Calculating Sensitivities in the SABR/LIBOR Market Model for European Swaptions

Calculating sensitivities in the SABR/LIBOR market model for European Swaptions

January 17, 2012

Moa Hållberg Umeå University This is the result of the Master’s Thesis in Engineering Physics, 30 ECTS. The document begins with a description of the background to the project together with the aims and objectives. Following is some theory behind the subject, continuing with the new method of calculating sensitivities under a stochastic volatility model, the SABR/LI- BOR market model. To verify the new method it will be applied to a complex ﬁnancial derivative, the European swaption. First, the SABR/LIBOR market model is calibrated to the market and then the new method is used to simulate the sensitivities with respect to different perturbations of the model. The results are analyzed and discussed at the end of this article and will lead to a conclusion about the method.

Course: Master’s Thesis in Engineering Physics, 30 ECTS. Student: Moa Hållberg, Umeå University. Tel. 070-628 44 58 Mentor: Thomas Önskog, Umeå University. Tel. 090-786 91 85 Client: Thomas Önskog, Umeå University. Tel. 090-786 91 85 Examiner: Marie Frentz, Umeå University. Tel. 090-786 60 27 i

Abstract

This article presents a new approach for calculating sensitivities of Euro- pean swaptions. The sensitivities are found by applying an adjoint method to a stochastic volatility model, namely the SABR/LIBOR market model. This market model predicts the volatility smile and follows the market fluctuations more accurately than earlier used deterministic volatility market models for complex derivatives. The new adjoint method involves not only sensitivity calculations, it also presents a way of estimating the time discretization error using an a posteriori approach. The error calculation is described in this document but not investigated further. The first step in order to calculate the sensitivities is to calibrate the SABR/LIBOR market model to some market data. In our calculations we used data from June 15 2011 with 6 month intervals between the maturity times. When this calibration is complete all of the parameters in the SABR/LIBOR market model are specified and we can continue with the sensitivity calculations using the new adjoint method. The results from these calculations show that the method is a good choice for estimating sensitivities if we consider a complex financial derivative like the European swaption. The method is quite computational so we recommend that it is only used on a small number of securities with respect to a large number of parameters. The method provides more market-driven price and sensitivity estimations than earlier used methods and can benefit hedging of portfolios. ii

Sammanfattning

Denna artikel presenterar en ny metod för att beräkna känsligheter hos swaptioner. Känsligheterna beräknas genom att applicera en adjungerad metod på en stokastisk volatilitetsmodell, nämligen SABR/LIBOR- modellen. Denna modell förutspår volatilitetssmilet bättre än tidigare an- vända deterministiska volatilitetsmodeller samtidigt som den också beskriver marknadens svängningar mer korrekt för komplexa kontrakt. Den nya metoden ger inte bara ett sätt att beräkna känsligheten utan medför också ett sätt att uppskatta diskritiseringsfelet med hjälp av en a posteriori beräkn- ing så att det kan begränsas till en viss toleransnivå. Beräkningen av diskritiseringsfelet förklaras i detta dokument men undersöks inte när- mare. Första steget i att beräkna känsligheterna är att kalibrera SABR/LIBOR- modellen till marknadsdata. I vårt fall använder vi data från den 15 Juni 2011 med 6 månaders intervall mellan löptiderna. När denna kali- brering är klar så är alla parametrarna i SABR/LIBOR modellen speciﬁ- cerade och vi kan fortsätta med känslighetsberäkningarna med hjälp av den nya metoden. Resultaten från dessa beräkningar visar att metoden är ett bra val för att beräkna känsligheter om vi studerar ett komplext ﬁ- nansiellt derivat. Metoden är beräkningskrävande så vi rekommenderar att den endast användas på ett litet antal värdepapper med avseende på ett stort antal parametrar. Metoden ger mer marknadsorienterade priser och känslighetsuppskattningar, som i sin tur kan minimera riskerna vid portföljutformning. iii

Regulation

My master thesis in Engineering physics is finished and I would like to take the opportunity to thank everyone that have supported and helped me during this project, a specially my mentor Thomas Önskog. He has been a great support and lent me a helping hand whenever I have needed. He is passionate about his work and a great inspiration who have learned me a lot about financial mathematics. I am very grateful that I got the opportunity to work with Thomas Önskog in this project. I also want to thank my family and friends who have supported me during this project when the motivation was tight. Personally I am very pleased with the result and feel that I have accomplished all of my goals with the project. I hope that my work will benefit and inspire others to continue develop the method of calculating sensitivities.

Moa Hållberg Umeå, January 17, 2012 iv

Contents

1 Introduction 1 1.1 Background ...... 1 1.2 Aims & Objectives ...... 2 1.3 Expected utility ...... 2 1.4 Approach ...... 2 1.5 Personal goals ...... 3

2 Preliminaries 5 2.1 Notations ...... 5 2.2 Stochastic Differential equations ...... 6 2.3 The Euler scheme ...... 8 2.4 Dual functions ...... 8 2.5 The Euler scheme for an extended system ...... 9 2.6 Monte Carlo Methods ...... 10 2.7 A posteriori and a priori estimates ...... 11 2.8 Greeks ...... 11 2.9 Volatility Smile/Skew ...... 12 2.10 Grid Search ...... 13

3 The European swaption 15 3.1 Pricing formula ...... 15 3.2 Choice of measure ...... 16 3.3 Pricing under the forward measure ...... 17

4 Market Models 19 4.1 LIBOR Market Model ...... 19 4.2 SABR Market Model ...... 19 4.3 The SABR/LIBOR Market Model ...... 20 4.3.1 Perturbations ...... 22 4.3.2 Discrete dual functions ...... 23

5 The Adjoint Method 27 5.1 Deﬁning entities ...... 27 5.2 The sensitivity estimation ...... 29 5.3 Expansion of the error terms ...... 31 5.4 Summary of the method ...... 33

6 Sensitivities 35 6.1 Sensitivities in the SABR/LIBOR market model ...... 35 6.2 Sensitivities of the European swaption ...... 36 6.2.1 Drift and diffusion coefﬁcients ...... 36 6.2.2 The explicit dual functions ...... 37 6.3 Calculation of sensitivities ...... 39 v

7 Simulations 41 7.1 Calibration data ...... 41 7.2 Calibration of the SABR/LIBOR Market Model ...... 41 7.3 Simulating the sensitivities ...... 43 7.3.1 Perturbations ...... 44 7.3.2 Algorithm ...... 45 7.4 Simulating the error terms ...... 46

8 Results 47 8.1 Calibration results ...... 47 8.2 Simulation results ...... 53

9 Discussion 61 9.1 The Calibration ...... 61 9.2 The Sensitivities ...... 62

10 Conclusion 65

11 References 66

A Appendix 69 A.1 Market data ...... 69

1 Introduction

1.1 Background Methods used to price options has long been based on market models with deterministic volatilities, such as the Black Scholes model or the Dupire’s lo- cal volatility model. The problem with these models are that the so called volatility smile that the models predict does not follow the real market for certain complex derivatives such as swaptions. This flaw also affects the estimates of Greeks/sensitivities and can imply unstable and misleading results. To solve this, one has in recent years developed a model that combines the advantages of two known models, the SABR market model, a stochastic volatility model that follows market fluctuations, and the LIBOR market model, an interest rate model that allows us to calculate a number of forward rates simultaneously. This new model is called the SABR/LIBOR market model and can model future interest rates for complex derivatives in a more market-driven way. The goal of the model is to create an opportunity to reduce uncertainty in pricing derivatives to an acceptable level by using the information available on the market. 1 This discovery is relatively new and the model needs to be calibrated to match the market. This can be done by first examining the correlation between the volatility of forward rates and the exercise prices. Then simulate future interest rates based on the model and compare the results with observed data. Conclusions can then be drawn on how accurate the model is and what corrections that need to be implemented. When the model is calibrated one can also calculate more market-driven sensitivities. There are also many different methods to calculate sensitivities, for example finite difference approximations, pathwise derivative estimates and the likelihood ratio method. All these have different advantages and disadvantages but they all tend to be heuristically. Some of them are also based on the technique that the calculation in the next step depend on the calculations up to the present step, like for the pathwise derivative estimation, where we need the discounted payoff to be continues. In the likelihood ratio method, there is no smoothness required in the discounted payoff, but instead we need to know the density function. These forward looking methods are advantageous when calculating sensitivities for many securities with respect to a small number of parameters. However, an adjoint formulation for calculating sensitivities has been found and has its advantages when calculating sensitivities of a small number of securities with respect to a large number of parameters. So, using the new SABR/LIBOR market model and applying the adjoint method to calculate the sensitivities of a complex derivative one could establish more market-driven and stable results than for other models.

1 The SABR/LIBOR market model is presented in [10] which is published in 2009. 2 1 Introduction

1.2 Aims & Objectives We would like to generalize a method for estimating sensitivities of option prices from a ﬁxed model with deterministic volatility to an interest rate model with stochastic volatility, with focus on applying the method on European swaptions. The goal is to calibrate the SABR/LIBOR model to market data and then calculate the sensitivities with respect to changes in the volatility, called vega, and the volatility of volatility, called vomma, for a European swaption under the forward measure, using the adjoint method described in [6]. After simulating the sensitivities we want to draw conclusions about the new method when it is applied on the SABR/LIBOR market model.

