion (1. Then he value of an asse-or-nohing binary opion on S is he same, afer a change of numeraire, as he value of a cash-or-nohing binary opion on S: ( E Q e τ rsds S τ 1 { Sτ K } = S E Q (e τ δsds 1 { S S τ }, for any sopping ime τ T. Anoher inerpreaion is obained if τ min[t, inf{: S K }] and S has coninuous sample pahs. Then he lef-hand side is he value of a barrier, or firs-ouch digial, opion paying K dollars when he asse price S rises o K ; and he righ-hand side is he value of a barrier opion paying S when S falls o S (he evens {S τ K } and {S S τ } are idenical. 8 When δ, Corollary 2 can be obained from Theorem 2 in Geman, El Karoui, and Roche (1995. The resul is derived independenly under he assumpions of geomeric Brownian moion and consan r and δ by Carr (1993, Dufresne, Keirsead, and Ross (1997, and Ingersoll (1997. When r and δ are deerminisic (exenions o he sochasic case are sraighforward, Corollary 2 shows ha any European opion price can be derived from he probabiliies Q(S T K and Q(S S T [see also Theorem 2 in Geman, El Karoui, and Roche (1995]. The nex corollary presens a new fuures price expression. Le F (T denoe he ime-zero fuures price for delivery of asse S a ime T. Wih coninuous marking o marke, he fuures price equals he risk-neural expecaion of he spo price a delivery [see Duffie (1992: chap. 7]: F (T = E Q (S T. (3 When he ineres rae and he payou rae are deerminisic, he fuures price is simply F (T = S exp[ T (r s δ s ds]. When eiher he ineres rae or he payou rae is sochasic, however, a change of measure o Q gives an expression ha is ofen easier o compue and also more clearly emphasizes he role of he cos of carry in fuures pricing. 8 Reiner and Rubinsein (1991 price a variey of binary and one-sided barrier opions assuming he asse price follows geomeric Brownian moion. 1149
8 The Review of Financial Sudies/v12n51999 Corollary 3. The fuures price is he produc of he spo price and he expecaion, under Q, of he exponenial of he cos of carry: ( T F (T = S E Q e (r s δ s ds. In he general diffusion model in he appendix, for example, he Q-expecaion on he righ-hand side doesn depend on he sock price process if he insananeous covariance beween asse reurns and changes in he sae variable is no a funcion of he asse price. Example 5 shows ha he compuaion of he fuures price using Corollary 3 is paricularly simple wih a consan volailiy sock reurn process and an Ornsein Uhlenbeck sae variable driving eiher r or δ. The nex corollary presens a new forward price expression. Le B (T denoe he ime dollar price of a discoun bond paying $1 a T : ( B (T = E Q e T r s ds F, T. (4a Le B (T denoe he ime price, measured in unis of asse, of a discoun bond paying one uni of he asse a T : ( B (T = S 1 E Q e T r s ds S T F = E Q (e T δ s ds F, T. (4b Leing G (T denoe he ime-zero forward price for delivery of asse S a ime T, Duffie (1992: chap. 7 shows ha ( G (T = E Q e T / rsds S T B (T. (5 Corollary 4 follows from Equaions (4b and (5. Corollary 4. The forward price is given by he produc of he spo price and he raio of asse and dollar denominaed discoun bond prices: G (T = S B (T /B (T, where B (T and B (T are defined by Equaion (4. The main advanage of Corollary 4 is in a model where boh he shor rae and payou rae are sochasic. If he shor rae is deerminisic, hen forward and fuures prices are equal and Corollary 3 can be used. If he payou rae is deerminisic, hen Corollary 4 holds rivially. When we se δ r and reinerpre S as a fuures price wih delivery dae T (which implies ha he forward on he fuures conrac is equivalen o a forward on he asse underlying he fuures conrac, hen Corollary 4 provides a 115
9 Changes of Numeraire for Pricing Fuures, Forwards, and Opions simple expression for he raio of forward and fuures prices on he same underlying asse. The final applicaion of Proposiion 1 is o he valuaion of exchange opions. Le S a and S b denoe wo asse prices and δ a and δ b denoe heir corresponding payou raes. Then ds i = S i (r δ i d + dmi, i {a, b}, where M i is a Q-local maringale. Corollary 5 expresses he value of an exchange opion as an ordinary call opion by changing he numeraire o he reinvesed price of asse a. Corollary 5. Define S b = S bsa /Sa and d Q/dQ = e (δa s rsds ST a /Sa. Then he value of an opion o receive one uni of asse b in exchange for one uni of asse a is he same, afer a change of numeraire, as he value of a call opion on S b : ( E Q e τ rsds max[sτ b Sa τ, ] = E Q (e τ δa s ds max[ S τ b Sa, ], for any sopping ime τ T. Furhermore d S b = S b (δa δ b d + d M b, S b = Sb, where M b is a local maringale under Q. The righ-hand side of he firs equaion is he value of an ordinary call opion wih underlying asse process S b, shor rae process δ a, and fixed srike price S a. Corollary 5 exends a similar resul in Geman, El Karoui, and Roche (1995 o dividend-paying asses and American-syle exercise (under Assumpion 2. To apply Proposiion 1 o American opions, we need o assume ha he price of an American opion is he supremum, over all sopping imes τ,of he risk-neural expeced discouned payoff from exercising a τ. Assumpion 2. Le p be he ime zero price of an American opion allowing he holder o exercise and receive, a any sopping ime τ [, T ], he payoff P τ, where P is an adaped process. Then ( p = sup E Q e τ rsds P τ (6 τ [,T ] Karazas (1988 proves Equaion (6 in a complee markes diffusion seing for American opions on asses. When markes are incomplee, his characerizaion is problemaic [see Duffie (1992: chap. 2]. Because of possible ineracion beween he sae price densiy and he choice of exercise policy, he wo-sep procedure of firs deermining he risk-neural T 1151
10 The Review of Financial Sudies/v12n51999 measure and hen compuing Equaion (6 may no be valid. Neverheless, i is common in he lieraure o ignore his ineracion and firs assign a marke price of risk o he relevan sae variables (in effec, deermining he risk-neural measure, hen price opions as in Equaion (6. 2. Examples The examples in his secion are all special cases of he general jumpdiffusion model presened in he appendix. Throughou he remainder of he aricle, I le W [W 1,...,W d ] and W [ W 1,..., W d ] be vecors of d independen sandard Brownian moions under he measures Q and Q, respecively. Example 2. Consan elasiciy of variance (CEV. The risk-neural asse price process is ds = (r δ d + νs ξ dw 1, ξ [ 1, 1], S where ν and ξ are consans, and r and δ are deerminisic. 9 Geomeric Brownian moion corresponds o ξ =. Closed-form soluions for European call and pu opions in his model have been derived by Cox (1975 [see also Schroder (1989]. Under he measure Q, S is also a CEV process: d S S = (δ r d + ν S ξ d W 1, S = K, wih an absorbing boundary a zero (see Foonoe 7, where ν ν(ks ξ and ξ ξ. Using Corollary 1, we obain he pricing formula for he American pu from he formula for he American call by exchanging S and K, exchanging r and δ, and replacing ν wih ν and ξ wih ξ. For he case of geomeric Brownian moion, he equivalence formula is paricularly simple because ν = ν and ξ = ξ. The nex example shows ha he Carr and Chesney (1996 pu-call symmery resul can be obained from Corollary 1. Example 3. Carr and Chesney (1996. Le he risk-neural asse price process saisfy ds = (r δ d + σ(s dw 1 S, where r and δ are deerminisic and σ( f ( log( / yk for some bounded funcion f and fixed y R +. The funcional form of σ saisfies 9 The resuls in his and all he succeeding examples are unchanged if any of he consan parameers are permied o be deerminisic funcions of ime. 1152
