Quanto Options. Uwe Wystup. MathFinance AG Waldems, Germany 19 September 2008

Size: px
Start display at page:

Download "Quanto Options. Uwe Wystup. MathFinance AG Waldems, Germany 19 September 2008"

Transcription

1 Quano Opions Uwe Wysup MahFinance AG Waldems, Germany 19 Sepember 2008

2 Conens 1 Quano Opions FX Quano Drif Adjusmen Exensions o oher Models Quano Vanilla Quano Forward Quano Digial Hedging of Quano Opions Vega Posiions of Quano Plain Vanilla Opions Applicaions Performance Linked Deposis Quano Opions A quano opion can be any cash-seled opion, whose payoff is convered ino a hird currency a mauriy a a pre-specified rae, called he quano facor. There can be quano plain vanilla, quano barriers, quano forward sars, quano corridors, ec. The Arbirage pricing heory and he Fundamenal heorem of asse pricing, also covered for example in [3] and [2], allow he compuaion of opion values. Oher references: Opions: basic definiions, Opion pricing: general principles, Foreign exchange marke erminology. 1.1 FX Quano Drif Adjusmen We ake he example of a Gold conrac wih underlying XAU/USD in XAU-USD quoaion ha is quanoed ino EUR. Since he payoff is in EUR, we le EUR be he numeraire or domesic or base currency and consider a Black-Scholes model XAU-EUR: ds (3) = (r EUR r XAU )S (3) d + σ 3 S (3) dw (3), (1) USD-EUR: d = (r EUR r USD ) d + σ 2 dw (2), (2) dw (3) dw (2) = ρ 23 d, (3) where we use a minus sign in fron of he correlaion, because boh S (3) and have he same base currency (DOM), which is EUR in his case. The scenario is displayed in Figure 1. The acual underlying is hen Using Iô s formula, we firs obain XAU-USD: S (1) = S(3). (4)

3 Quano Opions 3 XAU σ σ 3 1 ϕ 23 EUR π ϕ 23 σ 2 ϕ π ϕ12 12 USD Figure 1: XAU-USD-EUR FX Quano Triangle. The arrows poin in he direcion of he respecive base currencies. The lengh of he edges represens he volailiy. The cosine of he angles cos φ ij = ρ ij represens he correlaion of he currency pairs S (i) and S (j), if he base currency (DOM) of S (i) is he underlying currency (FOR) of S (j). If boh S (i) and S (j) have he same base currency (DOM), hen he correlaion is denoed by ρ ij = cos(π φ ij ). d 1 = 1 ( ) 2 ds(2) = (r USD r EUR + σ 2 2) 1 ( ) 3 (ds(2) ) 2 1 d σ 2 dw (2), (5)

4 4 Wysup and hence ds (1) = 1 ds (3) + S (3) d 1 = S(3) + S(3) (r EUR r XAU ) d + S(3) + ds (3) d 1 σ 3 dw (3) (r USD r EUR + σ2) 2 d S(3) σ 2 dw (2) + S(3) ρ 23 σ 2 σ 3 d = (r USD r XAU + σ ρ 23 σ 2 σ 3 )S (1) d + S (1) (σ 3 dw (3) σ 2 dw (2) ). Since S (1) is a geomeric Brownian moion wih volailiy σ 1, we inroduce a new Brownian moion W (1) and find ds (1) = (r USD r XAU + σ ρ 23 σ 2 σ 3 )S (1) d + σ 1 S (1) dw (1). (6) Now Figure 1 and he law of cosine imply which yields σ 2 3 = σ σ 2 2 2ρ 12 σ 1 σ 2, (7) σ 2 1 = σ σ ρ 23 σ 2 σ 3, (8) σ ρ 23 σ 2 σ 3 = ρ 12 σ 1 σ 2. (9) As explained in he currency riangle in Figure 1, ρ 12 is he correlaion beween XAU-USD and USD-EUR, whence ρ = ρ 12 is he correlaion beween XAU-USD and EUR-USD. Insering his ino Equaion (6), we obain he usual formula for he drif adjusmen ds (1) = (r USD r XAU ρσ 1 σ 2 )S (1) d + σ 1 S (1) dw (1). (10) This is he Risk Neural Pricing process ha can be used for he valuaion of any derivaive depending on S (1) which is quanoed ino EUR Exensions o oher Models The previous derivaion can be exended o he case of erm-srucure of volailiy and correlaion. However, inroducion of volailiy smile would disor he relaionships. Neverheless, accouning for smile effecs is imporan in real marke scenarios. See Foreign exchange smile: convenions and empirical facs and Foreign exchange smile modeling for deails. To do his, one could, for example, capure he smile for a muli-currency model wih a weighed Mone Carlo echnique as described by Avellaneda e al. in [1]. This would sill allow o use he previous resul.

