A Comparison of Alternative SABR Models in a Negative Interest Rate Environment

Master of Science Thesis

Esmée Winnubst 10338179

30th December 2016

Faculty of Economics and Business · University of Amsterdam

A Comparison of Alternative SABR Models in a Negative Interest Rate Environment

Master of Science Thesis

For the degree of Master of Science in Financial Econometrics at the University of Amsterdam

Esmée Winnubst

10338179

30th December 2016

Supervisor: Dr. N. Van Giersbergen

Second Reader: Dr. S.A. Broda

Faculty of Economics and Business · University of Amsterdam Statement of Originality: This document is written by Esmée Winnubst who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for supervision of completion of the work, not for the contents.

Acknowledgement: I would like to thank my supervisor Noud van Giersbergen for his as- sistance during the writing of this thesis. The work in this thesis was supported by EY. I would like to thank my colleagues at EY for their support and time.

The widespread used SABR model (P. S. Hagan, Kumar, Lesniewski, & Woodward, 2002) is not able to correctly price options, capture volatility curves and inter- and extrapolate market quotes in a negative interest rate environment. This thesis investigates two extensions of the SABR model that are able to work with negative rates: the normal SABR model and the free boundary SABR model. In theory both models are very promising, but both models are thus far not calibrated to actual market data. In this thesis first a closed form solution for both models is derived. The accurateness of this approximation is then tested using an arbitrage free PDE solution as benchmark. To test the two models with actual market data, first the normal ATM volatility formulas for both models are derived. Then the closed form solutions are calibrated to the swaption data for 01- 07-2005, 01-01-2008, 01-01-2016, 01-06-2016, 01-08-2016 and 01-12-2016 using the previously derived ATM formulas. The free boundary SABR model optimization ends up in a sub-optimum for the 2016 data (which contains negative rates). Therefore very inaccurate results were found around a strike price of zero. After grid searching for better start parameters without result, the calibration results of the normal SABR model were chosen as start parameters for the free boundary optimization. Now the free boundary SABR model gives better results than the normal SABR model. It can be concluded that, without using the normal SABR model for finding start parameters, the free boundary SABR model is not able to correctly model forward rates that are close to zero. Therefore this model is not a good model for financial practitioners, who need this model in all kind of situations, compared to the normal SABR model which works in every situation.

Contents

Abstract i

1 Introduction1 1-1 Economic Background...... 2 1-2 Problem Statement...... 2 1-3 Thesis Outline...... 2

2 Interest Rate Options5 2-1 Interest rate derivatives...... 5 2-2 Bond options...... 6 2-3 Caps and Floors...... 6 2-4 Swaptions...... 6 2-5 Implied Volatility...... 6

3 Literature Review9 3-1 The Black model...... 9 3-2 The SABR Model...... 10 3-2-1 Martingale Reresentation Theorem...... 11 3-2-2 Description of the Model...... 11 3-2-3 Option pricing with the SABR model...... 12 3-2-4 SABR approximation formula...... 13 3-2-5 SABR in practice: the classical SABR model...... 13 3-2-6 Conclusion...... 15

4 Negative Interest Rate Models 17 4-1 Displaced model...... 17 4-1-1 Drawback Displaced Model...... 18 4-2 Stochastic Normal Model...... 18 4-3 The Free Boundary SABR Model...... 19 iv Contents

5 Partial Diﬀerential Equation Solution 23 5-1 Solving the PDE...... 23 5-1-1 Arbitrage-free pricing within the SABR model...... 23 5-1-2 Boundary conditions...... 25 5-2 Option pricing with positive rates...... 27 5-2-1 Transformation of the PDE...... 27 5-2-2 Discretization of the PDE...... 29 5-2-3 Pricing options...... 31 5-3 Option pricing with negative rates...... 31 5-3-1 Modeling negative interest rates with the normal SABR model...... 31 5-3-2 Modeling negative interest rates with the free boundary SABR model.. 33

6 Closed Form Solution Results 35 6-1 PDF plotting...... 35 6-2 Performance of the Approximation Formula...... 38 6-2-1 Test 1...... 38 6-2-2 Test 2...... 40 6-2-3 Conclusion approximation formulas...... 42

7 Methodology 43 7-1 Dataset...... 43 7-2 Calibration...... 50 7-2-1 Hagan calibration...... 50 7-2-2 Calibration in this thesis...... 50 7-3 Measurement of ﬁt...... 51

8 Market Data Results 53 8-1 Parameter Calibration...... 53 8-2 Market Fit...... 54 8-2-1 Normal market conditions...... 54 8-2-2 Excited market conditions...... 57 8-2-3 Negative rate conditions...... 57

9 Conclusion 65

A Parameter Calibration Results 67 List of Figures

1-1 Central bank’s interest rates...... 1

3-1 Implied volatility smile...... 11

5-1 Y (z) and transformed F (z) ...... 29

6-1 Probability density function obtained by solving the PDE for the classical SABR. 36 6-2 Probability density function obtained by solving the PDE for the normal SABR. 36 6-3 Probability density function obtained by solving the PDE for the free boundary SABR 37 6-4 Probability density function obtained by solving the PDE for the free boundary SABR with varying β ...... 37 6-5 Probability density function obtained by solving the PDE, test 1...... 39 6-6 Volatility smile, test 1...... 39 6-7 Absolute diﬀerence option prices PDE and approximation, test 1...... 40 6-8 Probability density function obtained by solving the PDE, test 2...... 41 6-9 Volatility smile, test 2...... 41 6-10 Absolute diﬀerence option prices PDE and approximation, test 2...... 42

7-1 Volatility surface for a EUR swaption on 01-07-2005...... 44 7-2 Volatility surface for a EUR swaption on 01-01-2008...... 45 7-3 Volatility surface for a EUR swaption on 01-01-2016...... 46 7-4 Volatility surface for a EUR swaption on 01-06-2016...... 47 7-5 Volatility surface for a EUR swaption on 01-08-2016...... 48 7-6 Volatility surface for a EUR swaption on 01-12-2016...... 49 7-7 Strike surface for a EUR swaption on 01-08-2016...... 49

8-1 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-07-2005...... 55 vi List of Figures

8-2 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-01-2008...... 56 8-3 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-01-2016...... 58 8-4 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-06-2016...... 59 8-5 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-08-2016...... 60 8-6 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-12-2016...... 61 List of Tables

8-1 RMSE compared for normal SABR and free boundary SABR for 01-07-2005... 54 8-2 RMSE compared for normal SABR and free boundary SABR for 01-01-2008... 57 8-3 RMSE compared for normal SABR and free boundary SABR for 01-01-2016... 62 8-4 RMSE compared for normal SABR and free boundary SABR for 01-06-2016... 62 8-5 RMSE compared for normal SABR and free boundary SABR for 01-08-2016... 63 8-6 RMSE compared for normal SABR and free boundary SABR for 01-12-2016... 63

A-1 Swaption quotes for EUR, 01-07-2005, maturity: 1Y Normal SABR...... 68 A-2 Swaption quotes for EUR, 01-07-2005, maturity: 5Y Normal SABR...... 68 A-3 Swaption quotes for EUR, 01-07-2005, maturity: 10Y Normal SABR...... 68 A-4 Swaption quotes for EUR, 01-07-2005, maturity: 1Y Free Boundary SABR.... 69 A-5 Swaption quotes for EUR, 01-07-2005, maturity: 5Y Free Boundary SABR.... 69 A-6 Swaption quotes for EUR, 01-07-2005, maturity: 10Y Free Boundary SABR... 69 A-7 Swaption quotes for EUR, 01-01-2008, maturity: 1Y Normal SABR...... 70 A-8 Swaption quotes for EUR, 01-01-2008, maturity: 5Y Normal SABR...... 70 A-9 Swaption quotes for EUR, 01-01-2008, maturity: 10Y Normal SABR...... 70 A-10 Swaption quotes for EUR, 01-01-2008, maturity: 1Y Free Boundary SABR.... 71 A-11 Swaption quotes for EUR, 01-01-2008, maturity: 5Y Free Boundary SABR.... 71 A-12 Swaption quotes for EUR, 01-01-2008, maturity: 10Y Free Boundary SABR... 71 A-13 Swaption quotes for EUR, 01-01-2016, maturity: 1Y Normal SABR...... 72 A-14 Swaption quotes for EUR, 01-01-2016, maturity: 5Y Normal SABR...... 72 A-15 Swaption quotes for EUR, 01-01-2016, maturity: 10Y Normal SABR...... 72 A-16 Swaption quotes for EUR, 01-01-2016, maturity: 1Y Free Boundary SABR.... 73 A-17 Swaption quotes for EUR, 01-01-2016, maturity: 5Y Free Boundary SABR.... 73 A-18 Swaption quotes for EUR, 01-01-2016, maturity: 10Y Free Boundary SABR... 73 viii List of Tables

A-19 Swaption quotes for EUR, 01-06-2016, maturity: 1Y Normal SABR...... 74 A-20 Swaption quotes for EUR, 01-06-2016, maturity: 5Y Normal SABR...... 74 A-21 Swaption quotes for EUR, 01-06-2016, maturity: 10Y Normal SABR...... 74 A-22 Swaption quotes for EUR, 01-06-2016, maturity: 1Y Free Boundary SABR.... 75 A-23 Swaption quotes for EUR, 01-06-2016, maturity: 5Y Free Boundary SABR.... 75 A-24 Swaption quotes for EUR, 01-06-2016, maturity: 10Y Free Boundary SABR... 75 A-25 Swaption quotes for EUR, 01-08-2016, maturity: 1Y Normal SABR...... 76 A-26 Swaption quotes for EUR, 01-08-2016, maturity: 5Y Normal SABR...... 76 A-27 Swaption quotes for EUR, 01-08-2016, maturity: 10Y Normal SABR...... 76 A-28 Swaption quotes for EUR, 01-08-2016, maturity: 1Y Free Boundary SABR.... 77 A-29 Swaption quotes for EUR, 01-08-2016, maturity: 5Y Free Boundary SABR.... 77 A-30 Swaption quotes for EUR, 01-08-2016, maturity: 10Y Free Boundary SABR... 77 A-31 Swaption quotes for EUR, 01-12-2016, maturity: 1Y Normal SABR...... 78 A-32 Swaption quotes for EUR, 01-12-2016, maturity: 5Y Normal SABR...... 78 A-33 Swaption quotes for EUR, 01-12-2016, maturity: 10Y Normal SABR...... 78 A-34 Swaption quotes for EUR, 01-12-2016, maturity: 1Y Free Boundary SABR.... 79 A-35 Swaption quotes for EUR, 01-12-2016, maturity: 5Y Free Boundary SABR.... 79 A-36 Swaption quotes for EUR, 01-12-2016, maturity: 10Y Free Boundary SABR... 79 Chapter 1

Introduction

On the 11th of June 2014 the European Central bank (ECB) introduced for the ﬁrst time in history a negative interest rate of minus 0.1. Nowadays negative interest rates are a very important topic of the economy. When searching for articles with a topic on negative interest rates at the Financial Times1, more than 100 hits appear in the period from January 2016 until end of September 2016. The European Central bank (ECB), the Swiss national bank, Denmark’s national bank, Sveriges Riksbank (Sweden) and the bank of Japan have proposed negative rates on reserves. The development of these rates can be seen in Figure 1-1.

Figure 1-1: Central bank’s interest rates in %, Source: Thomson Reuters Datastream

In this introduction ﬁrst some theoretical economic background will be given on negative interest rates: why they are introduced and what the idea behind negative interest rates is. The subsequent section sets out why negative rates are a problem to the interest rate models used today. In the last section the structure of this thesis is covered. 1https://www.ft.com/negative-interest-rates 2 Introduction

1-1 Economic Background

Central banks set negative interest rates in order to stimulate economic growth. Negative rates improve the economic growth by encourage borrowing and spur inflation (Kennedy, 2016). The main objective of low and negative interest rates is to motivate lending due to the lower costs of borrowing. Households and companies are both encouraged to demand for loans. The second goal of imposing negative rates is to spur inflation, which is caused by two effects. When the demand for loans increases, more money is available to spend on services and goods, leading to higher outputs and higher inflation. The second way inflation occurs is caused by investors changing currency to countries with a higher yield on their government bonds. Therefore the exchange rate will drop, leading to higher import prices, leading to a higher inflation. According to Bloomberg ‘proposing negative rates is an unorthodox move that has distorted financial markets and triggered complaints that the strategy is backfiring’. They think that negative rates will either cause a new era for the world’s central banks or make the economic situation even worse. By imposing negative rates, new limits are explored by central banks. This is a signal that traditional policy options have proved ineffective.

1-2 Problem Statement

Derivative pricing models assume interest rates to be positive and high, and are therefore not capable to work correctly with negative interest rates (Hull, 2003). As a result, exist- ing interest rate pricing models do not work correctly anymore: they give a wrong price (P. S. Hagan, Kumar, Lesniewski, & Woodward, 2014), cause arbitrage possibilities, or do not give an output at all. These models are not only incapable of pricing and forecasting during negative interest rate periods, they also are not able to give a positive probability to a negative rate in the future, while the current situation has a positive interest rate. The aim of this thesis is to compare diﬀerent interest rate option models that are suitable with negative interest rates. It is of main importance that these models are appropriate for modeling risk, and especially hedging risk, in practice. The SABR model, a stochastic volatility model, is the most commonly used interest rate model among ﬁnancial institutions, mainly to price interest rate derivatives (P. S. Hagan et al., 2002). The research question of this thesis is the following: Which extension of the SABR model is the most suitable for modelling option prices in a negative interest rate environment?

1-3 Thesis Outline

In Chapter 2 a short introduction to the diﬀerent types of commonly traded interest rate options is given. Also the implied volatility used in this thesis is examined in order to better understand further chapters. Readers with a large knowledge on interest rate models can skip this chapter with no problem. A literature review on the most commonly used interest rate models before the introduction of negative rates is given in Chapter 3. Subsequently the interest rate models that work with negative rates, the normal SABR and the free boundary 1-3 Thesis Outline 3

SABR, are explained in Chapter 4. The two methods used in this thesis to calculate option prices are also briefly discussed in this chapter. The first method is solving a partial differential equation, which will further be explained in Chapter 5. Chapter 6 treats the second method, solving a closed form solution, the so-called approximation formula. In Chapter 7 the data is described and the model used for testing the quality of fit of the SABR approximation formulas is given. The results of comparing the negative interest rate option models are presented in Chapter 8. Finally a conclusion is given in Chapter 9.

Chapter 2

Interest Rate Options

Before explaining the interest rate option models that are compared and tested in this thesis, first basic interest rate theory needed to understand the models is set out. The most popular interest rate derivatives are bond options, caps/floors and swaptions (Hull, 2003, p. 508). The pricing of interest rate derivatives such as caps, floors and swaptions requires the modelling of a series of forward LIBOR rates (Chiarella, He, & Nikitopoulos, 2015, p. 580). More theory on option pricing models can be found in Chapter3. In this chapter it is first explained what an interest rate derivative is. Also the size of the interest rate derivative market is emphasized. Finally three different interest rate options are discussed, concluded with information on the used implied volatility in this thesis.