11 Changes of Numeraire for Pricing Fuures, Forwards, and Opions Carr and Chesney s symmery condiion which ensures ha σ(s = σ(ŝ,, where Ŝ yk/s. 1 The dynamics of Ŝ under Q are herefore d Ŝ Ŝ = (δ r d + σ(ŝ d W 1, Ŝ = yk/s. When S represens a fuures price process (and δ = r, he reurn disribuions of S and Ŝ are idenical. Applying Corollary 1 and rearranging, we obain E Q ( e τ rsds max[s τ K, ] S K = E Q (e τ δsds max[y Ŝ τ, ]. Ŝ y The numeraor on he lef-hand side is he price of a call opion on S wih srike price K. The numeraor on he righ-hand side is he price of a pu opion on Ŝ wih srike price y, and wih he roles of r and δ swiched. These call and pu opions have he same moneyness in he sense ha Ŝ /y = K/S. For he case of geomeric Brownian moion, where f is a consan funcion, we le y S o reconcile he resul wih Example 2. Example 4. Sochasic volailiy model of Heson (1991. The risk-neural asse price and volailiy processes are ds = (r δ d + ν dw 1 S, dν = (µ κν d + ψ ν (ρdw ρ 2 dw 2, where r and δ are deerminisic, and µ, κ, ψ and ρ [ 1, 1] are consans. Recall ha W 1 and W 2 are independen sandard Brownian moions under Q, and herefore ρw ρ 2 W 2 is sandard Brownian moion wih insananeous correlaion of ρ wih W 1. Under he numeraire change, d S = (δ r d + ν d W 1 S, S = K, dν = (µ [κ ρψ]ν d ψ ν (ρd W ρ 2 d W 2. The numeraire change resuls in a modificaion o he mean reversion parameer of he volailiy process, and reverses he sign of he covariance beween insananeous asse reurns and volailiy changes. 1 Proposiion 1 implies ha he ime insananeous reurn volailiy of Ŝ is σ(s = σ(yk/ŝ. The funcional form of σ implies σ(yk/ŝ = σ(ŝ. 1153
12 The Review of Financial Sudies/v12n51999 The nex example includes a one-dimensional sochasic sae variable driving eiher he shor rae or he dividend rae. Example 5. Sochasic dividend and sochasic shor rae models. Le he risk-neural asse price and one-dimensional sae variable saisfy ds S = (r δ d + σ dw 1, dx = (µ κ X d + ν(ρdw ρ 2 dw 2, where he coefficiens σ, µ, κ, ν, and ρ [ 1, 1] are consans. Under he numeraire change, d S S = (δ r d + σ d W 1, S = K, dx = (µ + ρσν κ X d ν(ρd W ρ 2 d W 2. The sae variable sill is Ornsein Uhlenbeck under Q, bu wih a differen drif parameer. (a Sochasic shor rae. Le δ be deerminisic and r = f (X for some f : R R. This model includes Ornsein Uhlenbeck ( f (x = x, x R and lognormal ( f (x = e x shor rae processes. From Corollary 3, he price of a fuures conrac on S for delivery a T is F (T = S exp ( T δ s ds E Q [ exp ( T This is simpler han evaluaing he sandard Equaion (3: ( T F (T = E Q (S T = S exp δ s ds ] r s ds. [ T E Q (exp r s ds 1 ] 2 σ 2 T + σ WT 1. Corollary 1 implies ha he price of a call opion on S is equal, afer he numeraire change, o he price of a pu opion on S wih a deerminisic shor rae and a sochasic dividend rae. (b Sochasic dividend rae. Le r be deerminisic and δ = f (X.We obain he fuures and forward prices from eiher Corollary 3 or Corollary 4: ( T F (T = G (T = S exp r s ds E Q [ exp ( T δ s ds ]. 1154
13 Changes of Numeraire for Pricing Fuures, Forwards, and Opions Again, his is simpler han evaluaing he sandard Equaion (3: ( T ( [ T F (T = S exp r s ds E Q exp δ s ds 1 ] 2 σ 2 T + σ WT 1. Corollary 1 implies ha a call opion on S is equal, afer he numeraire change, o he price of a pu opion on S wih a deerminisic dividend rae and a sochasic shor rae. Example 5 shows ha opion pricing models for sochasic dividend models can be obained from sochasic ineres rae models and vice versa. The European call opion formula in Jamshidian and Fein s (199 Ornsein Uhlenbeck δ and consan r model can be obained, via some parameer changes, from he European pu opion formula in Rabinovich s (1989 Ornsein Uhlenbeck r and consan δ model. The example also illusraes how Corollary 3 can simplify fuures pricing by reducing a wo-facor problem o a one-facor problem when he volailiy erms of he asse reurns and he sae variable do no depend on he asse price. Example 6. Exchange opions. The risk-neural price processes of asses a and b saisfy ds a S a ds b S b = (r δ a d + σ a dw 1, = (r δ b d + σ b (ρdw ρ 2 dw 2, where he volailiy coefficiens, σ a and σ b, and he insananeous correlaion beween asse reurns, ρ [ 1, 1], are consans. The shor rae, r, and he dividend raes, δ a and δ b, are funcions of he m-dimensional sae variable vecor, X, which saisfies dx = µ(x d + φ(x dw, where µ is m 1, φ is m d, and again W [W 1,...,W d ] is sandard Brownian moion under Q. The price raio S b S bsa /Sa under Q is he consan volailiy process d S b / S b = (δ a δ b d + (σ a ρσ b d W 1 1 ρ 2 σ b d W 2, S b = Sb, where again W [ W 1,..., W d ] is sandard Brownian moion under Q. The drif of X under Q is modified by adding he produc of he volailiy of asse a and he firs column of he sae-variable volailiy marix: dx = [µ(x + σ a φ(x e]d φ(x d W, 1155
14 The Review of Financial Sudies/v12n51999 where e = [1,,...,]. From Corollary 5, he valuaion of an exchange opion can be reduced o he compuaion of an ordinary call opion on S wih shor rae δ a and dividend rae δ b. The nex example is he jump-diffusion model of Meron (1976. Example 7. Meron (1976. As in Example 1, he asse price follows a Poisson jump process wih inensiy λ under he risk-neural probabiliy measure, Q. A jump ime τ i,i = 1, 2,...,he sock price raio is lognormally disribued: log[s(τ i /S(τ i ] (α, γ 2, where (m,vdenoes a normal disribuion wih mean m and variance v. Beween jumps he sock price saisfies ds = [r δ λ(e α+γ 2 /2 1] d S + σ dw 1, τ i < <τ i+1, i =, 1,..., where τ. Using he same calculaion as in Example 1 (or he general resuls in he appendix, he inensiy under Q is equal o he produc of he inensiy under Q and he expeced price raio a jumps: λ = λ exp(α γ 2. The appendix shows ha he disribuion funcions under Q and Q of he sock price raio, denoed by ( and (, respecively, saisfy (dy = (dy exp( α 1 2 γ 2 y. From [ 1 (dy = exp 1 ( ] log(y α 2 dy, 2πγy 2 γ i is sraighforward o show ha he logarihm of he sock price raio under Q is sill normally disribued wih variance γ 2, bu wih mean α + γ 2. The dynamics of S KS /S are herefore log[ S(τ i / S(τ i ] ( α γ 2,γ 2, i = 1, 2,..., d S S = [ ] δ r λ(e α γ 2 /2 1 d + σ d W 1, τ i < <τ i+1, i =, 1,... When he mean sock reurn a jumps is zero, ha is α = γ 2 /2, hen λ = λ and he jump reurns of S under Q and S under Q have he same 1156