1.3 Expected utility This is an ongoing research topic and the client has developed the method during the last year. The results of this project will, together with Thomas Önskog and Kaj Nyström’s research be published in the article Calculating sensitivites in the SABR/LIBOR market model. For a successful outcome, this method could contribute to more efﬁ- cient and safer pricing and hedging of options than earlier. Our ambition is that this method will beneﬁt the multi-stakeholders work and provide better results than for earlier used methods.

1.4 Approach The starting point is to study existing research material on the subject and specify how to calculate the sensitivities using the stochastic interest model and the adjoint method. Continuing, we need to calibrate the SABR/LIBOR market model to the market, before the sensitivity calculations can be performed. The adjoint method for calculating sensitivities is described in article [6] for the LIBOR market model. We will try to apply this method to the SABR/LIBOR market model and hope that it will create more market-oriented sensitivities. The ﬁrst half of the project consists of theory and derivation of the mathematical method by which the sensitivities of option prices can be estimated. The second half of the work consists of calibrating the SABR/LI- BOR model to the market and then numerically calculate the sensitivities by the adjoint method derived earlier and draw conclusions about this method. The work is limited to the study of European swaptions under the forward measure, although the same method could be applied to other types of ﬁnancial derivatives and measures. 1.5 Personal goals 3

1.5 Personal goals My personal goal is first and foremost to develop my ability to analyze and solve problems. By independently perform the work in this project, I will most likely find new weaknesses and strengths in myself that I can continue to develop. I am aware of that this project will not be easy, but I am also confident that I will successfully complete it, and hopefully with good results. Besides that I want to develop my analytic skills during the project, I also have a goal to pass on what I have accomplished in a clear and interesting way to the audience. I see the presentation as an important part of the project and will spend some extra time on preparing the presentation. The third and final goal is that I want to accomplish something that will benefit the client or someone else. That my work either leads to useful results for research purposes or that it inspire others to continue develop the method. I hope I manage to meet these three goals and become satisfied with the project, not just the results but also all the work during the project.

2 Preliminaries

In this chapter we introduce some important notations used in this article. Furthermore, we give the theory of some useful mathematical methods, terms and models.

2.1 Notations ∂ Derivatives is quite common in this article and will be denoted as i f 2 ∂ f ∂ f n for ∂ , ∂ f for ∂ ∂ and so on. If f = f (t, x), when (t, x) ∈ R+ × R , xi ij xi xj ∂ ∂ then i, ij and so on will refer to differentiation with respect to the space variable x. Other common notations used in the article are M(n, R), for the set of all n × n-matrices with real valued entries. Given a matrix b ∈ M(n, R), ∗ we denote the transpose b . One important remark is that we use the Einstein summation conven- tion for indices representing spatial directions, implying that when an index occurs more than once in the same expression, the expression is summed over all possible values for that index. Perturbations of variables a and b are denoted as a˜ and b˜, respectively and we will use the decomposition

a(t, x) = a(t, x, θa) = a(t, x) + θaa˜(t, x) (2.1) θ θ ˜ b(t, x) = b(t, x, b) = b(t, x) + bb(t, x)

R × Rn → Rn ˜ R × Rn → R θ ∈ R θ ∈ R where a, a˜ : + , b, b : + M(n, ), a , b and |θ | ≤ ϵ |θ | ≤ ϵ ϵ a , b , for some small > 0. ∈ Rn k Consider an open set U , then we denote Cb(U) as the space of k times continuously differentiable functions f : U → R which have k Rn bounded derivatives. Similarly, we let Cp( ) denote the space of k times continuously differentiable functions g : Rn → R, which satisfy

|∂αg(x)| ≤ cα(1 + |x|qα ), whenever x ∈ Rn, |α| ≤ k (2.2)

∈ Z+ ∞ Rn for some constants cα, qα . Furthermore, we let C0 ( ) be the space of inﬁnitely differentiable functions with compact support and let k R × Rn ∈ Z+ Cb( + ), k , be the space of bounded and continuous functions n on R+ × R whose partial derivatives up to order k in both space and time, are bounded and continuous. More commonly used notations in this article are 1 ∈ { } . Maturity times: Ti, i 1, 2, ..., n + 1 2 δ − . Spacing between maturity times, called tenor: i = Ti+1 Ti 6 2 Preliminaries

3 ∈ { } . Forward rates: fi(t), i 1, 2, ..., n 4 ∈ { } . Stochastic volatilities of the forward rates: Ki(t), i 1, 2, ..., n { 1, for i = j, 5. Kronecker delta : δ = ij 0, for i ̸= j.

We also use the so-called super-correlation matrix, P, defined in [10] as ( ) ρ R P = ∗ (2.3) R r where ρ, r and R are the correlation matrices. The matrix ρ contains the forward rate/forward rate correlations, r contains the volatility/volatility correlations and R contains the forward rate/volatility correlations. In the simulations we will calculate the Root Mean Square error. This is an estimator that quantifies the difference between values implied by an estimator and the true values of the quantity. Hence, it is the square root of the estimated variance and is often referred to as the standard deviation. When we describe the theory behind the new method of estimating sensitivities some uncommon terms may occur so we will explain these here. One of these terms is the implied volatility of an option. This is the volatility of the price of the underlying security that is implied by the market price of the option based on a pricing model. Hence, using the implied volatility in a pricing model yields to a theoretical value of the option based on the current market price. Two other used terms is the numeraire and the measure. The numeraire is the basic standard of how values are measured. The most common numeraire in our everyday life is money as it measures the worth of different goods and services relative to one another. In finance, one often use another numeraire than the money market. In our case we will use the forward measure and the numeraire as a zero- coupon bond with maturity T. The forward measure is used when calculating price of options and is a systematic way to assign the size of a subset. One can think of a measure as a generalization of the concepts of length, area and volume.

2.2 Stochastic Differential equations A stochastic differential equation can describe the behavior of many different random processes, for example fluctuations of stock prices or thermal fluctuations. Considering a stochastic differential equation, one can divide the equation into two parts, the drift and the diffusion part. The drift co- 2 4 θ efficient in equation ( . ) is ai(s, X(s), ) and describes how the stochastic process X changes in average. The diffusion coefficient in equation (2.4) is 2.2 Stochastic Differential equations 7

θ bij(s, X(s), ) and it, together with the stochastic element dWj(s), describes the fluctuations of the change in X. ∗ Consider the stochastic process X = (X1, ..., Xn) with initial condition θ θ θ θ X(t, ) = x and define = ( a, b). Let X be given by the stochastic differential equation θ θ θ 2 4 dXi(s, ) = ai(s, X(s), )ds + bij(s, X(s), )dWj(s) ( . ) ∈ ∈ ∗ whenever s [t, T] and i 1, ..., n. Here W(s) = (W1(s), ..., Wn(s)) is a standard Brownian motion in Rn defined on the probability space Ω ( , F , (Fs)0≤s≤T, P). The standard Brownian motion is a process whose components are independent one-dimensional Brownian motions. In [9], a Brownian motion or Wiener process is characterized by three properties, namely

1. W0 = 0

2. The function t → Wt is almost surely continuous

3. Wt has independent increments with Wt − Ws ∼ N(0, t − s) for (0 ≤ s ≤ t), where N(µ, σ2) denotes the normal distribution with expected value µ and variance σ2. Assuming appropriate growth and regularity conditions on the coef- 2 4 ﬁcients ai and bij, the stochastic differential equation ( . ) has a unique θ θ θ |θ | ≤ ϵ |θ | ≤ ϵ strong solution for all parameter values = ( a, b), a , b . There is a well-known connection between stochastic differential equations and second order parabolic partial differential equations which is used to show this. We refer to [7] for a detail description of this and only presents the following classical result. ∈ ∞ Rn Theorem 1. Let T > 0 be given and let g Cp ( ). Assume that (2.1) holds |θ | ≤ ϵ |θ | ≤ ϵ ϵ with a and b , for some small > 0 and that ˜ ∈ ∞ R × Rn 2 5 ai, a˜i, bij, bij Cb ( + ). ( . ) Let X be the stochastic process given by (2.4) and let u be the unique solution to the Cauchy problem { ∂ u(t,x) + Lu(t,x) = 0, whenever (t, x) ∈ (0,T) × (R)n, t (2.6) u(T,x) = g(x), whenever x ∈ (R)n, where L is deﬁned as the second order parabolic operator

n n 1 ∗ ∂ ∂ 2 7 L = ∑ [bb ]ij(t, x) ij + ∑ ai(t, x) i. ( . ) 2 i,j=1 i=1 8 2 Preliminaries

Then u is uniquely determined by

θ θ θ t,x θ 2 8 u(t, x) = u(t, x, ) = u(t, x, ( a, b)) = E [g(X(T, ))], ( . ) where the subscript t, x means that X satisﬁes the initial condition X(t, θ) = x.

In order to approximate the solution u(t, x) one can use the Euler scheme, which is described in the next section.