5 Quano Opions Quano Vanilla Common among Foreign exchange opions is a quano plain vanilla paying Q[φ(S T K)] +, (11) where K denoes he srike, T he expiraion ime, φ he usual pu-call indicaor aking he value +1 for a call and 1 for a pu, S he underlying in FOR-DOM quoaion and Q he quano facor from he domesic currency ino he quano currency. We le µ = r d r f ρσ σ, (12) be he adjused drif, where r d and r f denoe he risk free raes of he domesic and foreign underlying currency pair respecively, σ = σ 1 he volailiy of his currency pair, σ = σ 2 he volailiy of he currency pair DOM-QUANTO and ρ = σ2 3 σ 2 σ 2 (13) 2σ σ he correlaion beween he currency pairs FOR-DOM and DOM-QUANTO in his quoaion. Furhermore we le r Q be he risk free rae of he quano currency. Wih he same principles as in Pricing formulae for foreign exchange opions we can derive he formula for he value as v = Qe r QT φ[s 0 e µt N (φd + ) KN (φd )], (14) d ± = ln S 0 + ( µ ± 1σ2) T K 2 σ, (15) T where N denoes he cumulaive sandard normal disribuion funcion and n is densiy. 1.3 Quano Forward Similarly, we can easily deermine he value of a quano forward paying Q[φ(S T K)], (16) where K denoes he srike, T he expiraion ime, φ he usual long-shor indicaor, S he underlying in FOR-DOM quoaion and Q he quano facor from he domesic currency ino he quano currency. Then he formula for he value can be wrien as v = Qe r QT φ[s 0 e µt K]. (17) This follows from he vanilla quano value formula by aking boh he normal probabiliies o be one. These normal probabiliies are exercise probabiliies under some measure. Since a forward conrac is always exercised, boh hese probabiliies mus be equal o one.

6 6 Wysup 1.4 Quano Digial A European syle quano digial pays QII {φst φk}, (18) where K denoes he srike, S T he spo of he currency pair FOR-DOM a mauriy T, φ akes he values +1 for a digial call and 1 for a digial pu, and Q is he pre-specified conversion rae from he domesic o he quano currency. The valuaion of European syle quano digials follows he same principle as in he quano vanilla opion case. The value is v = Qe r QT N (φd ). (19) We provide an example of European syle digial pu in USD/JPY quano ino EUR in Table 1. Noional Mauriy European syle Barrier Theoreical value Fixing source 100,000 EUR 3 monhs (92days) USD-JPY 71,555 EUR ECB Table 1: Example of a quano digial pu. The buyer receives 100,000 EUR if a mauriy, he ECB fixing for USD-JPY (compued via EUR-JPY and EUR-USD) is below Terms were creaed on Jan wih he following marke daa: USD-JPY spo ref , USD-JPY ATM vol 8.55%, EUR-JPY ATM vol 6.69%, EUR-USD ATM vol 10.99% (corresponding o a correlaion of % for USD-JPY agains JPY-EUR), USD rae 2.5%, JPY rae 0.1%, EUR rae 4%. 1.5 Hedging of Quano Opions Hedging of quano opions can be done by running a muli-currency opions book. All he usual Greeks can be hedged. Dela hedging is done by rading in he underlying spo marke. An excepion is he correlaion risk, which can only be hedged wih oher derivaives depending on he same correlaion. This is normally no possible. In FX he correlaion risk can be ranslaed ino a vega posiion as shown in [4] or in he secion on Foreign exchange baske opions. We illusrae his approach for quano plain vanilla opions now.

7 Quano Opions Vega Posiions of Quano Plain Vanilla Opions Saring from Equaion (14), we obain he sensiiviies v [ σ = QS 0e ( µ r Q)T n(d + ) ] T φn (φd + )ρ σt, v σ = QS 0e ( µ r Q)T φn (φd + )ρσt, v ρ = QS 0e ( µ r Q)T φn (φd + )σ σt, v = v ρ σ 3 ρ σ 3 = v σ 3 ρ σ σ = QS 0 e ( µ r Q)T φn (φd + )σ σt σ 3 σ σ = QS 0 e ( µ r Q)T φn (φd + )σ 3 T = QS 0 e ( µ r Q)T φn (φd + ) σ 2 + σ 2 + 2ρσ σt. Noe ha he compuaion is sandard calculus and repeaedly using he ideniy S 0 e µt n(φd + ) = Kn(φd ). (20) The undersanding of hese Greeks is ha σ and σ are boh risky parameers, independen of each oher. The hird independen risk is eiher σ 3 or ρ, depending on wha is more likely o be known. This shows exacly how he hree vega posiions can be hedged wih plain vanilla opions in all hree legs, provided here is a liquid vanilla opions marke in all hree legs. In he example wih XAU-USD-EUR he currency pairs XAU-USD and EUR-USD are raded, however, here is no liquid vanilla marke in XAU-EUR. Therefore, he correlaion risk remains unhedgeable. Similar saemens would apply for quanoed socks or sock indices. However, in FX, here are siuaions wih all legs being hedgeable, for insance EUR-USD-JPY. The signs of he vega posiions are no uniquely deermined in all legs. The FOR-DOM vega is smaller han he corresponding vanilla vega in case of a call and posiive correlaion or pu and negaive correlaion, larger in case of a pu and posiive correlaion or call and negaive correlaion. The DOM-Q vega akes he sign of he correlaion in case of a call and is opposie sign in case of a pu. The FOR-Q vega akes he opposie sign of he pu-call indicaor φ. We provide an example of pricing and vega hedging scenario in Table 2, where we noice, ha dominaing vega risk comes from he FOR-DOM pair, whence mos of he risk can be hedged.

