Cashsettled swaptions How wrong are we?


 Frank Holland
 1 years ago
 Views:
Transcription
1 Cashsettled swaptions How wrong are we? Marc Henrard Quantitative Research  OpenGamma 7th Fixed Income Conference, Berlin, 7 October 2011
2 ... Based on: Cashsettled swaptions: How wrong are we? Available at SSRN Working Paper series, , November 2010.
3 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
4 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
5 Introduction Cashsettled swaptions are the most liquid swaptions in the interbank market for EUR and GBP. The standard pricing formulas used are not properly justified (they require untested approximations). The standard pricing formulas with standard smile description leads to arbitrage: see F. Mercurio, Cashsettled swaptions and noarbitrage, Risk, Local coherence between Market pricing formula for cashsettled and physical delivery swaption has not been analyzed.
6 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
7 Description The analyses are done for receiver swaptions. Swap start date t 0, fixed leg payment dates (t i ) 1 i n, coupon rate K and the accrual fraction (δ i ) 1 i n. Floating leg payment dates ( t i ) 1 i ñ. Swaption expiry date: θ t 0. The cash annuity for a swap rate S is (for a frequency m) C(S) = n i=1 1 m (1 + 1 m S)i. The cashsettled swaption payment is (in t 0 ) C(S)(K S) +.
8 Swap rate We work in a multicurve setup. The discounting curve is denoted P D (s, t) and the forward curve is denoted P j (s, t) where j is the Libor tenor. The physical annuity (also called PVBP or level) is A t = n δ i P D (t, t i ). i=1 The swap rate in t is ñ ( ) i=1 PD (t, t i ) P j (t, t i ) P S t = j (t, t i+1 ) 1. A t
9 Market formulas (physical delivery) The value at expiry of the delivery swaption is The generic price is A θ (K S θ ) +. N 0 E N [N 1 θ A θ (K S θ ) + ] With the numeraire A t, S t is a martingale and under lognormal model ds t = σs t dw t, the price can be computed explicitly A 0 Black(K, S 0, σ).
10 Market formulas (cashsettled) The value at expiry of the cashsettle swaption is P D (θ, t 0 )C(S θ )(K S θ ) +. With the numeraire P D (t, t 0 )C(S t ),the price becomes P D (0, t 0 )C(S 0 ) E C [(K S θ ) + ]. But S θ is not a martingale anymore. The market standard formula is to substitute C by A as numeraire and approximate the price by P(0, t 0 )C(S 0 ) E C [(K S θ ) + ] P D (0, t 0 )C(S 0 ) E A [(K S θ ) + ] = P D (0, t 0 )C(S 0 ) Black(K, S 0, σ).
11 Market formulas (cashsettled) The value at expiry of the cashsettle swaption is P D (θ, t 0 )C(S θ )(K S θ ) +. With the numeraire P D (t, t 0 )C(S t ),the price becomes P D (0, t 0 )C(S 0 ) E C [(K S θ ) + ]. But S θ is not a martingale anymore. The market standard formula is to substitute C by A as numeraire and approximate the price by P(0, t 0 )C(S 0 ) E C [(K S θ ) + ] P D (0, t 0 )C(S 0 ) E A [(K S θ ) + ] = P D (0, t 0 )C(S 0 ) Black(K, S 0, σ).
12 Market formulas (cashsettled) The value at expiry of the cashsettle swaption is P D (θ, t 0 )C(S θ )(K S θ ) +. With the numeraire P D (t, t 0 )C(S t ),the price becomes P D (0, t 0 )C(S 0 ) E C [(K S θ ) + ]. But S θ is not a martingale anymore. The market standard formula is to substitute C by A as numeraire and approximate the price by P(0, t 0 )C(S 0 ) E C [(K S θ ) + ] P D (0, t 0 )C(S 0 ) E A [(K S θ ) + ] = P D (0, t 0 )C(S 0 ) Black(K, S 0, σ).
13 Market formulas (cashsettled) The price is A 0 E A [ A 1 θ PD (θ, t 0 )C(S θ )(K S θ ) +] Consider that the annuity/annuity ratio P D (θ, t 0 )C(S θ ) A θ has a low variance and it can be replaced by its initial value (initial freeze technique) A 0 E A [ A 1 θ PD (θ, t 0 )C(S θ )(K S θ ) +] A 0 E A [ A 1 0 PD (0, t 0 )C(S 0 )(K S θ ) +] = P D (0, t 0 )C(S 0 ) E A [ (K S θ ) +].