2.3 The Euler scheme The Euler scheme for the stochastic process X in (2.4) is according to [9] { }N deﬁned as follows. Given a time horizon T, let tm m=0 deﬁne a partition ∆ of the interval [t, T], i.e. t = t0 < t1 < .... < tN−1 < tN = T, and let ∆ − ∈ { − } { ∈ } tm = tm+1 tm, for m 0, ..., N 1 . Let X(s) : s [t, T] be the solution to equation (2.4) for parameter values θ = (0, 0) and consider the function ϕ(s) = sup{tm : tm ≤ s} for s ∈ [t, T]. The continuous Euler ∆ approximation Xi (s) is then given recursively by the relation

∫ ( ) s ∆ ∆ ϕ ϕ ∆ ϕ Xi (s) =Xi ( (s)) + ai (r), X ( (r)) dr+ ϕ(s) ∫ ( ) (2.9) s ϕ ∆ ϕ bij (r), X ( (r)) dWj(r), ϕ(s)

∈ { } ∆ for i 1, ..., n , with initial condition Xi (t0) = xi. ∆ The set {X (s) : s ∈ {t0, ..., tN}} is often referred to as the associated discrete Euler approximation and it satisﬁes the difference equation ( ) ∆ ∆ ∆ ∆ Xi (tm+1) =Xi (tm) + ai tm, X (tm) tm+ ( ) (2.10) ∆ ∆ bij tm, X (tm) Wj(tm),

∆ − where Wj(tm) = Wj(tm+1) Wj(tm) is the Wiener increment during time step [tm, tm+1].

2.4 Dual functions Given an Euler approximation for each time step during a trajectory, then the dual function gives a measure of how sensitive the approximation is for a small error at this speciﬁc time step. Hence, those time steps where the dual functions are large is in some sense where the error is large, during the assumption that the drift and diffusion coefﬁcients are nearly constant. 2.5 The Euler scheme for an extended system 9

The dual functions solve certain backward stochastic differential equations. In this article we consider the function ∆ ∆ 2 11 cβ(tm,x) = xβ + aβ(tm,x) tm + bβj(tm,x) Wj(tm), ( . ) whenever β ∈ {1, ..., n}, m ∈ {0, ..., N − 1} and x ∈ Rn. The explicit expressions for the dual functions can then be calculated by recursive relations involving derivatives of the function c. n We deﬁne the ﬁrst order dual function as ψ : {t0, ..., tN} → R and the (1) second order dual function as ψ : {t0, ..., tN} → M(n, R), associated to (2.4).

2.5 The Euler scheme for an extended system When calculating the sensitivities later on, we will consider variations up to fourth order of the stochastic process X in (2.4) and the associated Euler scheme. The ﬁrst variation process of X, denoted as X(1), is according to [5], deﬁned as ∂X(s|X(t) = x) (1)( ) = 2 12 Xik t, s ∂ , ( . ) xik for i ∈ {1, ..., n} and solves the stochastic differential equation

(1) (1) dX (t, s) = ∂αa (s, X(s))X (t, s) ds+ ik | i {z αk } ( ) A 1 (t,s) ik 2 13 (1) ( . ) ∂αb (s, X(s))X (t, s) dW (s), | ij {z αk } j (1) Bikj (t,s)

(1) δ δ with initial value Xik (t,t) = ik, where ik is the Kronecker delta deﬁned in Section 2.1. Similarly, the second variation of X solves the stochastic differential equation [ ] (2) (2) (1),t (1) ∂α ∂αγ dXikl (t, s) = ai(s, X(s))Xαkl (t, s) + ai(s, X(s))Xαk (t, s)Xγl (t, s) ds | {z } ( ) A 2 (t,s) [ ikl ] (2) (1) (1) ∂α ∂αγ + bij(s, X(s))Xαkl (t, s) + bij(s, X(s))Xαk (t, s)Xγl (t, s) dWj(s), | {z } (2) Biklj(t,s) (2.14)

≤ ≤ ≤ (2) 5 for 0 t s T, with initial value Xikl (t, t) = 0. We refer to [ ] for further details. 10 2 Preliminaries

Next we introduce the vector Z(t, s) as an enumeration of the process X and its first and second variations. Note that Z(t, s) contains nˆ := n + n2 + n3 elements. The vector Z(t, s) satisfies the initial condition Z(t, t) = xˆ and a system of stochastic differential equations A B ≤ ≤ ≤ 2 15 dZ(t, s) = (s, Z(t, s))ds + j(s, Z(t, s))dWj(s), 0 t s T, ( . ) for some matrices A and B with dimensions nˆ × 1 and nˆ × n, respectively. The notation xˆ denotes that the parameter x belongs to the extended system. The Euler scheme for the system in (2.15) is given in analogy with 2 10 ∈ { − } ( . ) but first we specify some new notations. Fix th, h 0, ..., N 1 , as { ∆ the starting point of Z(th, s). The descrete Euler approximation Z (th, s) : ∈ { }} { ≤ ≤ } s th, ..., tN of Z(th, s) : th s T is now given by the equation ∆ ∆ ∆ ∆ Z (th, tm+1) =Z (th, tm) + A(tm, Z (th, tm)) tm+ (2.16) ∆ ∆ Bj(tm, Z (th, tm)) Wj(tm) ∈ { − } ∆ for m h, ..., N 1 , with initial condition Z (th, th) = xˆ. Combining equation (2.11) with (2.13)-(2.14), we can see that the first and second variation of X is given by ( ) (1),∆ (1),∆ ∆ (1),∆ ∂β Xik (th, tm+1) =Xik (th, tm) + ci tm, X (tm) Xβk (th, tm) ( ) (2),∆ (2),∆ ∆ (1),∆ ∂βγ 2 17 Xikl (th, tm+1) =Xikl (th, tm) + ci tm, X (tm) Xβk (th, tm) ( . ) ( ) (1),∆ ∆ (2),∆ · ∂β Xγl (th, tm) + ci th, tm, X (tm) Xβkl (tm)

∆ Thus, the Euler approximation Z can be calculated by means of recursive relations involving derivatives of the function c described in Section 2.4.

2.6 Monte Carlo Methods The price of a derivative security can often be represented as an expected value, as in (2.8). Thus, valuing derivatives reduces to computing an expected value, which according to [2] can be done by Monte Carlo methods. The Monte Carlo method involves simulating paths of a stochastic process that describes the evolution of the underlying asset prices, interest rates and other factors that can affect the security. By generating the Brownian motion W1,W2, ..., Wn and calculating X1(T), X2(T), ..., Xn(T) say k number of times, we get k realizations of the derivative value at time T. Depending on what kind of contract we are considering we can calculate the price uˆ by discounting the payoff and then take the mean of all replicates. If we consider more complicated derivative securities, which are 2.7 A posteriori and a priori estimates 11 path-dependent, we must divide the partition [0, T] into m subintervals and simulate transitions using the discrete Euler approximation. In each step we use independent draws from the normal distribution, which together produces a value of the option, whose distribution approximates the exact distribution when m becomes large. Since this is an estimation we would like to calculate the variance and a conﬁdence interval for our result. An estimation of the standard deviation of u1, u2, ..., un is √ n 1 − 2 2 18 su = − ∑(ui uˆn) , ( . ) n 1 i=1 − δ where uˆn denotes the mean of u1, u2, ..., un. By denoting zδ to be the 1 quantile of the standard normal distribution we get the conﬁdence interval

√su √su uˆn − zδ < uˆn < uˆn + zδ . (2.19) /2 n /2 n A short scheme to ﬁnd the price of a derivative using the Monte Carlo method is: 1. Simulate paths of the process X in equation (2.4)

2. Calculate the payoff of the derivative security for each path

3. Discount payoffs using the risk-free rate r

4. Calculate the average over all paths

2.7 A posteriori and a priori estimates The expressions a priori and a posteriori are Latin idioms where the ”a” in these expressions is the Latin preposition meaning ”from”. So a priori means from before observation and a posteriori means from after observation. Hence, the expressions a priori and a posteriori describe how we know the truth or falsity of a statement, i.e. before or after observations. In this article we will consider the a posteriori error estimate of the sensitivity approximation, which means that this estimate can only be found after some observations.

2.8 Greeks Greeks are sensitivities of the price of ﬁnancial derivatives with respect to perturbations of the parameters in the underlying model. Greeks are a vital tool in risk management and are used to hedge portfolios against changes in market conditions, see [1] for further details. There are different types of greeks, the most common are delta, vega and theta. Delta measures the rate of change of the option price with respect to changes in 12 2 Preliminaries the underlying asset’s price. Vega measures the sensitivity of the option price with respect to volatility changes and theta measures the sensitivity of the value of the derivative with respect to the passage of time. In this article we will consider greeks under the name sensitivities, with respect to perturbations in the volatility, vega, and the volatility of volatility, called vomma. Note that vomma is only relevant when we are using a stochastic volatility model.

2.9 Volatility Smile/Skew The volatility smile/skew refers to how the implied volatility depends on the strike price K for a ﬁxed forward rate f . The market smile/skew gives a snapshot of the market prices for different strike prices K at a given time, when the forward rate f has a speciﬁc value. One example of a volatility smile is shown in Figure 2.1. In [11] the authors describe how deterministic volatility models do not follow the dynamics of the volatility smile/skew but also how stochastic volatility models do.

Figure 2.1: One example of the volatility smile, which describes how the implied volatility depends on strike price for a ﬁxed forward rate. This graph is made using the data in Table 1.1 in Appendix A.1. 2.10 Grid Search 13

2.10 Grid Search Grid search is a method used to ﬁnd optimal values of parameters from a dataset. Choosing a dataset for the parameters and then looping over all combinations of the parameters one can ﬁnd the combination that gives the smallest root mean square error. Those values are then the optimal choice for the parameters. In this article we will use this method when we calibrate the SABR/LIBOR market model to market data.

