Term Structure of Prices of Asian Options


 Magnus Osborne
 2 years ago
 Views:
Transcription
1 Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy Nojihigashi, Kusasu, Shiga , Japan {akahori, rp4992, yasuomi, Absrac In he presen paper, we will sudy he pricing of Asian opions. The main conribuion is he model consrucion; our model is compaible wih marke convenions, can be calibraed o observable marke daa, and compuaionally racable, while i is 2facor and he ineres rae is sochasic. The model akes i ino accoun ha he ineres rae risks can influence he prices of he opions in he long run. Thus our model can be used o analyse he erm srucure, from shor o long erm, of he prices of he opions 1 Inroducion 1.1 Backgrounds In 1998, he Financial Accouning Sandards Board (FASB) issued Saemen No.133, Accouning for Derivaive Insrumens and Hedging Aciviies[3], which is known as FAS133. Since hen, pricing Derivaives is geing more and more imporan because he FAS133 requires a firm o be aware of he risk exposure of Derivaives in is porfolio. When a Derivaive is raded Over The Couner (OTC), however, pricing becomes difficul since he marke does no quoe he price. This is ofen he case wih Exoic Derivaives such as knockou opions or Asian (average) opions. To evaluae he fair price of an OTC Derivaive, one hen needs o rely on a mahemaical model which inerpolaes he price of he Derivaive from marke observables. The model should be
2 compaible wih marke convenions, capable of calibraion o marke quoes, compuaionally racable. To consruc such a model deserves a careful sudy from a pracical poin of view. 1.2 The problem In he presen paper, we will sudy he pricing of Asian opions, a ypical example of Exoic opions. The payoff of he Asian call opion we will sudy is 1 N max S Tk K, N, (1) k=1 where S denoe he price a ime of an asse, T 1 < < T N are he daes for aking average, and K is he srike price. We le T( T N ) be he mauriy dae, and he fair price a ime (< T) of he opion will be denoed π(, T), a funcion of T; we consider he cases where T 1,..., T N is compleely deermined by he choice of T. By a sandard argumen of mahemaical finance [2, e.g.], he price should be π (T, K) = P T ET max 1 N S Tk K, N F, (2) where P T is he price a ime of a zero coupon bond which maures a T, he (condiional) expecaion E T is aken in erms of a probabiliy measure ha makes S/P T a maringale (which is usually called forward measure and will be denoed P T ) wih respec o he given filraion {F }. In he nex secion, we specify he sochasic dynamics of S and P T o mee he requiremens of he las paragraph of secion 1.1. We sress ha k=1 he ineres raes are sochasic in our model. This means ha we ake he ineres rae risks ino accoun. This is new in he conex of pricing of exoics, and is why we use he word erm srucure, which is usually associaed wih ineres rae. Then in secion 3, we will discuss how o calibrae he model o marke observables, and in secion 4, a numerical sudy will be presened using MoneCarlo mehod. Remark. The pricing of Asian opions has been inensively sudied, especially by M. Yor and his coworkers (see [6] and references herein). They 2
3 derived explici formulas for Asian opions, which may be useful in our cases. Bu in his paper our sress is on modelling, no in numerical analysis. So we ook an easy way o use MoneCarlo mehod. 2 The Model 2.1 Marke convenion In he conex of opion pricing, he marke convenion usually means he convenional consensus ha he acual price of a plain opion follows he BlackScholes formula (see e.g.[4]). In he sandard BlackScholes model, he ineres rae is nonsochasic, while we wan o consruc a model including ineresrae risks. We can overcome his dilemma by using he forward measure P T menioned in he previous secion; he price a ime of a plain call opion mauring a ime T is P T ET [max(s T K, ) F ] (3) where r is he (consan) ineres rae. To read his as a BlackScholes formula, log S T S is Gaussian under P T (4) is required. Our goal is o price he Asian opions, so we impose a lile sronger condiion: log S is Gaussian under P T for all (, T) (5) 2.2 Ineres rae risks aken ino accoun The convenion discussed above implicily require ha he model should be based on Brownian moions. In he presen paper, we use wo independen Brownian moion (W, B) as he risk facors; we use a 2facor model. The firs Brownian moion W is mean o represen he inheren risk of he asse, while W represens he ineres rae risk. The sae price densiy is hen, in is fully general form, { D = P exp f (s, ) db s 1 } f (s, ) 2 ds (6) 2 3