14 Market formulas (cashsettled) The price is A 0 E A [ A 1 θ PD (θ, t 0 )C(S θ )(K S θ ) +] Consider that the annuity/annuity ratio P D (θ, t 0 )C(S θ ) A θ has a low variance and it can be replaced by its initial value (initial freeze technique) A 0 E A [ A 1 θ PD (θ, t 0 )C(S θ )(K S θ ) +] A 0 E A [ A 1 0 PD (0, t 0 )C(S 0 )(K S θ ) +] = P D (0, t 0 )C(S 0 ) E A [ (K S θ ) +].
15 Mercurio s result Mercurio presented no arbitrage conditions for cashsettled swaptions. One relates to the density deduced from option prices: C(S 0 ) x 2 Black(x, S 0, σ) C(x) dx = 1. If not a digital cashsettled option with strike 0 has a value different from its constant payoff Black implied volatility
16 Cashsettle and physical swaptions Difference between cashsettled and physical delivery swaptions for different curve environments O/N rate (%) Slope (%) 3 4
17 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
18 Data The tests are done for 5Yx10Y swaptions. For all the model used, explicit formula exists for the delivery swaptions. The market data are from 30Sep The smile is obtained with a Hagan et al. SABR approach. When a calibration is required, it is done on physical delivery swaption. At least the swaption of same maturity and strike is in the calibration basket (and calibrated perfectly).
19 Linear Terminal Swap Rate model This is not a term structure model but an approximation technique to treat the swap rate as the only state variable for the yield curve. Applied in our case, the quantity P D (θ, t 0 )A 1 θ is approximated by a swap rate function α(s θ ). It is an approximation technique more sophisticated than the initial freeze approach. It is presented in Andersen, L. and Piterbarg, V. Interest Rate Modeling, Section The cashsettled swaptions price is A 0 E A [ α(s θ )C(S θ )(K S θ ) +]. In the simple linear TSR, function is α(s) = α 1 S + α 2 and the coefficients are 1 α 2 = n i=1 δ and α 1 = 1 ( P D ) (0, t p ) α 2. i S 0 A 0
20 Linear Terminal Swap Rate model In this framework, the cashsettled swaption are priced by a replication formula + ) A 0 (k(k) Swpt(S 0, K) + (k (x)(x K) + 2k (x)) Swpt(S 0, x)dx K with k(x) = (α 1 x + α 2 )C(x). Note that the linear TSR is such that α(s 0 ) = P D (0, t 0 )A 1 0 (the approximation is exact in 0 and ATM).
21 Linear Terminal Swap Rate model Strike Mrkt DC TSR Diff. Vega Payer Receiver
22 HullWhite: mean reversion Strike Mean reversion 0.1% 1% 2% 5% 10% Payer Receiver
23 HullWhite: mean reversion Strike Mean reversion 0.1% 1% 2% 5% 10% Payer Receiver
24 HullWhite: delta Tenor Delta DSC Delta FWD Mrkt HW Diff. Mrkt HW Diff. 5Y Y Y Y Y Y Y Y Y Y Y Total Strike: ATM+1.5% Notional: 100m.
25 HullWhite: curves Differences for different curve shapes O/N rate (%) Slope (%) 3 4
26 G2++: correlation Calibration on one swaption, imposed mean reversions (1% and 30%), imposed volatility ratio (4). Pricing: numerical integration. Strike Correlation 90% 45% 0% 45% 90% Payer Receiver
27 G2++: term structure of volatility Calibration to 10Y and 1Y swaptions. Missing numbers due to not perfect calibrations. Strike Correlation 90% 45% 0% 45% 90% Payer Receiver
28 LMM: multifactor Calibration to one swaptions (tenor 10Y); displacement is 10%. Pricing by MonteCarlo. Strike Angle 0 π/4 π/2 3π/4 π Payer Receiver
29 LMM: multifactor Calibration to swaptions with tenor 1Y to 10Y (10 calibrating instruments); displacement is 10%. Pricing by MonteCarlo. Strike Angle 0 π/4 π/2 3π/4 π Payer Receiver
30 LMM: skew Calibration to swaptions with tenor 10Y (one calibrating instrument); angle is π/2. Strike Displacement Payer Receiver
31 Multimodels: summary Market data as of 30Sep Strike Price difference (bps) All figures computed with OpenGamma open source OGAnalytics library.