15 Changes of Numeraire for Pricing Fuures, Forwards, and Opions disribuions. In his special case, Corollary 1 implies ha simply swiching he roles of he srike price and he curren asse price, and swiching he roles of he shor rae and dividend rae in he American call price formula gives he American pu price. Baes (1991 derives his special case for fuures opions from he parial differenial equaion for he opion price. When applied o he case of fuures opions, several previous examples conain special cases in which he disribuions of he reurns of F under Q and F under Q are idenical. 11 In such cases he equivalence relaionship in Corollary 1 akes a paricularly simple form: he American pu price is obained from he American call price formula by simply swiching he roles of he srike price and curren fuures price. A change of variables can hen be used o relae American call and pu prices on he same underlying fuures price process. Example 8 shows general condiions under which Corollary 1 can be used o relae call and pu prices on he same underlying fuures price process, and, using he ideas of Baes (1991, shows how hese condiions can be esed. I is easy o show ha he same condiions imply ha he geomeric average of he early exercise boundaries of oherwise idenical American calls and pus is equal o he srike price. Example 8. Empirical implicaions for fuures opions. Le F denoe he fuures price for delivery a D, where D T. The fuures price is assumed o follow a jump-diffusion process under Q wih inensiy λ(x and jumps a τ i,i = 1, 2,...,when he fuure price raio has he disribuion Q (F(τ i /F(τ i y = (y, y, and beween jumps, df = λ (X (1 µ d + σ(x dw 1 F τ i < <τ i+1, i =, 1,..., where τ and µ y R + yd (y is he expeced price raio a jumps. The m-dimensional sae variable vecor, X, saisfies dx = µ(x d + φ(x dw, where W [W 1,...,W d ] is sandard Brownian moion under Q, and he coefficiens µ and φ have he appropriae dimensions. The shor rae may also be a funcion of X. We obain an equivalence formula for calls and pus wih he same ime zero underlying price by defining ˆF = F 2 /F (= F F /K. Using he resuls in he appendix, ˆF is a jump-diffusion process under Q wih inensiy 11 See Example 1 when u = d 1 and µ = 1; Example 2 when ξ = ; Examples 4 and 5 when ρ = ; and Example 7 when α = σ 2 /
16 The Review of Financial Sudies/v12n51999 λ ( = λ ( µ and dynamics Q ( ˆF(τ i / ˆF(τ i y 1 = (y, y >, i = 1, 2,..., d ˆF ˆF = λ (X (1 µ 1 d + σ(x d W 1, τ i < <τ i+1, i =, 1,..., ˆF = F, dx = [µ(x + σ(x φ(x e]d φ(x d W, where (dy = (dyyµ 1, and, as earlier, e [1,,...,] and W [ W 1,..., W d ] is sandard Brownian moion under Q. Sufficien condiions for he disribuions of F under Q and ˆF under Q o be idenical are (a φ( e = (he insananeous changes in he sae variables and fuures price are uncorrelaed, and (b λ (no jumps or (y = [y 1, u (du, y >. Noe ha he resricion on he disribuion funcion in (b implies ha µ = 1. When he fuures price raio a jumps has a discree disribuion, as in Example 1, hen he resricion on is equivalen o (y = y 1 (y 1, y > [ (y and (y 1 are he Q-probabiliies of oucomes y and y 1, respecively]. When he jump disribuion funcion is differeniable, as in Example 7, hen he resricion is equivalen o (y = (y 1 y 3, y >. Defining x = K/F, hen Corollary 1 implies he following relaionship beween calls and pus on fuures prices wih he same iniial value: ( E Q e τ rsds max[f τ F x, ] = xe Q (e τ rsds max[f /x ˆF τ, ], (7 for any x >. Under condiions (a and (b, Equaion (7 relaes call and pu prices on he same underlying fuures price process. We can es hese condiions by comparing he relaive prices of American calls and pus. For example, if boh condiions hold, hen oherwise idenical a-he-money calls and pus should be priced he same. Baes (1991 proves Equaion (7, using parial differenial equaion mehods, for he cases of geomeric Brownian moion, Meron s (1976 jumpdiffusion model wih zero-mean jump reurns (α = γ 2 /2 in Example 7 above, and for he case of a diffusion sock price process and an uncorrelaed one-dimensional sae variable represening sochasic volailiy. 1158
17 Changes of Numeraire for Pricing Fuures, Forwards, and Opions Appendix: Price and Sae-Variable Dynamics This appendix derives price and sae-variable processes under a change of numeraire and corresponding change of measure for a general class of diffusion and jump-diffusion processes. Le W [W 1,...,W d ] be a vecor of d independen sandard Brownian moions under he risk-neural measure Q. The asse price, S, and he sae variables, X [X 1,...,X m ], saisfy ds S = (r δ d + σ(s, X dw 1 dx = µ(x d + φ(x dw, where µ is m 1, and φ is m d. To simplify noaion (and wihou loss of generaliy, he differenial of S is defined as a funcion of he differenial of W 1 only, and hus he volailiy process σ is a scalar. The shor rae, r, and payou rae, δ, are given by r = β(x and δ = κ(x, where β and κ are real-valued funcions. I is easy o generalize he model o allow he parameers of X, as well as r and δ, o depend on S also. The Radon Nikodym derivaive [Equaion (2] has he explici soluion [ Z = exp 1 ] σ(s s, X s 2 ds + σ(s s, X s dw 1 s. 2 Define he d-lengh column vecor e = [1,,...,]. By Girsanov s heorem, W = e σ(s s, X s ds W is d-dimensional sandard Brownian moion under Q, where d Q/dQ = Z T. Defining S = KS /S and applying Iô s lemma and Girsanov s heorem, we obain, d S = (δ r d + σ(ks / S, X d W 1 S, S = K, dx = [µ(x + φ(x σ (KS / S, X e] d φ(x d W. The modificaion o he drif, φ(x σ (KS / S, X e, represens he insananeous covariance beween asse reurns and he incremens in he vecor of sae variables. We now inroduce d jump processes, each indexed by i, i = 1,...d. Each jump process is characerized by he double sequence (Tn i, J n i; n = 1, 2,..., where T n represens he ime and Jn i he amoun of he nh jump.12 Le B(R denoe he Borel σ -algebra of subses of he real line. For each se A B(R and i {1,...d}, he couning process N i (A represens he number of jumps wih a magniude in he se A by ime. The jump processes are assumed o be independen of W and are assumed o saisfy [N i ((,, N 1 ((, ] =, a.s., i 1; ha is, he jumps of he firs process do no 12 See Brémaud (1981, for all he needed resuls on poin processes. This discussion borrows heavily from chaper VIII. 1159