4 where f (, ) is an adaped process defined for every. Under he NoArbirage hypohesis, he bond prices are P u = E[D F u ]/D u ( = P u P u exp { f (s, ) f (s, u)} db s 1 u ) { f (s, ) 2 f (s, u) 2 } ds, 2 (7) and he process {D S } mus be a maringale (see e.g. [1]). As a consequence, a fully general expression of S is as follows: { S = D 1 S D exp 1 2 σ(s) dw s + λ(s) db s } ( σ(s) 2 + λ(s) 2 ) ds, (8) where σ, λ are adaped processes. 2.3 Specificaion of he model The forward measure is defined via is densiy on F T wih respec o he physical measure P; dp T dp = D T /E[D T ], (9) or dp T dp = E[D T F ]/E[D T ] = P T D /E[D T ]. (1) F Then, by MaruyamaGirsanov Theorem B T := B + f (s, T) ds (11) 4
5 is Brownian moion under P T (see e.g. [5]). Since we have S = 1 { S u P exp σ(s) dw s u u {λ(s) f (s, )} db s } { σ(s) 2 + λ(s) 2 f (s, ) 2 } ds u (subsiuing (11)) = 1 { P exp σ(s) dw s + u u 1 2 u u { σ(s) 2 + λ(s) f (s, T) 2 f (s, T) f (s, ) 2 } ds } u {λ(s) f (s, )} db T s (12) Consequenly, o be consisen wih (5), σ, λ, f mus be nonrandom. (13) If his is he case, hen under he forward measure we have log S S ( u N log P u 1 2 u u { σ(s) 2 + λ(s) f (s, T) 2 f (s, T) f (s, ) 2 } ds, { σ(s) 2 + λ(s) f (s, ) 2} ) ds. (14) 3 Calibraions In his secion we will presen a calibraion scheme for he model we gave in he previous secion. 3.1 A Convenion Firsly, we inroduce an approximaion which faciliae he calibraion scheme. For k =, 1,, N 1, se X k := log S T k S Tk 1. 5
6 Here we le T 1 =. By (12), we have X k = log P T k T k 1 { Tk + σ(s) dw s + T k Tk Tk T k 1 {λ(s) f (s, T k )} db T s { σ(s) 2 + λ(s) f (s, T k ) 2 T k 1 } f (s, T) f (s, T k ) 2 } ds, (15) so ha S Tk = S exp k k X j = S j= 1 j= P T j T j 1 where Y k = log P T k T k 1 + X k. Noe ha Y, Y 1,..., Y N 1 are muually independen. e k j= Y j, By he approximaion of k 1 j= P T j T j 1 1 P T k, (16) he value of he Asian opion in quesion is given by he following muliple inegral: π (T, K) P T 1 max N N 1 k= + S P T k + (2π) N/2 e 1 2 Nk=1 x 2 k e k j= (α j x j+1 +β j ) K, dx 1 dx N. (17) Here we le α k be he sandard variaion of Y k, and β k be he mean of Y k. More precisely, and α k = ( Tk β k = 1 2 { σ(s) 2 + λ(s) f (s, T k )) 2} ) 1/2 ds (18) T k 1 Tk { σ(s) 2 + λ(s) f (s, T k ) 2 T k 1 (19) f (s, T) f (s, T k ) 2 } ds. 6
7 3.2 Parameer esimaion To ge a value of he inegral (17), we need o esimae he followings., T, K, N : No problem! S /P T k is (heoriically) equal o he foward price, which is usully quoed in he marke, P T α k is quoed in he marke as ineres rae, β k (k =, 1,..., N 1); see he followings. (As a maer of course, he acual marke quoaions are he discree daa, and so we need o inerpolae hem o ge coninuous daa.) Following he marke convenion we menioned above, we can observe he volailiy of he asse price S : v(, T) = T { σ(s) 2 + λ(s) f (s, T) 2} ds (2) from he price of he plain opion mauring a T. Noe ha a ime we can observe v(s, T) for s and T. We have α 2 k = v(t k 1, T k ) = v(, T k ) Tk 1 { σ(s) 2 + λ(s) f (s, T k )) 2} ds, and he hird erm can be seen as he implied volailiy of he plain opion whose selemen dae is T k 1 and he paymen dae is T k. Inuiively, a leas when T is enough large, he difference of he wo dae does no effec he opion price so much. So, we claim ha he following convenion is appropriae: Tk 1 { σ(s) 2 + λ(s) f (s, T k )) 2} ds v(, T k 1 ) (21) Consequenly, we can esimae he value of α k by α k v(, T k ) v(, T k 1 ). (22) 7