32 Multimodels: summary Market data as of 30Sep Strike Price difference (bps) All figures computed with OpenGamma open source OGAnalytics library.
33 Multimodels: summary Market data as of 30Sep HullWhite TSR G2++ LMM 7.98 Strike Price difference (bps) All figures computed with OpenGamma open source OGAnalytics library.
34 Multimodels: summary Market data as of 23Apr Strike Hull White G2++ LMM Price difference
35 Multimodels: summary Market data as of 30Apr Strike Hull White G2++ LMM Price difference
36 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
37 Conclusion The standard market formula is arbitrable with Black or SABR smile (Mercurio result). We presented multimodels analyses of cashsettled swaption prices and delta. The model prices can be very far away from standard market formula prices. Within one model, the price range from noncalibrated parameters can be large. Risk uncertainty between different models for cashsettled swaption is not large but usually applies to nonnetting positions. The cashsettled swaption are liquid but not as simple as they may look!
38 Conclusion The standard market formula is arbitrable with Black or SABR smile (Mercurio result). We presented multimodels analyses of cashsettled swaption prices and delta. The model prices can be very far away from standard market formula prices. Within one model, the price range from noncalibrated parameters can be large. Risk uncertainty between different models for cashsettled swaption is not large but usually applies to nonnetting positions. The cashsettled swaption are liquid but not as simple as they may look!
39 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
40 Approximation in HullWhite model In the case of the HullWhite model, it is possible to obtain an efficient approximated formula for the cashsettled swaptions. All variable can be written as function of a normally distributed random variable X. Exercise boundary: S θ (X ) < K which is equivalent to X < κ where κ is defined by S θ (κ) = K. The rate S θ is roughly linear in X and the annuity is also roughly linear. The result is more parabola shape than a straight line. A third order approximation is required to obtain a precise enough formula. The payoff expansion around a reference point X 0 is C(S θ (X ))(K S θ (X )) U 0 +U 1 (X X 0 )+ 1 2 U 2(X X 0 ) ! U 3(X X 0 ) 3.
41 Approximation in HullWhite model Theorem In the extended Vasicek model, the price of a cashsettled receiver swaption is given to the third order by [ ( P(0, t 0 ) E exp α 0 X 1 ) 2 α2 0 (U 0 + U 1 (X X 0 )+ 1 2 U 2(X X 0 ) )] 3! U 3(X X 0 ) 3 ( U 0 U 1 α U 2(1 + α0) 2 1 ) 3! U 3( α α 0 ) N( κ) ( + U U 2( 2 α 0 + κ) + 1 ) 3! U 3( 3 α κ α 0 κ 2 2) 1 exp ( 12 ) κ2. 2π where κ is given above, U i are the payoff expansion coefficients, κ = κ + α 0 and α 0 = α 0 + X 0.
42 Approximation in HullWhite model Strike Int. Appr. 2 IntAp. 2 Approx. 3 IntAp. 3 Payer Receiver
SWAPTION PRICING OPENGAMMA QUANTITATIVE RESEARCH
SWAPTION PRICING OPENGAMMA QUANTITATIVE RESEARCH Abstract. Implementation details for the pricing of European swaptions in different frameworks are presented.. Introduction This note describes the pricing
More informationOpenGamma Quantitative Research Adjoint Algorithmic Differentiation: Calibration and Implicit Function Theorem
OpenGamma Quantitative Research Adjoint Algorithmic Differentiation: Calibration and Implicit Function Theorem Marc Henrard marc@opengamma.com OpenGamma Quantitative Research n. 1 November 2011 Abstract
More informationNoarbitrage conditions for cashsettled swaptions
Noarbitrage conditions for cashsettled swaptions Fabio Mercurio Financial Engineering Banca IMI, Milan Abstract In this note, we derive noarbitrage conditions that must be satisfied by the pricing function
More informationSWAP AND CAP/FLOORS WITH FIXING IN ARREARS OR PAYMENT DELAY
SWAP AND CAP/FLOORS WITH FIXING IN ARREARS OR PAYMENT DELAY OPENGAMMA QUANTITATIVE RESEARCH Abstract. Pricing methods for swaps and caps/floors with fixing in arrears or payment delays are proposed. The
More informationINTEREST RATES AND FX MODELS
INTEREST RATES AND FX MODELS 4. Convexity and CMS Andrew Lesniewski Courant Institute of Mathematical Sciences New York University New York February 20, 2013 2 Interest Rates & FX Models Contents 1 Introduction