18 The Review of Financial Sudies/v12n51999 coincide wih he jumps of he oher processes. 13 The couning measure p i (d dy is defined as p i ((, ] A = N i (A, A B(R, i = 1,...,d. Le λ i (dy denoe he inensiy kernel of pi (d dy, i = 1,...,d; for each A B(R, λ i (A is an F -predicable process. Wrie he inensiy kernel as λ i (dy = λi i (dy, i = 1,...,d, where λ i λ i (R and i (dy = λi (dy/λi on {λ i > }. The process i is a disribuion funcion for each. Loosely speaking, λ i d can be inerpreed as he probabiliy, condiional on F, of a jump in he nex d unis of ime; i (A can be inerpreed as he probabiliy of a jump wih magniude in he se A condiional on F and given ha a jump occurs a. Define he compensaed poin processes q [q 1,...,q d ], where q i (d dy = p i (d dy λ i i (dyd, i = 1,...,d. For any bounded and F -predicable process f (, y, he process M i defined by M i = f (s, yq i (ds dy, >, i = 1,...,d R is a maringale [see Brémaud (1981 for less resricive condiions on f ]. A he jump imes, M i (T i n = Mi (T i n + f i (T i n, J i n, n = 1, 2,..., and beween jumps, dm i = f i (, yλ i i (dyd, T i n 1 < < T i n, n = 1, 2,... R The asse price and he sae variables saisfy ds = (r δ d + σ(s, X dw 1 + g(s, X, yq 1 (d dy, S R dx = µ(x d + φ(x dw + G(X, yq(d dy, R where σ and g are real-valued funcions, µ is m 1, and φ and G are each m d. We allow λ i and i o be funcions of S and X. 13 The Brownian moion W inroduced above is defined on he probabiliy space ( W, F W, P W. The jump processes are defined on he probabiliy space ( p, F p, P p where he filraion is ha generaed by he hisory of he processes: F p = σ (N i s (A; s [, ], A B(R, i {1,...d}. On he produc space, (, F, P ( W p, F W he couning jump processes and W are independen. F p, P W P p, 116
19 Changes of Numeraire for Pricing Fuures, Forwards, and Opions The Radon Nikodym derivaive [Equaion (2] is [ Z = exp 1 ] σ 2 s 2 ds + σ s dw 1 s g 1 (S s, X s, yλ 1 s 1 (dyds R [1 + g(s(t 1 1 n, X (Tn, J 1 n ]1 { Tn 1 }. n 1 The Girsanov Meyer heorem implies W = e σ(s s, X s ds W is d-dimensional sandard Brownian moion under Q. The inensiy kernel of p 1 under Q is characerized by λ 1 = λ 1 [1 + g(s, X, y] 1 (dy, R and 1 (dy = 1 (dy 1 + g(s, X, y R [1 + g(s, X, y] 1 (dy. The following simple heurisic derivaion can be used o obain he inensiy kernel under Q. Suppose here have been exacly n 1 jumps in he asse price before ime. Then λ 1 (dyd = Q ( T 1 n [, + d], J 1 n [y, y + dy] F = Z 1 E Q ( Z+d 1 { T 1 n [,+d], J 1 n [y,y+dy] } F, where he maringale propery of Z and ieraed expecaions have been { used o ge he second equaliy. Now subsiue Z +d = Z [1 + g(s, X, y]on Tn 1 [, + d]} (ignoring smaller-order erms o ge λ ( 1 (dyd = [1 + g(s, X, y]q T 1 n [, + d], J 1 n [y, y + dy] F = [1 + g(s, X, y]λ 1 (dyd. The inensiy kernels of (p 2,...,p d are unalered by he measure change. The compensaed poin processes under Q are herefore q [ q 1,..., q d ], where q 1 (d dy = p 1 (d dy λ 1 1 (dyd and q i = q i, i = 2,...,d. The processes under he numeraire change saisfy d S = (δ r d + σ(ks / S, X d W 1 S g(ks / S, X, y R 1 + g(ks / S, X, y q1 (d dy, 1161
20 The Review of Financial Sudies/v12n51999 where dx = µ (KS / S, X d φ(x d W + G(X, y q(d dy, R µ (KS / S, X µ(x + φ(x σ (KS / S, X e + G(X, ye[ λ 1 1 (dy λ1 1 (dy]. R References Baes, D., 1991, Opion Pricing Under Asymmeric Processes, wih Applicaions o Opions on Deuschemark Fuures, working paper, Universiy of Pennsylvania. Bjerksund, P., and G. Sensland, 1993, American Exchange Opions and a Pu-Call Transformaion: A Noe, Journal of Business Finance and Accouning, 2, Brémaud, P., 1981, Poin Processes and Queues, Springer-Verlag, New York. Byun, S., and I. Kim, 1996, Relaionships Beween American Pus and Calls on Fuures Conracs, working paper, Korea Advanced Insiue of Science and Technology. Carr, P., 1993, Deriving Derivaives of Derivaive Securiies, working paper, Cornell Universiy. Carr, P., and M. Chesney, 1996, American Pu Call Symmery, working paper, Morgan Sanley, Groupe H.E.C. Chesney, M., and R. Gibson, 1993, Sae Space Symmery and Two Facor Opion Pricing Models, in J. Janssen and C. H. Skiadas (eds., Applied Sochasic Models and Daa Analysis, World Scienific Publishing, River Edge, N.J. Cox, J., 1975, Noes on Opion Pricing I: Consan Elasiciy of Variance Diffusions, working paper, Sanford Universiy. Duffie, D., 1992, Dynamic Asse Pricing Theory, Princeon Universiy Press, Princeon, N.J. Dufresne, P., W. Keirsead, and M. Ross, 1997, Maringale Pricing: A Do-I-Yourself Guide o Deriving Black-Scholes, in Equiy Derivaives: Applicaions in Risk Managemen and Invesmen, Risk Books, London. Geman, H., N. El Karoui, and J. Roche, 1995, Changes of Numeraire, Changes of Probabiliy Measure and Opion Pricing, Journal of Applied Probabiliy, 32, Grabbe, J., 1983, The Pricing of Call and Pu Opions on Foreign Exchange, Journal of Inernaional Money and Finance, 2, Heson, S., 1991, A Closed-Form Soluion for Opions wih Sochasic Volailiy, wih Applicaion o Bond and Currency Opions, working paper, Yale Universiy. Ingersoll, J., 1997, Digial Conracs: Simple Tools for Pricing Complex Derivaives, working paper, Yale Universiy. Jamshidian, F., and M. Fein, 199, Closed Form Soluions for Oil Fuures and European Opions in he Gibson Schwarz Model: A Noe, working paper, Merrill Lynch Capial Markes. Karazas, I., 1988, On he Pricing of American Opions, Applied Mahemaics and Opimizaion, 17,
21 Changes of Numeraire for Pricing Fuures, Forwards, and Opions Kholodnyi, V., and J. Price, 1998, Foreign Exchange Symmery, World Scienific Publishing, River Edge, N.J. McDonald, R., and M. Schroder, 199, A Pariy Resul for American Opions, working paper, Norhwesern Universiy. Meron, R., 1976, Opion Pricing when Underlying Sock Reurns are Disconinuous, Journal of Financial Economics, 3, Olsen, T., and G. Sensland, 1991, Invarian Conrols in Sochasic Allocaion Problems, in D. Lund and B. B. Øksendal (eds., Sochasic Models and Opion Values, Elsevier Science, New York. Proer, P., 1992, Sochasic Inegraion and Differenial Equaions, Springer-Verlag, New York. Rabinovich, R., 1989, Pricing Sock and Bond Opions when he Defaul-Free Rae is Sochasic, Journal of Financial and Quaniaive Analysis, 24, Reiner, E., and M. Rubinsein, 1991, Unscrambling he Binary Code, RISK, 4, Schroder, M., 1989, Compuing he Consan Elasiciy of Variance Opion Pricing Formula, Journal of Finance, 44, Schroder, M., 1992, Some Opion Pricing Resuls Obained Using a Change of Numeraire, Ph.D. summer paper, Norhwesern Universiy. 1163
The Mathematics Of Stock Option Valuation - Part Four Deriving The Black-Scholes Model Via Partial Differential Equations
The Mahemaics Of Sock Opion Valuaion - Par Four Deriving The Black-Scholes Model Via Parial Differenial Equaions Gary Schurman, MBE, CFA Ocober 1 In Par One we explained why valuing a call opion as a sand-alone
More informationMAFS Quantitative Modeling of Derivative Securities
MAFS 5030 - Quaniaive Modeling of Derivaive Securiies Soluion o Homework Three 1 a For > s, consider E[W W s F s = E [ W W s + W s W W s Fs We hen have = E [ W W s F s + Ws E [W W s F s = s, E[W F s =
More informationMatematisk statistik Tentamen: kl FMS170/MASM19 Prissättning av Derivattillgångar, 9 hp Lunds tekniska högskola. Solution.