8 8 Wysup daa se 1 daa se 2 daa se 3 FX pair FOR-DOM XAU-USD XAU-USD XAU-USD spo FOR-DOM srike FOR-DOM quano DOM-Q volailiy FOR-DOM 10.00% 10.00% 10.00% quano volailiy DOM-Q 12.00% 12.00% 12.00% correlaion FOR-DOM DOM-Q 25.00% 25.00% % domesic ineres rae DOM % % % foreign ineres rae FOR % % % quano currency rae Q % % % ime in years T =call -1=pu FOR quano vanilla opion value quano vanilla opion vega FOR-DOM quano vanilla opion vega DOM-Q quano vanilla opion vega FOR-Q quano vanilla opion correlaion risk quano vanilla opion vol FOR-Q % % % vanilla opion value vanilla opion vega Table 2: Example of a quano plain vanilla. 1.6 Applicaions The sandard applicaion are performance linked deposi or performance noes as in [5]. Any ime he performance of an underlying asse needs o be convered ino he noional currency invesed, and he exchange rae risk is wih he seller, we need a quano produc. Naurally,

9 Quano Opions 9 an underlying like gold, which is quoed in USD, would be a defaul candidae for a quano produc, when he invesmen is in a currency oher han USD Performance Linked Deposis A performance linked deposi is a deposi wih a paricipaion in an underlying marke. The sandard is ha a GBP invesor waives her coupon ha he money marke would pay and insead buys a EUR-GBP call wih he same mauriy dae as he coupon, srike K and noional N in EUR. These parameers have o be chosen in such a way ha he offer price of he EUR call equals he money marke ineres rae plus he sales margin. The srike is ofen chosen o be he curren spo. The noional is ofen a percenage p of he deposi amoun A, such as 50% or 25%. The annual coupon paid o he invesor is hen a pre-defined minimum coupon plus he paricipaion p max[s T S 0, 0] S 0, (21) which is he reurn of he exchange rae viewed as an asse, where he invesor is proeced agains negaive reurns. So, obviously, he invesor buys a EUR call GBP pu wih srike K = S 0 and noional N = pa GBP or N = pa/s 0 EUR. Thus, if he EUR goes up by 10% agains he GBP, he invesor ges a coupon of p 10% p.a. in addiion o he minimum coupon. Example. We consider he example shown in Table 3. In his case, if he EUR-GBP spo fixing is , he addiional coupon would be % p.a. The break-even poin is a , so his produc is advisable for a very srong EUR bullish view. For a weakly bullish view an alernaive would be o buy an up-and-ou call wih barrier a and 75% paricipaion, where we would find he bes case o be wih an addiional coupon of 4.275% p.a., which would lead o a oal coupon of 6.275% p.a. Composiion ˆ From he money marke we ge 49, GBP a he mauriy dae. ˆ The invesor buys a EUR call GBP pu wih srike and wih noional 1.5 Million GBP. ˆ The offer price of he call is 26, GBP, assuming a volailiy of 8.0% and a EUR rae of 2.50%. ˆ The deferred premium is 24, GBP. ˆ The invesor receives a minimum paymen of 24, GBP.

10 10 Wysup Noional 5,000,000 GBP Sar dae 3 June 2005 Mauriy Number of days 91 Money marke reference rae EUR-GBP spo reference Minimum rae Addiional coupon S T Fixing source 2 Sepember 2005 (91 days) 4.00% ac/ % ac/365 30% 100 max[s T ,0] ac/365 EUR-GBP fixing on 31 Augus 2005 (88 days) ECB Table 3: Example of a performance linked deposi, where he invesor is paid 30% of he EUR-GBP reurn. Noe ha in GBP he daycoun convenion in he money marke is ac/365 raher han ac/360. ˆ Subracing he deferred premium and he minimum paymen from he money marke leaves a sales margin of GBP (awfully poor I admi). ˆ Noe ha he opion he invesor is buying mus be cash-seled. Variaions. There are many variaions of he performance linked noes. Of course, one can hink of European syle knock-ou calls or window-barrier calls. For a paricipaion in a downward rend, he invesor can buy pus. One of he frequen issues in Foreign Exchange, however, is he deposi currency being differen from he domesic currency of he exchange rae, which is quoed in FOR-DOM (foreign-domesic), meaning how many unis of domesic currency are required o buy one uni of foreign currency. So if we have a EUR invesor who wishes o paricipae in a EUR-USD movemen, we have a problem, he usual quano confusion ha can drive anybody up he wall in FX a various occasions. Wha is he problem? The payoff of he EUR call USD pu [S T K] + (22) is in domesic currency (USD). Of course, his payoff can be convered ino he foreign currency (EUR) a mauriy, bu a wha rae? If we conver a rae S T, which is wha we could do in he spo marke a no cos, hen he invesor buys a vanilla EUR call. Bu here, he invesor receives a coupon given by