More informationOpenGamma Quantitative Research Multicurves Framework with Stochastic Spread: A Coherent Approach to STIR Futures and Their Options
OpenGamma Quantitative Research Multicurves Framework with Stochastic Spread: A Coherent Approach to STIR Futures and Their Options Marc Henrard marc@opengamma.com OpenGamma Quantitative Research n. 11
More informationThe Irony In The Derivatives Discounting
The Irony In The Derivatives Discounting Marc Henrard Head of Quantitative Research, Banking Department, Bank for International Settlements, CH4002 Basel (Switzerland), email: Marc.Henrard@bis.org Abstract
More informationOptions On Credit Default Index Swaps
Options On Credit Default Index Swaps Yunkang Liu and Peter Jäckel 20th May 2005 Abstract The value of an option on a credit default index swap consists of two parts. The first one is the protection value
More informationOn the Valuation of PowerReverse Duals and EquityRates Hybrids
On the Valuation of PowerReverse Duals and EquityRates Hybrids Oliver Caps oliver.caps@dkib.com RMT Model Validation Rates Dresdner Bank Examples of Hybrid Products Pricing of Hybrid Products using a
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationGuaranteed Annuity Options
Guaranteed Annuity Options Hansjörg Furrer Marketconsistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Guaranteed Annuity Options Contents A. Guaranteed Annuity Options B. Valuation and Risk
More informationThe TwoFactor HullWhite Model : Pricing and Calibration of Interest Rates Derivatives
The TwoFactor HullWhite Model : Pricing and Calibration of Interest Rates Derivatives Arnaud Blanchard Under the supervision of Filip Lindskog 2 Abstract In this paper, we study interest rate models
More informationCredit volatility : Options and beyond
Credit volatility : Options and beyond Jerome.Brun@sgcib.com joint work with F. Rhazal, L. Luciani, T. Bounhar Société Générale Corporate and Investment Banking 25 June 2008, Credit Risk Workshop EVRY
More informationInterest Rate Volatility
Interest Rate Volatility I. Volatility in fixed income markets Andrew Lesniewski Baruch College and Posnania Inc First Baruch Volatility Workshop New York June 1618, 2015 Outline Linear interest rate
More informationIL GOES OCAL A TWOFACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 // MAY // 2014
IL GOES OCAL A TWOFACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 MAY 2014 2 MarieLan Nguyen / Wikimedia Commons Introduction 3 Most commodities trade as futures/forwards Cash+carry arbitrage
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationMiscellaneous. Simone Freschi freschi.simone@gmail.com Tommaso Gabriellini tommaso.gabriellini@mpscs.it. Università di Siena
Miscellaneous Simone Freschi freschi.simone@gmail.com Tommaso Gabriellini tommaso.gabriellini@mpscs.it Head of Global MPS Capital Services SpA  MPS Group Università di Siena ini A.A. 20142015 1 / 1 A
More informationDIGITAL FOREX OPTIONS
DIGITAL FOREX OPTIONS OPENGAMMA QUANTITATIVE RESEARCH Abstract. Some pricing methods for forex digital options are described. The price in the GarhmanKohlhagen model is first described, more for completeness
More informationMaster s Thesis. Pricing Constant Maturity Swap Derivatives
Master s Thesis Pricing Constant Maturity Swap Derivatives Thesis submitted in partial fulfilment of the requirements for the Master of Science degree in Stochastics and Financial Mathematics by Noemi
More informationThe interest volatility surface
The interest volatility surface David Kohlberg Kandidatuppsats i matematisk statistik Bachelor Thesis in Mathematical Statistics Kandidatuppsats 2011:7 Matematisk statistik Juni 2011 www.math.su.se Matematisk
More informationRichard@opengamma.com
Richard@opengamma.com Abstract This paper is a followup to The Pricing and Risk Management of Credit Default Swaps, with a Focus on the ISDA Model [Whi13]. Here we show how to price indices (portfolios
More informationBootstrapping the interestrate term structure
Bootstrapping the interestrate term structure Marco Marchioro www.marchioro.org October 20 th, 2012 Bootstrapping the interestrate term structure 1 Summary (1/2) Market quotes of deposit rates, IR futures,
More informationPublished in Journal of Investment Management, Vol. 13, No. 1 (2015): 6483
