# Cash-settled swaptions How wrong are we?

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Cash-settled swaptions How wrong are we? Marc Henrard Quantitative Research - OpenGamma 7th Fixed Income Conference, Berlin, 7 October 2011

2 ... Based on: Cash-settled 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 Multi-models analysis 4 Conclusion 5 Appendix: approximation in Hull-White model

4 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multi-models analysis 4 Conclusion 5 Appendix: approximation in Hull-White model

5 Introduction Cash-settled 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, Cash-settled swaptions and no-arbitrage, Risk, Local coherence between Market pricing formula for cash-settled and physical delivery swaption has not been analyzed.

6 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multi-models analysis 4 Conclusion 5 Appendix: approximation in Hull-White 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 cash-settled swaption payment is (in t 0 ) C(S)(K S) +.

8 Swap rate We work in a multi-curve set-up. 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 log-normal model ds t = σs t dw t, the price can be computed explicitly A 0 Black(K, S 0, σ).

10 Market formulas (cash-settled) The value at expiry of the cash-settle 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 (cash-settled) The value at expiry of the cash-settle 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 (cash-settled) The value at expiry of the cash-settle 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 (cash-settled) 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 (cash-settled) 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 cash-settled 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 cash-settled option with strike 0 has a value different from its constant pay-off Black implied volatility

16 Cash-settle and physical swaptions Difference between cash-settled 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 Multi-models analysis 4 Conclusion 5 Appendix: approximation in Hull-White 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 30-Sep 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 cash-settled 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 cash-settled 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 D-C TSR Diff. Vega Payer Receiver

22 Hull-White: mean reversion Strike Mean reversion 0.1% 1% 2% 5% 10% Payer Receiver

23 Hull-White: mean reversion Strike Mean reversion 0.1% 1% 2% 5% 10% Payer Receiver

24 Hull-White: 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 Hull-White: 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: multi-factor Calibration to one swaptions (tenor 10Y); displacement is 10%. Pricing by Monte-Carlo. Strike Angle 0 π/4 π/2 3π/4 π Payer Receiver

29 LMM: multi-factor Calibration to swaptions with tenor 1Y to 10Y (10 calibrating instruments); displacement is 10%. Pricing by Monte-Carlo. 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 Multi-models: summary Market data as of 30-Sep Strike Price difference (bps) All figures computed with OpenGamma open source OG-Analytics library.

32 Multi-models: summary Market data as of 30-Sep Strike Price difference (bps) All figures computed with OpenGamma open source OG-Analytics library.

33 Multi-models: summary Market data as of 30-Sep Hull-White TSR G2++ LMM 7.98 Strike Price difference (bps) All figures computed with OpenGamma open source OG-Analytics library.

34 Multi-models: summary Market data as of 23-Apr Strike Hull White G2++ LMM Price difference

35 Multi-models: summary Market data as of 30-Apr Strike Hull White G2++ LMM Price difference

36 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multi-models analysis 4 Conclusion 5 Appendix: approximation in Hull-White model

37 Conclusion The standard market formula is arbitrable with Black or SABR smile (Mercurio result). We presented multi-models analyses of cash-settled swaption prices and delta. The model prices can be very far away from standard market formula prices. Within one model, the price range from non-calibrated parameters can be large. Risk uncertainty between different models for cash-settled swaption is not large but usually applies to non-netting positions. The cash-settled 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 multi-models analyses of cash-settled swaption prices and delta. The model prices can be very far away from standard market formula prices. Within one model, the price range from non-calibrated parameters can be large. Risk uncertainty between different models for cash-settled swaption is not large but usually applies to non-netting positions. The cash-settled 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 Multi-models analysis 4 Conclusion 5 Appendix: approximation in Hull-White model

40 Approximation in Hull-White model In the case of the Hull-White model, it is possible to obtain an efficient approximated formula for the cash-settled 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 pay-off 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 Hull-White model Theorem In the extended Vasicek model, the price of a cash-settled 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 pay-off expansion coefficients, κ = κ + α 0 and α 0 = α 0 + X 0.