11 Quano Opions 11 p max[s T S 0, 0] S T. (23) If he invesor wishes o have performance of Equaion (21) raher han Equaion (23), hen he payoff a mauriy is convered a a rae of ino EUR, and his rae is se a he beginning of he rade. This is he quano facor, and he vanilla is acually a self-quano vanilla, i.e., a EUR call USD pu, cash-seled in EUR, where he payoff in USD is convered ino EUR a a rae of This self quano vanilla can be valued by invering he exchange rae, i.e., looking a USD-EUR. This way he valuaion can incorporae he smile of EUR-USD. Similar consideraions need o be aken ino accoun if he currency pair o paricipae in does no conain he deposi currency a all. A ypical siuaion is a EUR invesor, who wishes o paricipae in he gold price, which is measured in USD, so he invesor needs o buy a XAU call USD pu quanoed ino EUR. So he invesor is promised a coupon as in Equaion (21) for a XAU-USD underlying, where he coupon is paid in EUR, his implicily means ha we mus use a quano plain vanilla wih a quano facor of References [1] Avellaneda, M., Buff, R., Friedman, C., Grandechamp, N., Kruk, L. and Newman, j. (2001). Weighed Mone Carlo: A new Technique for Calibraing Asse- Pricing Models. Inernaional Journal of Theoreical and Applied Finance, vol 4, No. 1, pp [2] Hakala, J. and Wysup, U. (2002). Foreign Exchange Risk. Risk Publicaions, London. [3] Shreve, S.E. (2004). Sochasic Calculus for Finance I+II. Springer. [4] Wysup, U. (2001). How he Greeks would have hedged Correlaion Risk of Foreign Exchange Opions, Wilmo Research Repor, Augus Also in Foreign Exchange Risk, Risk Publicaions, London [5] Wysup, U. (2006). FX Opions and Srucured Producs. Wiley Finance Series.

12 Index correlaion, FX, 5 currency riangle, 4 law of cosine, 4 performance linked deposi, 8 quano digial, 6 quano drif adjusmen, 2 quano facor, 2 quano forward, 5 quano opions, 2 quano plain vanilla, 8 quano vanilla, 4 self-quano, 10 vega, quano plain vanilla, 6 12

Pricing formula for power quanto options with each type of payoffs at maturity

Pricing 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 information

INSTITUTE OF ACTUARIES OF INDIA

INSTITUTE 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 information

The Mathematics Of Stock Option Valuation - Part Four Deriving The Black-Scholes Model Via Partial Differential Equations

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 information

Foreign Exchange, ADR s and Quanto-Securities

Foreign 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 information

Pricing FX Target Redemption Forward under. Regime Switching Model

Pricing 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 information

INSTITUTE OF ACTUARIES OF INDIA

INSTITUTE 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 information

May 2007 Exam MFE Solutions 1. Answer = (B)

May 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 information

MAFS Quantitative Modeling of Derivative Securities

MAFS 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 information

Equivalent Martingale Measure in Asian Geometric Average Option Pricing

Equivalent 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 information

Tentamen i 5B1575 Finansiella Derivat. Måndag 27 augusti 2007 kl Answers and suggestions for solutions.

Tentamen 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 information

A pricing model for the Guaranteed Lifelong Withdrawal Benefit Option

A 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 information

Matematisk statistik Tentamen: kl FMS170/MASM19 Prissättning av Derivattillgångar, 9 hp Lunds tekniska högskola. Solution.

Matematisk 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 information

Market Models. Practitioner Course: Interest Rate Models. John Dodson. March 29, 2009

Market 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 information

Black-Scholes Model and Risk Neutral Pricing

Black-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 information

Black-Scholes and the Volatility Surface

Black-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 information

CURRENCY TRANSLATED OPTIONS

CURRENCY 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 information

(1 + Nominal Yield) = (1 + Real Yield) (1 + Expected Inflation Rate) (1 + Inflation Risk Premium)

(1 + Nominal Yield) = (1 + Real Yield) (1 + Expected Inflation Rate) (1 + Inflation Risk Premium) 5. Inflaion-linked bonds Inflaion is an economic erm ha describes he general rise in prices of goods and services. As prices rise, a uni of money can buy less goods and services. Hence, inflaion is an

More information

Introduction to Black-Scholes Model

Introduction 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 information

t=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

t=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 information

Description of the CBOE S&P 500 2% OTM BuyWrite Index (BXY SM )

Description of the CBOE S&P 500 2% OTM BuyWrite Index (BXY SM ) Descripion of he CBOE S&P 500 2% OTM BuyWrie Index (BXY SM ) Inroducion. The CBOE S&P 500 2% OTM BuyWrie Index (BXY SM ) is a benchmark index designed o rack he performance of a hypoheical 2% ou-of-he-money

More information

Origins of currency swaps

Origins of currency swaps Origins of currency swaps Currency swaps originally were developed by banks in he UK o help large cliens circumven UK exchange conrols in he 1970s. UK companies were required o pay an exchange equalizaion

More information

Proceedings of the 48th European Study Group Mathematics with Industry 1

Proceedings 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 information

Description of the CBOE Russell 2000 BuyWrite Index (BXR SM )

Description of the CBOE Russell 2000 BuyWrite Index (BXR SM ) Descripion of he CBOE Russell 2000 BuyWrie Index (BXR SM ) Inroducion. The CBOE Russell 2000 BuyWrie Index (BXR SM ) is a benchmark index designed o rack he performance of a hypoheical a-he-money buy-wrie

More information

FINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004

FINAL 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 information

Balance of Payments. Second quarter 2012

Balance of Payments. Second quarter 2012 Balance of Paymens Second quarer 2012 Balance of Paymens Second quarer 2012 Saisics Sweden 2012 Balance of Paymens. Second quarer 2012 Saisics Sweden 2012 Producer Saisics Sweden, Balance of Paymens and

More information

IJRSS Volume 2, Issue 2 ISSN:

IJRSS 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 information

Available online at ScienceDirect

Available 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 information

1 Purpose of the paper

1 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 information

Portfolio investments accounted for the largest outflow of SEK 77.5 billion in the financial account, which gave a net outflow of SEK billion.