Published in Journal of Investment Management, Vol. 13, No. 1 (2015): 6483 OIS Discounting, Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads John Hull and Alan White Joseph
More informationVannaVolga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013
VannaVolga Method for Foreign Exchange Implied Volatility Smile Copyright Changwei Xiong 011 January 011 last update: Nov 7, 01 TABLE OF CONTENTS TABLE OF CONTENTS...1 1. Trading Strategies of Vanilla
More informationFX Options and Smile Risk_. Antonio Castagna. )WILEY A John Wiley and Sons, Ltd., Publication
FX Options and Smile Risk_ Antonio Castagna )WILEY A John Wiley and Sons, Ltd., Publication Preface Notation and Acronyms IX xiii 1 The FX Market 1.1 FX rates and spot contracts 1.2 Outright and FX swap
More informationMarkovian projection for volatility calibration
cutting edge. calibration Markovian projection for volatility calibration Vladimir Piterbarg looks at the Markovian projection method, a way of obtaining closedform approximations of Europeanstyle option
More informationConvexity overview. Didier Faivre
Convexity overview Didier Faivre Convexity on CMS : explanation by static hedge CMS pricing by replication Convexity on Libor in arrears : explanation by static hedge Quanto effect : correlation impact
More informationINTEREST RATES AND FX MODELS
INTEREST RATES AND FX MODELS 8. Portfolio greeks Andrew Lesniewski Courant Institute of Mathematical Sciences New York University New York March 27, 2013 2 Interest Rates & FX Models Contents 1 Introduction
More informationRisk and Investment Conference 2013. Brighton, 17 19 June
Risk and Investment Conference 03 Brighton, 7 9 June 0 June 03 Acquiring fixed income assets on a forward basis Dick Rae, HSBC and Neil Snyman, Aviva Investors 8 June 0 Structure of Presentation Introduction
More informationSome Practical Issues in FX and Equity Derivatives
Some Practical Issues in FX and Equity Derivatives Phenomenology of the Volatility Surface The volatility matrix is the map of the implied volatilities quoted by the market for options of different strikes
More informationConsistent Pricing of FX Options
Consistent Pricing of FX Options Antonio Castagna Fabio Mercurio Banca IMI, Milan In the current markets, options with different strikes or maturities are usually priced with different implied volatilities.
More informationCVA on an ipad Mini Part 2: The Beast
CVA on an ipad Mini Part 2: The Beast Aarhus Kwant Factory PhD Course January 2014 Jesper Andreasen Danske Markets, Copenhagen kwant.daddy@danskebank.com The Beast... was initially developed for the pricing
More informationA Day in the Life of a Trader
Siena, April 2014 Introduction 1 Examples of Market Payoffs 2 3 4 Sticky Smile e Floating Smile 5 Examples of Market Payoffs Understanding risk profiles of a payoff is conditio sine qua non for a mathematical
More informationInterest Rate Models: Paradigm shifts in recent years
Interest Rate Models: Paradigm shifts in recent years Damiano Brigo QSCI, Managing Director and Global Head DerivativeFitch, 101 Finsbury Pavement, London Columbia University Seminar, New York, November
More informationConsistent pricing and hedging of an FX options book
Consistent pricing and hedging of an FX options book L. Bisesti, A. Castagna and F. Mercurio 1 Introduction In the foreign exchange (FX) options market awayfromthemoney options are quite actively traded,
More informationEquity forward contract
Equity forward contract INRODUCION An equity forward contract is an agreement between two counterparties to buy a specific number of an agreed equity stock, stock index or basket at a given price (called
More informationOpenGamma Documentation Bond Pricing
OpenGaa Docuentation Bond Pricing Marc Henrard arc@opengaa.co OpenGaa Docuentation n. 5 Version 2.0  May 2013 Abstract The details of the ipleentation of pricing for fixed coupon bonds and floating rate
More informationExotic Options Trading
Exotic Options Trading Frans de Weert John Wiley & Sons, Ltd Preface Acknowledgements 1 Introduction 2 Conventional Options, Forwards and Greeks 2.1 Call and Put Options and Forwards 2.2 Pricing Calls
More informationOptions On Credit Default Index Swaps *
Options On Credit Default Index Swaps * Yunkang Liu and Peter Jäckel Credit, Hybrid, Commodity and Inflation Derivative Analytics Structured Derivatives ABN AMRO 250 Bishopsgate London EC2M 4AA Abstract:
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationF3 Excel Edition Release Notes
F3 Excel Edition Release Notes F3 Excel Edition Release Notes Software Version: 4.3 Release Date: July 2014 Document Revision Number: A Disclaimer FINCAD makes no warranty either express or implied, including,
More informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationUsing the SABR Model
Definitions Ameriprise Workshop 2012 Overview Definitions The Black76 model has been the standard model for European options on currency, interest rates, and stock indices with it s main drawback being
More informationEquityindexlinked swaps
Equityindexlinked swaps Equivalent to portfolios of forward contracts calling for the exchange of cash flows based on two different investment rates: a variable debt rate (e.g. 3month LIBOR) and the
More informationLecture 6: Option Pricing Using a Onestep Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a Onestep Binomial Tree An oversimplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationSensitivity analysis of utility based prices and risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationBond Options, Caps and the Black Model
Bond Options, Caps and the Black Model Black formula Recall the Black formula for pricing options on futures: C(F, K, σ, r, T, r) = Fe rt N(d 1 ) Ke rt N(d 2 ) where d 1 = 1 [ σ ln( F T K ) + 1 ] 2 σ2
More informationFORWARDS AND EUROPEAN OPTIONS ON CDO TRANCHES. John Hull and Alan White. First Draft: December, 2006 This Draft: March 2007
FORWARDS AND EUROPEAN OPTIONS ON CDO TRANCHES John Hull and Alan White First Draft: December, 006 This Draft: March 007 Joseph L. Rotman School of Management University of Toronto 105 St George Street
More informationThe properties of interest rate swaps An investigation of the price setting of illiquid interest rates swaps and the perfect hedging portfolios.
The properties of interest rate swaps An investigation of the price setting of illiquid interest rates swaps and the perfect hedging portfolios. Max Lindquist 12/23/211 Abstract The main purpose of this
More informationLIBOR Market Models with Stochastic Basis
LIBOR Market Models with Stochastic Basis Fabio Mercurio Bloomberg, New York Swissquote Conference on Interest Rate and Credit Risk Lausanne, 28 October 2010 Stylized facts Before the credit crunch of
More informationAdvanced Fixed Income Analytics Lecture 6
Advanced Fixed Income Analytics Lecture 6 Backus & Zin/April 28, 1999 Fixed Income Models: Assessment and New Directions 1. Uses of models 2. Assessment criteria 3. Assessment 4. Open questions and new
More informationMarket Value of Insurance Contracts with Profit Sharing 1
Market Value of Insurance Contracts with Profit Sharing 1 Pieter Bouwknegt NationaleNederlanden Actuarial Dept PO Box 796 3000 AT Rotterdam The Netherlands Tel: (31)10513 1326 Fax: (31)10513 0120 Email:
More informationFinance 350: Problem Set 6 Alternative Solutions
Finance 350: Problem Set 6 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas
More informationDisclosure of European Embedded Value (summary) as of March 31, 2012
May 25, 2012 SUMITOMO LIFE INSURANCE COMPANY Disclosure of European Embedded Value (summary) as of 2012 This is the summarized translation of the European Embedded Value ( EEV ) of Sumitomo Life Insurance
More informationIntroduction to ArbitrageFree Pricing: Fundamental Theorems
Introduction to ArbitrageFree Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 810, 2015 1 / 24 Outline Financial market
More informationBlackScholes Equation for Option Pricing
BlackScholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there
More informationCaps and Floors. John Crosby
Caps and Floors John Crosby Glasgow University My website is: http://www.johncrosby.co.uk If you spot any typos or errors, please email me. My email address is on my website Lecture given 19th February
More informationOPTIONS, FUTURES, & OTHER DERIVATI
Fifth Edition OPTIONS, FUTURES, & OTHER DERIVATI John C. Hull Maple Financial Group Professor of Derivatives and Risk Manage, Director, Bonham Center for Finance Joseph L. Rotinan School of Management
More informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.