8 3.3 Convenions for he implied volailiy of ineres rae To esimae β k = 1 2 α2 k + Tk we use he implied volailiy δ(, T 1, T 2 ) := T k 1 { f (s, T) f (s, T k )} 2 ds, (23) T1 of he ineres rae of he period [T 1, T 2 ]: { f (s, T 2 ) f (s, T 1 )} 2 ds. (24) L(T 1, T 2 ) = P T 2 T 1. (25) The daa of (24) are implied by he prices of he plain opion on L i.e. he caple, which is no OTC. The problem is ha he usual marke convenion assumes ha L is lognormally disribued while in our model under (13), L+1 is lognormal random variable under P T. The difference is no ignorable, since he laer allows he negaive ineres raes, while he former does no. We overcome his hardship by inroducing anoher convenion. We assume ha he marke convenion is doing a momen maching. Namely (under he hypohesis of (13)) he marke assumes ha for each L(T 1, T 2 ) e R on F where R is a Gaussian random variable such ha and E[e R ] = E T 2 [L(T 1, T 2 ) F ] (26) E[e 2R ] = E T 2 [ L(T 1, T 2 ) 2 F ]. (27) In oher words, he marke convenion under (13) is jus E T [max(l(t 1, T 2 ) K, ) F ] E[max(e R K, )]. (28) The new convenion implies ha on F, e R = 1 + PT 1 ecx 1 2 c2 (29) for an X N(, 1) and a consan c c(, T 1, T 2 ). In paricular, c 2 is he (convenional) implied volailiy. 8 P T 2
9 On he oher hand, since we have we can insis ha E T [ L(T 1, T 2 ) 2 F ] = 1 2 PT 1 + exp PT 1 P T 2 2 { T1 P T 2 { T1 exp { f (s, T 2 ) f (s, T 1 )} 2 ds } { f (s, T 2 ) f (s, T 1 )} 2 ds 1 + T1 { f (s, T 2 ) f (s, T 1 )} 2 ds, }, provided ha T 2 T 1 is small enough. If his is he case, we have { T1 } { f (s, T 2 ) f (s, T 1 )} 2 ds 1 + PT 2 P T 1 2 c 2 (, T 1, T 2 ). (3) (31) (32) Hence our convenion here is ( ) 2 L (T 1, T 2 ) δ(, T 1, T 2 ) = c(, T 1, T 2 ) 2 (33) 1 + L (T 1, T 2 ) where L (T 1, T 2 ) = 1 + PT 1 4 Numerical Analysis P T 2. (34) The acual daa is no coninuous bu discree. Mos common daa is monhly, like Fig 1. One can ge daily daa, or even shorer, bu anyway he/she mus inerpolae he daa o some exen. In his secion we exhibi he resuls of several numerical calculaions using he monhly daa Fig 1 o illusrae how our model works. We se he period of aking average is one monh, aking he average of daily daa; T k T k 1 = 1 (day) and N = 2 (day) = 1 (monh). Looking a he inegral (17), we need a daily erm srucure of Forward prices, volailiies of he asse, spo ineres raes, and he volailiies of forward raes of a fixed lengh. The sample daa (1) is missing he las of he 9
10 Figure 1: Sample Daa 1
11 four, So we se hem zero for simpliciy. I has firs hree of he four, which we inerpolae o ge daily daa as follows: Suppose ha we are given Forward prices F T[i] and volailiies V T[i] for i N. Here T[i] N and T[i] T[i 1] = 2 (days = 1 monh). We se F and V for T[i 1] < < T[i], N as and F[i] F = F[i 1]( F[i 1] ) T[i] T[i] T[i 1], T[i] V = V[i 1] + (V[i] V[i 1]) T[i] T[i 1]. And we se S for T[i 1] < T[i] as T[i 1] S = F exp (a[h]n h + b[h]), where N h are muually independen Gaussian random variables, and T[i 1] a[] = V T[i 1] Y b[] = 1 2 V T[i 1] T[i 1] Y 1 a[h] = V T[i 1]+h Y b[h] = 1 2 V 1 T[i 1]+h Y. h= Here we have se Y = 24; he number of days in a year. We used a sandard and simples MoneCarlo mehod, using RAND as quasirandom number generaor. The firs example Fig 2 shows he prices of Asian Pu and Call when T = 4 (days). he inersecion poin indicaes he PuCall pariy for Asian opion. The second one Fig 3 shows he prices for a lile longer erm Asian opions. The las one Fig 4 shows a erm srucure of prices of Asian opions. I is parallel o ha of plain opions. This is because we have se he volailiy of ineres raes as zero. 11
12 Figure 2: Asian opion pu and call price wih T=4 5 Conclusions As have been shown, we have successfully consruced a proper model for pricing Asian opions and describing heir erm srucure. A he same ime we have inroduced a calibraion scheme which acually works quie well. References [1] D. Duffie: Dynamic Asse Pricing Theory, Princeon Universiy Press, [2] R. J.Ellio, and P.E. Kopp: Mahemaics of Financial Markes, Springer,
13 Figure 3: Asian opion pu and call price wih T=46 [3] Financial Accouning Sandards Board: he URL of is web pages: hp://www.fasb.org/ [4] M. Musiela, and M. Rukowski Maringale Mehods in Financial Modelling, Springer, [5] B. Oksendal: Sochasic Differenial Equaions; an inroducion wih applicaions, 5h ediion, Springer, [6] M. Yor: Exponenial funcionals of Brownian moion and relaed processes, Springer Finance. SpringerVerlag, Berlin,
14 Figure 4: Term srucure of Asian opions 14
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationForeign Exchange and Quantos
IEOR E4707: Financial Engineering: ConinuousTime Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationValuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate
Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationStochastic Calculus, Week 10. Definitions and Notation. TermStructure Models & Interest Rate Derivatives