42 Approximation in Hull-White model Strike Int. Appr. 2 Int-Ap. 2 Approx. 3 Int-Ap. 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

### OpenGamma 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

### No-arbitrage conditions for cash-settled swaptions

No-arbitrage conditions for cash-settled swaptions Fabio Mercurio Financial Engineering Banca IMI, Milan Abstract In this note, we derive no-arbitrage conditions that must be satisfied by the pricing function

### SWAP 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

### INTEREST 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

### OpenGamma Quantitative Research Multi-curves Framework with Stochastic Spread: A Coherent Approach to STIR Futures and Their Options

OpenGamma Quantitative Research Multi-curves Framework with Stochastic Spread: A Coherent Approach to STIR Futures and Their Options Marc Henrard marc@opengamma.com OpenGamma Quantitative Research n. 11

### The Irony In The Derivatives Discounting

The Irony In The Derivatives Discounting Marc Henrard Head of Quantitative Research, Banking Department, Bank for International Settlements, CH-4002 Basel (Switzerland), e-mail: Marc.Henrard@bis.org Abstract

### Options 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

### On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids

On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids Oliver Caps oliver.caps@dkib.com RMT Model Validation Rates Dresdner Bank Examples of Hybrid Products Pricing of Hybrid Products using a

### Hedging 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

### Guaranteed Annuity Options

Guaranteed Annuity Options Hansjörg Furrer Market-consistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Guaranteed Annuity Options Contents A. Guaranteed Annuity Options B. Valuation and Risk

### The Two-Factor Hull-White Model : Pricing and Calibration of Interest Rates Derivatives

The Two-Factor Hull-White 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

### IL GOES OCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 // MAY // 2014

IL GOES OCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 MAY 2014 2 Marie-Lan Nguyen / Wikimedia Commons Introduction 3 Most commodities trade as futures/forwards Cash+carry arbitrage

### Interest 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 16-18, 2015 Outline Linear interest rate

### Credit 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

### The Black-Scholes Formula

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

### Miscellaneous. 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. 2014-2015 1 / 1 A

### DIGITAL FOREX OPTIONS

DIGITAL FOREX OPTIONS OPENGAMMA QUANTITATIVE RESEARCH Abstract. Some pricing methods for forex digital options are described. The price in the Garhman-Kohlhagen model is first described, more for completeness

### Master 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

### Richard@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

### Convexity 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

### Markovian projection for volatility calibration

cutting edge. calibration Markovian projection for volatility calibration Vladimir Piterbarg looks at the Markovian projection method, a way of obtaining closed-form approximations of European-style option

### The 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

### Bootstrapping the interest-rate term structure

Bootstrapping the interest-rate term structure Marco Marchioro www.marchioro.org October 20 th, 2012 Bootstrapping the interest-rate term structure 1 Summary (1/2) Market quotes of deposit rates, IR futures,

### Published in Journal of Investment Management, Vol. 13, No. 1 (2015): 64-83

Published in Journal of Investment Management, Vol. 13, No. 1 (2015): 64-83 OIS Discounting, Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads John Hull and Alan White Joseph

### FX 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

### INTEREST 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

### Some 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

### Risk 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

### CVA 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

### Consistent 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.

### A 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

### Interest Rate Models: Paradigm shifts in recent years

Interest Rate Models: Paradigm shifts in recent years Damiano Brigo Q-SCI, Managing Director and Global Head DerivativeFitch, 101 Finsbury Pavement, London Columbia University Seminar, New York, November

### F3 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,

### Consistent 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 away-from-the-money options are quite actively traded,

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

### Equity 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

### Options 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:

### Using the SABR Model

Definitions Ameriprise Workshop 2012 Overview Definitions The Black-76 model has been the standard model for European options on currency, interest rates, and stock indices with it s main drawback being

### OpenGamma 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

Equity-index-linked 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. 3-month LIBOR) and the

### Chapter 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

### Advanced 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

### Sensitivity analysis of utility based prices and risk-tolerance wealth processes

Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,

### The 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

### LIBOR 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

### Lecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12

Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond

### Moreover, 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

### The Black-Scholes-Merton Approach to Pricing Options

he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining

### Bond 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

### FORWARDS 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

### OPTIONS, 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

### Market Value of Insurance Contracts with Profit Sharing 1

Market Value of Insurance Contracts with Profit Sharing 1 Pieter Bouwknegt Nationale-Nederlanden Actuarial Dept PO Box 796 3000 AT Rotterdam The Netherlands Tel: (31)10-513 1326 Fax: (31)10-513 0120 E-mail:

### Black-Scholes Equation for Option Pricing

Black-Scholes 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

### Finance 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

### Disclosure 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

### Properties 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

### Caps and Floors. John Crosby

Caps and Floors John Crosby Glasgow University My website is: http://www.john-crosby.co.uk If you spot any typos or errors, please email me. My email address is on my website Lecture given 19th February

### Introduction to Arbitrage-Free Pricing: Fundamental Theorems

Introduction to Arbitrage-Free Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 8-10, 2015 1 / 24 Outline Financial market

### LIBOR 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

### Jorge 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

### Derivatives: 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

### MONEY 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

### Explaining 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

### The LIBOR Market Model

The LIBOR Market Model Master Thesis submitted to Prof. Dr. Wolfgang K. Härdle Prof. Dr. Ostap Okhrin Humboldt-Universität zu Berlin School of Business and Economics Ladislaus von Bortkiewicz Chair of

### Eurodollar 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

### GN47: 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

### The Impact of the Financial Crisis on the Pricing and Hedging of Interest Rate Swaps

The Impact of the Financial Crisis on the Pricing and Hedging of Interest Rate Swaps Caroline Backas and Åsa Höijer Graduate School Master of Science in Finance Master Degree Project No. 2012: Supervisor:

### Introduction 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

### Collateral 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

### Model 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

### 24. 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),

### Lecture 4: The Black-Scholes model

OPTIONS and FUTURES Lecture 4: The Black-Scholes model Philip H. Dybvig Washington University in Saint Louis Black-Scholes option pricing model Lognormal price process Call price Put price Using Black-Scholes

### Pricing 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

### YIELD 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

### Some Applications of Mathematics in Finance

Some Applications of Mathematics in Finance Robert Campbell Department of Mathematics and Statistics wrc@mcs.st-and.ac.uk Ravenscourt Capital rcampbell@ravenscourtcapital.com 7 th November, 2008 Role of

### Important 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

### On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.

### INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION

INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION 14 April 2016 (pm) Subject ST6 Finance and Investment Specialist Technical B Time allowed: Three hours 1. Enter all the candidate and examination details

### Pricing of an Exotic Forward Contract

Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail: {akahori,

### Disclosure 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

### Lecture 6 Black-Scholes PDE

Lecture 6 Black-Scholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the risk-neutral measure Q by If the contingent

### Pricing and managing life insurance risks

University of Bergamo Faculty of Economics Department of Mathematics, Statistics, Computer Science and Applications Ph.D. course in Computational Methods for Forecasting and Decisions in Economics and

### 1 The Black-Scholes model: extensions and hedging

1 The Black-Scholes 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

### MULTIPLE DEFAULTS AND MERTON'S MODEL L. CATHCART, L. EL-JAHEL

ISSN 1744-6783 MULTIPLE DEFAULTS AND MERTON'S MODEL L. CATHCART, L. EL-JAHEL Tanaka Business School Discussion Papers: TBS/DP04/12 London: Tanaka Business School, 2004 Multiple Defaults and Merton s Model

### European 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

### Analytical 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

### Affine 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, 18-20 February 2016 18-20.02.2016

### 1. 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

### Potential 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

### Fixed 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 market-neutral strategies intended to exploit

### Swaps: complex structures

Swaps: complex structures Complex swap structures refer to non-standard swaps whose coupons, notional, accrual and calendar used for coupon determination and payments are tailored made to serve client

### International 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

### CDS Market Formulas and Models

CDS Market Formulas and Models Damiano Brigo Credit Models Banca IMI Corso Matteotti 6 20121 Milano, Italy damiano.brigo@bancaimi.it Massimo Morini Università di Milano Bicocca Piazza dell Ateneo Nuovo

### How 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

### Risk Management and Governance Reminder on Derivatives. Prof. Hugues Pirotte

Risk Management and Governance Reminder on Derivatives Prof. Hugues Pirotte Quick presentation made in class of the main categories of derivatives and short reminder Prof. Hugues Pirotte 2 3 Commitments