Maemaisk saisik Tenamen: 8 5 8 kl 8 13 Maemaikcenrum FMS17/MASM19 Prissäning av Derivaillgångar, 9 hp Lunds ekniska högskola Soluion. 1. In he firs soluion we look a he dynamics of X using Iôs formula.
More informationBlack-Scholes Model and Risk Neutral Pricing
Inroducion echniques Exercises in Financial Mahemaics Lis 3 UiO-SK45 Soluions Hins Auumn 5 eacher: S Oriz-Laorre Black-Scholes Model Risk Neural Pricing See Benh s book: Exercise 44, page 37 See Benh s
More informationModels of Default Risk
Models of Defaul Risk Models of Defaul Risk 1/29 Inroducion We consider wo general approaches o modelling defaul risk, a risk characerizing almos all xed-income securiies. The srucural approach was developed
More informationMay 2007 Exam MFE Solutions 1. Answer = (B)
May 007 Exam MFE Soluions. Answer = (B) Le D = he quarerly dividend. Using formula (9.), pu-call pariy adjused for deerminisic dividends, we have 0.0 0.05 0.03 4.50 =.45 + 5.00 D e D e 50 e = 54.45 D (
More informationPricing FX Target Redemption Forward under. Regime Switching Model
In. J. Conemp. Mah. Sciences, Vol. 8, 2013, no. 20, 987-991 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/10.12988/ijcms.2013.311123 Pricing FX Targe Redempion Forward under Regime Swiching Model Ho-Seok
More informationIntroduction to Black-Scholes Model
4 azuhisa Masuda All righs reserved. Inroducion o Black-choles Model Absrac azuhisa Masuda Deparmen of Economics he Graduae Cener, he Ciy Universiy of New York, 365 Fifh Avenue, New York, NY 6-439 Email:
More informationMarket Models. Practitioner Course: Interest Rate Models. John Dodson. March 29, 2009
s Praciioner Course: Ineres Rae Models March 29, 2009 In order o value European-syle opions, we need o evaluae risk-neural expecaions of he form V (, T ) = E [D(, T ) H(T )] where T is he exercise dae,
More informationTentamen i 5B1575 Finansiella Derivat. Måndag 27 augusti 2007 kl Answers and suggestions for solutions.
Tenamen i 5B1575 Finansiella Deriva. Måndag 27 augusi 2007 kl. 14.00 19.00. Answers and suggesions for soluions. 1. (a) For he maringale probabiliies we have q 1 + r d u d 0.5 Using hem we obain he following
More informationEquivalent Martingale Measure in Asian Geometric Average Option Pricing
Journal of Mahemaical Finance, 4, 4, 34-38 ublished Online Augus 4 in SciRes hp://wwwscirporg/journal/jmf hp://dxdoiorg/436/jmf4447 Equivalen Maringale Measure in Asian Geomeric Average Opion ricing Yonggang
More informationJarrow-Lando-Turnbull model
Jarrow-Lando-urnbull model Characerisics Credi raing dynamics is represened by a Markov chain. Defaul is modelled as he firs ime a coninuous ime Markov chain wih K saes hiing he absorbing sae K defaul
More informationOption pricing and hedging in jump diffusion models
U.U.D.M. Projec Repor 21:7 Opion pricing and hedging in jump diffusion models Yu Zhou Examensarbee i maemaik, 3 hp Handledare och examinaor: Johan ysk Maj 21 Deparmen of Mahemaics Uppsala Universiy Maser
More informationProceedings of the 48th European Study Group Mathematics with Industry 1
Proceedings of he 48h European Sudy Group Mahemaics wih Indusry 1 ADR Opion Trading Jasper Anderluh and Hans van der Weide TU Delf, EWI (DIAM), Mekelweg 4, 2628 CD Delf jhmanderluh@ewiudelfnl, JAMvanderWeide@ewiudelfnl
More informationINSTITUTE OF ACTUARIES OF INDIA
INSIUE OF ACUARIES OF INDIA EAMINAIONS 23 rd May 2011 Subjec S6 Finance and Invesmen B ime allowed: hree hours (9.45* 13.00 Hrs) oal Marks: 100 INSRUCIONS O HE CANDIDAES 1. Please read he insrucions on
More informationINSTITUTE OF ACTUARIES OF INDIA
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 05 h November 007 Subjec CT8 Financial Economics Time allowed: Three Hours (14.30 17.30 Hrs) Toal Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1) Do no wrie your
More informationPricing formula for power quanto options with each type of payoffs at maturity
Global Journal of Pure and Applied Mahemaics. ISSN 0973-1768 Volume 13, Number 9 (017, pp. 6695 670 Research India Publicaions hp://www.ripublicaion.com/gjpam.hm Pricing formula for power uano opions wih
More informationChange of measure and Girsanov theorem
and Girsanov heorem 80-646-08 Sochasic calculus I Geneviève Gauhier HEC Monréal Example 1 An example I Le (Ω, F, ff : 0 T g, P) be a lered probabiliy space on which a sandard Brownian moion W P = W P :
More informationIJRSS Volume 2, Issue 2 ISSN:
A LOGITIC BROWNIAN MOTION WITH A PRICE OF DIVIDEND YIELDING AET D. B. ODUOR ilas N. Onyango _ Absrac: In his paper, we have used he idea of Onyango (2003) he used o develop a logisic equaion used in naural
More informationUCLA Department of Economics Fall PhD. Qualifying Exam in Macroeconomic Theory
UCLA Deparmen of Economics Fall 2016 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and you are o complee each par. Answer each par in a separae bluebook. All
More informationPricing Vulnerable American Options. April 16, Peter Klein. and. Jun (James) Yang. Simon Fraser University. Burnaby, B.C. V5A 1S6.
Pricing ulnerable American Opions April 16, 2007 Peer Klein and Jun (James) Yang imon Fraser Universiy Burnaby, B.C. 5A 16 pklein@sfu.ca (604) 268-7922 Pricing ulnerable American Opions Absrac We exend
More informationA UNIFIED PDE MODELLING FOR CVA AND FVA
AWALEE A UNIFIED PDE MODELLING FOR CVA AND FVA By Dongli W JUNE 2016 EDITION AWALEE PRESENTATION Chaper 0 INTRODUCTION The recen finance crisis has released he counerpary risk in he valorizaion of he derivaives
More informationVALUATION OF THE AMERICAN-STYLE OF ASIAN OPTION BY A SOLUTION TO AN INTEGRAL EQUATION
Aca Universiais Mahiae Belii ser. Mahemaics, 16 21, 17 23. Received: 15 June 29, Acceped: 2 February 21. VALUATION OF THE AMERICAN-STYLE OF ASIAN OPTION BY A SOLUTION TO AN INTEGRAL EQUATION TOMÁŠ BOKES
More informationOptimal Early Exercise of Vulnerable American Options
Opimal Early Exercise of Vulnerable American Opions March 15, 2008 This paper is preliminary and incomplee. Opimal Early Exercise of Vulnerable American Opions Absrac We analyze he effec of credi risk
More informationFinal Exam Answers Exchange Rate Economics
Kiel Insiu für Welwirhschaf Advanced Sudies in Inernaional Economic Policy Research Spring 2005 Menzie D. Chinn Final Exam Answers Exchange Rae Economics This exam is 1 ½ hours long. Answer all quesions.