Portfolio investments accounted for the largest outflow of SEK 77.5 billion in the financial account, which gave a net outflow of SEK billion. BALANCE OF PAYMENTS DATE: 27-11-27 PUBLISHER: Saisics Sweden Balance of Paymens and Financial Markes (BFM) Maria Falk +46 8 6 94 72, maria.falk@scb.se Camilla Bergeling +46 8 6 942 6, camilla.bergeling@scb.se

More information

Models of Default Risk

Models 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 information

Volatility and Hedging Errors

Volatility and Hedging Errors Volailiy and Hedging Errors Jim Gaheral Sepember, 5 1999 Background Derivaive porfolio bookrunners ofen complain ha hedging a marke-implied volailiies is sub-opimal relaive o hedging a heir bes guess of

More information

Pricing Vulnerable American Options. April 16, Peter Klein. and. Jun (James) Yang. Simon Fraser University. Burnaby, B.C. V5A 1S6.

Pricing 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 information

Online Appendix. Using the reduced-form model notation proposed by Doshi, el al. (2013), 1. and Et

Online Appendix. Using the reduced-form model notation proposed by Doshi, el al. (2013), 1. and Et Online Appendix Appendix A: The concep in a muliperiod framework Using he reduced-form model noaion proposed by Doshi, el al. (2013), 1 he yearly CDS spread S c,h for a h-year sovereign c CDS conrac can

More information

Tentamen i 5B1575 Finansiella Derivat. Torsdag 25 augusti 2005 kl

Tentamen 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 information

Synthetic CDO s and Basket Default Swaps in a Fixed Income Credit Portfolio

Synthetic CDO s and Basket Default Swaps in a Fixed Income Credit Portfolio Synheic CDO s and Baske Defaul Swaps in a Fixed Income Credi Porfolio Louis Sco June 2005 Credi Derivaive Producs CDO Noes Cash & Synheic CDO s, various ranches Invesmen Grade Corporae names, High Yield

More information

Computations in the Hull-White Model

Computations 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 information

Lecture Notes to Finansiella Derivat (5B1575) VT Note 1: No Arbitrage Pricing

Lecture 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 information

Option Valuation of Oil & Gas E&P Projects by Futures Term Structure Approach. Hidetaka (Hugh) Nakaoka

Option 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 information

MORNING SESSION. Date: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES

MORNING 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 information

Coupling Smiles. November 18, 2006

Coupling 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 information

a. If Y is 1,000, M is 100, and the growth rate of nominal money is 1 percent, what must i and P be?

a. If Y is 1,000, M is 100, and the growth rate of nominal money is 1 percent, what must i and P be? Problem Se 4 ECN 101 Inermediae Macroeconomics SOLUTIONS Numerical Quesions 1. Assume ha he demand for real money balance (M/P) is M/P = 0.6-100i, where is naional income and i is he nominal ineres rae.

More information

Term 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 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 information

Single Premium of Equity-Linked with CRR and CIR Binomial Tree

Single 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 information

Exotic FX Swap. Analytics. ver 1.0. Exotics Pricing Methodology Trading Credit Risk Pricing

Exotic FX Swap. Analytics. ver 1.0. Exotics Pricing Methodology Trading Credit Risk Pricing Exoic FX Swap Analyics ver 1. Exoics Pricing Mehodology Trading Credi Risk Pricing Exoic FX Swap Version: ver 1. Deails abou he documen Projec Exoics Pricing Version ver 1. Dae January 24, 22 Auhors Deparmen

More information

Principles of Finance CONTENTS

Principles of Finance CONTENTS Principles of Finance CONENS Value of Bonds and Equiy... 3 Feaures of bonds... 3 Characerisics... 3 Socks and he sock marke... 4 Definiions:... 4 Valuing equiies... 4 Ne reurn... 4 idend discoun model...

More information

Improving the Jarrow-Yildirim Inflation Model

Improving the Jarrow-Yildirim Inflation Model Improving he Jarrow-Yildirim Inflaion Model Rober Hardy May 19, 2013 1 Inroducion The mos liquid inflaion markes are hose of he US, UK, France and Eurozone. Each is suppored by a regular supply of governmen-issued

More information

Optimal Early Exercise of Vulnerable American Options

Optimal 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 information

Standard derivatives pricing theory (see, for example, Hull,

Standard derivatives pricing theory (see, for example, Hull, Cuing edge Derivaives pricing Funding beyond discouning: collaeral agreemens and derivaives pricing Sandard heory assumes raders can lend and borrow a a risk-free rae, ignoring he inricacies of he repo

More information

Financial Econometrics (FinMetrics02) Returns, Yields, Compounding, and Horizon

Financial Econometrics (FinMetrics02) Returns, Yields, Compounding, and Horizon Financial Economerics FinMerics02) Reurns, Yields, Compounding, and Horizon Nelson Mark Universiy of Nore Dame Fall 2017 Augus 30, 2017 1 Conceps o cover Yields o mauriy) Holding period) reurns Compounding