3 The European swaption

First we introduce the swap before explaining the European swaption. A swap is a financial contract where the two parts agree to exchange cash flows. The most common type of swap is a Plain Vanilla interest rate swap where the two parts agree to exchange a fixed rate loan with a floating rate loan. A swaption is a financial contract that gives the owner the right but not the obligation to enter an underlying swap. The underlying swap in the contract can be many different trades but swaptions typically refer to options on interest rate swaps. There are two types of swaption contracts, a payer swaption and a receiver swaption, see [1]. The payer swaption gives the owner the right to enter into a swap where he/she pays the fixed forward rate to the other part and receives the floating interest rate from the other part. In opposite, the receiver swaption owner will, if one decides to enter the swap, receive the fixed forward rate and pay the floating interest rate. When the contract is made, the buyer and seller will agree on the following

1. The premium/price of the swaption.

2. The ﬁxed forward rate of the underlying swap.

3. The term of the underlying swap, hence the length of the option period.

4. Notional amount, used to calculate the payments in the swap.

5. Frequency of settlement of payments on the underlying swap.

3.1 Pricing formula

A swap over the time interval [Tp, Tq] entered at time t ≤ Tp is an agree- δ ment to switch payments on a loan; i−1 fi−1(t) in ﬂoating LIBOR rates δ κ with ﬁxed payments i−1 at payment dates Ti, i = p + 1, ..., q. The payoff of such a swaption, discounted at time t, equals

q δ − κ 3 1 ∑ Bi(t) i−1( fi−1(t) ), ( . ) i=p+1 where Bi(t) is the value at time t of a zero-coupon bond maturing at Ti.A well-known relation between the forward rates and zero-coupon bonds is

δ Bi(t) 3 2 1 + i fi(t) = ( . ) Bi+1(t) 16 3 The European swaption

and then we obtain that the swap rate κp,q(t) is given as q δ ∑ Bi(t) i−1 fi−1(t) i=p+1 Bp(t) − Bq(t) κ (t) = = . (3.3) p,q q q−1 δ δ ∑ Bi(t) i−1 ∑ iBi+1(t) i=p+1 i=p As mentioned above, a European swaption gives the right, not the obligation, to enter a swap over the time interval [Tp,Tq] at time t with a pre- speciﬁed swap-rate κ. The payoff at time Tp of a European swaption is χκ denoted as p,q and deﬁned as ( )+ q χκ δ − κ p,q = ∑ Bi(t) i−1( fi−1(Tp) ) = i=p+1 (3.4) q−1 κ − κ + δ ( p,q(Tp) ) ∑ iBi+1(Tp). i=p

We denote the price of the swaption at time zero as u(0, f0), where the forward rates equal f0. When using standard numeraire techniques, the price of a European swaption is given by an expected value like this [ ] χκ p,q u(0, f0) = M(0)E |F0 , (3.5) M(Tp) where F0 is a σ-algebra containing information about f0 and M(t) is the numeraire of the chosen measure.

3.2 Choice of measure The numeraire and measure affects both the form of the swaption payoff and the arbitrage-free drift coefﬁcient. One can choose between the spot measure, the swap measure and the forward measure. We will use the forward measure since both the payoff and drift corrections are relatively simple in this measure. So choosing the numeraire as the zero-coupon bond maturing at Tp, i.e f ω 3 6 Mp (t) = Bp(t), ( . ) then the price u(0, f0) equals [ ] χκ ω p,q(Tp) ( ) = f ( ) |F = u 0, f0 Mp 0 E f ω 0 Mp (Tp) [ ] (3.7) q−1 f ω κ − κ + δ | Bp(0)E ( p,q(Tp) ) ∑ iBi+1(Tp) f (0) . i=p 3.3 Pricing under the forward measure 17

3.3 Pricing under the forward measure From equation (3.2) we can easily deduce that the relation

− k 1 1 B (T ) = ∏ (3.8) k l + δ ( ) i=l 1 i fi Tl holds whenever 0 ≤ l < k ≤ q. We can also note that Bp(0) is F0- measurable and can be moved inside the conditional expectation in (3.7). Using this, the swaption price can be written as

u(0, f0) = E[gp,q( f (Tp))|F0], (3.9) where ( )+ Np,q gp,q(F(Tp)) = Qp Mp,q − κ (3.10) Mp,q and ( ) − q 1 i 1 M = ∑ δ ∏ , (3.11) p,q i + δ ( ) i=1 j=p 1 j fj Tp − q 1 1 N = 1 − ∏ , (3.12) p,q + δ ( ) i=p 1 i fi Tp p 1 Q = ∏ . (3.13) p + δ ( ) i=1 1 i fi T0

Conclusively, given the values of f (Tp), equations (3.9)-(3.13) provide a way to calculate the swaption price. However, in order to calculate the ( )+ N sensitivities we need to differentiate the function x+ = p,q − κ in Mp,q (3.10), which is not differentiable at x = 0. So, we need to approximate + + the function x with a function ϕϵ smooth enough and very close to x . Since the derivatives of the pay-off up to fourth order are required in order to determine the dual functions, which is described in Subsection 4.3.2, we choose ϕϵ to be the four times continuously differential piecewise 2 polynomial ,

1 ϕϵ(x) = (−5x8 + 28x6ϵ2 − 70x4ϵ4 + 140x2ϵ6 + 128xϵ7 256ϵ7 (3.14) ϵ8 χ χ + 35 ) (−ϵ,ϵ)(x) + x [ϵ,inf)(x),

2 The polynomial is piecewise deﬁned and continuously differentiable up to the fourth order. 18 3 The European swaption

3 where χI (x) denotes the indicator function for the interval I ⊂ R. Hence, to calculate the price u(0, f0), we use ϵ ϵ | 3 15 u(0, f0) = E[gp,q(F(Tp)) F0], ( . ) where ( ) ϵ ϕ Np,q − κ 3 16 gp,q(F(Tp)) = Qp Mp,q ϵ ( . ) Mp,q for some small ϵ > 0, instead of considering (3.9) and (3.10) above. In [12] it is shown that

∆ ∆ ∆ 3 17 u(0, f0) = u(0, x, (0, 0)) = u (0, x) + u(0, x) + Ru , ( . ) ∆ ∆ where the term u (0,(x) denotes) the expected value, u(0, x) is the er- ∆∗ ∆ ror terms of order O ( N) and Ru denotes the error terms of order ( ) ∆∗ 2 O ( N) . All these terms will be explained more explicitly in Section ∆ ∆ 5.2. Using this result together with the Euler approximation ( f , K ) we ϵ ﬁnd an approximation of the price u (0, f0) by [ ] ϵ ϵ ∆ 0, f0 ∆ u (0, f0, (0,0)) = E gp,q( f (Tp)) + u(0, f0) (3.18) ∆∗ 2 + O(( N) ) + statistical error, where the ﬁrst order error ∆u(0, f0) in the Euler approximation is computable in a posteriori form and will be explained in Section 5.2.

To summarize, we now have a formula for approximating the price ϵ of a European swaption. This formula, and especially the function gp,q in (3.16), is used later on when we derive the method for calculating the sensitivities in Section 6.2.

3 An indicator function is a function deﬁned on the set R that has the value 1 for all elements of subset I ∈ R and the value 0 for all elements of R not in I. 19

4 Market Models

In this chapter we describe the market models that gives rise to the stochastic volatility market model, which we will consider in the adjoint method to calculate the sensitivities.

4.1 LIBOR Market Model The ﬁrst thing to note about the LIBOR market model according to [10] is that, despite the name, it is in fact not a model but rather a set of no- arbitrage conditions of the forward rates that depends on the numeraire. Hence, the LIBOR market model describes the arbitrage free dynamics of a set of forward rates. In the setting of a LIBOR market model, the volatility is a deterministic function of time and forward rate. The evolution of these forward rates is described in [10] by the following equation.

d fi(t) µ { } {σ } ρ σ 4 1 = i( f(t) , (t) , , t)dt + i(t,Ti)dzi(t), ( . ) fi(t) with [ ] E ρ 4 2 dzi(t)dzj(t) = (t,Ti, Tj)dt. ( . )

Here f(t) is the vector of spanning forward rates fi(t) that constitute σ σ ρ the yield curve, (t) the vector of associated volatilities i and is the matrix of the associated correlations. The drift term in (4.1) is denoted µ { } {σ } ρ σ i( f(t) , (t) , , t) and the diffusion is denoted i(t,Ti). For further details about the terms in the LIBOR market model we refer to Section 2.1 in [10]. This model is commonly used for pricing interest rate derivatives of complex form, like swaptions, spread options, caps and ﬂoors, among many others. However, since this model assumes a deterministic volatility it is unable to reproduce many of the volatility smiles that are found in the market and, as a consequence, a variety of models with stochastic volatility has been developed in recent years, for example the SABR market model.