2-1 Interest rate derivatives

Interest rate derivatives are instruments of which the value decreases or increases with movements of interest rates (Hull, 2003). Interest rate derivatives are particularly popular for investors with customized cashﬂow needs. They can be used to reduce or increase interest rate exposure and therewith risk. Interest rate derivatives are used to control cashﬂows, as a hedge to protect against changes in market interest rates. Therefore interest rate derivatives are mostly used by (large) companies, banks and institutional investors. Interest rate derivatives are both traded in over-the-counter (OTC) and exchange-traded markets, but since they are mostly used by investors they are particularly traded on the OTC market (Hull, 2003). The interest rate derivatives market is the largest derivatives market in the world. According to the BIS1 the total derivative notional amount outstanding in the second half of 2015 is $492.911 billion US dollar. Interest rate contracts are the largest part of this enormous amount with a notional amount outstanding of $384.025 billion US dollar.

1Established on 17 May 1930, the Bank for International Settlements (BIS) is the world’s oldest international ﬁnancial organisation. The BIS has 60 member central banks, representing countries from around the world that together make up about 95% of world GDP. http://www.bis.org/statistics/d5_1.pdf 6 Interest Rate Options

2-2 Bond options

A bond option is an option to buy or sell a bond by a certain date at a certain price. Bond options are typically traded OTC. Bond options are also frequently embedded in bonds (Hull, 2003, p. 511). It is assumed that the bond price of the option at maturity is lognormal. Generally, there are two main types of bond options: a call option and put option. The call options are for hedging against decreases in interest rates, causing an increase in bond price. Likewise, the put options are for hedging against increases in interest rates.

2-3 Caps and Floors

An interest rate cap is a type of interest rate derivative which protects the holder from increasing interest rates (Hull, 2003). It guarantees the buyer that otherwise floating rates, such as LIBOR, will not exceed a certain level, the cap rate. Because of this, the cap is a commonly used risk management tool. A floor is defined analogously to a cap. The buyer receives payments at the end of each period in which the interest rate falls below an agreed strike price.

2-4 Swaptions

Swaptions are options on interest rate swaps (Hull, 2003). They give the holder right, but not the obligation for a certain amount of time (the expiry), to enter an interest rate swap at a certain moment for a period of time (the maturity). The notation of swaptions follows the forward rate agreement notation. For example a 1Y x 5Y swaption gives the holder an option that expires in 1 year to enter a swap with the length of 5 years. A swaption can be used in two ways: as an option to pay fixed and receive floating, or to pay floating and receive fixed. Swaptions give holders the guarantee that the fixed interest rate they will pay on a loan at a certain time in the future will not exceed a certain level. Products offered by life insurance companies and pension funds are highly affected by changes in interest rates (Grosen & Jørgensen, 2000). Insurance companies and pension funds hold large investments in interest rate sensitive assets, e.g. bonds, as well as sell long-term products whose present value depends on interest rates. To overcome the risks that come with interest rate changes, life insurers and pension funds take options on offsetting positions in interest rate derivatives, such as bond options, caps and floors, and swaptions. The most common type of derivative used by these companies is a swaption (Berends, McMenamin, Plestis, & Rosen, 2013).

2-5 Implied Volatility

The definition of implied volatility is the basis points per year volatility (Hull, 2003). A more standard definition is that an option’s implied volatility is the volatility that is needed to 2-5 Implied Volatility 7 match the market price of the option (P. S. Hagan et al., 2002). Prices are often quoted in terms of the implied volatility. There are two different implied volatilities quoted in the market: implied lognormal volatility, often referred to as Black volatility, σB and implied normal volatility, σN . The implied Black volatility comes from Black’s option model, which is further explained in Section 3-1. Implied Black volatility and implied normal volatility closely relate to each other (P. S. Hagan et al., 2014). In Black’s model the forward asset price, F (t) is modeled by

dFB = σBFBdWB, (2-1) and in the normal model by

dFN = σN dWN , (2-2) where dWB and dWN are standard Brownian motions. The following rule of thumb is a useful translation between the two volatilities

σN σB = . (2-3) dFN

Since the implied Black volatility is lognormal distributed and therefore not able to work with negative interest rates, the implied normal volatility σN is used in this thesis.

Chapter 3

Literature Review

In this section an overview of some commonly used (interest rate) option models is given. Basic stochastic calculus knowledge is assumed to be known. For more details on stochastic calculus is referred to Etheridge (2002).

3-1 The Black model

The fundamental model for valuing interest rate options is Black’s model (1976). It is an extension of the Black-Scholes option pricing model (1973), used to price options on interest rates (Hull, 2003, p. 508). The Black model is similar to the Black-Scholes model except that the spot price of the underlying is replaced by a forward rate F (t). No assumption on the geometric Brownian motion for the dynamics of the forward rate is needed for the Black model, which makes this model easy to understand and use. In interest rate models, two different times t are used. The first one is the exercise date and the second one the maturity date. Consider for example a 5Y X 10Y swaption. This swaption exercises in 5 years, and has a maturity of 10 years. This means that the owner of this swaption has the possibility to enter a swap in 5 years with a maturity of 10 years. Define

τex Time to exercise: expiration date minus current date, τex = tex − t

τmat Time to maturity: maturity date minus current date, τmat = tmat − t

B(t) The value of $1 today delivered on time t, f The forward rate at time zero, Fˆ(0) = f,

K The strike price. 10 Literature Review

The value of a call option under the Black model is given by

Vcall = B(τex)[fN (d1) − KN (d2)] (3-1) and the value of the put option by

Vput = B(τex)[KN (−d2) − fN (−d1)] (3-2) where

N the standard normal cumulative distribution function,

2 ln(f/K)+σ τex/2 d1 = √ , σB τex

2 ln(f/K)−σ τex/2 √ d2 = √ = d1 − σB τex. σB τex

This model is very basic (no diﬃcult mathematical equations and all parameters, but the volatility, are know) and therefore easy to understand and use. Risk-managers like this model a lot as it allows for quick pricing.

3-2 The SABR Model

The limitation of Black’s interest rate option model is that it does not show how interest rates change over time. As a result, the model is not able to price complex interest rate derivatives. As a solution the LIBOR market model (LMM) was presented by Brace et al. (1997). Before the introduction of the SABR model in 2002, a commonly used model for pricing and hedging European interest rate options was the LMM (P. S. Hagan et al., 2002). Using the LIBOR market option pricing model, the volatility of an underlying can be calculated (Hull, 2012) by plugging in the market prices for the options. In theory, options with the same expiration date have the same implied volatility, regardless of the strike price. In reality the volatility used to price an option also depends on its strike price (see Figure 3-1). Therefore the implied volatility is different across various strike prices for a certain option. This deviation is called the volatility skew or volatility smile. Correctly handling these market skews is essential to exchange desks with a large exposure across a wide range of strikes (e.g. foreign exchange desks or fixed income desks). The LMM model is not able to capture the volatility skew. Therefore a new model was needed. The SABR, stochastic-αβρ, model was presented by Hagan et al. in 2002 to overcome the volatility skew. The SABR model is a stochastic volatility model in which a correlation is assumed between the asset price and volatility. This model is however not completely arbitrage-free. Therefore Hagan et al. proposed an improvement of the SABR model in 2014. The SABR model is the most used model for interpolations of volatilities in the financial market because of two features: It is a stochastic volatility model and can therefore fit the volatility skew noticed in the market, and the SABR model can be written in the form of an explicit approximation formula that expresses the implied volatility in terms of the model parameters. This explicit formula is what makes the SABR model favored by financial practitioners over other stochastic volatility models. 3-2 The SABR Model 11

Figure 3-1: Implied volatility for the June 99 Eurodollar option. Shown are close-of-day values along with the volatilities predicted by the SABR model. Source: (P. S. Hagan et al., 2002).

In this section, ﬁrst the SABR model will be derived, using the martingale representation theorem, and explained. Thereafter is set out how the SABR model can be used in order to calculate option prices.

3-2-1 Martingale Reresentation Theorem

According to the Martingale representation theorem the forward price, F (t), can be written in terms of an Itô integral with respect to the Brownian motion dW (t) (Etheridge, 2002, p.100). The value of a call option under the risk neutral measure Q, written as the expectation of its payoﬀ, is h i Q + Vcall = B(tex)E [F (tex) − K] |F0 . (3-3)

Therefore, using the Martingale representation theorem, F (t) is a Martingale so that the dynamics of F (t) are

dF = C(F )dW, F0 = f. (3-4)

This is the basic form of the SABR model. The function C(F ) must be determined by using a mathematical model for C(F ). A solution for C(F ) will be given later.

3-2-2 Description of the Model

The goal of the SABR model is to ascertain the dynamics of a single forward rate F (t) (P. Hagan & Lesniewski, 2008). This forward rate could for example be a forward LIBOR, a 12 Literature Review forward swap rate or the forward yield on a bond. Hagan et al. (2002) show that smile risk cannot be correctly modeled by using one Brownian motion. Therefore they add an extra Brownian motion, so the model becomes a two-factor model. Because volatility is a random function of time, Hagan et al. (2002) added the volatility process α to the model. Under the SABR model the forward rate, F (t), follows the stochastic diﬀerential equations

dF = αC(F )dW1,F (0) = f

dα = ναdW2, α(0) = α (3-5) E[dW1dW2] = ρdt, with C(F ) = f(F, β) F > 0, α > 0, ρ ∈ [−1, 1], ν ≥ 0. Here α is the volatility of forward rate F , ν is the volatility of the volatility and ρ the correlation between those two processes. Since α and ν are volatilities, they should be non- negative. Because of market observations, the volatilities are also set to be non-zero.

3-2-3 Option pricing with the SABR model

In order to understand how an option price can be derived using the SABR model, the value of a European call and put option are examined Z ∞ Vcall(τex,K) = (F − K)Q(τex,F )dF, (3-6) K

Z K Vput(τex,K) = (K − F )Q(τex,F )dF. (3-7) −∞

Here Q(τex,F ) is the risk-neutral probability density at the exercise date τex. Q(T,F ) is the probability density that F (T ) = F at time T , without taking the value of α(T ) into account. Although Q is also a function of α, f and t, in this thesis those variables are omitted because of clarity of the equations. Except for the special cases of C(F ), no closed form expression for the probability density Q(T,F ) is known (P. S. Hagan et al., 2002). A way to get an arbitrage-free option price is to write Q(T,F ) in the form of a partial diﬀerential equation (PDE), and solve this PDE. How this is done is shown in Chapter5. Another way to obtain an option price is by using an approximation formula. This approximation formula, derived by Hagan et al. (2002), and adjusted by Hagan et al. (2014) is very accurate. The solution has an easy form, which makes it easy it implement in Excel. Because of the accurateness and simplicity of the SABR approximation formula, SABR is the most widespread used option pricing model. The SABR model is especially popular for the fact that its approximation formula gives a solution in the form of the Black implied volatility, as a function of the four parameters α, β, ρ and ν. The remaining part of this section will show the explicit SABR formula as used in practice. 3-2 The SABR Model 13

3-2-4 SABR approximation formula

The explicit formula for the general SABR model is derived by Hagan et al. (2002 & 2014) using perturbation methods α(f − K) ξ σ (K) = · N R f dF χ(ξ) K C(F ) " # ! (3-8) 2γ − γ2 1 2 − 3ρ2 · 1 + 2 2 α2C2(f ) + ρναγ C(f ) + ν2 τ + ... 24 av 4 1 av 24 ex this gives the value of an European option (P. S. Hagan et al., 2014). Here 0 00 p C (fav) C (fav) fav = fK, γ1 = , γ2 = , C(fav) C(fav) ! ν f − K p1 − 2ρξ + ξ2 − ρ + ξ ξ = , χ(ξ) = log , α C(fav) 1 − ρ There are several variants of Equation (3-8), which are all a good approximation (P. S. Hagan et al., 2014). The most robust one according to Hagan et al. (2014) is the following α(f − K) ξ σ (K) = · N R f dF χ(ξ) K C(F ) " # ! −β(2 − β) C(f)C(K) 1 C(f) − C(K) 2 − 3ρ2 · 1 + α2 + ρνα + ν2 τ + ... . 24 fK 4 f − K 24 ex (3-9)

Here f = F (0), α = α(0) are today’s values of the forward price and volatility, and τex is the time to exercise, ! ν Z f dF p1 − 2ρξ + ξ2 − ρ + ξ ξ = , χ(ξ) = log . α K C(F ) 1 − ρ

3-2-5 SABR in practice: the classical SABR model

When referring to the SABR model, often the SABR model with C(F ) = F β is meant. The dynamics become β dF = αF dW1,F (0) = f

dα = ναdW2, α(0) = α (3-10) E[dW1dW2] = ρdt, with F > 0, α > 0, β ≥ 0, ρ ∈ [−1, 1], ν ≥ 0. 14 Literature Review

R f dF The integral K C(F ) can now be reduced to Z f dF Z f 1 = β dF K C(F ) K F Z f = F −βdF K #f (3-11) F 1−β = 1 − β K f 1−β − K1−β = 1 − β

The explicit implied normal volatility formula in Equation (3-9) now is α(f − K)(1 − β) ξ σ (K) = · N f 1−β − K1−β χ(ξ) " # ! (3-12) −β(2 − β) (fK)β 1 f β − Kβ 2 − 3ρ2 · 1 + α2 + ρνα + ν2 τ + ... , 24 fK 4 f − K 24 ex

where ! ν f 1−β − K1−β p1 − 2ρξ + ξ2 − ρ + ξ ξ = , χ(ξ) = log . α 1 − β 1 − ρ

In order to calibrate the SABR model to market data (more on this in Section 7-2), the at-the-market (ATM) volatility is needed. This is the volatility where the strike price K is equal to the forward rate F . In Hagan et al. (2002) is shown how the ATM volatility is derived for the implied black volatility. In this thesis, these steps are followed in order to come up with an ATM volatility for the implied normal volatility case. To see what happens to the implied volatility when the forward rate and strike price are equal, parts of the above mentioned equation for the SABR model need to be expanded using a Taylor series 1 1 f − K = pfK log(f/K) · 1 + log2(f/K) + log4(f/K) + ... , (3-13) 24 1920

f 1−β − K1−β = (1 − β)(fK)(1−β)/2 log(f/K) " (1 − β)2 (1 − β)4 (3-14) · 1 + log2(f/K) + log4(f/K) + ... , 24 1920

" β2 β4 f β − Kβ = β(fK)β/2 log(f/K) · 1 + log2(f/K) + log4(f/K) + ... (3-15) 24 1920 and neglecting terms higher than fourth order. The normal implied volatility reduces to 1 2 1 4 β/2 1 + 24 log (f/K) + 1920 log (f/K) + ... ξ σN (K) = α(fK) · (1−β)2 2 (1−β)4 4 χ(ξ) 1 + 24 log (f/K) + 1920 log (f/K) + ... " # ! (3-16) −β(2 − β) 1 f β − Kβ 2 − 3ρ2 · 1 + α2 + ρνα + ν2 τ + ... , 24(fK)1−β 4 f − K 24 ex 3-2 The SABR Model 15 where ! ν p1 − 2ρξ + ξ2 − ρ + ξ ξ = (fK)(1−β)/2 log(f/K), χ(ξ) = log . α 1 − ρ

f β −Kβ Further expanding f−K gives

h 2 4 β β β/2 β 2 β 4 f − K β(fK) log(f/K) · 1 + 24 log (f/K) + 1920 log (f/K) + ... = √ f − K h 1 2 1 4 fK log(f/K) · 1 + 24 log (f/K) + 1920 log (f/K) + ... h 2 4 β/2 β 2 β 4 β(fK) 1 + 24 log (f/K) + 1920 log (f/K) + ... = √ (3-17) fK h 1 2 1 4 1 + 24 log (f/K) + 1920 log (f/K) + ... h β2 2 β4 4 1 + 24 log (f/K) + 1920 log (f/K) + ... = β(fK)(β−1)/2 h 1 2 1 4 1 + 24 log (f/K) + 1920 log (f/K) + ...

