Term Structure of Prices of Asian Options

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Term Structure of Prices of Asian Options"

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 2-facor 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 knock-ou 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 pay-off 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 Mone-Carlo mehod. Remark. The pricing of Asian opions has been inensively sudied, especially by M. Yor and his co-workers (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 Mone-Carlo 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 Black-Scholes formula (see e.g.[4]). In he sandard Black-Scholes model, he ineres rae is non-sochasic, while we wan o consruc a model including ineres-rae 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 Black-Scholes 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 2-facor 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 No-Arbirage 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 Maruyama-Girsanov 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 non-random. (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 log-normal 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 Mone-Carlo mehod, using RAND as quasi-random number generaor. The firs example Fig 2 shows he prices of Asian Pu and Call when T = 4 (days). he inersecion poin indicaes he Pu-Call 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. Springer-Verlag, 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 INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

The option pricing framework

The 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 Ornstein-Uhlenbeck or Vasicek process,

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process, Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ

More information

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

More information

Stochastic Optimal Control Problem for Life Insurance

Stochastic 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 information

A general decomposition formula for derivative prices in stochastic volatility models

A 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, 5-7 85 Barcelona Absrac We see ha he price of an european call opion

More information

ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT

ON THE PRICING OF EQUITY-LINKED 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 94-9(5)634-4 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE

More information

Foreign Exchange and Quantos

Foreign Exchange and Quantos IEOR E4707: Financial Engineering: Coninuous-Time 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 information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT 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 information

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC 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 information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal 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 Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

Option Pricing Under Stochastic Interest Rates

Option Pricing Under Stochastic Interest Rates I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp://www.mecs-press.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecs-press.ne/ijem Opion ricing Under Sochasic Ineres

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 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 information

Hedging with Forwards and Futures

Hedging 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 buy-side of a forward/fuures

More information

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling

Modeling 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 information

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate

Valuation 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 information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics 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 information

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives Sochasic Calculus, Week 10 Term-Srucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Term-srucure models 3. Ineres rae derivaives Definiions and Noaion Zero-coupon

More information

ARCH 2013.1 Proceedings

ARCH 2013.1 Proceedings Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM 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 information

12. Market LIBOR Models

12. 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 information

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...

Conceptually 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 Black-Scholes Shl Ml Moel... pricing opions an calculaing

More information

Duration 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.

Duration 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 UVA-F-38 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 information

Pricing Futures and Futures Options with Basis Risk

Pricing Futures and Futures Options with Basis Risk Pricing uures and uures Opions wih Basis Risk Chou-Wen ang Assisan professor in he Deparmen of inancial Managemen Naional Kaohsiung irs niversiy of cience & Technology Taiwan Ting-Yi Wu PhD candidae in

More information

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Option Put-Call Parity Relations When the Underlying Security Pays Dividends Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

More information

Dynamic Option Adjusted Spread and the Value of Mortgage Backed Securities

Dynamic 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 information

European option prices are a good sanity check when analysing bonds with exotic embedded options.

European 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. Arbirage-free economy where ZCB prices are driven 1-D BM, i.e. dp (, T ) = r()p (,

More information

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

Optimal 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 information

Dependent Interest and Transition Rates in Life Insurance

Dependent 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 information

An accurate analytical approximation for the price of a European-style arithmetic Asian option

An accurate analytical approximation for the price of a European-style arithmetic Asian option An accurae analyical approximaion for he price of a European-syle 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 information

The Transport Equation

The 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 information

Risk Modelling of Collateralised Lending

Risk Modelling of Collateralised Lending Risk Modelling of Collaeralised Lending Dae: 4-11-2008 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 information

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table 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 information

Economics Honors Exam 2008 Solutions Question 5

Economics 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 information

UNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.

UNIVERSITY 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 information

Stochastic Calculus and Option Pricing

Stochastic 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 Black-Scholes

More information

I. Basic Concepts (Ch. 1-4)

I. Basic Concepts (Ch. 1-4) (Ch. 1-4) 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 information

PRICING and STATIC REPLICATION of FX QUANTO OPTIONS

PRICING 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 information

IMPLICIT 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 ** 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 so-called implici or embedded opions.

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\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 information

Fourier Series Solution of the Heat Equation

Fourier 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 information

ABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION

ABSTRACT 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 information

Modelling of Forward Libor and Swap Rates

Modelling 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 information

Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance

Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance Finance Leers, 003, (5), 6- Skewness and Kurosis Adjused Black-Scholes Model: A Noe on Hedging Performance Sami Vähämaa * Universiy of Vaasa, Finland Absrac his aricle invesigaes he dela hedging performance

More information

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

Principal 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 information

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets A Generalized Bivariae Ornsein-Uhlenbeck 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 information

FX OPTION PRICING: RESULTS FROM BLACK SCHOLES, LOCAL VOL, QUASI Q-PHI AND STOCHASTIC Q-PHI MODELS

FX OPTION PRICING: RESULTS FROM BLACK SCHOLES, LOCAL VOL, QUASI Q-PHI AND STOCHASTIC Q-PHI MODELS FX OPTION PRICING: REULT FROM BLACK CHOLE, LOCAL VOL, QUAI Q-PHI AND TOCHATIC Q-PHI MODEL Absrac Krishnamurhy Vaidyanahan 1 The paper suggess a new class of models (Q-Phi) o capure he informaion ha he

More information

THE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS

THE 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 information

New Pricing Framework: Options and Bonds

New Pricing Framework: Options and Bonds arxiv:1407.445v [q-fin.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 information

HANDOUT 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.