4.2 SABR Market Model The SABR market model is a stochastic volatility model that follows market ﬂuctuations but can only be used for a single forward rate. Hence, it is only applicable to options with a ﬁxed expiration and tenor of the underlying. As mentioned in [10] the underlying in the SABR model, in our 20 4 Market Models

case the forward rate that expiry at T, fT(t), follows the dynamics: β d fT(t) = ( fT(t)) σT(t)dzT(t), (4.3)

dσT(t) = νTdwT(t), (4.4) σT(t) QT E [dzT(t)dwT(t)] = ρT(t)dt. (4.5)

Here the terms is deﬁned in the same way as for the LIBOR market model in Section 4.1 above but with an index T, denoting that the forward rate expiry at T. The model is complete when we add the initial conditions fT(0) and σT(0) to the equations above. We refer to Section 3.2 of [10] for deeper knowledge. 4 The SABR market model is a CEV model augmented by a stochastic volatility. A few observations are in order.

1. The parameters νT, β and ρT are constants.

2. All parameters of the model, νT, β and ρT, are speciﬁc to a particular forward rate.

3. The increments dzT and dwT are increments of standard Brownian 5 motions under the terminal measure QT by appending the same superscript T.

4. Within the SABR market model there is no way for the various forward rates to interact with each other. For instance, we cannot use the SABR model to determine the payoff of a path-dependent option. Each forward rate live in its own measure and does not know anything about the other forward rates. The SABR model cannot describe the dynamics of a yield curve.

5. There is no mean reversion in the process for the volatility. Points 1, 2 and 4 are crucial to the development of the SABR/LIBOR market model, described in the next section.

4.3 The SABR/LIBOR Market Model The SABR/LIBOR market model described in [10] states that the joint dynamics of the forward rates f (t) = ( f1(t), f2(t), ..., fn(t)) and their volatilities K(t) = (K1(t), K2(t), ..., Kn(t)) is given by

( ) = µ ( ( ) ( )) + σˆ ( ( )) ( ) ( ) d fi t î t, f t , K t dt i t, f t Ki t dzi t , 4 6 η ν ( . ) dKi(t) = î(t, f (t), K(t))dt + î(t, K(t))dwi(t),

4 CEV = Constant Elasticity of Volatility model 5 See further explanation in page 26, Section 3.2 in [10]. 4.3 The SABR/LIBOR Market Model 21

≤ ≤ ∈ { } µ η σ ν when 0 t Ti, i 1,..., n for some functions ˆ, ˆ, ˆ and ˆ. The terms dz(t) and dw(t) are n-dimensional Wiener increments, under the risk-adjusted measure and satisﬁes the correlation relations ρ E[dzi(t)dzj(t)] = ijdt, 4 7 E[dwi(t)dwj(t)] = rijdt, ( . )

E[dzi(t)dwj(t)] = Rijdt, for i, j ∈ {1,..., n}. In other words, the correlation between forward rates ρ are denoted as ij, the correlation between volatilities are rij and the mixed forward rate/volatility correlations are Rij. Some remarks from [10]: Under the assumption of no possibility of arbitrage the drift terms µˆ and ηˆ are, under a given measure, deterministic functions of the parameters and processes in the model. The explicit expressions for µˆ and ηˆ are given by expressions (6.5) and (6.6) stated in Section 6.2 when we consider the European swaption. The diffusion terms σˆ and νˆ are assumed, using the standard procedure in [10] and [3] to be

β σˆ (t, f (t)) = σ (t)( f (t)) i , i i i (4.8) ν ν ˆi(t,K(t)) = i(t)Ki(t), σ ν β ∈ ∈ { } for some choice of functions i, i and exponent i [0, 1], i 1,..., n . The exponent β has three main effects on the volatility smile:

1. A progressive steepening of the smile when β goes from 1 to 0.

2. Lowering the level of the smile when β increases.

3. Introduction of curvature to the smile as β goes from 1 to 0.

A deeper discussion about the influence of β is found in Subsection 3.5.3 in [10]. In Section 3.9 in [10] they argue for the choice of β = 0.5 but in this 6 article we will choose β = 1. Recent results indicates that β = 0 is appropriate in normal conditions, whereas β = 1 is preferable when the forward rates are either very low or very high. In our market data the forward rates are very low. In contrast, β = 0.5 is often used in the market, but for low rates, β < 1 is infeasible as it makes the barrier f (t) = 0 attainable. Hence we have chosen to fix β = 1. In order to remove the forward rate dependence in the diffusion coefficient we make a transformation like this 4 9 Fi(t) = log( fi(t)), ( . ) 6 See Section 8.6 in [10]. 22 4 Market Models

∈ { } for i 1,...,n and define F(t) = (F1(t),F2(t), ..., Fn(t)). For notational convenience, we also introduce the notation µ µ î(t, f (t),K(t)) i(t,F(t),K(t)) = , fi(t) (4.10) η η i(t,F(t),K(t)) = î(t, f (t),K(t)).

If we then apply Ito¯’s formula to F(t) we obtain the following system of stochastic differential equations for the forward rates and their volatilities ( ) ρ σ 2 µ − ii( i(t),Ki(t)) σ dFi(t) = i(t,F(t),K(t)) dt + i(t)Ki(t)dzi(t), 2 (4.11) η ν dKi(t) = i(t,F(t),K(t))dt + i(t)Ki(t)dwi(t), ≤ ≤ whenever 0 t Ti, i = 1, ..., n.

4.3.1 Perturbations According to Section 2.8 sensitivities is deﬁned as how much the option price changes when there are perturbations in the underlying variables. So in this subsection we will consider perturbations of some of the parameters in equation (4.11), namely perturbations of the drift and diffusion parameters σ and ν. It is also possible to consider perturbations of the correlation parameters ρ, r and R but in Chapter 9 we explain why these are not considered here. We start by writing (4.11) in matrix form where we temporarily sup- press the time dependence of the variables.         F FF ··· FF FK ··· FK dW1 F1 a b b b b      1   11 1n 11 1n   .   .   .   ......   .   .   .   ......       F   FF FF FK FK   .   Fn  a  b ··· b b ··· b   .  d   =  n  dt +  n1 nn n1 nn    K  aK bKF ··· bKF bKK ··· bKK  .   1  1   11 1n 11 1n   .   .   .   ......    ......  .  K KF ··· KF KK ··· KK . Kn an bn1 bnn bn1 bnn dW2n where W is a 2n-dimensional Wiener process. We denote the perturbations with a tilde and get the decompositions of σ and ν as

σ = σ + θσσ˜ i i i 4 12 ν ν θ ν ( . ) i = i + ν ˜i where θσ and θν are real numbers whose absolute values do not exceed ϵ > 0. Inserting these decompositions into (4.11) and neglecting higher 4.3 The SABR/LIBOR Market Model 23

order terms in θσ and θν, we find that the drift and diffusion coefficients when perturbing σ or ν can be written as u u θ u σ ai = ai + σa˜i ( ) uv uv θ ˜uv σ bij = bij + σbij ( ) (4.13) u u θ u ν ai = ai + νa˜i ( ) uv uv θ ˜uv ν bij = bij + νbij ( ) respectively, for i, j ∈ {1,..., n} and u, v ∈ {F, K}. These decompositions follow the same definition as in (2.1). Explicitly, the non-perturbed drift and diffusion coefficients equal to

ρ σ2 2 ij i Ki FF KF aF = µ − b = σ K ρ b = ν K R i i 2 ij i i ij ij i i ij (4.14) K η FK σ KK ν ai = i bij = iKiRij bij = iKirij where µ and η are obtained from the explicit expression (6.9) in Section 6.2 where we consider the European swaption. In Section 6.2 we also see that the drift coefficient µ is a function of σ, ρ, F and K and that the diffusion coefficient is a function of σ, ν, ρ, R, F and K. Hence, the explicit expressions for the perturbed drift coefficients are F σ µ σ − ρ σ σ 2 F ν a˜i ( ) = ˜i( ) ii i ˜iKi a˜i ( ) = 0 K σ η σ K ν η ν 4 15 a˜i ( ) = ˜i( ) a˜i ( ) = ˜i( ), ( . ) where µ˜ and η˜ are obtained from the explicit expression (6.10) in Section 6.2. In the same way we find the diffusion coefficients to be ˜ FF σ σ ρ ˜ FF ν bij ( ) = ˜iKi ij bij ( ) = 0 ˜ FK σ σ ˜ FK ν bij ( ) = ˜iKiRij bij ( ) = 0 ˜KF σ ˜KF ν ν ¯ 4 16 bij ( ) = 0 bij ( ) = ˜iKiRij ( . ) ˜KK σ ˜KK ν ν bij ( ) = 0 bij ( ) = ˜iKir¯ij where µ˜ and η˜ are found from (6.10). These explicit expressions of the perturbed drift and diffusion coefficients are used to calculate the discrete dual functions that we need in order to calculate the sensitivities.