When calculating the ATM volatility this term becomes (f = K)

f β − Kβ = β(fK)(β−1)/2, f − K (3-18) β = , f 1−β since log(f/K) = 0. Finally the ATM volatility (f = K) is given by " # ! −β(2 − β) α2 1 ρναβ 2 − 3ρ2 σ = αf β · 1 + + + ν2 τ + ... . (3-19) AT M 24 f 2−2β 4 f 1−β 24 ex

This normal ATM volatility only has one small diﬀerence compared to the black ATM volatility derived by Hagan et al. (2002). The ﬁrst term, αf β used to be α in the implied f (1−β) black volatility situation.

3-2-6 Conclusion

Because of the different implied volatilities for options with different strikes, a new option model was needed. The most used option model by financial institutions for interpolations of volatilities is the SABR model (P. S. Hagan et al., 2002). This model is famous for its easy, accurate and closed form approximation formula. This section first showed the SABR model, consisting of two stochastic differential equations to model the dynamics of forward rate F (t). Thereafter we explained how options can be priced using the SABR model. Two methods were presented: writing the risk-neutral probability density Q(T,F ) in the form of a PDE and solve it, or use an approximation formula. The approximation formula for the SABR model was then given. The next chapter will show how to derive the approximation formulas for the normal SABR model and the free boundary SABR model. In Chapter 5 is derived how option prices can be determined solving the PDE.

Chapter 4

Negative Interest Rate Models

Chapter 3 showed how options prices can be determined using an explicit formula. Unfortu- nately this approximation formula does not cope with negative interest rates, since the SABR model gives a probability of zero to negative rates. This chapter gives three solutions to overcome this problem, of which the last two are further investigated in this thesis. These three solutions are the displaced SABR model, the normal SABR model and the free boundary SABR model. Closed form formulas are derived for each model, following the same steps Hagan et al. (2014) did for their classical SABR model (presented in Chapter 3).

4-1 Displaced model

The ﬁrst solution that comes in mind to overcome negative rates is a displaced model, or a shifted model. In these type of models the forward rate F in the SABR process is shifted

F 0 = F + s, (4-1) so that in the original SABR model C(F ) = (F + s)β. Filling this into Equation (3-5) gives the dynamics of the displaced SABR model (P. S. Hagan et al., 2014)

β dF = α(F + s) dW1,F (0) = f

dα = ναdW2, α(0) = α (4-2)

E[dW1dW2] = ρdt, with

F + s > 0. 18 Negative Interest Rate Models

The explicit (approximated) implied volatility is derived using the same steps as for the classical SABR model and is given by α(f − K)(1 − β) ξ σ (K) = · N (f + s)1−β − (K + s)1−β χ(ξ) " # ! (4-3) 1 (f + s)β − (K + s)β 2 − 3ρ2 · 1 + gα2 + ρνα + ν2 τ + ... , 4 f − K 24 ex where ! ν (f + s)1−β − (K + s)1−β p1 − 2ρξ + ξ2 − ρ + ξ ξ = , χ(ξ) = log , α 1 − β 1 − ρ and −β(2 − β) (f + s)β(K + s)β g = (4-4) 24 fK

4-1-1 Drawback Displaced Model

The diﬃculty with this is the selection of the shift size. In advance it is not known how low interest rates can or will go (Antonov, Konikov, & Spector, 2015). A risk manager may end up in a situation where the shift needs to be adjusted, and the entire portfolio risk recalculated. Therefore this model is not suitable for use in practice. Although this model is used by companies that have not found a solution to overcome the negative rate problem yet, this model is not covered in this thesis.

4-2 Stochastic Normal Model

The stochastic normal model, or normal SABR model, is a simpliﬁed version of the classical SABR model. In this SABR model C(F ) = 1. The dynamics of Equation (3-5) become

dF = αdW1,F (0) = f

dα = ναdW2, α(0) = α (4-5) E[dW1dW2] = ρdt. Equation (3-8) is simpliﬁed so that the explicit implied volatility becomes " # ! ξ 2 − 3ρ2 σ (K) = α · 1 + ν2 τ + ... , (4-6) N χ(ξ) 24 ex where ! ν p1 − 2ρξ + ξ2 − ρ + ξ ξ = (f − K), χ(ξ) = log . (4-7) α 1 − ρ

Using Equation (3-19) the ATM volatility for the normal SABR model can easily be derived by setting β equal to zero " # ! 2 − 3ρ2 σN (K) = α · 1 + ν2 τ + ... . (4-8) AT M 24 ex 4-3 The Free Boundary SABR Model 19

4-3 The Free Boundary SABR Model

The free boundary SABR model is another extension of the classic SABR model. It allows for negative rates by including absolute signs around F . According to Antonov et al. (2015) a more natural solution to overcome the negative rate problem (compared to the normal SABR model) is to use the free boundary SABR method. The forward rates in this model follow the following dynamics

β dF = α|F | dW1,F (0) = f, (4-9)

dα = ναdW2, α(0) = α, (4-10) E[dW1dW2] = ρdt. (4-11)

When there is zero-correlation between the two standard Brownian motions dW1 and dW2 the free boundary SABR model can be solved exactly. In cases of general correlation the option price needs to be approximated (Antonov et al., 2015). Antonov et al. do this based on a numerical integration and Markovian projection, which they call the mapping technique. Since this method is very time consuming, it is not a useful method for practitioners where fast calibration and approximation performances are essential (Kienitz, 2015). Kienitz (2015) shows that an approximation for the free boundary model based on Hagan (2014) gives nearly the same results as using the mapping technique. C(F ) is chosen as |F |β so that the explicit implied normal volatility (Equation 3-9) becomes

α(f − K) ξ σ (K) = · N R f dF χˆ(ξ) K C(F ) " # ! (4-12) −β(2 − β) |f|β|K|β ρνα |f|β − |K|β 2 − 3ρ2 · 1 + α2 + + ν2 τ , 24 fK 4 f − K 24 ex

where ! ν Z f dF p1 − 2ρξ + ξ2 − ρ + ξ ξ = , χˆ(ξ) = log . (4-13) α K C(F ) 1 − ρ

R f dF The integral K C(F ) can again be solved analytical (Equation 3-11), but now with three diﬀerent solutions

 1−β 1−β −(−f) +(−K) for K < 0, f < 0, f  1−β Z dF  1−β 1−β = f +(−K) (4-14) 1−β for K < 0, f > 0, K C(F )  1−β 1−β  f −K 1−β for K > 0, f > 0.

The ATM volatility for the free boundary SABR model is obtained in this thesis using the same method as Hagan et al. (2002) for the classical SABR model. However we have to consider two diﬀerent situations: both f and K larger than zero, or both smaller than zero. 20 Negative Interest Rate Models

The ﬁrst situation gives us the same normal ATM volatility as the classical SABR model. To analyze the eﬀect of a negative forward rate and strike price on the normal ATM volatility, parts of the equation for the implied volatility (Equation 4-12) of the free boundary SABR model need to be expanded using a Taylor series 1 1 f − K = pfK log(f/K) · 1 + log2(f/K) + log4(f/K) + ... , (4-15) 24 1920

" # β2 β4 |f|β − |K|β = β(|f||K|)β/2 log(f/K) · 1 + log2(f/K) + log4(f/K) + ... , (4-16) 24 1920

−(−f)1−β + (−K)1−β = −1 · (−f)1−β − (−K)1−β = −(1 − β)(−f)(1−β)/2(−K)(1−β)/2 log(f/K) (4-17) " # (1 − β)2 (1 − β)4 · 1 + log2(f/K) + log4(f/K) + ... , 24 1920

and neglecting terms higher than fourth order. The normal implied volatility then reduces to

1 2 1 4 β/2 β/2 1 + 24 log (f/K) + 1920 log (f/K) + ... ξ σN (K) = −α(−f) (−K) · (1−β)2 2 (1−β)4 4 χ(ξ) 1 + 24 log (f/K) + 1920 log (f/K) + ... " # ! (4-18) −β(2 − β) |f|β|K|β ρνα |f|β − |K|β 2 − 3ρ2 · 1 + α2 + + ν2 τ , 24 fK 4 f − K 24 ex where ! ν p1 − 2ρξ + ξ2 − ρ + ξ ξ = (fK)(1−β)/2 log(f/K), χ(ξ) = log . α 1 − ρ

|f|β −|K|β Further expanding f−K gives

h 2 4 β β β/2 β/2 β 2 β 4 |f| − |K| β|f| |K| log(f/K) · 1 + 24 log (f/K) + 1920 log (f/K) + ... = √ f − K h 1 2 1 4 fK log(f/K) · 1 + 24 log (f/K) + 1920 log (f/K) + ... h 2 4 (4-19) β/2 β/2 β 2 β 4 β|f| |K| 1 + 24 log (f/K) + 1920 log (f/K) + ... = √ fK h 1 2 1 4 1 + 24 log (f/K) + 1920 log (f/K) + ...

When calculating the ATM volatility this term becomes (f = K),

β|f|β/2|K|β/2 β|f|β √ = √ (4-20) fK fK since log(f/K) = 0. 4-3 The Free Boundary SABR Model 21

Finally the normal ATM volatility (f = K) for both f and K below zero is given by " # ! −β(2 − β) |f|2β 1 ρναβ|f|β 2 − 3ρ2 σ = αf β · 1 + α2 + + ν2 τ + ... . (4-21) AT M 24 f 2 4 f 24 ex

The normal ATM volatility for both f and K positive is given by " # ! −β(2 − β) α2 1 ρναβ 2 − 3ρ2 σ = αf β · 1 + + + ν2 τ + ... . (4-22) AT M 24 f 2−2β 4 f 1−β 24 ex

Chapter 5

Partial Diﬀerential Equation Solution

Recall the theory on pricing options as described in Chapter3: in order to obtain a completely arbitrage-free option price, the risk-neutral probability density Q(τex,F ) needs to be derived. This is done by solving a partial diﬀerential equation (PDE). It is ﬁrst explained how Hagan et al. (2014) solved the PDE in order to obtain an arbitrage- free option price. This chapter then concludes with how the same method could be applied for the two negative rate SABR models compared in this thesis.

5-1 Solving the PDE

5-1-1 Arbitrage-free pricing within the SABR model

There are two key requirements for obtaining an arbitrage-free option price (P. S. Hagan et al., 2014). First, the put-call parity must hold. Second the probability density implied by the call and put option prices needs to be non-negative. The ﬁrst requirement automatically holds since the the same implied volatility σN is used for both the call and put options. To see in what situation the second condition holds we take a closer look at the value for the call and put price

Z ∞ Vcall(τex,K) = (F − K)Q(τex,F )dF, (5-1) K

Z K Vput(τex,K) = (K − F )Q(τex,F )dF, (5-2) −∞ where Q(τex,F ) is the risk-neutral probability density at the exercise date τex. Q(T,F ) is the probability density that F (T ) = F at time T , without taking the value of α(T ) into account. Although Q is also a function of α, f and t, in this thesis those variables are omitted because 24 Partial Diﬀerential Equation Solution of clarity of the equations. Since the risk-neutral probability Q is unknown, we need to ﬁnd an equation for Q. One can see from above equations that

∂2 ∂2 V (τ ,K) = V (τ ,K) = Q(τ ,F ) ≥ 0 (5-3) ∂K2 call ex ∂K2 put ex ex

since a probability density cannot go below zero. Therefore the explicit implied volatility formula (Equation 3-12) can only be arbitrage-free if

∂2 ∂2 V (τ ,K) = V (τ ,K) ≥ 0 for all K. (5-4) ∂K2 call ex ∂K2 put ex

In Hagan et al. (2014) is derived that the marginal probability density that satisﬁes the two requirements for arbitrage-free pricing is given by the PDE

∂Qc ∂2M(T,F )Qc(T,F ) (T,F ) = , (5-5) ∂T ∂F 2

and

∂QL ∂M(T,F )Qc(T,F ) ∂T (T ) = limF →Fmin ∂F ∂QR ∂M(T,F )Qc(T,F ) (5-6) ∂T (T ) = limF →Fmax ∂F

Here Qc is the central and continuous part of the density function. QL and QR are the extreme left and right limits of the density.

1 M(T,F ) = D2(F )E(T,F ), 2

E(T,F ) = eρναΓ(F )(T −t),

D2(F ) = α2[1 + 2ρνy(F ) + ν2y2(F )]C2(F ), and where y(F ) and Γ(F ) are given by

Z F du y(F ) = , C(u) f (5-7) F 1−β − f 1−β = 1 − β

C(F ) − C(f) Γ(F ) = , F − f and F (0) = f, C(F ) = F β as usual. The superscript c is added to Q(T,F ) since this is the continuous part of the density Q(T,F ), which will be further explained in the following R f dF section. The derivation of y(F ) is the same as the derivation of the integral K C(F ) in Equation (3-11). 5-1 Solving the PDE 25

5-1-2 Boundary conditions

One aspect of the SABR model is that C(F ) cannot go below zero. Therefore the barrier of the PDE ∂Qc ∂2M(T,F )Qc(T,F ) (T,F ) = , ∂T ∂F 2 is were D(F ) = 0. A common barrier for the PDE is at F = 0 with C(F ) = F β. This is not useful for negative rates, therefore this barrier should be adjusted for both the normal SABR model and the free boundary SABR model.

Since the PDE needs to be solved analytically, a finite domain Fmin < F < Fmax is required. The lower limit can be set as the barrier, but this is not essential (P. S. Hagan et al., 2014). The upper limit, Fmax is chosen such that this boundary does not affect the pricing. That is, sufficiently large that the probability of reaching the boundary is almost zero. Since there still is a probability that this boundary is reached, and the forward rate F (T ) should not be able to take off from the domain, the boundaries are allowed to expand. Therefore there is a δ-function included in the probability density. The probability density Q(T,F ) is given by

 L  Q (T )δ(F − Fmin) at F = Fmin,  c Q(T,F ) = Q (T,F ) for Fmin < F < Fmax, (5-8)  R  Q (T )δ(F − Fmax) at F = Fmax.