More informationCompleteness of a General Semimartingale Market under Constrained Trading
Compleeness of a General Semimaringale Marke under Consrained Trading Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 666, USA Monique Jeanblanc Déparemen de
More informationA pricing model for the Guaranteed Lifelong Withdrawal Benefit Option
A pricing model for he Guaraneed Lifelong Wihdrawal Benefi Opion Gabriella Piscopo Universià degli sudi di Napoli Federico II Diparimeno di Maemaica e Saisica Index Main References Survey of he Variable
More informationTentamen i 5B1575 Finansiella Derivat. Torsdag 25 augusti 2005 kl
Tenamen i 5B1575 Finansiella Deriva. Torsdag 25 augusi 2005 kl. 14.00 19.00. Examinaor: Camilla Landén, el 790 8466. Tillåna hjälpmedel: Av insiuionen ulånad miniräknare. Allmänna anvisningar: Lösningarna
More informationAn Analytical Implementation of the Hull and White Model
Dwigh Gran * and Gauam Vora ** Revised: February 8, & November, Do no quoe. Commens welcome. * Douglas M. Brown Professor of Finance, Anderson School of Managemen, Universiy of New Mexico, Albuquerque,
More informationLecture Notes to Finansiella Derivat (5B1575) VT Note 1: No Arbitrage Pricing
Lecure Noes o Finansiella Deriva (5B1575) VT 22 Harald Lang, KTH Maemaik Noe 1: No Arbirage Pricing Le us consider a wo period marke model. A conrac is defined by a sochasic payoff X a bounded sochasic
More informationBrownian motion. Since σ is not random, we can conclude from Example sheet 3, Problem 1, that
Advanced Financial Models Example shee 4 - Michaelmas 8 Michael Tehranchi Problem. (Hull Whie exension of Black Scholes) Consider a marke wih consan ineres rae r and wih a sock price modelled as d = (µ
More informationExtended MAD for Real Option Valuation
Exended MAD for Real Opion Valuaion A Case Sudy of Abandonmen Opion Carol Alexander Xi Chen Charles Ward Absrac This paper exends he markeed asse disclaimer approach for real opion valuaion. In sharp conras
More informationSystemic Risk Illustrated
Sysemic Risk Illusraed Jean-Pierre Fouque Li-Hsien Sun March 2, 22 Absrac We sudy he behavior of diffusions coupled hrough heir drifs in a way ha each componen mean-revers o he mean of he ensemble. In
More informationCompleteness of a General Semimartingale Market under Constrained Trading
1 Compleeness of a General Semimaringale Marke under Consrained Trading Tomasz R. Bielecki, Monique Jeanblanc, and Marek Rukowski 1 Deparmen of Applied Mahemaics, Illinois Insiue of Technology, Chicago,
More informationPDE APPROACH TO VALUATION AND HEDGING OF CREDIT DERIVATIVES
PDE APPROACH TO VALUATION AND HEDGING OF CREDIT DERIVATIVES Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6066, USA Monique Jeanblanc Déparemen de Mahémaiques
More informationPricing options on defaultable stocks
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
More informationAvailable online at ScienceDirect
Available online a www.sciencedirec.com ScienceDirec Procedia Economics and Finance 8 ( 04 658 663 s Inernaional Conference 'Economic Scienific Research - Theoreical, Empirical and Pracical Approaches',
More informationErratic Price, Smooth Dividend. Variance Bounds. Present Value. Ex Post Rational Price. Standard and Poor s Composite Stock-Price Index
Erraic Price, Smooh Dividend Shiller [1] argues ha he sock marke is inefficien: sock prices flucuae oo much. According o economic heory, he sock price should equal he presen value of expeced dividends.
More informationLIDSTONE IN THE CONTINUOUS CASE by. Ragnar Norberg
LIDSTONE IN THE CONTINUOUS CASE by Ragnar Norberg Absrac A generalized version of he classical Lidsone heorem, which deals wih he dependency of reserves on echnical basis and conrac erms, is proved in
More informationComputations in the Hull-White Model
Compuaions in he Hull-Whie Model Niels Rom-Poulsen Ocober 8, 5 Danske Bank Quaniaive Research and Copenhagen Business School, E-mail: nrp@danskebank.dk Specificaions In he Hull-Whie model, he Q dynamics
More informationYou should turn in (at least) FOUR bluebooks, one (or more, if needed) bluebook(s) for each question.
UCLA Deparmen of Economics Spring 05 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and each par is worh 0 poins. Pars and have one quesion each, and Par 3 has
More informationThe Binomial Model and Risk Neutrality: Some Important Details
The Binomial Model and Risk Neuraliy: Some Imporan Deails Sanjay K. Nawalkha* Donald R. Chambers** Absrac This paper reexamines he relaionship beween invesors preferences and he binomial opion pricing
More informationOption Valuation of Oil & Gas E&P Projects by Futures Term Structure Approach. Hidetaka (Hugh) Nakaoka
Opion Valuaion of Oil & Gas E&P Projecs by Fuures Term Srucure Approach March 9, 2007 Hideaka (Hugh) Nakaoka Former CIO & CCO of Iochu Oil Exploraion Co., Ld. Universiy of Tsukuba 1 Overview 1. Inroducion
More informationA Note on Forward Price and Forward Measure
C Review of Quaniaive Finance and Accouning, 9: 26 272, 2002 2002 Kluwer Academic Publishers. Manufacured in The Neherlands. A Noe on Forward Price and Forward Measure REN-RAW CHEN FOM/SOB-NB, Rugers Universiy,
More informationForeign Exchange, ADR s and Quanto-Securities
IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2013 c 2013 by Marin Haugh Foreign Exchange, ADR s and Quano-Securiies These noes consider foreign exchange markes and he pricing of derivaive
More informationFAIR VALUATION OF INSURANCE LIABILITIES. Pierre DEVOLDER Université Catholique de Louvain 03/ 09/2004
FAIR VALUATION OF INSURANCE LIABILITIES Pierre DEVOLDER Universié Caholique de Louvain 03/ 09/004 Fair value of insurance liabiliies. INTRODUCTION TO FAIR VALUE. RISK NEUTRAL PRICING AND DEFLATORS 3. EXAMPLES
More informationEXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION
EXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION VICTOR GOODMAN AND KYOUNGHEE KIM Absrac. We find a simple expression for he probabiliy densiy of R exp(b s s/2ds in erms of is disribuion funcion
More informationAN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM MODEL WITH STOCHASTIC INTEREST RATES
Inernaional Journal of Pure and Applied Mahemaics Volume 76 No. 4 212, 549-557 ISSN: 1311-88 (prined version url: hp://www.ijpam.eu PA ijpam.eu AN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM
More informationContinuous-time term structure models: Forward measure approach
Finance Sochas. 1, 261 291 (1997 c Springer-Verlag 1997 Coninuous-ime erm srucure models: Forward measure approach Marek Musiela 1, Marek Rukowski 2 1 School of Mahemaics, Universiy of New Souh Wales,
More informationOnce we know he probabiliy densiy funcion (pdf) φ(s ) of S, a European call wih srike is priced a C() = E [e r d(s ) + ] = e r d { (S )φ(s ) ds } = e
Opion Basics Conens ime-dependen Black-Scholes Formula Black-76 Model Local Volailiy Model Sochasic Volailiy Model Heson Model Example ime-dependen Black-Scholes Formula Le s begin wih re-discovering he
More informationDYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Krzysztof Jajuga Wrocław University of Economics
DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń 2006 Krzyszof Jajuga Wrocław Universiy of Economics Ineres Rae Modeling and Tools of Financial Economerics 1. Financial Economerics
More informationModeling of Tradeable Securities with Dividends
Modeling of Tradeable Securiies wih Dividends Michel Vellekoop 1 & Hans Nieuwenhuis 2 June 15, 26 Absrac We propose a generalized framework for he modeling of radeable securiies wih dividends which are
More informationarxiv:math/ v2 [math.pr] 26 Jan 2007
arxiv:mah/61234v2 [mah.pr] 26 Jan 27 EXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION VICTOR GOODMAN AND KYOUNGHEE KIM Absrac. We find a simple expression for he probabiliy densiy of R exp(bs
More informationSTOCHASTIC METHODS IN CREDIT RISK MODELLING, VALUATION AND HEDGING
STOCHASTIC METHODS IN CREDIT RISK MODELLING, VALUATION AND HEDGING Tomasz R. Bielecki Deparmen of Mahemaics Norheasern Illinois Universiy, Chicago, USA T-Bielecki@neiu.edu (In collaboraion wih Marek Rukowski)
More informationOn Monte Carlo Simulation for the HJM Model Based on Jump
On Mone Carlo Simulaion for he HJM Model Based on Jump Kisoeb Park 1, Moonseong Kim 2, and Seki Kim 1, 1 Deparmen of Mahemaics, Sungkyunkwan Universiy 44-746, Suwon, Korea Tel.: +82-31-29-73, 734 {kisoeb,
More informationRisk-Neutral Probabilities Explained
Risk-Neural Probabiliies Explained Nicolas Gisiger MAS Finance UZH ETHZ, CEMS MIM, M.A. HSG E-Mail: nicolas.s.gisiger @ alumni.ehz.ch Absrac All oo ofen, he concep of risk-neural probabiliies in mahemaical
More informationConstructing Out-of-the-Money Longevity Hedges Using Parametric Mortality Indexes. Johnny Li
1 / 43 Consrucing Ou-of-he-Money Longeviy Hedges Using Parameric Moraliy Indexes Johnny Li Join-work wih Jackie Li, Udiha Balasooriya, and Kenneh Zhou Deparmen of Economics, The Universiy of Melbourne
More informationMORNING SESSION. Date: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES
SOCIETY OF ACTUARIES Quaniaive Finance and Invesmen Core Exam QFICORE MORNING SESSION Dae: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Insrucions 1. This examinaion
More informationVaR and Low Interest Rates
VaR and Low Ineres Raes Presened a he Sevenh Monreal Indusrial Problem Solving Workshop By Louis Doray (U de M) Frédéric Edoukou (U de M) Rim Labdi (HEC Monréal) Zichun Ye (UBC) 20 May 2016 P r e s e n
More informationAn Incentive-Based, Multi-Period Decision Model for Hierarchical Systems
Wernz C. and Deshmukh A. An Incenive-Based Muli-Period Decision Model for Hierarchical Sysems Proceedings of he 3 rd Inernaional Conference on Global Inerdependence and Decision Sciences (ICGIDS) pp. 84-88
More informationCoupling Smiles. November 18, 2006
Coupling Smiles Valdo Durrleman Deparmen of Mahemaics Sanford Universiy Sanford, CA 94305, USA Nicole El Karoui Cenre de Mahémaiques Appliquées Ecole Polyechnique 91128 Palaiseau, France November 18, 2006
More informationProblem 1 / 25 Problem 2 / 25 Problem 3 / 11 Problem 4 / 15 Problem 5 / 24 TOTAL / 100
Deparmen of Economics Universiy of Maryland Economics 35 Inermediae Macroeconomic Analysis Miderm Exam Suggesed Soluions Professor Sanjay Chugh Fall 008 NAME: The Exam has a oal of five (5) problems and
More informationResearch Article A General Gaussian Interest Rate Model Consistent with the Current Term Structure
Inernaional Scholarly Research Nework ISRN Probabiliy and Saisics Volume 212, Aricle ID 67367, 16 pages doi:1.542/212/67367 Research Aricle A General Gaussian Ineres Rae Model Consisen wih he Curren Term
More informationEconomic Growth Continued: From Solow to Ramsey
Economic Growh Coninued: From Solow o Ramsey J. Bradford DeLong May 2008 Choosing a Naional Savings Rae Wha can we say abou economic policy and long-run growh? To keep maers simple, le us assume ha he
More informationMA Advanced Macro, 2016 (Karl Whelan) 1
MA Advanced Macro, 2016 (Karl Whelan) 1 The Calvo Model of Price Rigidiy The form of price rigidiy faced by he Calvo firm is as follows. Each period, only a random fracion (1 ) of firms are able o rese
More informationCURRENCY TRANSLATED OPTIONS
CURRENCY RANSLAED OPIONS Dr. Rober ompkins, Ph.D. Universiy Dozen, Vienna Universiy of echnology * Deparmen of Finance, Insiue for Advanced Sudies Mag. José Carlos Wong Deparmen of Finance, Insiue for
More informationOn the multiplicity of option prices under CEV with positive elasticity of variance
Rev Deriv Res (207) 20: 3 DOI 0.007/s47-06-922-2 On he mulipliciy of opion prices under CEV wih posiive elasiciy of variance Dirk Veesraeen Published online: 4 April 206 The Auhor(s) 206. This aricle is
More information(c) Suppose X UF (2, 2), with density f(x) = 1/(1 + x) 2 for x 0 and 0 otherwise. Then. 0 (1 + x) 2 dx (5) { 1, if t = 0,
:46 /6/ TOPIC Momen generaing funcions The n h momen of a random variable X is EX n if his quaniy exiss; he momen generaing funcion MGF of X is he funcion defined by M := Ee X for R; he expecaion in exiss
More informationFIXED INCOME MICHAEL MONOYIOS
FIXED INCOME MICHAEL MONOYIOS Absrac. The course examines ineres rae or fixed income markes and producs. These markes are much larger, in erms of raded volume and value, han equiy markes. We firs inroduce
More informationResearch Article On Option Pricing in Illiquid Markets with Jumps
ISRN Mahemaical Analysis Volume 213, Aricle ID 56771, 5 pages hp://dx.doi.org/1.1155/213/56771 Research Aricle On Opion Pricing in Illiquid Markes wih Jumps Youssef El-Khaib 1 and Abdulnasser Haemi-J 2
More informationModeling of Tradeable Securities with Dividends
Modeling of Tradeable Securiies wih Dividends Michel Vellekoop 1 & Hans Nieuwenhuis 2 April 7, 26 Absrac We propose a generalized framework for he modeling of radeable securiies wih dividends which are
More informationHEDGING OF CREDIT DERIVATIVES IN MODELS WITH TOTALLY UNEXPECTED DEFAULT
HEDGING OF CREDIT DERIVATIVES IN MODELS WITH TOTALLY UNEXPECTED DEFAULT Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6616, USA Monique Jeanblanc Déparemen
More informationThe Investigation of the Mean Reversion Model Containing the G-Brownian Motion
Applied Mahemaical Sciences, Vol. 13, 219, no. 3, 125-133 HIKARI Ld, www.m-hikari.com hps://doi.org/1.12988/ams.219.918 he Invesigaion of he Mean Reversion Model Conaining he G-Brownian Moion Zixin Yuan
More informationAlexander L. Baranovski, Carsten von Lieres and André Wilch 18. May 2009/Eurobanking 2009
lexander L. Baranovski, Carsen von Lieres and ndré Wilch 8. May 2009/ Defaul inensiy model Pricing equaion for CDS conracs Defaul inensiy as soluion of a Volerra equaion of 2nd kind Comparison o common
More informationOn multicurve models for the term structure.