Week 9 Binomial Trees : Hull, Ch. 12. 1 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral
More informationLIBOR and swap market models. Lectures for the Fixed Income course
LIBOR and swap market models Lectures for the Fixed Income course Università Bocconi, Milano Damiano Brigo Head of Credit Models Department of Financial Engineering, San Paolo IMI Group Corso Matteotti
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationExplaining the Lehman Brothers Option Adjusted Spread of a Corporate Bond
Fixed Income Quantitative Credit Research February 27, 2006 Explaining the Lehman Brothers Option Adjusted Spread of a Corporate Bond Claus M. Pedersen Explaining the Lehman Brothers Option Adjusted Spread
More informationDerivatives: Principles and Practice
Derivatives: Principles and Practice Rangarajan K. Sundaram Stern School of Business New York University New York, NY 10012 Sanjiv R. Das Leavey School of Business Santa Clara University Santa Clara, CA
More informationThe LIBOR Market Model
The LIBOR Market Model Master Thesis submitted to Prof. Dr. Wolfgang K. Härdle Prof. Dr. Ostap Okhrin HumboldtUniversität zu Berlin School of Business and Economics Ladislaus von Bortkiewicz Chair of
More informationGN47: Stochastic Modelling of Economic Risks in Life Insurance
GN47: Stochastic Modelling of Economic Risks in Life Insurance Classification Recommended Practice MEMBERS ARE REMINDED THAT THEY MUST ALWAYS COMPLY WITH THE PROFESSIONAL CONDUCT STANDARDS (PCS) AND THAT
More informationIntroduction Pricing Effects Greeks Summary. Vol Target Options. Rob Coles. February 7, 2014
February 7, 2014 Outline 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity The Basic Idea Basket split between risky asset and cash Chose weight of risky asset w to keep volatility
More informationCollateral consistent derivatives pricing
Collateral consistent derivatives pricing FRIC Practitioner Seminar, CBS Martin D. Linderstrøm, marlin@danskebank.dk Counterparty Credit & Funding Risk Agenda The intuition behind collateral consistent
More informationMONEY MARKET SUBCOMMITEE(MMS) FLOATING RATE NOTE PRICING SPECIFICATION
MONEY MARKET SUBCOMMITEE(MMS) FLOATING RATE NOTE PRICING SPECIFICATION This document outlines the use of the margin discounting methodology to price vanilla money market floating rate notes as endorsed
More informationProperties of the SABR model
U.U.D.M. Project Report 2011:11 Properties of the SABR model Nan Zhang Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Juni 2011 Department of Mathematics Uppsala University ABSTRACT
More informationModel Independent Greeks
Model Independent Greeks Mathematical Finance Winter School Lunteren January 2014 Jesper Andreasen Danske Markets, Copenhagen kwant.daddy@danskebank.com Outline Motivation: Wots me Δelδa? The short maturity
More information24. Pricing Fixed Income Derivatives. through Black s Formula. MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture:
24. Pricing Fixed Income Derivatives through Black s Formula MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: John C. Hull, Options, Futures & other Derivatives (Fourth Edition),
More informationLecture 4: The BlackScholes model
OPTIONS and FUTURES Lecture 4: The BlackScholes model Philip H. Dybvig Washington University in Saint Louis BlackScholes option pricing model Lognormal price process Call price Put price Using BlackScholes
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationEurodollar Futures, and Forwards
5 Eurodollar Futures, and Forwards In this chapter we will learn about Eurodollar Deposits Eurodollar Futures Contracts, Hedging strategies using ED Futures, Forward Rate Agreements, Pricing FRAs. Hedging
More informationYIELD CURVE GENERATION
1 YIELD CURVE GENERATION Dr Philip Symes Agenda 2 I. INTRODUCTION II. YIELD CURVES III. TYPES OF YIELD CURVES IV. USES OF YIELD CURVES V. YIELD TO MATURITY VI. BOND PRICING & VALUATION Introduction 3 A
More information1 The BlackScholes model: extensions and hedging
1 The BlackScholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes
More informationFINANCE 1. DESS Gestion des Risques/Ingéniérie Mathématique Université d EVRY VAL D ESSONNE EXERCICES CORRIGES. Philippe PRIAULET
FINANCE 1 DESS Gestion des Risques/Ingéniérie Mathématique Université d EVRY VAL D ESSONNE EXERCICES CORRIGES Philippe PRIAULET 1 Exercise 1 On 12/04/01 consider a fixed coupon bond whose features are
More informationDisclosure of European Embedded Value (summary) as of March 31, 2015
May 28, 2015 SUMITOMO LIFE INSURANCE COMPANY Disclosure of European Embedded Value (summary) as of March 31, 2015 This is the summarized translation of the European Embedded Value ( EEV ) of Sumitomo Life
More informationImportant Exam Information:
Important Exam Information: Exam Registration Candidates may register online or with an application. Order Study Notes Study notes are part of the required syllabus and are not available electronically