Since Q(T,F ) is a probability density, the total integrated sum has to be 1 for all T ,

Z Fmax QL(T ) + Qc(T,F )dF + QR(T ) = 1. (5-9) Fmin Taking the derivative with respect to T gives " # ∂ Z Fmax QL(T ) + Qc(T,F )dF + QR(T ) = 0. (5-10) ∂T Fmin Substituting

∂Qc ∂2M(T,F )Qc(T,F ) (T,F ) = , ∂T ∂F 2 for Qc(T,F ) gives

dQL 1 ∂ Fmax dQR + D2(F )E(T,F ) + = 0. (5-11) dT 2 ∂F Fmin dT As a consequence the probability density function requires that

dQL 1 ∂ = lim D2(F )E(T,F ), (5-12) dT + 2 ∂F F →Fmin

dQR 1 ∂ = lim − D2(F )E(T,F ). (5-13) − dT F →Fmax 2 ∂F 26 Partial Diﬀerential Equation Solution

Recall that F (T ) needs to be a martingale in order to correctly price options (Section 3-2-1)). For F (T ) to be a martingale the expected value must satisfy the following condition

E [F (T )|F (t) = f, α(t) = α] = Z Fmax L c R (5-14) FminQ (T ) + FQ (T,F )dF + FmaxQ (T ) = f. Fmin Therefore

L Z Fmax R dQ (T ) c dQ (T ) Fmin + FQ (T,F )dF + Fmax = 0. (5-15) dT Fmin dT

Substituting ∂Qc ∂2M(T,F )Qc(T,F ) (T,F ) = , ∂T ∂F 2 for Qc(T,F ), integrating by parts twice and using Equation (5-12) and (5-13) gives

h iFmax D2(F )E(T,F )Qc(T,F ) = 0. (5-16) Fmin So in order for F (T ) to be a martingale, the following boundary conditions are required

2 c + D (F )E(T,F )Q (T,F ) → 0 as F → Fmin, (5-17)

2 c − D (F )E(T,F )Q (T,F ) → 0 as F → Fmax. (5-18)

Thus in summary, in order to numerically solve the PDE ∂Qc ∂2M(T,F )Qc(T,F ) (T,F ) = for F < F < F , ∂T ∂F 2 min max boundary conditions

2 c + D (F )E(T,F )Q (T,F ) → 0 as F → Fmin, 2 c − D (F )E(T,F )Q (T,F ) → 0 as F → Fmax, are needed for t < T < τex. The probabilities at the boundary are given by dQL 1 ∂ = lim D2(F )E(T,F ), dT + 2 ∂F F →Fmin dQR 1 ∂ = lim − D2(F )E(T,F ), − dT F →Fmax 2 ∂F and the initial conditions are

QL(0) = 0,

Qc(T,F ) → δ(F − f), QR(0) = 0. 5-2 Option pricing with positive rates 27

5-2 Option pricing with positive rates

c In order to compute option prices for the derived SABR model, the PDE ∂Q (τex,F ) needs to be ∂τex c solved numerically. By doing so, the probability density Q (τex,F ) is derived. One diﬃculty with this is the choice of Fmin and particularly Fmax. The formula proposed by Hagan et al. 2014 is not suitable for solving Fmax and Fmin for all SABR parameter combinations (Floc’h & Kennedy, 2014). Another problem with their formula is that Fmax can become very large for long term deals. As a consequence a very large number of discretization points is needed in order to obtain an accurate probability density. This problem is not new and can be solved by a transformation: a change in variables (Floc’h & Kennedy, 2014). This subsection ﬁrst treats this transformation.

c After the PDE transformation, we can solve the PDE so that the probability density Q (τex,F ) becomes known. This is done by using a discretization of the PDE. In the second part of this subsection, it is explained how the discretization of the PDE is done. Hagan et al. (2014) use a Crank-Nicolson scheme for the discretization of the PDE. Le Floc’h & Kennedy (2014) explore alternative formulations for the transformation of the PDE. They ﬁnd that the Lawson-Swayne scheme stands out on this problem in terms of stability and speed. Accuracy of option pricing is of course always important, but since pricing a 30 year cap on a 3M LIBOR requires 120 PDEs to be solved, speed is also an important factor. Therefore in this thesis the Lawson-Swayne scheme as proposed in Le Floc’h & Kennedy (2014) is used for transforming the PDE in order to obtain a numerical solution for the PDE. This section is concluded with a way to price options with the derived probability density c Q (τex,F ). Knowing the solution of the PDE, option prices can then easily be computed using

Z Fmax c R Vcall(T,K) = (F − K)Q (τex,F )dF + (Fmax − K)Q (T ), (5-19) K

Z K c L Vput(T,K) = (K − F )Q (τex,F )dF + (K − Fmin)Q (T ). (5-20) Fmin

5-2-1 Transformation of the PDE

We transform the PDE from Q to θ in order to solve the PDE numerically fast and eﬃcient. This transformation is a change of variable F to z(F ) that preserves the moments (Floc’h & Kennedy, 2014). The transformation is given by

Z F 1 z(F ) = du, (5-21) f D(u) with f = F (0) and D(u) = αp1 + 2ρνy(u) + ν2y2(u)C(u). Therefore

Z y(F ) dy0 z(y) = p (5-22) y(f) α 1 + 2ρνy0 + ν2y02 28 Partial Diﬀerential Equation Solution

This leads to a new PDE in θc(T, z). θc(T, z) is given by θc(T, z) := Qc(T,F (z))D(F (z)) (5-23) = Qc(T,F (z))C(z) with C(z) = D(F (z)) and  PL(T )δ(z − z−) for z ≤ z−,  θc(T, z) = θc(T, z) for z− < z < z+ (5-24)  R +  P (T )δ(F − Fmax) for z ≥ z , with − + z = z(Fmin), z = z(Fmax), and PL and PR are the probability masses at z− and z+ respectively. z is a grid used for the uniform discretization in the next subsection. The transformed PDE is given by (Floc’h & Kennedy, 2014) ∂θc 1 ∂ 1 ∂C(z)E(T, z)θc(T, z) (T, z) = , (5-25) ∂T 2 ∂z C(z) ∂z with E(T, z) = E(T,F (z)), α y(z) = [sinh(νz) + ρ(cosh(νz) − 1)] , ν (5-26) h i 1 F (y) = f 1−β + (1 − β)y 1−β . In the above equation sinh and cosh are the hyperbolic sinus and cosinus. The probabilities at the boundaries accumulate accordingly to ∂PL 1 1 ∂C(z)E(T, z)θc(T, z) (T ) = lim , (5-27) ∂T z→z− 2 C(z) ∂(z)

∂PR 1 1 ∂C(z)E(T, z)θc(T, z) (T ) = lim − , (5-28) ∂T z→z+ 2 C(z) ∂(z)

with the values for z+ and z− chosen to be z+ = −z− + √ z = nsd τex, − √ (5-29) z = −nsd τex. √ Here nsd is the number of standard deviations ( τex) above or below the forward located at z = 0. The eﬀect of the transformation from the PDE of risk-neutral probability density Q to θ can best be seen when looking at Figure 5-1. There is a higher concentration of points around the forward, and lower around the boundaries, after the probability density is transformed. This change results in a higher accuracy for the same number of points when calculating option prices. Uniform discretization of Q in F requires approximately 1000 times more points to reach a similar accuracy (Floc’h & Kennedy, 2014) than the discretization in the transformed variable θ. 5-2 Option pricing with positive rates 29

104 1 y(z) 0.8 f(y(z))

0.6

0.4

0.2

y(z) / f(y(z)) -0.2

-0.4

-0.6

-0.8

-1 -15 -10 -5 0 5 10 15 z

Figure 5-1: Y (z) (Equation 5-26) and transformed F (z): α = 1, β = 0.3, ρ = -0.1, ν = 1, f = 1, τex = 1

5-2-2 Discretization of the PDE

In order to solve the PDE numerically, the PDE is discretized, using finite difference discretization. This basically means that the derivatives in the PDE are replaced with a finite difference approximations on a discretized domain. These resulting algebraic equations are then solved one at a time.

The time τex is uniformly discretized for n = 0, ..., N − 1 τ t = nδ, with δ = ex . n N For j = 1, ..., J, the uniform grid for z is given by − zj = z + jh, 1 yˆ = y z − h , j j 2

Fˆj = F (ˆyj),

Cˆj = D(Fˆj), ˆβ β Fj − f Γˆj = , Fˆj − f

ρναΓˆj T Eˆj(T ) = e , n θj = θ(zj, tn). 30 Partial Diﬀerential Equation Solution

The PDE (Equation 5-25) is discretized in z as (P. S. Hagan et al., 2014)

∂θ (z , t ) = Lnθ(z , t ) (5-30) ∂T j n j j n

n for j = 1, ..., J with Lj the discrete operator deﬁned by (Floc’h & Kennedy, 2014)

ˆ n 1 Cj−1 ˆ Lj θ(zj, tn) = Ej−1(tn)θ(zj−1, tn) h Fˆj − Fˆj−1 ! 1 Cˆj Cˆj − + Eˆj(tn)θ(zj, tn) (5-31) h Fˆj+1 − Fˆj Fˆj − Fˆj−1

1 Cˆj+1 + Eˆj+1(tn)θ(zj+1, tn) h Fˆj+1 − Fˆj

Diﬀerent discretization schemes are analyzed in Le Floc’h et al. (2014). In their research the Lawson-Schwayne scheme stands out in terms of speed and stability. Therefore this discretization scheme is also used in this thesis. √ 2 Let b = 1 − 2 . The Lawson-Schwayne scheme is a combination of applying two implicit Euler steps with time-step bδ and an extrapolation on the values at those steps (Floc’h & Kennedy, 2014). First stage:

n+b n n+b n+b θj − θj = bδLj θj ˆ C1 ˆ n+b PL(tn+b) − PL(tn) = bδ E1(tn+b)θ1 Fˆ1 − Fˆ0 (5-32) ˆ CJ ˆ n+b PR(tn+b) − PR(tn) = bδ EJ (tn+b)θJ FˆJ+1 − FˆJ

Second stage:

n+2b n+b n+2b n+2b θj − θj = bδLj θj ˆ C1 ˆ n+2b PL(tn+2b) − PL(tn+b) = bδ E1(tn+2b)θ1 Fˆ1 − Fˆ0 (5-33) ˆ CJ ˆ n+2b PR(tn+2b) − PR(tn+b) = bδ EJ (tn+2b)θJ FˆJ+1 − FˆJ

Final stage: √ √ n+1 n+2b n+b θj = ( 2 + 1)θj − 2θj √ √ PL(tn+1) = ( 2 + 1)PL(tn+2b) − 2PL(tn+b) (5-34) √ √ PR(tn+1) = ( 2 + 1)PR(tn+2b) − 2PR(tn+b) for j = 1, ..., J and n = 0, ..., N − 1. 5-3 Option pricing with negative rates 31

5-2-3 Pricing options

Now that the PDE is solved numerically, an option price needs to be derived form the probab- c ility density θ (τex, z). Call and put prices are computed from this probability density using mid-point integration (Floc’h & Kennedy, 2014). Let z∗ = z(y(K)). The value of a call option is given by

 S − K, for z∗ ≤ z−,  0  h 2 M  (sk − K) θ 4(sk−sˆk) k Vcall = J−1 M − ∗ + (5-35)  +Σ (ˆsj − K)hθj + (Smax − K)PR for z < z < z  j=k+1  0 for z∗ ≥ z+, .

5-3 Option pricing with negative rates

Using the same method as described in Section 5-2, this section will show how option prices are determined for negative rates. Recall that the diﬀerence between the classical SABR model and the normal SABR and free boundary SABR lies in the change of C(F ) = F β. For the normal SABR model this change is C(F ) = 1 and for the free boundary SABR model C(F ) = |F |β. First is shown how option prices can be determined using the normal SABR model, followed by the free boundary SABR model.

5-3-1 Modeling negative interest rates with the normal SABR model

This subsection ﬁrst describes the diﬀerences in the formulation of the PDE for the normal SABR model. Thereafter the transformation of the normal SABR PDE is shown. For the normal SABR model the probability density Q is the solution of ∂Qc ∂2M(T,F )Qc(T,F ) (T,F ) = , (5-36) ∂T ∂F 2 and

∂QL ∂M(T,F )Qc(T,F ) ∂T (T ) = limF →Fmin ∂F ∂QR ∂M(T,F )Qc(T,F ) (5-37) ∂T (T ) = limF →Fmax ∂F with 1 M(T,F ) = D2(F )E(T,F ), 2 E(T,F ) = eρναΓ(F )(T −t), D2(F ) = α2[1 + 2ρνy(F ) + ν2y2(F )], and where y(F ) and Γ(F ) are given by

Z F y(F ) = du, f 32 Partial Diﬀerential Equation Solution

Γ(F ) = 0,

So only C(F ), Γ and y have changed compared to the classical SABR model which is described in the previous section. Instead of placing the left boundary Fmin at 0, the left boundary should now be placed far enough on the negative side. A choice could be Fmin = −Fmax. But since this PDE is transformed, just like in the classical SABR situation, the choice of Fmin has no inﬂuence on the option price. The transformation of the normal SABR PDE Q to θ is the same transformation of variable F to z(F ) as in the case of the standard SABR PDE Z F 1 z(F ) = du. (5-38) f D(u) This leads to the PDE in θ(z) = Q(T,F (z))D(F (z)) = Q(T,F (z))C(z) with C(z) = D(F (z))

∂θc 1 ∂ 1 ∂C(z)E(T, z)θc(T, z) (T, z) = , (5-39) ∂T 2 ∂z C(z) ∂z and

( − θ(T, z) = 0 as z → z = z(Fmin) + (5-40) θ(T, z) = 0 as z → z = z(Fmax) with E(T, z) = E(T,F (z)), α (5-41) y(z) = [sinh(νz) + ρ(cosh(νz) − 1)] . ν Since z(y) remains unchanged after changing C(F ) = F β to C(F ) = 1, y(z) also remains unchanged. F (y) however is modiﬁed by the normal SABR model. Therefore Equation (5-7), Z F du y(F ) = , f C(u) F 1−β − f 1−β y(F ) = , 1 − β is inverted to h i 1 F (y) = f 1−β + (1 − β)y 1−β with β = 0 this gives

F (y) = f + y. (5-42)

Again the boundary should be chosen as z− = z+

+ √ z = nsd τex, − √ (5-43) z = −nsd τex, with nsd the number of standard deviations. 5-3 Option pricing with negative rates 33

5-3-2 Modeling negative interest rates with the free boundary SABR model

This subsection ﬁrst describes the diﬀerences in the formulation of the PDE for the free boundary SABR model. Thereafter the transformation of the free boundary SABR PDE is shown. For the free boundary SABR model the probability density Q is the solution of ∂Qc ∂2M(T,F )Qc(T,F ) (T,F ) = , (5-44) ∂T ∂F 2 and ∂QL ∂M(T,F )Qc(T,F ) ∂T (T ) = limF →Fmin ∂F ∂QR ∂M(T,F )Qc(T,F ) (5-45) ∂T (T ) = limF →Fmax ∂F with 1 M(T,F ) = D2(F )E(T,F ), 2 E(T,F ) = eρναΓ(F )(T −t), D2(F ) = α2[1 + 2ρνy(F ) + ν2y2(F )]C2(F ), and where y(F ) and Γ(F ) are given by Z F du y(F ) = f C(u)  1−β 1−β −(−F ) +(−f) for f < 0, F < 0, F  1−β Z du  1−β 1−β = F +(−f) 1−β for f < 0, F > 0, f C(u)  1−β 1−β  F −f 1−β for f > 0, F > 0,

sign(F )|F |1−β − sign(f)|f|1−β y(F ) = , (5-46) 1 − β

C(F ) − C(f) Γ(F ) = , F − f |F |β − |f|β = . F − f So only C(F ), Γ and y have changed compared to the classical SABR model, which is described in the previous section. Instead of placing the left boundary Fmin at 0, the left boundary should now be chosen far enough on the negative side. A choice could be Fmin = −Fmax. But since this PDE is transformed, just like in the classical SABR situation, the choice of Fmin will have no inﬂuence on the option price. The transformation of the free boundary SABR PDE Q to θ is the same transformation of variable F to z(F ) as in the case of the standard SABR PDE Z F 1 z(F ) = du. (5-47) f D(u) 34 Partial Diﬀerential Equation Solution

This leads to the PDE in θ(z) = Q(T,F (z))D(F (z)) = Q(T,F (z))C(z) with C(z) = D(F (z))

∂θc 1 ∂ 1 ∂C(z)E(T, z)θc(T, z) (T, z) = , (5-48) ∂T 2 ∂z C(z) ∂z and

( − θ(T, z) = 0 as z → z = z(Fmin) + (5-49) θ(T, z) = 0 as z → z = z(Fmax) with E(T, z) = E(T,F (z)), α (5-50) y(z) = [sinh(νz) + ρ(cosh(νz) − 1)] . ν Since z(y) remains unchanged after changing C(F ) = F β to C(F ) = |F |β, y(z) also remains unchanged. F (y) however is modiﬁed for the free boundary SABR model. Equation (5-46) becomes

h i 1 F (y) = sign y − y0 (1 − β)|y − y0| 1−β

0 sign(f)|f|1−β with y = y(F = 0) = − 1−β Again the boundary should be chosen as z− = z+

+ √ z = nsd τex, − √ (5-51) z = −nsd τex, with nsd the number of standard deviations. Chapter 6

Closed Form Solution Results

In this chapter the accuracy of the approximation formulas as derived in Chapter 3 are tested. For testing the arbitrage-free approximation formulas, the PDE is taken as a benchmark, since this solution is arbitrage-free. For different SABR parameters both models are estimated in order to calculate an option price. Before estimating options prices for different parameter combinations for both models, first some PDFs are plotted using the PDE method, to get familiar with the models.