4.3.2 Discrete dual functions In this subsection, we derive the dual functions that are used in the sensitivity calculations. 24 4 Market Models

∈ { } { }N Let p, q 1, ..., n , p < q, be given and let tm m=0 define a partition ∆ of the interval [0, Tp], i.e. 0 = t0 < t1 < .... < tN−1 < tN = Tp with time ∆ − ∈ { − } 4 3 1 steps tm = tm+1 tm for m 0, ..., N 1 . Consider the system ( . . ) and let (F, K) denote the solution to the non-perturbed system, i.e when the coefficients a and b are replaced by a and b. Then we can approximate ∆ ∆ the solution (F, K) with the Euler approximation (F , K ) defined as in Section 2.3, so that ∆ ∆ FF FK F (t ) = F (t ) + aF∆t + b ∆W (t ) + b ∆W (t ), i m+1 i m i m ij j m ij n+j m 4 17 ∆ ∆ ( . ) K∆ KF∆ KK∆ Ki (tm+1) = Ki (tm) + ai tm + bij Wj(tm) + bij Wn+j(tm), ( ) ∆ ∆ for i ∈ {1, ..., n}, where a and b are evaluated at tm, F (tm), K (tm) and the initial values F(0) and K(0) are defined according to the market data. Based on the two equations in (4.17) and the non-perturbed coefficient in (4.14) we can define two counterparts, backwards stochastic differential equations which the dual functions solve, as ( ) ρ σ2 2 ij i Ki cF(t , xF, xK) =xF + µ − ∆t + σ xKρ ∆W (t )+ i m i i 2 m i i ij j m σ ∆ iKiRij Wn+j(tm), (4.18) K F K K η ∆ ν ∆ ci (tm, x , x ) =xi + i tm + iKiRij Wj(tm)+ ν ∆ iKirij Wn+j(tm). The first derivatives of these counterparts equal ∂F F F K δ ∂Fµ ∆ i cv (tm, x , x ) = iv + i v tm, ∂K F F K ∂Kµ − ρ σ2 Kδ ∆ i cv (tm, x , x ) = ( i v vv vxv iv) tm+ δ σ ρ ∆ δ σ ∆ iv v vj Wj(tm) + iv vRvj Wn+j(tm), ∂F K F K ∂Fη ∆ 4 19 i cv (tm, x , x ) = i v tm, ( . ) ∂K K F K δ ∂Kη ∆ δ ν ∆ i cv (tm, x , x ) = iv + i v tm + iv vRvj Wj(tm)+ δ ν ∆ iv vrvj Wn+j(tm),

× where (xF, xK) ∈ Rn n and the second order derivatives equal ∂FF F F K ∂FFµ ∆ ik cv (tm, x , x ) = ik v tm, ∂FK F F K ∂KF F F K ∂FKµ ∆ ik cv (tm, x , x ) = ik cv (tm, x , x ) = ik v tm, ∂KK F F K ∂KKµ ∆ − ρ σ2δ δ ∆ cv (tm, x , x ) = tm v kv iv tm, ik ik v vv (4.20) ∂FF K F K ∂FFη ∆ ik cv (tm, x , x ) = ik v tm, ∂FK K F K ∂KF K F K ∂FKη ∆ ik cv (tm, x , x ) = ik cv (tm, x , x ) = ik v tm, ∂KK K F K ∂KKη ∆ ik cv (tm, x , x ) = ik v tm. 4.3 The SABR/LIBOR Market Model 25

Even higher order derivatives of cv are given by the corresponding higher order derivatives of µ and η multiplied with ∆tm. These higher order derivatives are needed in the time-discretization error terms described in Section 5.3. We are now ready to deﬁne the dual functions. We introduce 2n × 2n- matrices Cm, m ∈ {0, ..., N − 1}, deﬁned as ( ) [∂FcF][∂FcK] C = (4.21) m [∂KcF][∂KcK]

∂r s ∈ { } × ∂r s ∂r s where [ c ] for r, s F, K are n n-matrices( defined as [ c ]iv) = i sv. ∆ ∆ F K The derivatives of c and c are evaluated at tm, F (tm), K (tm) . ϵ ∆ 3 16 The dual functions associated to gp,q(F (tN)) defined in equation ( . ) is denoted as ( ) ψF ψ = (4.22) ψK and ( ) (1),FF (1),FK (1) [ψ ][ψ ] ψ = (4.23) [ψ(1),KF][ψ(1),KK] for the first and second order dual functions, respectively. They satisfy the terminal conditions ψF ∂ ϵ ∆ ψ(1)FF ∂ ϵ ∆ 4 24 i (tN) = igp,q(F (tN)), ik (tN) = ikgp,q(F (tN)) ( . ) and K (1)FK (1)KK ψ (tN) = ψ (tN) = ψ (tN) = 0 (4.25) and also the recursive relation ψ ψ 4 26 (tm) = Cm (tm+1), ( . ) for m < N. So, using the expressions in (4.19) we find that

F F F K ψ (t ) =ψ (t ) + ∂Fµ ∆t ψ (t ) + ∂Fη ∆t ψ (t ), i m (i m+1 i v m v m+1 i v) m v m+1 ψK σ ∆ σ ∆ ψK i (tm) = 1 + iRij Wj(tm) + irij Wn+j(tm) i (tm+1)+ ( ν ρ ∆ ν ∆ − 4 27 i ij Wj(tm) + iRij Wn+j(tm) ( . ) ) ∆ ρ σ2 ∆ ψF ∂Kµ ∆ ψF ii i Ki (tm) tm i (tm+1) + i v tm v (tm+1)+

∂Kη ∆ ψK i v tm v (tm+1), 26 4 Market Models

( ) ∆ ∆ where µ, η are evaluated at tm, F (tm), K (tm) and σ, ν at tm. Furthermore, we have the recursive relation

ψ(1) ψ(1) ∗ 4 28 (tm) = Cm (tm+1)Cm + Pm, ( . ) for the second order dual function where

Pm = (4.29) ( ) ∂FF FψF ∂FF KψK ∂FK FψF ∂FK KψK ik cv v (tm+1) + ik cv v (tm+1) ik cv v (tm+1) + ik cv v (tm+1) ∂KF FψF ∂KF KψK ∂KK FψF ∂KK KψK ik cv v (tm+1) + ik cv v (tm+1) ik cv v (tm+1) + ik cv v (tm+1) and( the second order) derivatives of cF and cK are also evaluated at ∆ ∆ tm, F (tm), K (tm) . The explicit form of (4.28) is found by inserting the expressions in (4.20) but we omit further details here. To summarize, we now have explicit expression for the dual functions (1) ψ and ψ , which require that we determine the derivatives of the function ϵ gp,q up to second order to ﬁnd the terminal conditions. These derivations is done in Section 6.2, where we consider the European swaption. 27

5 The Adjoint Method

The new adjoint method of calculating sensitivities is derived in detail in [6] but in this chapter we present the main result. We start by deﬁning some useful terms and involving entities before stating the sensitivity expressions in the adjoint method.

5.1 Defining entities Consider the extended system (2.15) corresponding to the drift and diffu- ∆ ∆ ∆ sion coefficients a and b in equation (2.4). Let X , X(1), and X(2), be the Euler approximation of the solution to this system. Denote the drift coef- ∆ (1),∆ (2),∆ (0) ficients of X (th+1), X (tm, th+1) and X (tm, th+1) by Ai (tm,th+1), (1) (2) (0) Aik (tm,th+1) and Aikl (tm,th+1) and the diffusion coefficients by Bij (tm,th+1), (1) (2) 2 5 Bikj (tm,th+1) and Biklj(tm,th+1), according to Section . . Then we define

Γ(m,h) (0) − (0) i = Ai (tm, th+1) Ai (tm, th), Γ(m,h) (1) − (1) 5 1 ik = Aik (tm, th+1) Aik (tm, th), ( . ) Γ(m,h) (2) − (2) ikl = Aikl (tm, th+1) Aikl (tm, th) and ( ) (m,h) 1 (0) (0) (0) (0) Γ = B (t , t + )B (t , t + ) − B (t , t )B (t , t ) . (5.2) i,r 2 ij m h 1 rj m h 1 ij m h rj m h

Γ(m,h) Γ(m,h) Γ(m,h) Γ(m,h) Γ(m,h) Γ(m,h) Γ(m,h) Γ(m,h) We also define ik,r , ikl,r , i,rs , ik,rs , ikl,rs , i,rst , ik,rst and ikl,rst sim- (0) (1) (2) ilarly based on the other eight combinations of Bij , Bikj and Biklj. In the same way, we also define the ε functions in (5.18) and (5.19) as the difference between the drift coefficients and diffusion coefficients of the system (2.10), respectively as

ε(m) − = ai(tm, th+1) ai(tm, th), i (5.3) ε(m) − ij = bij(tm, th+1) bij(tm, th).