More information

GUIDELINE Solactive Gold Front Month MD Rolling Futures Index ER. Version 1.1 dated April 13 th, 2017

GUIDELINE Solactive Gold Front Month MD Rolling Futures Index ER. Version 1.1 dated April 13 th, 2017 GUIDELINE Solacive Gold Fron Monh MD Rolling Fuures Index ER Version 1.1 daed April 13 h, 2017 Conens Inroducion 1 Index specificaions 1.1 Shor name and ISIN 1.2 Iniial value 1.3 Disribuion 1.4 Prices

More information

Brownian motion. Since σ is not random, we can conclude from Example sheet 3, Problem 1, that

Brownian 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 information

Final Exam Answers Exchange Rate Economics

Final 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 information

7 pages 1. Hull and White Generalized model. Ismail Laachir. March 1, Model Presentation 1

7 pages 1. Hull and White Generalized model. Ismail Laachir. March 1, Model Presentation 1 7 pages 1 Hull and Whie Generalized model Ismail Laachir March 1, 212 Conens 1 Model Presenaion 1 2 Calibraion of he model 3 2.1 Fiing he iniial yield curve................... 3 2.2 Fiing he caple implied

More information

Funding beyond discounting: collateral agreements and derivatives pricing

Funding beyond discounting: collateral agreements and derivatives pricing cuing edge. DERIVAIVES PRICING Funding beyond discouning: collaeral agreemens and derivaives pricing Sandard heory assumes raders can lend and borrow a a risk-free rae, ignoring he inricacies of he repo

More information

Fundamental Basic. Fundamentals. Fundamental PV Principle. Time Value of Money. Fundamental. Chapter 2. How to Calculate Present Values

Fundamental Basic. Fundamentals. Fundamental PV Principle. Time Value of Money. Fundamental. Chapter 2. How to Calculate Present Values McGraw-Hill/Irwin Chaper 2 How o Calculae Presen Values Principles of Corporae Finance Tenh Ediion Slides by Mahew Will And Bo Sjö 22 Copyrigh 2 by he McGraw-Hill Companies, Inc. All righs reserved. Fundamenal

More information

Balance of Payments. Third quarter 2009

Balance of Payments. Third quarter 2009 Balance of Paymens Third quarer 2009 Balance of Paymens Third quarer 2009 Saisics Sweden 2009 Balance of Paymens. Third quarer 2009 Saisics Sweden 2009 Producer Saisics Sweden, Balance of Paymens and

More information

A UNIFIED PDE MODELLING FOR CVA AND FVA

A 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 information

Problem Set 1 Answers. a. The computer is a final good produced and sold in Hence, 2006 GDP increases by $2,000.

Problem Set 1 Answers. a. The computer is a final good produced and sold in Hence, 2006 GDP increases by $2,000. Social Analysis 10 Spring 2006 Problem Se 1 Answers Quesion 1 a. The compuer is a final good produced and sold in 2006. Hence, 2006 GDP increases by $2,000. b. The bread is a final good sold in 2006. 2006

More information

Hull-White one factor model Version

Hull-White one factor model Version Hull-Whie one facor model Version 1.0.17 1 Inroducion This plug-in implemens Hull and Whie one facor models. reference on his model see [?]. For a general 2 How o use he plug-in In he Fairma user inerface

More information

PRESS RELEASE EURO AREA ECONOMIC AND FINANCIAL DEVELOPMENTS BY INSTITUTIONAL SECTOR - FIRST QUARTER August 2012

PRESS RELEASE EURO AREA ECONOMIC AND FINANCIAL DEVELOPMENTS BY INSTITUTIONAL SECTOR - FIRST QUARTER August 2012 1 Augus 212 PRESS RELEASE EURO AREA ECONOMIC AND FINANCIAL DEVELOPMENTS BY INSTITUTIONAL SECTOR - FIRST QUARTER 212 In he firs quarer of 212, he annual growh rae 1 of households gross disposable income

More information

Calibrating and pricing with embedded local volatility models

Calibrating and pricing with embedded local volatility models CUING EDGE. OPION PRICING Calibraing and pricing wih embedded local volailiy models Consisenly fiing vanilla opion surfaces when pricing volailiy derivaives such as Vix opions or ineres rae/ equiy hybrids

More information

Constructing Out-of-the-Money Longevity Hedges Using Parametric Mortality Indexes. Johnny Li

Constructing 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 information

DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Krzysztof Jajuga Wrocław University of Economics

DYNAMIC 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 information

The Relationship between Money Demand and Interest Rates: An Empirical Investigation in Sri Lanka

The Relationship between Money Demand and Interest Rates: An Empirical Investigation in Sri Lanka The Relaionship beween Money Demand and Ineres Raes: An Empirical Invesigaion in Sri Lanka R. C. P. Padmasiri 1 and O. G. Dayarana Banda 2 1 Economic Research Uni, Deparmen of Expor Agriculure 2 Deparmen

More information

UCLA Department of Economics Fall PhD. Qualifying Exam in Macroeconomic Theory

UCLA 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 information

GUIDELINE Solactive Bitcoin Front Month Rolling Futures 5D Index ER. Version 1.0 dated December 8 th, 2017