6-1 PDF plotting

Using the PDE approach from the previous chapter, PDFs are obtained for the classical, normal and free boundary SABR model. In Figure 6-1, 6-2 and 6-3 the probability densities are plotted. The chosen SABR parameters are equal to those in Antonov et al. (2015): α = 0.6 ∗ f 1−β, β = 0.1, ν = 0.3, ρ = −0.3,T = 3 and f = 0.05. When looking at these graphs, one can see that the probability densities look similar, except for the free boundary SABR model. Trying diﬀerent βs, an interesting result occurs for the free boundary SABR model: a narrow large spike is observed around a forward rate of zero (Figure 6-4), for βs close to zero. Notice that this spike would have been larger as the steps used in computing the probability density would have been increased, or smaller if less steps would have been used. This spike is not the result of a programming error, but inherent to the free boundary model (Floc’h & Kennedy, 2014). The reason why the free boundary model shows this behaviour is because of sticky Brownian motion. A sticky Brownian motion has a large probability to be zero or close to zero, if the previous value of the stochastic process is also close to zero. This sticky Brownian motion can complicate the calibration of the free boundary model to market data. 36 Closed Form Solution Results

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06 Probability Density Classical SABR

0.04

0.02

0 0 0.05 0.1 0.15 0.2 0.25 Forward

Figure 6-1: Probability density function obtained by solving the PDE for the classical SABR with α = 0.6 ∗ f 1−β, β = 0.1, ν = 0.3, ρ = −0.3,T = 3, f = 0.05.

0.16

0.14

0.12

0.1

0.08

0.06 Probability Density Normal SABR 0.04

0.02

0 0 0.05 0.1 0.15 0.2 0.25 0.3 Forward

Figure 6-2: Probability density function obtained by solving the PDE for the normal SABR with α = 0.6 ∗ f 1−β, β = 0.1, ν = 0.3, ρ = −0.3,T = 3, f = 0.05. 6-1 PDF plotting 37

0.25

0.2

0.15

0.1 Probability Density Free Boundary

0.05

0 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Forward

Figure 6-3: Probability density function obtained by solving the PDE for the free boundary SABR with α = 0.6 ∗ f 1−β, β = 0.1, ν = 0.3, ρ = −0.3,T = 3, f = 0.05.

Figure 6-4: Probability density function obtained by solving the PDE for the free boundary SABR with varying β and α = 0.6 ∗ f 1−β, ν = 0.3, ρ = −0.3,T = 3, f = 0.05. 38 Closed Form Solution Results

6-2 Performance of the Approximation Formula

Using the exact PDE solutions derived in Chapter 5, the performance of the approximation formulas for the normal and free boundary SABR model are tested. The call prices derived from the PDE solution are used as a benchmark for the performance of the approximation formulas, since these are arbitrage free. The implied normal volatilities that result from the explicit solution are inserted in Bachelier’s model (which was presented in 1900) for normal volatilities to obtain an option price. Black’s model, Section 3-1, is based on the Bachelier’s model, and is adjusted in such a way that it works with implied lognormal volatilities instead of the implied normal volatilities which are used by Bachelier (Thomson, 2016). Black’s implied lognormal volatility model was considered better than the Bachelier model since it was assumed that negative interest rates were impossible. Since we try to price options with a negative forward rate, we have implied normal volatilities (Section 2-5). The value of a call option under Bachelier’s model is given by (Thomson, 2016) √ Vcall = B(τex)[(f − K)N (d1) + σN τexN (d1)] (6-1) and the value of the put option by √ Vput = B(τex)[(K − f)N (−d1) + σN τexN (d1)] (6-2) with f − K d1 = √ (6-3) σN τex and N the standard normal cumulative distribution function. The results of the accurateness test for both models are given in the next subsections. Next the results of two accurateness tests are given. Both tests use diﬀerent SABR parameters to test the diﬀerence between the PDE- and closed form solution.

6-2-1 Test 1

The chosen parameters for the first test equal to those in Antonov et al. (2015): α = 0.6 ∗ f 1−β, β = 0.1, ν = 0.3, ρ = −0.3,T = 3 and f = 0.05. In Figure 6-5 the probability densities of the normal and free boundary SABR models for this set of parameters are given. In Figure 6-6 the implied volatility, obtained via the approximation formulas, is shown for both the normal SABR model and the free boundary SABR model. For both models the volatility smile can clearly be seen from the figure. The implied volatility for the free boundary SABR model has a deviation around zero which is due to the sticky Brownian motion. This indicates that the free boundary approximation formula shows the same behaviour as the PDE around zero. In Figure 6-7 the accurateness of both approximation formulas is shown. These graphs show the absolute difference in option price. Because the difference in option price is automatically 6-2 Performance of the Approximation Formula 39

0.25

0.16

0.14 0.2

0.12

0.15 0.1

0.08

0.1 0.06 Probability Density Normal SABR Probability Density Free Boundary 0.04 0.05

0.02

0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Forward Forward

(a) Normal SABR (b) Free boundary SABR

Figure 6-5: Probability density function obtained by solving the PDE, test 1

0.09 0.09

0.08 0.08

0.07 0.07

0.06

0.05

Implied Normal Volatility 0.05 Implied Normal Volatility 0.04

0.04 0.03

0.03 0.02 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Strike Strike

(a) Normal SABR (b) Free boundary SABR

Figure 6-6: Volatility smile, test 1 40 Closed Form Solution Results small for small prices, the MAPE (mean absolute percentage error) is taken as criterion in stead of the RMSE (root mean square error). This makes the relative diﬀerence in option price more visible. One can see from Figure 6-7 that both approximation formulas give a larger error around zero. The approximation formula for the free boundary SABR model is a bit more accurate around zero than the normal SABR model. The MAPE of the normal SABR model is 0.1954% whereas the free boundary SABR approximation formula gives a MAPE of 0.0195%.

0.02 0.02

0.018 0.018

0.016 0.016

0.014 0.014

0.012 0.012

0.01 0.01

0.008 0.008

0.006 0.006

0.004 0.004 Absolute difference PDE and Approximation price Absolute difference PDE and Approximation price

0.002 0.002

0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Strike Strike

(a) Normal SABR (b) Free boundary SABR

Figure 6-7: Absolute diﬀerence option prices PDE and approximation, test 1

6-2-2 Test 2

The same test is executed for different SABR parameters in order to see the difference in accurateness for a different situation of both SABR models. For a small (close to zero) initial forward rate, the performances of the approximation formulas for the normal en free boundary SABR model are tested and compared. A small initial forward rate is chosen in order to test the performance of the free boundary approximation around zero (which is interesting because of the sticky Brownian motion). The chosen SABR parameters for the second test are based upon later in this thesis presented calibration results for the 2016 data. The parameters are α = 0.008, β = 0, ν = 3, ρ = 0.38,T = 3 and f = 0.005. In Figure 6-8 the probability densities of the normal and free boundary SABR models for this set of parameters are given. These probability densities are obtained via solving the PDE. Neither the normal nor the free boundary probability distributions shows abnormality. In Figure 6-9 the implied volatility, obtained via the approximation formulas, is shown for both models. For both models the volatility smile can clearly be seen from the figure. The volatility smile is skewed in a different direction compared to test 1. The implied volatility shows instead of a concave upward movement a concave downward movement for positive strikes. Although this form is uncommon, some options do show this behaviour. According to Hull (2003) this behaviour of the implied volatility is common when a large drop or rise in an asset price is expected. 6-2 Performance of the Approximation Formula 41

0.25 0.09

0.08 0.2 0.07

0.06 0.15

0.05

0.04 0.1

0.03 Probability Density Normal SABR Probability Density Free Boundary

0.02 0.05

0.01

0 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Forward Forward

(a) Normal SABR (b) Free boundary SABR

Figure 6-8: Probability density function obtained by solving the PDE, test 2

0.05 0.05

0.045 0.045

0.04 0.04

0.035 0.035

0.03 0.03

0.025 0.025

Implied Normal Volatility 0.02 Implied Normal Volatility 0.02

0.015 0.015

0.01 0.01

0.005 0.005 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Strike Strike

(a) Normal SABR (b) Free boundary SABR

Figure 6-9: Volatility smile, test 2 42 Closed Form Solution Results

In Figure 6-10 the accurateness of both approximation formulas is shown. These graphs show the absolute diﬀerence in option price. The MAPE of the normal SABR model approximation in test 2 is 0.0107%. The MAPE of the free boundary SABR model is 0.0138%. Again both approximation formulas show small deviation around a strike of zero.

10-3 10-3 4 4

3.5 3.5

3 3

2.5 2.5

2 2

1.5 1.5

1 1 Absolute difference PDE and Approximation price Absolute difference PDE and Approximation price 0.5 0.5

0 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Strike Strike

(a) Normal SABR (b) Free boundary SABR

Figure 6-10: Absolute diﬀerence option prices PDE and approximation, test 2

6-2-3 Conclusion approximation formulas

From these tests one can conclude that both approximation formulas perform quite good for accurately modelling negative forward rates. The main deviations from the true values for both models occur around a strike price of zero. This deviation should be investigated in further research. In order to investigate the performance of both models in practice, they are both calibrated to market data. The two following chapters will show how the calibration is done, and how accurate the models predict the market situations. Chapter 7

Methodology

This chapter describes the methodology behind comparing the accurateness of the approximation formulas for the two SABR models. First the data-set used in this thesis is analysed. Thereafter is stated how the parameter calibration is done. Finally the correctness of ﬁt measure is investigated.

7-1 Dataset

The market data that is used in this thesis is obtained via Bloomberg. The data is EURIBOR data on implied normal volatilities for swaptions ranging from 1Y X 1Y to 15Y x 10Y. This range is chosen so that it represents the swaption range insurers and pension funds use the most. 01-07-2005, 01-01-2008, 01-01-2016, 01-06-2016, 01-08-2016 and 01-12-2016 are chosen as testing dates. These dates are chosen so that they each represent a different market situation. 01-07-2005 is chosen to represent a stable non-negative market situation, since this date is before the financial crisis. Likewise 01-01-2008 represents a non-stable non-negative market situation, since this date is during the financial crisis. These two dates with non-negative strikes are included in the testing to see how the models perform in a stable market (01- 07-2005) and high-volatility market (01-01-2008), and to see whether the models perform differently in these situations. The 2016 data is added to test the performance of the models in a negative rate environment. Each of these four dates contains many negative forward rates. Four different negative rate dates were chosen to see whether the models perform different for different situations. When looking at the volatility surfaces for 01-07-2005 in Figure 7-1 one can see that for a maturity of 1 year, the implied volatility increases as the strike increases and the expiry decreases, as long as the expiry is not below 1 year. When the expiry is shorter then 1 year, the implied volatility decreases as the expiry decreases. For a maturity of 5 or 10 years, the implied volatility increases for higher strikes and lower expiries. 44 Methodology

0.9 1

0.8 0.9

0.7 0.8

0.6 0.7

0.5 0.6

0.4 0.5

Implied Normal volatility 0.3 Implied Normal volatility 0.4 200bps 1Mo 200bps 1Mo 50bps 9Mo 50bps 9Mo 3Yr 3Yr ATM 6Yr ATM 6Yr -50bps 9Yr -50bps 9Yr -200bps 15Yr -200bps 15Yr Moneyness 30Yr Expiry Moneyness 30Yr Expiry

(a) Maturity: 1Y (b) Maturity: 5Y

0.9

0.8

0.7

0.6

0.5

Implied Normal volatility 0.4 200bps 1Mo 50bps 9Mo 3Yr ATM 6Yr -50bps 9Yr -200bps 15Yr Moneyness 30Yr Expiry

Figure 7-1: Volatility surface for a EUR swaption on 01-07-2005 7-1 Dataset 45

0.9 1

0.8 0.9 0.8 0.7 0.7 0.6 0.6

0.5 0.5

Implied Normal volatility 0.4 Implied Normal volatility 0.4 200bps 1Mo 200bps 1Mo 50bps 9Mo 50bps 9Mo 3Yr 3Yr ATM 6Yr ATM 6Yr -50bps 9Yr -50bps 9Yr -200bps 15Yr -200bps 15Yr Moneyness 30Yr Expiry Moneyness 30Yr Expiry

(a) Maturity: 1Y (b) Maturity: 5Y

0.9

0.8

0.7

0.6

0.5

Implied Normal volatility 0.4 200bps 1Mo 50bps 9Mo 3Yr ATM 6Yr -50bps 9Yr -200bps 15Yr Moneyness 30Yr Expiry

Figure 7-2: Volatility surface for a EUR swaption on 01-01-2008

In a period of high volatility, 01-01-2008, again the expiry explains a large part of the implied volatility (Figure 7-2). The shorter the expiry, the higher the implied volatility. For a non- stable period the moneyness of the swaption becomes important, rather than a large strike price. Now also a swaption with a low strike price (compared to the at the money strike) has a high volatility.