HANDOUT 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 information

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A 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 information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 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 information

HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES?

HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Absrac. We compare he opion pricing formulas of Louis Bachelier and Black-Meron-Scholes

More information

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005 FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a

More information

2.5 Life tables, force of mortality and standard life insurance products

2.5 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 information

Why Did the Demand for Cash Decrease Recently in Korea?

Why 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 information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A 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 information

APPLICATION 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. 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 information

OpenGamma Quantitative Research Multi-curves: Variations on a Theme

OpenGamma Quantitative Research Multi-curves: Variations on a Theme OpenGamma Quaniaive Research Muli-curves: Variaions on a Theme Marc Henrard marc@opengamma.com OpenGamma Quaniaive Research n. 6 Ocober 2012 Absrac The muli-curves framework is ofen implemened in a way

More information

Variance Swap. by Fabrice Douglas Rouah

Variance 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 information

A Market Model of Interest Rates with Dynamic Basis Spreads in the presence of Collateral and Multiple Currencies

A Market Model of Interest Rates with Dynamic Basis Spreads in the presence of Collateral and Multiple Currencies CIRJE-F-698 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 information

The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1

The 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 ime-series smoohing forecasing mehods. Various models are discussed,

More information

A Note on Construction of Multiple Swap Curves with and without Collateral

A 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 information

Stochastic Volatility Models: Considerations for the Lay Actuary 1. Abstract

Stochastic 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, 19-1 June 005) Absrac Sochasic models for asse prices processes

More information

Fifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance

Fifth 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 bes-esimae provisions... 3 2.1

More information

Individual Health Insurance April 30, 2008 Pages 167-170

Individual Health Insurance April 30, 2008 Pages 167-170 Individual Healh Insurance April 30, 2008 Pages 167-170 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 information

Optimal Time to Sell in Real Estate Portfolio Management

Optimal Time to Sell in Real Estate Portfolio Management Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and Jean-Luc Prigen hema, Universiy of Cergy-Ponoise, Cergy-Ponoise, France E-mails: fabricebarhelemy@u-cergyfr; jean-lucprigen@u-cergyfr

More information

On the paper Is Itô calculus oversold? by A. Izmailov and B. Shay

On 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 information

LECTURE 7 Interest Rate Models I: Short Rate Models

LECTURE 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 information

Random Walk in 1-D. 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 1-D. 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 information

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b

LIFE 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 information

Collateral Posting and Choice of Collateral Currency

Collateral 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 KIER-TMU Inernaional Workshop on Financial Engineering

More information

Fixed Income Analysis: Securities, Pricing, and Risk Management

Fixed 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, DK-5230 Odense M,

More information

Carol Alexander ICMA Centre, University of Reading. Aanand Venkatramanan ICMA Centre, University of Reading

Carol 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 DP2007-11

More information

PREMIUM INDEXING IN LIFELONG HEALTH INSURANCE

PREMIUM 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 information

Efficient Pricing of Energy Derivatives

Efficient 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 information

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates

Pricing 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 information

Curve Building and Swap Pricing in the Presence of Collateral and Basis Spreads SIMON GUNNARSSON

Curve 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 information

A Probability Density Function for Google s stocks

A 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 information

Return Calculation of U.S. Treasury Constant Maturity Indices

Return 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 information

Pricing Black-Scholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension

Pricing Black-Scholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension Pricing Black-choles Opions wih Correlaed Ineres Rae Risk and Credi Risk: An Exension zu-lang Liao a, and Hsing-Hua Huang b a irecor and Professor eparmen of inance Naional Universiy of Kaohsiung and Professor

More information

EURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN HJM ONE-FACTOR MODEL

EURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN HJM ONE-FACTOR MODEL EURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN HJM ONE-FACTOR MODEL MARC HENRARD Absrac. In his noe we give pricing formlas for differen insrmens linked o rae fres ero-dollar fres. We provide

More information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION 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 information

Term 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 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 information

TEMPORAL 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 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 information

SPEC model selection algorithm for ARCH models: an options pricing evaluation framework

SPEC 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 information

Black Scholes Option Pricing with Stochastic Returns on Hedge Portfolio

Black 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 information

Introduction to Arbitrage Pricing

Introduction 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 information

The performance of popular stochastic volatility option pricing models during the Subprime crisis

The 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 information

Modeling a distribution of mortgage credit losses Petr Gapko 1, Martin Šmíd 2

Modeling 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 information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal 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 information

Options and Volatility

Options 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 information

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619

Working 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 Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS 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 information

A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul

A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul universiy of copenhagen Universiy of Copenhagen A Two-Accoun Life Insurance Model for Scenario-Based Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:

More information

Time-inhomogeneous Lévy Processes in Cross-Currency Market Models

Time-inhomogeneous Lévy Processes in Cross-Currency Market Models Time-inhomogeneous Lévy Processes in Cross-Currency Marke Models Disseraion zur Erlangung des Dokorgrades der Mahemaischen Fakulä der Alber-Ludwigs-Universiä Freiburg i. Brsg. vorgeleg von Naaliya Koval

More information

Inductance and Transient Circuits

Inductance 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 information

Credit Risk Modeling with Random Fields

Credit Risk Modeling with Random Fields Credi Risk Modeling wih Random Fields Inaugural-Disseraion zur Erlangung des Dokorgrades an den Naurwissenschaflichen Fachbereichen (Mahemaik der Jusus-Liebig-Universiä Gießen vorgeleg von Thorsen Schmid

More information