Furthermore, the sensitivity expressions contains third and fourth or- (2) (3) der dual functions, ψ and ψ respectively, which can be determined recursively by means of derivatives of the function c in (4.18), just like we did for the first and second order dual functions in Subsection 4.3.2. Finally, ξ are the first and second order dual functions of the extended system constructed to approximate the first and second order derivatives 28 5 The Adjoint Method of the functions ( ) m ∂ ∆ (1),∆ Vv = βg X (T) Xβv (tm,T), ( ) m ∂ ∆ (2),∆ 5 4 Vvw = βg X (T) Xβvw (tm,T)+ ( . ) ( ) ∂ ∆ (1),∆ (1),∆ βγg X (T) Xβv (tm,T)Xγv (tm,T).

m m For v, w and m ﬁxed, Vv and Vvw give rise to two and three sets of ﬁrst order dual functions, ( ) ( ) ξm 0 ξm 1 ξm 0 ξm 1 ξm 2 5 5 [ v ]i [ v ]ik and [ vw]i [ vw]ik [ vw]ikl , ( . )

∈ { } m m where i, k, l 1, ..., n . Similarly, for v, w and m ﬁxed, Vv and Vvw give rise to four and nine sets of second order dual functions, ( ) [ξm]00 [ξm]01 v i,r v i,rs (5.6) ξm 10 ξm 11 [ v ]ik,r [ v ]ik,rs and   [ξm ]00 [ξm ]01 [ξm ]02  vw i,r vw i,rs vw i,rst   ξm 10 ξm 11 ξm 12  5 7 [ vw]ik,r [ vw]ik,rs [ vw]ik,rst ( . ) ξm 20 ξm 21 ξm 22 [ vw]ikl,r [ vw]ikl,rs [ vw]ikl,rst, where i, k, l, r, s, t ∈ {1, ..., n}. To continue calculating the dual functions ξ, we let (x, x1, x2) denote an nˆ-dimensional vector describing the component of the solution to the extended system Z. With this notation we note that in the extended system, the counterpart to equation (2.11) are the three equations

0 1 2 wr (th, x, x , x ) = cr(th, x), 1 1 2 1 ∂ 1 5 8 wrs(th, x, x , x ) = xrs + αcr(th, x)xαs, ( . ) 1 1 2 2 ∂ 1 1 ∂ 1 2 wrst(th, x, x , x ) = xrst + αγcr(th, x)xαsxγt + αcr(th, x)xαsxαst. ξm 0 From these equations, the ﬁrst dual function [ vw]i is recursively de- ﬁned as ξm 0 ∂0 m [ vw]i (T) = i Vvw, ξm 0 ∂0 0 ξm 0 ∂0 1 ξm 1 5 9 [ vw]i (th) = i wr [ vw]r (th+1) + i wrs[ vw]rs(th+1)+ ( . ) ∂0 2 ξm 2 i wrst[ vw]rst(th+1), ∂0 for h < N where i represents the( differentiation with respect to xi and the) ∆ (1),∆ (2),∆ derivatives of w are evaluated at th, X (th), X (tm, th), X (tm, th) . 5.2 The sensitivity estimation 29

Analogous relations hold for the other ﬁrst order dual functions, see Sec- tion 3.1 in [13] for further details. We can also ﬁnd the second order dual ξm 00 function [ vw]i,r recursively by

ξm 00 ∂0∂0 m [ vw]i,r (T) = i r Vvw, ξm 00 ∂0 0 ∂0 0 ξm 00 ∂0 0 ∂0 1 ξm 01 [ vw]i,r (th) = i wβ r wζ [ vw]βζ (th+1) + i wβ r wζη[ vw]βζη(th+1) ∂0 0 ∂0 2 ξm 02 + i wβ r wζηϕ[ vw]βζηϕ(th+1) ∂0 1 ∂0 0 ξm 10 + i wβγ r wζ [ vw]βγζ (th+1) ∂0 1 ∂0 1 ξm 11 5 10 + i wβγ r wζη[ vw]βγ,ζη(th+1) ( . ) ∂0 1 ∂0 2 ξm 10 + i wβγ r wζηϕ[ vw]βγ,ζηϕ(th+1) ∂0 2 ∂0 2 ξm 22 + i wβγθ r wζηϕ[ vw]βγθ,ζηϕ(th+1) ∂0∂0 0 ξm 0 ∂0∂0 1 ξm 1 + i r wβ[ vw]β(th+1) + i r wβγ[ vw]βγ(th+1) ∂0∂0 2 ξm 2 + i r wβγθ[ vw]βγθ(th+1), for h < N where the derivatives of w are calculated at ( ) ∆ (1),∆ (2),∆ th, X (th), X (tm, th), X (tm, th) .

Analogous relations hold for the other second order dual functions, see Section 3.1 in [13] for further details. Now we are ready to specify the sensitivity estimations in the adjoint method.

5.2 The sensitivity estimation

∂θ ∂θ The sensitivities a u and b u, of perturbations in the drift and diffusion coefﬁcients respectively can be found by the expressions

∆ ∆ ∂θ u(0, x, (0,0)) = u (0, x) + ∆ (0, x) + R , a a a a 5 11 ∆ ∆ ( . ) ∂θ ( ( )) = ( ) + ∆ ( ) + b u 0, x, 0,0 ub 0, x b 0, x Rb .

∆ ∆ The ﬁrst term on the right hand side, ua (0, x) and ub (0, x), are approximations of the sensitivities. These approximations are given by [ ] N−1 ∆ 0,x ˜a ψ ∆ ua (0,x) = ∑ E Si (tm) i(tm) tm, = m 0 [ ] (5.12) N−1 ∆ 0,x ˜b ψ(1) ∆ ub (0,x) = ∑ E Sij(tm) ij (tm) tm, m=0 30 5 The Adjoint Method

ψ ψ(1) where i(tm) and ij (tm) are the ﬁrst and second order dual functions respectively and ( ) ∆ ˜a Si (tm) = a˜i tm, X (tm) , [ ] ( ) (5.13) ∗ ∆ ˜b 1 ˜∗ ˜ Sij(tm) = bb + bb tm, X (tm) . 2 ij

5 11 ∆ ∆ The second terms in ( . ), a(0, x) and b(0, x), are error terms of ∆∗ 7 ∆ order O( N) , that are computable( in) a posteriori form, whereas Ra and ∆ ∆∗ 2 ∆ ∆ Rb are error terms of order O ( N) . Explicitly, a(0, x) and b(0, x) are given by [ ] ∆ 0,x a a tm ∆ (0, x) =E S˜ (t + )ψ (t + ) − S˜ (t )ψ (t ) + a i m 1 i m 1 i m i m 2 [ − ] N 1 ∆ ∆ 0,x ∆ tm E ∑ Ta,m,h th + (5.14) h=m 2 [ − ] N 1 ∆ ∆ 0,x ∆ tm E ∑ Ua,m+1,h th h=m+1 2 and [ ] ∆ 0,x b (1) b (1) tm ∆ (0, x) =E S˜ (t + )ψ (t + ) − S˜ (t )ψ (t ) + b ij m 1 ij m 1 ij m ij m 2 [ − ] N 1 ∆ ∆ 0,x ∆ tm E ∑ Tb,m,h th + (5.15) h=m 2 [ − ] N 1 ∆ ∆ 0,x ∆ tm E ∑ Ub,m+1,h th , h=m+1 2 where the functions T and U are given by ( ∆ ˜a Γ(m,h) ξm 0 Γ(m,h) ξm 1 Ta,m,h =Sv(tm) i [ v ]i (th+1) + ik [ v ]ik(th+1)

(m,h) m (m,h) m + Γ [ξ ]00(t ) + Γ [ξ ]01 (t ) (5.16) i,r v i,r h+1 i,rs v i,rs h+1 ) Γ(m,h) ξm 10 Γ(m,h) ξm 11 + ik,r [ v ]ik,r(th+1) + ik,rs [ v ]ik,rs(th+1) ,

7∆∗ N denotes the biggest time step. 5.3 Expansion of the error terms 31

( ∆ ˜b Γ(m,h) ξm 0 Γ(m,h) ξm 1 Tb,m,h =Svw(tm) i [ vw]i (th+1) + ik [ vw]ik(th+1)

Γ(m,h) ξm 2 Γ(m,h) ξm 00 + ikl [ vw]ikl(th+1) + i,r [ vw]i,r (th+1) Γ(m,h) ξm 01 Γ(m,h) ξm 02 + i,rs [ vw]i rs(th+1) + i,rst [ vw]i rst(th+1) , , (5.17) Γ(m,h) ξm 10 Γ(m,h) ξm 11 + ik,r [ vw]ik,r(th+1) + ik,rs [ vw]ik,rs(th+1) (m,h) m (m,h) m + Γ [ξ ]12 (t ) + Γ [ξ ]20 (t ) ik,rst vw ik,rst h+1 ikl,r vw ikl,r h+1 ) Γ(m,h) ξm 21 Γ(m,h) ξm 22 + ikl,rs [ vw]ikl,rs(th+1) + ikl,rst[ vw]ikl,rst(th+1)

and ( ) ∆ ∆ ε(m) ∂ ã ψ ã ψ(1) (1), Ua,m+1,h = i vSr (th) r(th) + Sr (th) rv (th) Xvi (tm+1,th) ( ) ∆ ε(m) ∂ ã ψ ã ψ(1) (2), + ij vSr (th) r(th) + Sr (th) rv (th) Xvij (tm+1,th) ( (5.18) ε(m) ∂ ã ψ ∂ ã ψ(1) + ij vwSr (th) r(th) + 2 wSr (th) rv (th)+ ) ∆ ∆ ã ψ(2) (1), (1), Sr (th) rvw(th) Xvi (tm+1,th)Xwj (tm+1,th),

( ) ∆ ∆ ε(m) ∂ ˜b ψ(1) ˜b ψ(2) (1), Ub,m+1,h = i vSrs(th) rs (th) + Srs(th) rsv(th) Xvi (tm+1,th) ( ) ∆ ε(m) ∂ ˜b ψ(1) ˜b ψ(2) (2), + ij vSrs(th) rs (th) + Srs(th) rsv(th) Xvij (tm+1,th) ( (5.19) 1ε(m) ∂ ˜b ψ(1) ∂ ˜b ψ(2) + ij vwSrs(th) rs (th) + 2 wSr (th) rsv(th)+ 2 ) ∆ ∆ ˜b ψ(3) (1), (1), Srs(th) rsvw(th) Xvi (tm+1,th)Xwj (tm+1,th).

As mentioned in the beginning of this chapter, the method is derived in [6], so we refer the reader to this reference for further details about the method.