The volatility surfaces for negative strikes, 2016, (Figure 7-3 through 7-6) show a different behaviour compared to the volatility surfaces with positive strikes. Instead of showing an increasing implied volatility for lower expiries, the implied volatility becomes smaller or even close to zero. For expiries larger than 10 years, the implied volatility shows regular behaviour. By taking a closer look at the different strikes for 01-08-2016, we further investigate this behaviour. The strikes for the same combinations of moneyness, expiry and maturity er given in Figure 7-7. This figure shows that the strikes show normal behaviour: The strikes are increasing as the moneyness and expiry are increasing. The abnormal form of the volatility surface can therefore not be caused by the strike prices for these EURIBOR swaptions. There can be concluded that the implied volatility shows odd market behaviour for negative strikes. 46 Methodology

1 1.2

0.8 1

0.6 0.8

0.4 0.6

0.2 0.4

Implied Normal volatility 0 Implied Normal volatility 0.2 200bps 1Mo 200bps 1Mo 50bps 9Mo 50bps 9Mo 3Yr 3Yr ATM 6Yr ATM 6Yr -50bps 9Yr -50bps 9Yr -200bps 15Yr -200bps 15Yr Moneyness 30Yr Expiry Moneyness 30Yr Expiry

(a) Maturity: 1Y (b) Maturity: 5Y

1.4

1.2

0.8

0.6

Implied Normal volatility 0.4 200bps 1Mo 50bps 9Mo 3Yr ATM 6Yr -50bps 9Yr -200bps 15Yr Moneyness 30Yr Expiry

Figure 7-3: Volatility surface for a EUR swaption on 01-01-2016 7-1 Dataset 47

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

Implied Normal volatility 0 Implied Normal volatility 0 200bps 1Mo 200bps 1Mo 50bps 9Mo 50bps 9Mo 3Yr 3Yr ATM 6Yr ATM 6Yr -50bps 9Yr -50bps 9Yr -200bps 15Yr -200bps 15Yr Moneyness 30Yr Expiry Moneyness 30Yr Expiry

(a) Maturity: 1Y (b) Maturity: 5Y

0.8

0.6

0.4

Implied Normal volatility 0.2 200bps 1Mo 50bps 9Mo 3Yr ATM 6Yr -50bps 9Yr -200bps 15Yr Moneyness 30Yr Expiry

Figure 7-4: Volatility surface for a EUR swaption on 01-06-2016 48 Methodology

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 Implied Normal volatility

Implied Normal volatility 0 0 30Yr 200bps 1Mo 15Yr 50bps 9Mo 9Yr 200bps 3Yr ATM 6Yr 50bps 6Yr 3Yr ATM -50bps 9Yr 15Yr 9Mo -50bps -200bps 30Yr 1Mo -200bps Moneyness Expiry Expiry Moneyness

(a) Maturity: 1Y (b) Maturity: 1Y

1.2 1.2

1 1 0.8 0.8 0.6 0.6 0.4

0.2 0.4

Figure 7-5: Volatility surface for a EUR swaption on 01-08-2016 7-1 Dataset 49

1 1.2

0.8 1 0.8 0.6 0.6 0.4 0.4

0.2 0.2

(a) Maturity: 1Y (b) Maturity: 5Y

1.4

1.2

0.8

0.6

0.4

Implied Normal volatility 0.2 200bps 1Mo 50bps 9Mo 3Yr ATM 6Yr -50bps 9Yr -200bps 15Yr Moneyness 30Yr Expiry

Figure 7-6: Volatility surface for a EUR swaption on 01-12-2016

0.04 0.04 0.04

0.03 0.03 0.03

0.02 0.02 0.02 0.01 0.01 0.01 0 0 Strikes Strikes Strikes 0 -0.01 -0.01

-0.02 -0.02 -0.01

-0.03 -0.03 -0.02 30Yr 30Yr 30Yr 15Yr 15Yr 15Yr 9Yr 200bps 9Yr 200bps 9Yr 200bps 6Yr 50bps 6Yr 50bps 6Yr 50bps 3Yr ATM 3Yr ATM 3Yr ATM 9Mo -50bps 9Mo -50bps 9Mo -50bps -200bps -200bps -200bps Expiry 1Mo Moneyness Expiry 1Mo Moneyness Expiry 1Mo Moneyness

(a) Maturity: 1Y (b) Maturity: 5Y (c) Maturity: 10Y

Figure 7-7: Strike surface for a EUR swaption on 01-08-2016 50 Methodology

7-2 Calibration

In order to obtain implied volatilities from the SABR models, four parameters are needed. These parameters are obtained by calibrating the model to the market data. Calibration can be seen as a reversed process. Calibration is a technique that uses known observations of the dependent variables to predict corresponding parameters. In the SABR model case, the unknown parameters are α, β, ρ, and ν, and the observations are the implied normal volatilities given by Bloomberg. The parameter calibration of the model is what makes using the SABR model complicated. There are many diﬀerent ways to do this, but only one leads to a correct result. Unfortunately the inventors of the original SABR model (P. S. Hagan et al., 2002) do not give a clear explanation of how they calibrated the parameters. They give a short explanation however, which is used in this thesis to recover the used calibration method. In the remaining of this section is explained what Hagan et al. (2002) do to obtain the SABR parameters, and what method is used in this thesis to obtain them.

7-2-1 Hagan calibration

First Hagan et al. (2002) choose the β parameter, which does not have a large influence on the quality of the model fit in the original SABR model. They often set β to 0.5. The β parameter in the normal SABR model is set to zero, which corresponds to C(F ) = 1, so that it is able to handle negative interest rates correctly. The β parameter in the free boundary SABR model however has influence on the quality of the market fit (see Chapter6). Therefore the choice of β is important in this model, and cannot be set to a specific number in all situations. The β parameter thus needs to be calibrated for the free boundary model, which is not the case in the original SABR model. Thereafter Hagan et al. (2002) state they obtain the α parameter by inverting the implied volatility of ATM options. They now state they can easily fit ρ and ν with σatm and β given. How this is precisely done is not explained. Other papers do not mention this part either.

7-2-2 Calibration in this thesis

Based on the steps described in Hagan et al. (2002), we calibrated the α, ρ, and ν for the normal SABR model and the α, β, ρ, and ν for the free boundary SABR model. The calibration of both the SABR models is done by minimizing the difference between the observed and predicted implied volatility, fitted by the SABR model for each corresponding strike. It uses the ATM volatility to derive α from the obtained β, ρ and ν, so only β, ρ and ν need to be estimated. Using Expression (3-8) with the corresponding coefficient C(F ) for the model you want to calibrate, α can be calculated given the strike that corresponds to the ATM volatility and the parameters β, ρ and ν. The objective function is

2 M min Σi σi − σN (fi,Ki; α(ρ, ν, σAT M ), β, ρ, ν) , (7-1) βρν 7-3 Measurement of ﬁt 51

M where σi is the implied market volatility and σN the SABR implied normal volatility given the parameters, strike K and ATM forward price f. The constraints of this minimization are equal to the constraints in the SABR model

α > 0, β ≥ 0, ρ ∈ [−1, 1], ν ≥ 0.

The ATM equations that are used to solve α for the normal SABR and free boundary SABR models are derived earlier in this thesis ! (2 − 3ρ2) σN = α 1 + ν2τ , (7-2) AT M 24 ex

" 2 2 # FB β −β(2 − β)α ρναβ · sign(fav) (2 − 3ρ ) 2 σAT M = |fav| α 1 + 2−2β τex + 1−β + ν τex . (7-3) 24|fav| 4|fav| 24

So the optimization procedure becomes:

1. Assign initial values to β, ρ and ν;

2. Solve α using Equations (7-2) and (7-3);

3. Insert this α combined with β, ρ and ν to calculate σN for every strike K; 4. Minimize the objective function to obtain a new α, β, ρ and ν;

5. Repeat step 3 through 4 and compare the result of the new objective function with the old result.;

6. Perform the next iteration until the algorithm converges to a certain tolerance level (in this thesis set to 1.0e-06).

7-3 Measurement of ﬁt

A model is said to perform well if the valuation is arbitrage free. In other words, the model matches the quoted market volatilities and the interpolation between those market volatilities is smooth. In order to investigate the quality of the models the RMSE measure is used. Therefore the results of the models need to be compared to true market values. These true values are quoted market volatilities for swaptions with a certain strike price. The closer the ﬁt of the volatility for a certain strike, the better the model. The deviation from the true implied market volatility is hereby squared, in order to penalize high deviations.

Chapter 8

Market Data Results

This chapter shows the results of calibrating the two SABR models to market data. First the SABR parameters are calibrated to the market data, followed by a test on how well these models ﬁt to these market data.

8-1 Parameter Calibration

The SABR parameters are calibrated using the method explained in Section 7-2. The calibration of the free boundary model for negative rates resulted initially in dissatisfying results: the fit, as will be further explained in the next section, for the normal SABR model was better than the free boundary SABR model. This is a strange result since the free boundary SABR model is an extension of the normal SABR model. The model should always be able to set β equal to zero, and obtain equal or better results than the normal SABR model. An explanation for this behaviour is that the optimization routine found a sub-optimum. Whereas the start parameters for the normal SABR model calibration had no effect on the final calibration results, they do have a large effect on the free boundary model calibration. In order to overcome this problem, a grid search was performed to find optimal start parameters for the calibration procedure. Over 10000 different start parameters were tested for calibration of the free boundary model to the 01-08-2016 data. This performed grid search did not result in a better calibration result. Therefore the calibration results of the normal SABR model where chosen as a start for the free boundary SABR model calibration. As a consequence, the calibration results look a lot like each other for the normal- and free boundary SABR model. The parameter calibration results are stated in AppendixA. When looking at the results, the first thing that stands out is that for 01-07-2005 and 01-01- 2008 the calibration results are almost equal for the normal- and the free boundary SABR model. For 2016, the calibration results are often equal, but not in every situation (e.g. 01-12-2016, maturity 1Y). As the maturity increases, α increases, except for 01-12-2016. This can be explained by looking at the volatility surfaces for 01-12-2016. The volatility becomes smaller when increasing the 54 Market Data Results maturity from 5 years to 10 years for this date. This is in contrary to the effect we see in the other volatility surfaces, where the implied volatility increases as the maturity increases. No other clear effect from the results can be observed thusfar.

8-2 Market Fit

In this section the market fit for the two different models are compared for the different dates. First the non-negative rate market situations are examined, followed by the negative rate market situations.

8-2-1 Normal market conditions

Figure 8-1 shows the implied normal volatility smiles generated by the calibrated SABR models for the 4 different expiries (1Y, 5Y, 10Y, 15Y) and 3 different maturities (1Y, 5Y, 10Y). All graphs look nearly the same. The free boundary SABR and normal SABR both show an equally good fit. The smile curvature goes almost perfectly though the market values in most cases. For the 1Y x 1Y swaption both models do not match the market value for a strike of −100bps. Recall the volatility surface in Figure 7-1. For a 1Y x 1Y swaption, the implied volatility suddenly drops. This probably is the reason why both models are less accurate in fitting the market value in this case. When looking at RMSE in Table 8-2, the RMSE is almost equal for both models (rounded they are equal). This is a logical result when looking at the parameters resulting from the calibrations, which where also equal. The average RMSE for both models is 0.00420. The largest RMSE for both models is for the 1Y x 1Y swaption. This is in line with the fit shown in Figure 8-1. For the 1Y x 1Y swaption the RMSE for both models is 0.0147, which is both still small. Both models have the best fit for the 5Y x 10Y swaption (0.00184). Overall the RMSE is low for both models in all situations. Therefore there can be concluded that both models are able to handle non-stable (positive) interest rate volatility situations in a correct way.

Table 8-1: RMSE compared for normal SABR and free boundary SABR for 01-07-2005.

Expiry\Maturity 1Y 5Y 10Y 1Y N 0.014662 0.008052 0.005155 FB 0.014662 0.008052 0.005155 5Y N 0.002863 0.002159 0.001835 FB 0.002863 0.002159 0.001835 10Y N 0.002383 0.002193 0.002135 FB 0.002383 0.002193 0.002135 15Y N 0.003000 0.002909 0.002981 FB 0.003000 0.002909 0.002981 8-2 Market Fit 55

1Y x 1Y 1Y x 5Y 1Y x 10Y 0.9 0.9 0.85

N SABR 0.8 FB SABR 0.8 Market Value 0.8 0.75

0.7 0.7 0.7

Implied Volatility Implied Volatility Implied Volatility 0.65 0.6

0.6 0.6 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05 0.06 Strike Strike Strike 5Y x 1Y 5Y x 5Y 5Y x 10Y 0.85 0.8 0.8

0.8 0.75 0.75

0.75 0.7 0.7

0.7 0.65 0.65

Implied Volatility 0.65 Implied Volatility 0.6 Implied Volatility 0.6

0.6 0.55 0.55 0.01 0.02 0.03 0.04 0.05 0.06 0.01 0.02 0.03 0.04 0.05 0.06 0.02 0.03 0.04 0.05 0.06 0.07 Strike Strike Strike 10Y x 1Y 10Y x 5Y 10Y x 10Y 0.8 0.75 0.75

0.75 0.7 0.7

0.7 0.65 0.65 0.65

0.6 0.6 Implied Volatility 0.6 Implied Volatility Implied Volatility

0.55 0.55 0.55 0.02 0.03 0.04 0.05 0.06 0.07 0.02 0.03 0.04 0.05 0.06 0.07 0.02 0.03 0.04 0.05 0.06 0.07 Strike Strike Strike 15Y x 1Y 15Y x 5Y 15Y x 10Y 0.75 0.7 0.7

0.7 0.65 0.65

0.65 0.6 0.6

0.6 0.55 0.55 Implied Volatility Implied Volatility Implied Volatility

0.55 0.5 0.5 0.02 0.03 0.04 0.05 0.06 0.07 0.02 0.03 0.04 0.05 0.06 0.07 0.02 0.03 0.04 0.05 0.06 0.07 Strike Strike Strike

Figure 8-1: Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-07-2005 56 Market Data Results

1Y x 1Y 1Y x 5Y 1Y x 10Y

0.74 N SABR 0.72 0.66 FB SABR Market Value 0.65 0.72 0.7 0.64

0.63 0.7 0.68 Implied Volatility Implied Volatility Implied Volatility 0.62

0.68 0.66 0.61 0.02 0.03 0.04 0.05 0.06 0.07 0.02 0.03 0.04 0.05 0.06 0.07 0.02 0.03 0.04 0.05 0.06 0.07 Strike Strike Strike 5Y x 1Y 5Y x 5Y 5Y x 10Y 0.72 0.7 0.66

0.7 0.68 0.64

0.68 0.66 0.62 0.66 0.64

0.6 Implied Volatility 0.64 Implied Volatility 0.62 Implied Volatility

0.62 0.6 0.58 0.02 0.03 0.04 0.05 0.06 0.07 0.02 0.03 0.04 0.05 0.06 0.07 0.03 0.04 0.05 0.06 0.07 0.08 Strike Strike Strike 10Y x 1Y 10Y x 5Y 10Y x 10Y 0.64 0.64

0.62

0.62 0.62

0.6

0.6 0.6

0.58 Implied Volatility Implied Volatility Implied Volatility 0.58 0.58

0.56 0.03 0.04 0.05 0.06 0.07 0.08 0.03 0.04 0.05 0.06 0.07 0.08 0.03 0.04 0.05 0.06 0.07 0.08 Strike Strike Strike 15Y x 1Y 15Y x 5Y 15Y x 10Y 0.64 0.62 0.62

0.62 0.6 0.6

0.6 0.58 0.58

0.58 0.56 0.56

Implied Volatility 0.56 Implied Volatility 0.54 Implied Volatility 0.54

0.54 0.52 0.52 0.03 0.04 0.05 0.06 0.07 0.08 0.03 0.04 0.05 0.06 0.07 0.08 0.03 0.04 0.05 0.06 0.07 0.08 Strike Strike Strike

Figure 8-2: Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-01-2008 8-2 Market Fit 57

8-2-2 Excited market conditions

Figure 8-2 shows the implied normal volatility smiles generated by the calibrated SABR models for the 4 different expiries (1Y, 5Y, 10Y, 15Y) and 3 different maturities (1Y, 5Y, 10Y). Again all graphs look nearly the same. The free boundary SABR and normal SABR both show a very good fit. The smile curvature goes almost perfectly though the market values in most cases. There is a very clear volatility smile in every graph visible. This effect can also be seen in the volatility surface plot in Figure 7-2. The implied volatility is here clearly larger for swaptions that are far out of the money (swaptions with a large deviation of −200bps or 200bps difference with the at the money strike price). The effect seems larger for short expiries, when looking at Figure 7-2. This assumption is confirmed when taking a closer look at Figure 8-2. The volatility smile is clearly larger for swaptions with an expiry of one year. Another distinctive feature are the market volatilities that do not lie on a straight line. This is due to noise that comes from the unstable market as a result from the crisis. Both models do not try to capture this up-and-down going market volatility, but model the implied volatility in a perfect curve. When looking at the RMSE in Table 8-2, again the RMSE are almost equal (rounded they are equal) for both models in all situations. The average RMSE is 0.00191 for the normal SABR model, and 0.00190 for the free boundary SABR model. Overall the RMSE is very low for both models in all situations. Therefore there can be concluded that both models are able to handle non-stable (positive) interest rate volatility situations in a correct way.