Sochasic Calculus, Week 10 TermSrucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Termsrucure models 3. Ineres rae derivaives Definiions and Noaion Zerocoupon
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More information12. Market LIBOR Models
12. Marke LIBOR Models As was menioned already, he acronym LIBOR sands for he London Inerbank Offered Rae. I is he rae of ineres offered by banks on deposis from oher banks in eurocurrency markes. Also,
More informationConceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...
Normal (Gaussian) Disribuion Probabiliy De ensiy 0.5 0. 0.5 0. 0.05 0. 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0. 0 3.6 5. 6.8 8.4 0.6 3. 4.8 6.4 8 The BlackScholes Shl Ml Moel... pricing opions an calculaing
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationPricing Futures and Futures Options with Basis Risk
Pricing uures and uures Opions wih Basis Risk ChouWen ang Assisan professor in he Deparmen of inancial Managemen Naional Kaohsiung irs niversiy of cience & Technology Taiwan TingYi Wu PhD candidae in
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationDynamic Option Adjusted Spread and the Value of Mortgage Backed Securities
Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao, Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics 27 January 2008 Absrac We exend a reduced form model for
More informationEuropean option prices are a good sanity check when analysing bonds with exotic embedded options.
European opion prices are a good saniy check when analysing bonds wih exoic embedded opions. I s an old exam quesion. Arbiragefree economy where ZCB prices are driven 1D BM, i.e. dp (, T ) = r()p (,
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationAn accurate analytical approximation for the price of a Europeanstyle arithmetic Asian option
An accurae analyical approximaion for he price of a Europeansyle arihmeic Asian opion David Vyncke 1, Marc Goovaers 2, Jan Dhaene 2 Absrac For discree arihmeic Asian opions he payoff depends on he price
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationUNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.
UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL
More informationStochastic Calculus and Option Pricing
Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 BlackScholes
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationPRICING and STATIC REPLICATION of FX QUANTO OPTIONS
PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of
More informationIMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **
IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include socalled implici or embedded opions.
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable
More informationModelling of Forward Libor and Swap Rates
Modelling of Forward Libor and Swap Raes Marek Rukowski Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology, 661 Warszawa, Poland Conens 1 Inroducion 2 2 Modelling of Forward Libor
More informationSkewness and Kurtosis Adjusted BlackScholes Model: A Note on Hedging Performance
Finance Leers, 003, (5), 6 Skewness and Kurosis Adjused BlackScholes Model: A Noe on Hedging Performance Sami Vähämaa * Universiy of Vaasa, Finland Absrac his aricle invesigaes he dela hedging performance
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationFX OPTION PRICING: RESULTS FROM BLACK SCHOLES, LOCAL VOL, QUASI QPHI AND STOCHASTIC QPHI MODELS
FX OPTION PRICING: REULT FROM BLACK CHOLE, LOCAL VOL, QUAI QPHI AND TOCHATIC QPHI MODEL Absrac Krishnamurhy Vaidyanahan 1 The paper suggess a new class of models (QPhi) o capure he informaion ha he
More informationTHE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS
HE PERFORMANE OF OPION PRIING MODEL ON HEDGING EXOI OPION Firs Draf: May 5 003 his Version Oc. 30 003 ommens are welcome Absrac his paper examines he empirical performance of various opion pricing models
More informationNew Pricing Framework: Options and Bonds
arxiv:1407.445v [qfin.pr] 14 Oc 014 New Pricing Framework: Opions and Bonds Nick Laskin TopQuark Inc. Torono, ON, M6P P Absrac A unified analyical pricing framework wih involvemen of he sho noise random
More informationHANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.
Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationHOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES?
HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Absrac. We compare he opion pricing formulas of Louis Bachelier and BlackMeronScholes
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationAPPLICATION OF THE KALMAN FILTER FOR ESTIMATING CONTINUOUS TIME TERM STRUCTURE MODELS: THE CASE OF UK AND GERMANY. January, 2005
APPLICATION OF THE KALMAN FILTER FOR ESTIMATING CONTINUOUS TIME TERM STRUCTURE MODELS: THE CASE OF UK AND GERMANY Somnah Chaeree* Deparmen of Economics Universiy of Glasgow January, 2005 Absrac The purpose
More informationOpenGamma Quantitative Research Multicurves: Variations on a Theme
OpenGamma Quaniaive Research Mulicurves: Variaions on a Theme Marc Henrard marc@opengamma.com OpenGamma Quaniaive Research n. 6 Ocober 2012 Absrac The mulicurves framework is ofen implemened in a way
More informationVariance Swap. by Fabrice Douglas Rouah
Variance wap by Fabrice Douglas Rouah www.frouah.com www.volopa.com In his Noe we presen a deailed derivaion of he fair value of variance ha is used in pricing a variance swap. We describe he approach
More informationA Market Model of Interest Rates with Dynamic Basis Spreads in the presence of Collateral and Multiple Currencies
CIRJEF698 A Marke Model of Ineres Raes wih Dynamic Basis Spreads in he presence of Collaeral and Muliple Currencies Masaaki Fujii Graduae School of Economics, Universiy of Tokyo Yasufumi Shimada Capial
More informationThe naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1
Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces imeseries smoohing forecasing mehods. Various models are discussed,
More informationA Note on Construction of Multiple Swap Curves with and without Collateral
A Noe on Consrucion of Muliple Swap Curves wih and wihou Collaeral Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi Absrac There are now available wide variey
More informationStochastic Volatility Models: Considerations for the Lay Actuary 1. Abstract
Sochasic Volailiy Models: Consideraions for he Lay Acuary 1 Phil Jouber Coomaren Vencaasawmy (Presened o he Finance & Invesmen Conference, 191 June 005) Absrac Sochasic models for asse prices processes
More informationFifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance
Fifh Quaniaive Impac Sudy of Solvency II (QIS 5) Naional guidance on valuaion of echnical provisions for German SLT healh insurance Conens 1 Inroducion... 2 2 Calculaion of besesimae provisions... 3 2.1
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationOn the paper Is Itô calculus oversold? by A. Izmailov and B. Shay
On he paper Is Iô calculus oversold? by A. Izmailov and B. Shay M. Rukowski and W. Szazschneider March, 1999 The main message of he paper Is Iô calculus oversold? by A. Izmailov and B. Shay is, we quoe:
More informationLECTURE 7 Interest Rate Models I: Short Rate Models
LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia email: abreccia@emsbbkacuk
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationCollateral Posting and Choice of Collateral Currency
Collaeral Posing and Choice of Collaeral Currency Implicaions for derivaive pricing and risk managemen Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi KIERTMU Inernaional Workshop on Financial Engineering
More informationFixed Income Analysis: Securities, Pricing, and Risk Management
Fixed Income Analysis: Securiies, Pricing, and Risk Managemen Claus Munk This version: January 23, 2003 Deparmen of Accouning and Finance, Universiy of Souhern Denmark, Campusvej 55, DK5230 Odense M,
More informationCarol Alexander ICMA Centre, University of Reading. Aanand Venkatramanan ICMA Centre, University of Reading
Analyic Approximaions for Spread Opions Carol Alexander ICMA Cenre, Universiy of Reading Aanand Venkaramanan ICMA Cenre, Universiy of Reading 15h Augus 2007 ICMA Cenre Discussion Papers in Finance DP200711
More informationPREMIUM INDEXING IN LIFELONG HEALTH INSURANCE
Far Eas Journal of Mahemaical Sciences (FJMS 203 Pushpa Publishing House, Allahabad, India Published Online: Sepember 203 Available online a hp://pphm.com/ournals/fms.hm Special Volume 203, Par IV, Pages
More informationEfficient Pricing of Energy Derivatives