GUIDELINE Solactive Bitcoin Front Month Rolling Futures 5D Index ER. Version 1.0 dated December 8 th, 2017 GUIDELINE Solacive Bicoin Fron Monh Rolling Fuures 5D Index ER Version 1.0 daed December 8 h, 2017 Conens Inroducion 1 Index specificaions 1.1 Shor name and ISIN 1.2 Iniial value 1.3 Disribuion 1.4 Prices

More information

DEBT INSTRUMENTS AND MARKETS

DEBT INSTRUMENTS AND MARKETS DEBT INSTRUMENTS AND MARKETS Zeroes and Coupon Bonds Zeroes and Coupon Bonds Ouline and Suggesed Reading Ouline Zero-coupon bonds Coupon bonds Bond replicaion No-arbirage price relaionships Zero raes Buzzwords

More information

On the Impact of Inflation and Exchange Rate on Conditional Stock Market Volatility: A Re-Assessment

On the Impact of Inflation and Exchange Rate on Conditional Stock Market Volatility: A Re-Assessment MPRA Munich Personal RePEc Archive On he Impac of Inflaion and Exchange Rae on Condiional Sock Marke Volailiy: A Re-Assessmen OlaOluwa S Yaya and Olanrewaju I Shiu Deparmen of Saisics, Universiy of Ibadan,

More information

Systemic Risk Illustrated

Systemic 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 information

Alexander L. Baranovski, Carsten von Lieres and André Wilch 18. May 2009/Eurobanking 2009

Alexander 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 information

Spring 2011 Social Sciences 7418 University of Wisconsin-Madison

Spring 2011 Social Sciences 7418 University of Wisconsin-Madison Economics 32, Sec. 1 Menzie D. Chinn Spring 211 Social Sciences 7418 Universiy of Wisconsin-Madison Noes for Econ 32-1 FALL 21 Miderm 1 Exam The Fall 21 Econ 32-1 course used Hall and Papell, Macroeconomics

More information

ECONOMIC GROWTH. Student Assessment. Macroeconomics II. Class 1

ECONOMIC GROWTH. Student Assessment. Macroeconomics II. Class 1 Suden Assessmen You will be graded on he basis of In-class aciviies (quizzes worh 30 poins) which can be replaced wih he number of marks from he regular uorial IF i is >=30 (capped a 30, i.e. marks from

More information

Valuing Real Options on Oil & Gas Exploration & Production Projects

Valuing 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 information

Agenda. What is an ESG? GIRO Convention September 2008 Hilton Sorrento Palace

Agenda. What is an ESG? GIRO Convention September 2008 Hilton Sorrento Palace GIRO Convenion 23-26 Sepember 2008 Hilon Sorreno Palace A Pracical Sudy of Economic Scenario Generaors For General Insurers Gareh Haslip Benfield Group Agenda Inroducion o economic scenario generaors Building

More information

Macroeconomics II A dynamic approach to short run economic fluctuations. The DAD/DAS model.

Macroeconomics 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 information

FAIR VALUATION OF INSURANCE LIABILITIES. Pierre DEVOLDER Université Catholique de Louvain 03/ 09/2004

FAIR 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 information

CHAPTER CHAPTER18. Openness in Goods. and Financial Markets. Openness in Goods, and Financial Markets. Openness in Goods,

CHAPTER CHAPTER18. Openness in Goods. and Financial Markets. Openness in Goods, and Financial Markets. Openness in Goods, Openness in Goods and Financial Markes CHAPTER CHAPTER18 Openness in Goods, and Openness has hree disinc dimensions: 1. Openness in goods markes. Free rade resricions include ariffs and quoas. 2. Openness

More information

Bond Prices and Interest Rates

Bond Prices and Interest Rates Winer erm 1999 Bond rice Handou age 1 of 4 Bond rices and Ineres Raes A bond is an IOU. ha is, a bond is a promise o pay, in he fuure, fixed amouns ha are saed on he bond. he ineres rae ha a bond acually

More information

Exam 1. Econ520. Spring 2017

Exam 1. Econ520. Spring 2017 Exam 1. Econ520. Spring 2017 Professor Luz Hendricks UNC Insrucions: Answer all quesions. Clearly number your answers. Wrie legibly. Do no wrie your answers on he quesion shees. Explain your answers do

More information

EQUILIBRIUM ASSET PRICING MODELS

EQUILIBRIUM ASSET PRICING MODELS EQUILIBRIUM ASSET PRICING MODELS 2 Asse pricing derived rom heory o consumpion and invesmen behavior 2 Pricing equaions oen ake he orm o PV models: 4 Asse value equals expeced sum o discouned uure CFs

More information

Advanced Tools for Risk Management and Asset Pricing

Advanced 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 information

HEDGING VOLATILITY RISK

HEDGING VOLATILITY RISK HEDGING VOLAILIY RISK Menachem Brenner Sern School of Business New York Universiy New York, NY 00, U.S.A. Email: mbrenner@sern.nyu.edu Ernes Y. Ou ABN AMRO, Inc. Chicago, IL 60604, U.S.A. Email: Yi.Ou@abnamro.com