Table 8-2: RMSE compared for normal SABR and free boundary SABR for 01-01-2008.

Expiry\Maturity 1Y 5Y 10Y 1Y N 0.002257 0.002023 0.001815 FB 0.002257 0.002023 0.001815 5Y N 0.001277 0.001049 0.000925 FB 0.001277 0.001049 0.000925 10Y N 0.002296 0.002301 0.002302 FB 0.002283 0.002277 0.002286 15Y N 0.002290 0.002197 0.002153 FB 0.002290 0.002197 0.002153

8-2-3 Negative rate conditions

Figures 8-3 through 8-6 show the implied normal volatility smiles generated by the calibrated SABR models for the four different expiries (1Y, 5Y, 10Y, 15Y), three different maturities (1Y, 5Y, 10Y) and the four different dates (01-01-2016, 01-06-2016, 01-08-2016, 01-12-2016). These graphs show an almost identical fit for the two models. This is natural result of setting the start parameters for the free boundary optimization equal to the calibration results of the normal SABR model. From these graphs one can conclude that the more the implied 58 Market Data Results

1Y x 1Y 1Y x 5Y 1Y x 10Y 0.6

N SABR 1 1.2 0.5 FB SABR Market Value 0.4 0.8 1

0.3 0.6 0.8

Implied Volatility 0.2 Implied Volatility Implied Volatility

0.1 0.4 0.6 -0.02 -0.01 0 0.01 0.02 -0.02 -0.01 0 0.01 0.02 0.03 -0.01 0 0.01 0.02 0.03 0.04 Strike Strike Strike

5Y x 1Y 5Y x 5Y 5Y x 10Y 0.9 0.9 1

0.8 0.8 0.9

0.7 0.7 0.8

0.6 0.6 0.7

Implied Volatility 0.5 Implied Volatility 0.5 Implied Volatility 0.6

0.4 0.4 0.5 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 Strike Strike Strike

10Y x 1Y 10Y x 5Y 10Y x 10Y 0.78 0.84 0.82

0.76 0.82 0.8

0.74 0.8 0.78

0.72 0.78 0.76

0.7 0.76 0.74 Implied Volatility Implied Volatility Implied Volatility 0.68 0.74 0.72

0.66 0.72 0.7 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 Strike Strike Strike

15Y x 1Y 15Y x 5Y 15Y x 10Y 0.7 0.8 0.8

0.75 0.65 0.75

0.7 0.6 0.7 0.65

0.55 0.65 Implied Volatility Implied Volatility Implied Volatility 0.6

0.5 0.6 0.55 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 Strike Strike Strike

Figure 8-3: Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-01-2016 8-2 Market Fit 59

1Y x 1Y 1Y x 5Y 1Y x 10Y 0.1512 0.252 0.605 0.151 0.251

0.1508 N SABR 0.25 FB SABR 0.6 Market Value 0.1506 0.249

Implied Volatility 0.1504 Implied Volatility 0.248 Implied Volatility 0.595

0.1502 0.247 -0.03 -0.02 -0.01 0 0.01 0.02 -0.02 -0.01 0 0.01 0.02 0.03 -0.02 -0.01 0 0.01 0.02 0.03 Strike Strike Strike

5Y x 1Y 5Y x 5Y 5Y x 10Y 0.5 0.9 0.9

0.45 0.8 0.8 0.4 0.7 0.7 0.35 0.6 0.6 0.3 Implied Volatility Implied Volatility Implied Volatility 0.25 0.5 0.5

0.2 0.4 0.4 -0.02 -0.01 0 0.01 0.02 0.03 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 Strike Strike Strike

10Y x 1Y 10Y x 5Y 10Y x 10Y 0.8 0.8 0.8

0.75 0.75 0.75

0.7 0.7 0.7

0.65 0.65 0.65 Implied Volatility Implied Volatility Implied Volatility

0.6 0.6 0.6 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 Strike Strike Strike 15Y x 1Y 15Y x 5Y 15Y x 10Y 0.75 0.75 0.65

0.7 0.7 0.6 0.65

0.65 0.6 Implied Volatility 0.55 Implied Volatility Implied Volatility

0.6 0.55 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 Strike Strike Strike

Figure 8-4: Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-06-2016 60 Market Data Results

1Y x 1Y 1Y x 5Y 1Y x 10Y 0.14 0.15 0.65

N SABR 0.13 0.14 0.6 FB SABR Market Value 0.13 0.12 0.55 0.12 0.11 0.5 0.11 Implied Volatility Implied Volatility Implied Volatility 0.1 0.1 0.45

0.09 0.09 0.4 -0.03 -0.02 -0.01 0 0.01 0.02 -0.03 -0.02 -0.01 0 0.01 0.02 -0.02 -0.01 0 0.01 0.02 0.03 Strike Strike Strike

5Y x 1Y 5Y x 5Y 5Y x 10Y 1.2 1.1 0.4 1 1 0.3 0.9

0.8 0.8 0.2

Implied Volatility Implied Volatility Implied Volatility 0.7 0.6 0.1 0.6 -0.02 -0.01 0 0.01 0.02 0.03 -0.02 -0.01 0 0.01 0.02 0.03 -0.02 -0.01 0 0.01 0.02 0.03 Strike Strike Strike

10Y x 1Y 10Y x 5Y 10Y x 10Y 0.8 0.75

0.75 0.75

0.7 0.7 0.7

0.65 Implied Volatility 0.65 Implied Volatility Implied Volatility 0.65 0.6 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 Strike Strike Strike 15Y x 1Y 15Y x 5Y 15Y x 10Y 0.8 0.9 0.8

0.7 0.8 0.7

0.6 0.7 0.6

0.5 0.6 0.5

Implied Volatility 0.4 Implied Volatility 0.5 Implied Volatility 0.4

0.3 0.4 0.3 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 -0.02 -0.01 0 0.01 0.02 0.03 Strike Strike Strike

Figure 8-5: Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-08-2016 8-2 Market Fit 61

1Y x 1Y 1Y x 5Y 1Y x 10Y 0.35 0.7 1.1

N SABR 0.3 0.6 1 FB SABR Market Value 0.25 0.5 0.9

0.2 0.4 0.8

Implied Volatility 0.15 Implied Volatility 0.3 Implied Volatility 0.7

0.1 0.2 0.6 -0.03 -0.02 -0.01 0 0.01 0.02 -0.02 -0.01 0 0.01 0.02 0.03 -0.02 -0.01 0 0.01 0.02 0.03 Strike Strike Strike

5Y x 1Y 5Y x 5Y 5Y x 10Y 0.7 0.9 1.2 0.6 0.8

0.5 0.7 1

0.4 0.6 0.8

Implied Volatility 0.3 Implied Volatility 0.5 Implied Volatility

0.2 0.4 0.6 -0.02 -0.01 0 0.01 0.02 0.03 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 Strike Strike Strike

10Y x 1Y 10Y x 5Y 10Y x 10Y 1.1 1.1 1.3

1 1 1.2

0.9 0.9 1.1

0.8 0.8 1

0.7 0.7 0.9 Implied Volatility Implied Volatility Implied Volatility 0.6 0.6 0.8

0.5 0.5 0.7 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 Strike Strike Strike 15Y x 1Y 15Y x 5Y 15Y x 10Y 0.9 1 1.1

0.8 0.9 1 0.8 0.9 0.7 0.7 0.8 0.6 0.6 0.7 Implied Volatility Implied Volatility Implied Volatility 0.5 0.5 0.6

0.4 0.4 0.5 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 -0.01 0 0.01 0.02 0.03 0.04 Strike Strike Strike

Figure 8-6: Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-12-2016 62 Market Data Results volatilities lie on a straight line, the better the fit. This is especially true when looking at the results of 01-06-2016 in Figure 8-4. For an maturity of one year, the implied normal volatility shows a steep convex shape. This shape is poorly captured by the fit resulting from the calibration. In Table 8-3 through 8-6 the RMSE results for 2016 for both models are given. The RMSE are less equal than for the positive rate situations. The average RMSE for the normal SABR model are 0.011819, 0.007812, 0.011209 and 0.016295. For the free boundary model these are 0.010854, 0.005253, 0.010181 and 0.012991 respectively. For each negative rate situation, the free boundary model outperforms the normal SABR model. Overall the RMSE is very low for both models in all situations. Therefore there can be concluded that both models are able to handle negative interest rate volatility situations in a correct way. One has to take in mind that the results of the free boundary calibration were not available without first calibrating the normal SABR model and then use these results as start parameters for the free boundary optimization. Although the free boundary SABR model leads to a better fit, it takes a lot of more work to get this fit: First the normal SABR model has to be calibrated, then these results need to be set as start parameter for each different combination of expiry and maturity. This takes a lot of work compared to looping over each different expiry - maturity combination with the same set of start parameters for the normal SABR model. One has to consider whether this amount of extra work is worth the small improve in the fit.

Table 8-3: RMSE compared for normal SABR and free boundary SABR for 01-01-2016.

Expiry\Maturity 1Y 5Y 10Y 1Y N 0.027215 0.032812 0.027007 FB 0.026164 0.032037 0.026459 5Y N 0.008170 0.012218 0.008944 FB 0.007411 0.010336 0.007422 10Y N 0.003392 0.003468 0.003512 FB 0.002725 0.002769 0.002815 15Y N 0.004667 0.004961 0.005466 FB 0.003734 0.003976 0.004401

Table 8-4: RMSE compared for normal SABR and free boundary SABR for 01-06-2016.

Expiry\Maturity 1Y 5Y 10Y 1Y N 0.000582 0.003636 0.009029 FB 0.000205 0.001318 0.003898 5Y N 0.006405 0.024629 0.020918 FB 0.005743 0.019400 0.015150 10Y N 0.002617 0.002999 0.003193 FB 0.001483 0.001644 0.003521 15Y N 0.005948 0.006879 0.006911 FB 0.003156 0.003773 0.003739 8-2 Market Fit 63

Table 8-5: RMSE compared for normal SABR and free boundary SABR for 01-08-2016.

Expiry \Maturity 1Y 5Y 10Y 1Y N 0.002271 0.003021 0.012122 FB 0.001764 0.002482 0.010001 5Y N 0.009315 0.028996 0.018796 FB 0.009178 0.027873 0.017828 10Y N 0.001291 0.001177 0.001326 FB 0.001253 0.001149 0.001294 15Y N 0.017786 0.021440 0.016969 FB 0.015643 0.018716 0.014991

Table 8-6: RMSE compared for normal SABR and free boundary SABR for 01-12-2016.

Expiry\Maturity 1Y 5Y 10Y 1Y N 0.006210 0.009306 0.010465 FB 0.006210 0.009271 0.010439 5Y N 0.014978 0.018866 0.022877 FB 0.012329 0.014365 0.017003 10Y N 0.016954 0.017466 0.021777 FB 0.012525 0.014053 0.016397 15Y N 0.017765 0.018061 0.020808 FB 0.013550 0.013697 0.016051

Chapter 9

Conclusion

The SABR model by (P. S. Hagan et al., 2002) has become a widespread used interest rate model. Its approximation formula, first derived in 2002 and later adjusted in 2014 is very accurate and has a closed form solution. This SABR approximation formula is known for its ability to price options, capture volatility curves and inter- and extrapolate market quotes in a very easy and intuitive way. However, the SABR model is not able to work with negative interest rates, since a constraint of the classical SABR model is F > 0. Therefore this thesis investigated which extension of the SABR model is the most suitable for modelling rates in a negative interest rate environment. The two most promising models were examined in particular: the normal SABR model with C(F ) = 1 and the free boundary SABR model with C(F ) = |F |β. Both models are an altering of Hagan’s SABR model 2002. These models were tested before in theoretical situations, but not yet calibrated to market data. In this thesis two new approximation formulas were derived: one for the normal SABR model and one for the free boundary SABR model. In order to test the accurateness of these formulas, arbitrage-free PDE option prices for both models were calculated. These prices were then compared with the option prices derived from plugging in the implied normal volatilities derived from the approximation formulas in Bachelier’s option pricing formula. For two sets of parameters, both models were used to compute option prices for a large range of strikes. The PDE solution was hereby used as benchmark for the approximation formulas. Both closed form solutions showed very good (accurate) results. In the first test the MAPE for the normal SABR model was 0.1954%, compared to a MAPE of 0.0195% for the free boundary SABR model. For the second test, the normal SABR model MAPE was 0.0107%, and the free boundary SABR MAPE only 0.0138%. For all tested situations, the deviation from the true value arose around a strike of zero. Why this is the case should be investigated in further research. We can conclude that both approximation formulas give a good approximation. When plotting different PDF for different parameter combinations, the free boundary model showed a disturbing result though: for an initial forward close to zero, and a small beta, the PDF, derived using the PDE solution, showed stickiness around zero. The free boundary model has a sticky Brownian motion, and can therefore not well describe market movements in every situation. 66 Conclusion

In order to test the performance of both formulas in the market, both formulas were then calibrated to six different data sets. In order to do so, normal ATM volatility formulas were derived for the normal SABR model and the free boundary SABR model. The data was chosen so that it each describes a different market situation: a stable market at 01-07-2005, a non-stable market at 01-01-2008 and a negative rate market at 01-01-2016, 01-06-2016, 01-08-2016 and 01-12-2016. Both models were equally accurate in modeling market situations as long as the interest rates were positive. For the stable market at 01-07-2005, both models showed an average RMSE of 0.0147 and for the non-stable market at 01-01-2008 an average RMSE of 0.00191 for the normal SABR model and an average RMSE of 0.00190 for the free boundary SABR model. For the negative rate dates, 01-01-2016, 01-06-2016, 01-08-2016 and 01-12-2016, the average RMSE for the normal SABR model were 0.011819, 0.007812, 0.011209 and 0.016295, and for the free boundary SABR model these were 0.010854, 0.005253, 0.010181 and 0.012991 respectively. Thus the free boundary SABR model were better able to capture the volatility smiles in the market. One has to take in mind that these results of the free boundary calibration were not available without first calibrating the normal SABR model and then use these results as start parameters for the free boundary optimization. When using different start parameters for the optimization route, very inaccurate results for the free boundary SABR model were found around a strike price of zero, because of ending in a sub-optimum. The inaccurate results of the free boundary model are in line with the sticky results we saw earlier when plotting the PDF. It can therefore be concluded that the free boundary model is not able to correctly model forward rates that are close to zero. Therefore this model is not a good model for financial practitioners, who need this model in all kind of situations, compared to the normal SABR model which works in every situation. For further research, the SABR model can be further investigated in a negative interest rate environment. Examples of further research question are:

• Can the normal SABR model be adjusted so that it becomes completely arbitrage free?

• Can the free boundary SABR model be adjusted in such a way that it has no sticky Brownian motion?