On mulicurve models for he erm srucure. Wolfgang Runggaldier Diparimeno di Maemaica, Universià di Padova WQMIF, Zagreb 2014 Inroducion and preliminary remarks Preliminary remarks In he wake of he big crisis
More informationOPTIMUM FISCAL AND MONETARY POLICY USING THE MONETARY OVERLAPPING GENERATION MODELS
Kuwai Chaper of Arabian Journal of Business and Managemen Review Vol. 3, No.6; Feb. 2014 OPTIMUM FISCAL AND MONETARY POLICY USING THE MONETARY OVERLAPPING GENERATION MODELS Ayoub Faramarzi 1, Dr.Rahim
More informationPART. I. Pricing Theory and Risk Management
PART. I Pricing Theory and Risk Managemen CHAPTER. 1 Pricing Theory Pricing heory for derivaive securiies is a highly echnical opic in finance; is foundaions res on rading pracices and is heory relies
More informationBlack-Scholes and the Volatility Surface
IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2013 c 2013 by Marin Haugh Black-Scholes and he Volailiy Surface When we sudied discree-ime models we used maringale pricing o derive he Black-Scholes
More informationValuing Real Options on Oil & Gas Exploration & Production Projects
Valuing Real Opions on Oil & Gas Exploraion & Producion Projecs March 2, 2006 Hideaka (Hugh) Nakaoka Former CIO & CCO of Iochu Oil Exploraion Co., Ld. Universiy of Tsukuba 1 Overview 1. Inroducion 2. Wha
More informationPolicyholder Exercise Behavior for Variable Annuities including Guaranteed Minimum Withdrawal Benefits 1
Policyholder Exercise Behavior for Variable Annuiies including Guaraneed Minimum Wihdrawal Benefis 1 2 Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, 11h Floor; Alana, GA
More informationFINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004
FINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004 This exam has 50 quesions on 14 pages. Before you begin, please check o make sure ha your copy has all 50 quesions and all 14 pages.
More informationTerm Structure Models: IEOR E4710 Spring 2005 c 2005 by Martin Haugh. Market Models. 1 LIBOR, Swap Rates and Black s Formulae for Caps and Swaptions
Term Srucure Models: IEOR E4710 Spring 2005 c 2005 by Marin Haugh Marke Models One of he principal disadvanages of shor rae models, and HJM models more generally, is ha hey focus on unobservable insananeous
More informationMacroeconomics II A dynamic approach to short run economic fluctuations. The DAD/DAS model.
Macroeconomics II A dynamic approach o shor run economic flucuaions. The DAD/DAS model. Par 2. The demand side of he model he dynamic aggregae demand (DAD) Inflaion and dynamics in he shor run So far,
More informationValuation and Hedging of Correlation Swaps. Mats Draijer
Valuaion and Hedging of Correlaion Swaps Mas Draijer 4298829 Sepember 27, 2017 Absrac The aim of his hesis is o provide a formula for he value of a correlaion swap. To ge o his formula, a model from an
More informationON THE TIMING OPTION IN A FUTURES CONTRACT. FRANCESCA BIAGINI Dipartimento di Matematica, Università dibologna
Mahemaical Finance, Vol. 17, No. 2 (April 2007), 267 283 ON THE TIMING OPTION IN A FUTURES CONTRACT FRANCESCA BIAGINI Diparimeno di Maemaica, Universià dibologna TOMAS BJÖRK Deparmen of Finance, Sockholm
More informationt=1 C t e δt, and the tc t v t i t=1 C t (1 + i) t = n tc t (1 + i) t C t (1 + i) t = C t vi
Exam 4 is Th. April 24. You are allowed 13 shees of noes and a calculaor. ch. 7: 137) Unless old oherwise, duraion refers o Macaulay duraion. The duraion of a single cashflow is he ime remaining unil mauriy,
More informationLecture: Autonomous Financing and Financing Based on Market Values I
Lecure: Auonomous Financing and Financing Based on Marke Values I Luz Kruschwiz & Andreas Löffler Discouned Cash Flow, Secion 2.3, 2.4.1 2.4.3, Ouline 2.3 Auonomous financing 2.4 Financing based on marke
More informationINFORMATION ASYMMETRY IN PRICING OF CREDIT DERIVATIVES.
INFORMATION ASYMMETRY IN PRICING OF CREDIT DERIVATIVES. Join work wih Ying JIAO, LPMA, Universié Paris VII 6h World Congress of he Bachelier Finance Sociey, June 24, 2010. This research is par of he Chair
More informationNumerical probabalistic methods for high-dimensional problems in finance
Numerical probabalisic mehods for high-dimensional problems in finance The American Insiue of Mahemaics This is a hard copy version of a web page available hrough hp://www.aimah.org Inpu on his maerial
More informationInterest Rate Products
Chaper 9 Ineres Rae Producs Copyrigh c 2008 20 Hyeong In Choi, All righs reserved. 9. Change of Numeraire and he Invariance of Risk Neural Valuaion The financial heory we have developed so far depends
More informationSingle Premium of Equity-Linked with CRR and CIR Binomial Tree
The 7h SEAMS-UGM Conference 2015 Single Premium of Equiy-Linked wih CRR and CIR Binomial Tree Yunia Wulan Sari 1,a) and Gunardi 2,b) 1,2 Deparmen of Mahemaics, Faculy of Mahemaics and Naural Sciences,
More informationLeveraged Stock Portfolios over Long Holding Periods: A Continuous Time Model. Dale L. Domian, Marie D. Racine, and Craig A.
Leveraged Sock Porfolios over Long Holding Periods: A Coninuous Time Model Dale L. Domian, Marie D. Racine, and Craig A. Wilson Deparmen of Finance and Managemen Science College of Commerce Universiy of
More informationFinancial Markets And Empirical Regularities An Introduction to Financial Econometrics
Financial Markes And Empirical Regulariies An Inroducion o Financial Economerics SAMSI Workshop 11/18/05 Mike Aguilar UNC a Chapel Hill www.unc.edu/~maguilar 1 Ouline I. Hisorical Perspecive on Asse Prices
More informationCurrency Derivatives under a Minimal Market Model with Random Scaling
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 54 March 25 Currency Derivaives under a Minimal Marke Model wih Random Scaling David Heah and Eckhard Plaen ISSN
More informationAdvanced Tools for Risk Management and Asset Pricing
MSc. Finance/CLEFIN 214/215 Ediion Advanced Tools for Risk Managemen and Asse Pricing May 215 Exam for Non-Aending Sudens Soluions Time Allowed: 13 minues Family Name (Surname) Firs Name Suden Number (Mar.)
More informationdb t = r t B t dt (no Itô-correction term, as B has finite variation (FV), so ordinary Newton- Leibniz calculus applies). Now (again as B is FV)
ullin4b.ex pm Wed 21.2.2018 5. The change-of-numeraire formula Here we follow [BM, 2.2]. For more deail, see he paper Brigo & Mercurio (2001c) cied here, and H. GEMAN, N. El KAROUI and J. C. ROCHET, Changes
More information1 Purpose of the paper
Moneary Economics 2 F.C. Bagliano - Sepember 2017 Noes on: F.X. Diebold and C. Li, Forecasing he erm srucure of governmen bond yields, Journal of Economerics, 2006 1 Purpose of he paper The paper presens
More informationPricing corporate bonds, CDS and options on CDS with the BMC model
Pricing corporae bonds, CDS and opions on CDS wih he BMC model D. Bloch Universié Paris VI, France Absrac Academics have always occuled he calibraion and hedging of exoic credi producs assuming ha credi
More informationBasic Economic Scenario Generator: Technical Specications. Jean-Charles CROIX ISFA - Université Lyon 1
Basic Economic cenario Generaor: echnical pecicaions Jean-Charles CROIX IFA - Universié Lyon 1 January 1, 13 Conens Inroducion 1 1 Risk facors models 3 1.1 Convenions............................................
More informationAsymmetry and Leverage in Stochastic Volatility Models: An Exposition
Asymmery and Leverage in Sochasic Volailiy Models: An xposiion Asai, M. a and M. McAleer b a Faculy of conomics, Soka Universiy, Japan b School of conomics and Commerce, Universiy of Wesern Ausralia Keywords:
More information