More informationAnalytical Research Series
EUROPEAN FIXED INCOME RESEARCH Analytical Research Series INTRODUCTION TO ASSET SWAPS Dominic O Kane January 2000 Lehman Brothers International (Europe) Pub Code 403 Summary An asset swap is a synthetic
More informationMULTIPLE DEFAULTS AND MERTON'S MODEL L. CATHCART, L. ELJAHEL
ISSN 17446783 MULTIPLE DEFAULTS AND MERTON'S MODEL L. CATHCART, L. ELJAHEL Tanaka Business School Discussion Papers: TBS/DP04/12 London: Tanaka Business School, 2004 Multiple Defaults and Merton s Model
More informationPricing of an Exotic Forward Contract
Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 111 Nojihigashi, Kusatsu, Shiga 5258577, Japan Email: {akahori,
More informationBlackScholes and the Volatility Surface
IEOR E4707: Financial Engineering: ContinuousTime Models Fall 2009 c 2009 by Martin Haugh BlackScholes and the Volatility Surface When we studied discretetime models we used martingale pricing to derive
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationLecture 6 BlackScholes PDE
Lecture 6 BlackScholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the riskneutral measure Q by If the contingent
More informationFixed Income Arbitrage
Risk & Return Fixed Income Arbitrage: Nickels in Front of a Steamroller by Jefferson Duarte Francis A. Longstaff Fan Yu Fixed Income Arbitrage Broad set of marketneutral strategies intended to exploit
More information1. The forward curve
1. The forward curve Andrew Lesniewsi January 28, 2008 Contents 1 LIBOR and LIBOR based instruments 1 1.1 Forward rate agreements...................... 2 1.2 LIBOR futures........................... 3
More informationAffine LIBOR models with multiple curves:
Affine LIBOR models with multiple curves: John Schoenmakers, WIAS Berlin joint with Z. Grbac, A. Papapantoleon & D. Skovmand 9th Summer School in Mathematical Finance Cape Town, 1820 February 2016 1820.02.2016
More informationSome Applications of Mathematics in Finance
Some Applications of Mathematics in Finance Robert Campbell Department of Mathematics and Statistics wrc@mcs.stand.ac.uk Ravenscourt Capital rcampbell@ravenscourtcapital.com 7 th November, 2008 Role of
More informationEuropean Options Pricing Using Monte Carlo Simulation
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical
More informationInternational Swaps and Derivatives Association, Inc. Disclosure Annex for Interest Rate Transactions
International Swaps and Derivatives Association, Inc. Disclosure Annex for Interest Rate Transactions This Annex supplements and should be read in conjunction with the General Disclosure Statement. NOTHING
More informationPotential Future Exposure and Collateral Modelling of the Trading Book Using NVIDIA GPUs
Potential Future Exposure and Collateral Modelling of the Trading Book Using NVIDIA GPUs 19 March 2015 GPU Technology Conference 2015 Grigorios Papamanousakis Quantitative Strategist, Investment Solutions
More informationInterest Rate and Currency Swaps
Interest Rate and Currency Swaps Eiteman et al., Chapter 14 Winter 2004 Bond Basics Consider the following: ZeroCoupon ZeroCoupon OneYear Implied Maturity Bond Yield Bond Price Forward Rate t r 0 (0,t)
More informationSwaps: complex structures
Swaps: complex structures Complex swap structures refer to nonstandard swaps whose coupons, notional, accrual and calendar used for coupon determination and payments are tailored made to serve client
More informationPRICING OF SWAPS AND SWAPTIONS
MSC. FINANCE MASTER THESIS DEP. OF ECONOMICS AND BUSINESS AARHUS MARCH 2013 PRICING OF SWAPS AND SWAPTIONS MSC. FINANCE STUDENT: MIKKEL KRONBORG OLESEN STUDENT NR.: 283755 SUPERVISOR: PETER LØCHTE JØRGENSEN
More informationCallable Bonds  Structure
1.1 Callable bonds A callable bond is a fixed rate bond where the issuer has the right but not the obligation to repay the face value of the security at a preagreed value prior to the final original maturity
More informationHow the Greeks would have hedged correlation risk of foreign exchange options
How the Greeks would have hedged correlation risk of foreign exchange options Uwe Wystup Commerzbank Treasury and Financial Products Neue Mainzer Strasse 32 36 60261 Frankfurt am Main GERMANY wystup@mathfinance.de
More informationor enters into a Futures contract (either on the IPE or the NYMEX) with delivery date September and pay every day up to maturity the margin
CashFutures arbitrage processes Cash futures arbitrage consisting in taking position between the cash and the futures markets to make an arbitrage. An arbitrage is a trade that gives in the future some
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationOption Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013
Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed
More information