More information

where r() = r(s)e a( s) + α() α(s)e a( s) + σ e a( u) dw(u) s α() = f M (0, ) + σ a (1 e a ) Therefore, r() condiional on F s is normally disribued wi

where r() = r(s)e a( s) + α() α(s)e a( s) + σ e a( u) dw(u) s α() = f M (0, ) + σ a (1 e a ) Therefore, r() condiional on F s is normally disribued wi Hull-Whie Model Conens Hull-Whie Model Hull-Whie Tree Example: Hull-Whie Tree Calibraion Appendix: Ineres Rae Derivaive PDE Hull-Whie Model This secion is adaped from Brigo and Mercurio (006). As an exension

More information

Macroeconomics. Typical macro questions (I) Typical macro questions (II) Methodology of macroeconomics. Tasks carried out by macroeconomists

Macroeconomics. Typical macro questions (I) Typical macro questions (II) Methodology of macroeconomics. Tasks carried out by macroeconomists Macroeconomics Macroeconomics is he area of economics ha sudies he overall economic aciviy in a counry or region by means of indicaors of ha aciviy. There is no essenial divide beween micro and macroeconomics,

More information

AN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM MODEL WITH STOCHASTIC INTEREST RATES

AN 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 information

Jarrow-Lando-Turnbull model

Jarrow-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 information

2. Quantity and price measures in macroeconomic statistics 2.1. Long-run deflation? As typical price indexes, Figure 2-1 depicts the GDP deflator,

2. Quantity and price measures in macroeconomic statistics 2.1. Long-run deflation? As typical price indexes, Figure 2-1 depicts the GDP deflator, 1 2. Quaniy and price measures in macroeconomic saisics 2.1. Long-run deflaion? As ypical price indexes, Figure 2-1 depics he GD deflaor, he Consumer rice ndex (C), and he Corporae Goods rice ndex (CG)

More information

Some Remarks on Derivatives Markets (third edition, 2013)

Some Remarks on Derivatives Markets (third edition, 2013) Some Remarks on Derivaives Markes (hird ediion, 03) Elias S. W. Shiu. The parameer δ in he Black-Scholes formula The Black-Scholes opion-pricing formula is given in Chaper of McDonald wihou proof. A raher

More information

4452 Mathematical Modeling Lecture 17: Modeling of Data: Linear Regression

4452 Mathematical Modeling Lecture 17: Modeling of Data: Linear Regression Mah Modeling Lecure 17: Modeling of Daa: Linear Regression Page 1 5 Mahemaical Modeling Lecure 17: Modeling of Daa: Linear Regression Inroducion In modeling of daa, we are given a se of daa poins, and

More information

The Market for Volatility Trading; VIX Futures

The Market for Volatility Trading; VIX Futures he Marke for olailiy rading; IX uures Menachem Brenner ern chool of Business New York Universiy New York, NY, U..A. Email: mbrenner@sern.nyu.edu el: 998 33, ax: 995 473 Jinghong hu chool of Inernaional

More information

Option pricing and hedging in jump diffusion models

Option 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 information

Once 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

Once 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 information

Research Article A General Gaussian Interest Rate Model Consistent with the Current Term Structure

Research 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 information

CENTRO DE ESTUDIOS MONETARIOS Y FINANCIEROS T. J. KEHOE MACROECONOMICS I WINTER 2011 PROBLEM SET #6

CENTRO DE ESTUDIOS MONETARIOS Y FINANCIEROS T. J. KEHOE MACROECONOMICS I WINTER 2011 PROBLEM SET #6 CENTRO DE ESTUDIOS MONETARIOS Y FINANCIEROS T J KEHOE MACROECONOMICS I WINTER PROBLEM SET #6 This quesion requires you o apply he Hodrick-Presco filer o he ime series for macroeconomic variables for he

More information

Documentation: Philadelphia Fed's Real-Time Data Set for Macroeconomists First-, Second-, and Third-Release Values

Documentation: Philadelphia Fed's Real-Time Data Set for Macroeconomists First-, Second-, and Third-Release Values Documenaion: Philadelphia Fed's Real-Time Daa Se for Macroeconomiss Firs-, Second-, and Third-Release Values Las Updaed: December 16, 2013 1. Inroducion We documen our compuaional mehods for consrucing

More information

Valuation and Hedging of Correlation Swaps. Mats Draijer

Valuation 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 information

Change of measure and Girsanov theorem

Change 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 information

An Indian Journal FULL PAPER. Trade Science Inc. The principal accumulation value of simple and compound interest ABSTRACT KEYWORDS

An Indian Journal FULL PAPER. Trade Science Inc. The principal accumulation value of simple and compound interest ABSTRACT KEYWORDS [Type ex] [Type ex] [Type ex] ISSN : 0974-7435 Volume 0 Issue 8 BioTechnology 04 An Indian Journal FULL PAPER BTAIJ, 08), 04 [0056-006] The principal accumulaion value of simple and compound ineres Xudong

More information

Provide a brief review of futures markets. Carefully review alternative market conditions and which marketing

Provide a brief review of futures markets. Carefully review alternative market conditions and which marketing Provide a brief review of fuures markes. Carefully review alernaive marke condiions and which markeing sraegies work bes under alernaive condiions. Have an open and ineracive discussion!! 1. Sore or Wai

More information

Eris EURIBOR Interest Rate Future

Eris EURIBOR Interest Rate Future ICE Fuures Europe Jan 21, 2018 Eris EURIBOR Ineres Rae Fuure Conrac Specificaions Descripion 100,000 noional principal whose value is based upon he difference beween a sream of annual fixed ineres paymens

More information