• Why are the approximation formulas less accurate around a strike price of zero? Appendix A

Parameter Calibration Results

This appendix contains the parameters that result from the calibrations. They are sorted by date, maturity, and type of SABR model. The forward rate at time zero, F (0) = f is equal for each model at the same date and same maturity. When the expiration is 5 years or longer, T is often not an integer, because of leap years. 68 Parameter Calibration Results

Table A-1: Swaption quotes for EUR, 01-07-2005, maturity: 1Y Normal SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f 0.0033 0.0158 0.0231 0.0244 α 0.1097 0.0820 0.5976 0.5600 β 0 0 0 0 ρ -0.4540 -0.0304 -0.2283 -0.2246 ν 8.8910 3.9032 3.0171 2.7039 T 1.0000 5.0027 10.0055 15.0110

Table A-2: Swaption quotes for EUR, 01-07-2005, maturity: 5Y Normal SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f 0.0096 0.0193 0.0237 0.0244 α 0.1169 0.0871 0.0695 0.0564 β 0 0 0 0 ρ -0.2800 0.0105 -0.0592 -0.1523 ν 7.7469 3.7190 2.9571 2.6713 T 1.0000 5.0027 10.0055 15.0110

Table A-3: Swaption quotes for EUR, 01-07-2005, maturity: 10Y Normal SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0147 0.0213 0.0240 0.0238 α 0.1261 0.0901 0.0698 0.0557 β 0 0 0 0 ρ -0.2098 0.0305 -0.0552 -0.1587 ν 7.0749 3.6405 2.9396 2.6779 T 1.0000 5.0027 10.0055 15.0110 69

Table A-4: Swaption quotes for EUR, 01-07-2005, maturity: 1Y Free Boundary SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f 0.0033 0.0158 0.0231 0.0244 α 0.1097 0.0820 0.0699 0.0570 β 1.40E-15 2.11E-14 1.72E-14 1.12E-13 ρ -0.4540 -0.0304 -0.0638 -0.1519 ν 8.8910 3.9032 3.0171 2.7022 T 1.0000 5.0027 10.0055 15.0110

Table A-5: Swaption quotes for EUR, 01-07-2005, maturity: 5Y Free Boundary SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f 0.0096 0.0193 0.0237 0.0244 α 0.1169 0.0871 0.0695 0.0564 β 7.72E-15 9.95E-15 4.54E-14 3.00E-13 ρ -0.2800 0.0105 -0.0592 -0.1523 ν 7.7469 3.7190 2.9571 2.6713 T 1.0000 5.0027 10.0055 15.0110

Table A-6: Swaption quotes for EUR, 01-07-2005, maturity: 10Y Free Boundary SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0147 0.0213 0.0240 0.0238 α 0.1261 0.0901 0.0698 0.0557 β 3.94E-14 4.70E-14 3.06E-14 3.11E-14 ρ -0.2098 0.0305 -0.0552 -0.1587 ν 7.0749 3.6405 2.9396 2.6779 T 1.0000 5.0027 10.0055 15.0110 70 Parameter Calibration Results

Table A-7: Swaption quotes for EUR, 01-01-2008, maturity: 1Y Normal SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f 0.0238 0.0273 0.0318 0.0327 α 0.1944 0.1055 0.0865 0.0697 β 0 0 0 0 ρ -0.4313 -0.2583 -0.3369 -0.3671 ν 6.8271 3.6628 2.9442 2.7078 T 1.0027 5.0055 10.0082 15.0110

Table A-8: Swaption quotes for EUR, 01-01-2008, maturity: 5Y Normal SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f 0.0256 0.0293 0.0326 0.0317 α 0.1952 0.1064 0.0867 0.0685 β 0 0 0 0 ρ -0.4156 -0.2517 -0.3342 -0.3741 ν 6.5677 3.5786 2.9256 2.7053 T 1.0027 5.0055 10.0082 15.0110

Table A-9: Swaption quotes for EUR, 01-01-2008, maturity: 10Y Normal SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0276 0.0307 0.0322 0.0306 α 0.1925 0.1067 0.0863 0.0676 β 0 0 0 0 ρ -0.3961 -0.2464 -0.3352 -0.3827 ν 6.1498 3.5043 2.9234 2.7261 T 1.0027 5.0055 10.0082 15.0110 71

Table A-10: Swaption quotes for EUR, 01-01-2008, maturity: 1Y Free Boundary SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f 0.0238 0.0273 0.0318 0.0327 α 0.1944 0.1055 0.0870 0.0697 β 1.08E-14 1.14E-13 3.24E-05 7.50E-15 ρ -0.4313 -0.2583 -0.3283 -0.3671 ν 6.8271 3.6628 2.9198 2.7078 T 1.0027 5.0055 10.0082 15.0110

Table A-11: Swaption quotes for EUR, 01-01-2008, maturity: 5Y Free Boundary SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f 0.0256 0.0293 0.0326 0.0317 α 0.1952 0.1064 0.0874 0.0685 β 1.86E-15 2.16E-14 4.53E-05 7.08E-15 ρ -0.4156 -0.2517 -0.3223 -0.3741 ν 6.5677 3.5786 2.8924 2.7053 T 1.0027 5.0055 10.0082 15.0110

Table A-12: Swaption quotes for EUR, 01-01-2008, maturity: 10Y Free Boundary SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0276 0.0307 0.0322 0.0306 α 0.1925 0.1067 0.0869 0.0676 β 2.51E-15 2.32E-14 3.69E-05 9.50E-14 ρ -0.3961 -0.2464 -0.3255 -0.3827 ν 6.1498 3.5043 2.8959 2.7261 T 1.0027 5.0055 10.0082 15.0110 72 Parameter Calibration Results

Table A-13: Swaption quotes for EUR, 01-01-2016, maturity: 1Y Normal SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f -0.0001 0.0123 0.0223 0.0208 α 0.0364 0.9629 1.0971 0.9262 β 0 0 0 0 ρ 0.5066 0.8167 0.8188 0.8172 ν 9.3654 38.4528 8.3231 12.2860 T 1.0027 5.0055 10.0082 15.0110

Table A-14: Swaption quotes for EUR, 01-01-2016, maturity: 5Y Normal SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f 0.0058 0.0170 0.0231 0.0218 α 0.0485 1.0511 1.1732 1.0431 β 0 0 0 0 ρ 0.3811 0.8167 0.8187 0.8171 ν 11.2858 36.1714 8.5906 13.2892 T 1.0027 5.0055 10.0082 15.0110

Table A-15: Swaption quotes for EUR, 01-01-2016, maturity: 10Y Normal SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0122 0.0199 0.0225 0.0200 α 0.0660 1.1124 1.1507 1.0182 β 0 0 0 0 ρ 0.3388 0.8168 0.8187 0.8170 ν 11.1390 32.5204 8.6302 14.0389 T 1.0027 5.0055 10.0082 15.0110 73

Table A-16: Swaption quotes for EUR, 01-01-2016, maturity: 1Y Free Boundary SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f -0.0001 0.0123 0.0223 0.0208 α 0.0402 0.9576 1.0974 0.9220 β 0.0002 0.0001 0.0001 0.0001 ρ 0.5089 0.8167 0.8188 0.8172 ν 9.1681 38.6188 8.3486 12.2894 T 1.0027 5.0055 10.0082 15.0110

Table A-17: Swaption quotes for EUR, 01-01-2016, maturity: 5Y Free Boundary SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f 0.0058 0.0170 0.0231 0.0218 α 0.0506 1.0412 1.1700 1.0419 β 0.0004 0.0001 0.0004 0 ρ 0.3822 0.8167 0.8187 0.8171 ν 11.1851 36.3537 8.5910 13.3345 T 1.0027 5.0055 10.0082 15.0110

Table A-18: Swaption quotes for EUR, 01-01-2016, maturity: 10Y Free Boundary SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0122 0.0199 0.0225 0.0200 α 0.0676 1.1049 1.1475 1.0159 β 0.0002 0 0.0007 0 ρ 0.3402 0.8168 0.8187 0.8170 ν 11.0761 32.6020 8.6306 14.0787 T 1.0027 5.0055 10.0082 15.0110 74 Parameter Calibration Results

Table A-19: Swaption quotes for EUR, 01-06-2016, maturity: 1Y Normal SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f -0.0016 0.0057 0.0166 0.0149 α 0.1510 0.4923 0.1740 0.8876 β 0 0 0 0 ρ 1.0000 0.8170 0.6685 0.8176 ν 0.0094 24.7634 3.2525 9.8800 T 1.0000 5.0027 10.0055 15.0082

Table A-20: Swaption quotes for EUR, 01-06-2016, maturity: 5Y Normal SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f 0.0015 0.0108 0.0171 0.0158 α 539.5905 0.9903 0.2021 1.0226 β 0 0 0 0 ρ 0.8201 0.8167 0.7065 0.8174 ν 36.9105 38.8133 3.4605 10.7358 T 1.0000 5.0027 10.0055 15.0082

Table A-21: Swaption quotes for EUR, 01-06-2016, maturity: 10Y Normal SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0071 0.0138 0.0165 0.0146 α 0.6058 1.0551 0.1991 1.0058 β 0 0 0 0 ρ -0.0001 0.8168 0.7096 0.8173 ν 0.0001 34.0459 3.5113 11.4837 T 1.0000 5.0027 10.0055 15.0082 75

Table A-22: Swaption quotes for EUR, 01-06-2016, maturity: 1Y Free Boundary SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f -0.0016 0.0057 0.0166 0.0149 α 0.1505 0.4879 0.1503 0.8800 β 0.0002 0 0.3355 0.0006 ρ 0.9893 0.8170 0.6129 0.8176 ν 0.0095 25.0246 3.1044 9.8835 T 1.0000 5.0027 10.0055 15.0082

Table A-23: Swaption quotes for EUR, 01-06-2016, maturity: 5Y Free Boundary SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f 0.0015 0.0108 0.0171 0.0158 α 0.2484 0.5561 0.1624 1.0139 β 0.0003 0 0.3923 0.0040 ρ 0.9837 0.8162 0.6312 0.8174 ν 0.0144 22.6634 3.1588 10.7392 T 1.0000 5.0027 10.0055 15.0082

Table A-24: Swaption quotes for EUR, 01-06-2016, maturity: 10Y Free Boundary SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0071 0.0138 0.0165 0.0146 α 0.5977 1.0333 0.5512 0.9970 β 0.0002 0.0017 0 0.0020 ρ -0.0001 0.8168 0.8146 0.8173 ν 0.0001 34.3522 8.3994 11.4881 T 1.0000 5.0027 10.0055 15.0082 76 Parameter Calibration Results

Table A-25: Swaption quotes for EUR, 01-08-2016, maturity: 1Y Normal SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f -0.0023 0.0025 0.0116 0.0101 α 0.1604 0.0100 0.1287 0.8184 β 0 0 0 0 ρ 1.0000 0.2196 0.4537 0.8166 ν 2.5601 5.2279 2.7165 35.2031 T 1.0000 5.0027 10.0055 15.0082

Table A-26: Swaption quotes for EUR, 01-08-2016, maturity: 5Y Normal SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f -0.0005 0.0070 0.0122 0.0109 α 0.1851 0.0493 0.1350 1.0099 β 0 0 0 0 ρ 0.9560 0.4570 0.4538 0.8166 ν 3.2841 6.4451 2.6826 42.3138 T 1.0000 5.0027 10.0055 15.0082

Table A-27: Swaption quotes for EUR, 01-08-2016, maturity: 10Y Normal SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0041 0.0096 0.0116 0.0098 α 0.8313 0.0683 0.1298 0.7964 β 0 0 0 0 ρ 0.8230 0.5167 0.4563 0.8166 ν 15.7879 6.0008 2.7262 34.5794 T 1.0000 5.0027 10.0055 15.0082 77

Table A-28: Swaption quotes for EUR, 01-08-2016, maturity: 1Y Free Boundary SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f -0.0023 0.0025 0.0116 0.0101 α 0.1728 0.0103 0.1303 0.8246 β 0.0001 0.0002 0.0019 0 ρ 0.9997 0.2184 0.4589 0.8166 ν 2.8311 5.1967 2.7120 36.4834 T 1.0000 5.0027 10.0055 15.0082

Table A-29: Swaption quotes for EUR, 01-08-2016, maturity: 5Y Free Boundary SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f -0.0005 0.0070 0.0122 0.0109 α 0.1825 0.0513 0.1364 0.9944 β 0.0002 0.0001 0.0006 0 ρ 0.9557 0.4616 0.4586 0.8166 ν 3.2877 6.3967 2.6786 42.8272 T 1.0000 5.0027 10.0055 15.0082

Table A-30: Swaption quotes for EUR, 01-08-2016, maturity: 10Y Free Boundary SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0041 0.0096 0.0116 0.0098 α 0.8210 0.0708 0.1312 0.7842 β 0 0 0 0.0004 ρ 0.8230 0.5233 0.4613 0.8166 ν 15.8123 5.9745 2.7221 34.9977 T 1.0000 5.0027 10.0055 15.0082 78 Parameter Calibration Results

Table A-31: Swaption quotes for EUR, 01-12-2016, maturity: 1Y Normal SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f -0.0009 0.0094 0.0195 0.0167 α 0.3393 0.7062 1.2240 1.0439 β 0 0 0 0 ρ 0.8207 0.8168 0.8166 0.8166 ν 19.5797 30.6215 36.3751 36.1084 T 1.0000 5.0027 10.0055 15.0082

Table A-32: Swaption quotes for EUR, 01-12-2016, maturity: 5Y Normal SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f 0.0036 0.0141 0.0194 0.0175 α 0.7052 1.0029 1.2322 1.0866 β 0 0 0 0 ρ 0.8175 0.8168 0.8166 0.8166 ν 39.6968 35.2060 36.7535 36.7545 T 1.0000 5.0027 10.0055 15.0082

Table A-33: Swaption quotes for EUR, 01-12-2016, maturity: 10Y Normal SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0097 0.0167 0.0185 0.0164 α 0.1800 1.4715 1.4851 1.2540 β 0 0 0 0 ρ 0.6582 0.8167 0.8166 0.8166 ν 10.4942 45.0011 46.3829 43.7453 T 1.0000 5.0027 10.0055 15.0082 79

Table A-34: Swaption quotes for EUR, 01-12-2016, maturity: 1Y Free Boundary SABR

1Y → 1Y 5Y → 1Y 10Y → 1Y 15Y → 1Y f -0.0009 0.0094 0.0195 0.0167 α 0.3393 0.1687 1.2068 1.0265 β 0.0142 0 0.1869 0.0069 ρ 0.8207 0.7895 0.8166 0.8166 ν 19.5775 7.9800 36.5663 36.4238 T 1.0000 5.0027 10.0055 15.0082

Table A-35: Swaption quotes for EUR, 01-12-2016, maturity: 5Y Free Boundary SABR

1Y → 5Y 5Y → 5Y 10Y → 5Y 15Y → 5Y f 0.0036 0.0141 0.0194 0.0175 α 0.7038 0.1547 135.9855 1.0688 β 0.0078 0 0.0028 0.0262 ρ 0.8175 0.7351 0.8165 0.8166 ν 39.7960 6.3736 3902.7045 37.0551 T 1.0000 5.0027 10.0055 15.0082

Table A-36: Swaption quotes for EUR, 01-12-2016, maturity: 10Y Free Boundary SABR

1Y → 10Y 5Y → 10Y 10Y → 10Y 15Y → 10Y f 0.0097 0.0167 0.0185 0.0164 α 0.1811 1.4484 1.4635 1.2339 β 0.0173 0.1402 0.1534 0.0292 ρ 0.6597 0.8167 0.8166 0.8166 ν 10.5039 45.2796 46.6653 44.1202 T 1.0000 5.0027 10.0055 15.0082

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