Efficien Pricing of Energy Derivaives Anders B. Trolle EPFL and Swiss Finance Insiue March 1, 2014 Absrac I presen a racable framework, firs developed in Trolle and Schwarz (2009), for pricing energy derivaives
More informationPricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates
Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear
More informationCurve Building and Swap Pricing in the Presence of Collateral and Basis Spreads SIMON GUNNARSSON
Curve Building and Swap Pricing in he Presence of Collaeral and Basis Spreads SIMON GUNNARSSON Maser of Science Thesis Sockholm, Sweden 2013 Curve Building and Swap Pricing in he Presence of Collaeral
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationReturn Calculation of U.S. Treasury Constant Maturity Indices
Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion
More informationPricing BlackScholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension
Pricing Blackcholes Opions wih Correlaed Ineres Rae Risk and Credi Risk: An Exension zulang Liao a, and HsingHua Huang b a irecor and Professor eparmen of inance Naional Universiy of Kaohsiung and Professor
More informationEURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN HJM ONEFACTOR MODEL
EURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN HJM ONEFACTOR MODEL MARC HENRARD Absrac. In his noe we give pricing formlas for differen insrmens linked o rae fres erodollar fres. We provide
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationTerm Structure Models: IEOR E4710 Spring 2010 c 2010 by Martin Haugh. Market Models. 1 LIBOR, Swap Rates and Black s Formulae for Caps and Swaptions
Term Srucure Models: IEOR E4710 Spring 2010 c 2010 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 informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationSPEC model selection algorithm for ARCH models: an options pricing evaluation framework
Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,
More informationBlack Scholes Option Pricing with Stochastic Returns on Hedge Portfolio
EJTP 3, No. 3 006 9 8 Elecronic Journal of Theoreical Physics Black Scholes Opion Pricing wih Sochasic Reurns on Hedge Porfolio J. P. Singh and S. Prabakaran Deparmen of Managemen Sudies Indian Insiue
More informationIntroduction to Arbitrage Pricing
Inroducion o Arbirage Pricing Marek Musiela 1 School of Mahemaics, Universiy of New Souh Wales, 252 Sydney, Ausralia Marek Rukowski 2 Insiue of Mahemaics, Poliechnika Warszawska, 661 Warszawa, Poland
More informationThe performance of popular stochastic volatility option pricing models during the Subprime crisis
The performance of popular sochasic volailiy opion pricing models during he Subprime crisis Thibau Moyaer 1 Mikael Peijean 2 Absrac We assess he performance of he Heson (1993), Baes (1996), and Heson and
More informationModeling a distribution of mortgage credit losses Petr Gapko 1, Martin Šmíd 2
Modeling a disribuion of morgage credi losses Per Gapko 1, Marin Šmíd 2 1 Inroducion Absrac. One of he bigges risks arising from financial operaions is he risk of counerpary defaul, commonly known as a
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationOptions and Volatility
Opions and Volailiy Peer A. Abken and Saika Nandi Abken and Nandi are senior economiss in he financial secion of he Alana Fed s research deparmen. V olailiy is a measure of he dispersion of an asse price
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationA TwoAccount Life Insurance Model for ScenarioBased Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul
universiy of copenhagen Universiy of Copenhagen A TwoAccoun Life Insurance Model for ScenarioBased Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:
More informationTimeinhomogeneous Lévy Processes in CrossCurrency Market Models
Timeinhomogeneous Lévy Processes in CrossCurrency Marke Models Disseraion zur Erlangung des Dokorgrades der Mahemaischen Fakulä der AlberLudwigsUniversiä Freiburg i. Brsg. vorgeleg von Naaliya Koval
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationCredit Risk Modeling with Random Fields
Credi Risk Modeling wih Random Fields InauguralDisseraion zur Erlangung des Dokorgrades an den Naurwissenschaflichen Fachbereichen (Mahemaik der JususLiebigUniversiä Gießen vorgeleg von Thorsen Schmid
More information