5.3 Expansion of the error terms Using the Monte Carlo method and the discrete Euler approximation, one can expand the sensitivity expressions in (5.11) a bit more explicitly as ∆ ∆ ∆ ∆ ∂ ∆,M ,M ,M ,M ( θa u)(0,x,(0,0)) = ua (x) + Ea d + Ea d s + Ea s + Ra d, , , , , , (5.20) ∆,M ∆,M ∆,M ∆,M ∆ (∂θ )( ( )) = ( ) + + + + b u 0,x, 0,0 ub x Eb,d Eb,d,s Eb,s Rb,d, 32 5 The Adjoint Method

∆,M ∆,M ∆,M ∆,M where Ea,s , Ea,d,s, Eb,s and Eb,d,s are statistical errors, from the use of the ∆ ∆ ∆ ∆ { }M ,M ,M finite set wr r=1, while Eb,d , Ea,d , Ra,d and Rb,d represent time- discretization errors due to the discrete Euler scheme. In general, the statistical errors can be controlled using the central limit theorem so we will focus mainly on the time-discretization error terms. The time discretization errors have the following important features 6 ∆∗ {∆ ∆ ∆ } according to [ ]. Let N = max t0, t1, ..., tN−1 . Then 1 ∆,M ∆,M ∆∗ . Ea,d and Eb,d are of order ( N). 2 ∆ ∆ ∆∗ 2 . Ra,d and Rb,d are of order (( N) ). 3 ∆,M ∆,M . Ea,d and Eb,d are computable in a posteriori form. Hence, the leading order terms in the expansion of the error terms are ∆,M ∆,M Ea,d and Eb,d . They are the naturally defined Monte Carlo estimators ∆ ∆ associated to the time-discretization errors Ea,d and Eb,d and are explicitly given by [ ] ∆ M ∆ ,M ã ψ − ã ψ tk Ea,d = ∑ Si (tk+1) i(tk+1) Si (tk) i(tk) m=1 2M [ − ] M N 1 ∆ ∆ ∆ tk 5 21 + ∑ ∑ Ta,k,h th ( . ) = 2M m 1 [ h=k ] M N−1 ∆ ∆ ∆ tk + ∑ ∑ Ua,k+1,h th m=1 h=k+1 2M and [ ] ∆ M ∆ ,M ˜b ψ(1) − ˜b ψ(1) tk Eb,d = ∑ Si (tk+1) ij (tk+1) Si (tk) ij (tk) m=1 2M [ − ] M N 1 ∆ ∆ ∆ tk 5 22 + ∑ ∑ Tb,k,h th ( . ) = 2M m 1 [ h=k ] M N−1 ∆ ∆ ∆ tk + ∑ ∑ Ub,k+1,h th . m=1 h=k+1 2M

∆,M ∆,M ∆,M ∆,M ∆ The other two terms Ea,d,s, Ea,s and Eb,d,s, Eb,s , respectively for a(0,x) ∆ and b(0,x), can not be calculated in an a posteriori form but can be estimated by analyzing the result from the sensitivity calculations. 5.4 Summary of the method 33

5.4 Summary of the method The sensitivities with respect to perturbations are calculated by the expressions in (5.11). Approximating the expectations in (5.12) by sample means, ∆ ∆ the terms ua (0,x) and ub (0,x) can be used as Monte Carlo estimators of ∂θ ∂θ the sensitivities ( a u)(0,x,(0,0)) and ( b u)(0,x,(0,0)), respectively. These estimators contains the drift and diffusion coefficients a, a˜, b and b˜, and the (1) first and second order dual functions ψ and ψ . Moreover, the formulas in (5.20) give an expansion of the error arising ∆ ∆ in the approximation. The first order error terms a(0,x) and b(0,x) de- (1) (2) pend on the first, second, third and fourth order dual functions, ψ, ψ , ψ (3) ∆ (1),∆ and ψ respectively, as well as on the Euler discretizations X , X and (2),∆ (1) (2) X of X and its first and second variation process X and X . Fur- ∆ ∆ thermore, the expressions for a(0,x) and b(0,x) also contains derivatives of a, a˜, b and b˜ up to second order. To estimate the sensitivities by (5.11) it requires a lot of calculations to find all the functions involved. These computations can be very costly, which implies that this method only should be used when the{ aim is to} cal- N ˜ culate the sensitivities with respect to a set of perturbations (a˜l, bl) l=1 where N is large.

In the next chapter we will use the adjoint method above to calculate the sensitivities in the setting of the SABR/LIBOR market model.

6 Sensitivities

In this part of the article we focus on the sensitivity calculations of the European swaption using the adjoint method in Section 5.2.

6.1 Sensitivities in the SABR/LIBOR market model Using the dual functions derived in Subsection 4.3.2 and the sensitivity expression (5.11)-(5.13) we can ﬁnd the sensitivities in the setting of the SABR/LIBOR market model. Assume that we are interested in calculating the sensitivities of a ﬁnancial derivative whose price can be represented as

0,x ϵ 6 1 u(0,x) = E [gp,q(F(TN))] ( . ) ∗ where x is the 2n-dimensional vector x = ( f (0),K(0)) . We introduce matrices for the drift and diffusion coefﬁcients in (4.14)-(4.16) as ( ) ( ) aF a˜F a = , a˜ = (6.2) m aK m a˜K and ( ) FF FK ( ) b b b˜ FF b˜ FK = ˜ = 6 3 bm KF KK , bm ˜KF ˜KK , ( . ) b b b b ( ) ∆ ∆ which are evaluated at tm, F (tm), K (tm) . Then the sensitivities with respect to perturbations in the drift and diffusion coefﬁcients of the SABR/ LIBOR market model can be approximated by

N−1 0,x ∗ ∗ ∂θ ψ ∆ ∆ ( a u)(0,x,(0,0)) = ∑ E [a˜m i(tm)] tm + O( N), m=0 − [ ( )] N 1 ∗ 1 0,x ∗ (1) (1) ∂θ ˜ ψ ˜ ψ ∆ ( b u)(0,x,(0,0)) = ∑ E trace bmbm (tm) + bmbm (tm) tm 2 m=0 ∆∗ + O( N), (6.4) ∆∗ where O( N) denotes the ﬁrst order discretization error. 36 6 Sensitivities

6.2 Sensitivities of the European swaption Now we can begin our calculation of the sensitivities of the European ϵ 3 16 swaption. The function gp,q(F(Tp)) in ( . ) can be used to approximate the sensitivity of the swaption price (3.7), with respect to perturbations of the parameters σ and ν in (4.11). We will use the sensitivity expressions, according to the adjoint method described in Section 5.2. As noted in Section 5.4, the sensitivity expressions involve the drift and diffusion coefficients and the first and second order dual functions. We will start this section with the calculations of the drift and diffusion coefficients and then continue with the explicit dual functions before we can state the sensitivities for the European swaption under the SABR/LI- BOR market model.

6.2.1 Drift and diffusion coefficients In this subsection we state the explicit expressions for the drift and diffusion coefficients, µ and η, for the forward measure, that we need in order to determine the non-perturbed drift coefficients a and the perturbed drift coefficients a˜, used in the sensitivity expressions. The drift corrections under the forward measure is presented in Sub- 4 10 4 10 3 µ section . . in [ ] . The first, î(t, f (t),K(t)), equals to  i ρ σ δ  ij ˆ j(t, f (t))Kj(t) j  σˆ (t, f (t))K (t) ∑ for i > p − 1,  i i + δ ( )  j=p 1 j fj t = − 6 5  0 if i p 1, ( . )  p−1 ρ σ δ  ij ˆ j(t, f (t))Kj(t) j  −σˆ (t, f (t))K (t) ∑ for i < p − 1, i i + δ ( ) j=i+1 1 j fj t η and the second, î(t, f (t),K(t)), equals to  i ρ σ δ  ij ˆ j(t, f (t))Kj(t) j  R νˆ (t,K(t)) ∑ for i > p − 1,  ii i + δ ( )  j=p 1 j fj t = − 6 6  0 if i p 1, ( . )  p−1 ρ σ δ  ij ˆ j(t, f (t))Kj(t) j  −R νˆ (t,K(t)) ∑ for i < p − 1. ii i + δ ( ) j=i+1 1 j fj t

Using the relations in (4.10) together with (6.5) and (6.6) one can easily see that ν ν η Rii î(t, K(t)) µ η Rii i(t) µ 6 7 î = σ î or, equivalently i = σ i ( . ) î(t, f (t))Ki(t) i(t) 3 This calculation is algebraically tedious, but conceptually straightforward, so we refer the reader to [10]. 6.2 Sensitivities of the European swaption 37 holds for the forward measure. Now we introduce the auxiliary function  i ρ σ δ  ij j(t) exp(Fj(t))Kj(t) j  ∑ for i > p − 1  + δ ( ( ))  j=p 1 j exp Fj t = − 6 8 s¯i(t,F(t),K(t)) =  0 if i p 1 ( . )  p−1 ρ σ ( ) ( ( )) ( )δ  ij j t exp Fj t Kj t j  ∑ for i < p − 1 + δ ( ( )) j=i+1 1 j exp Fj t so we can write the drift coefficients µ and η in equation (4.14) more com- pactly as µ σ η ν 6 9 i = iKisi and i = Rii iKisi. ( . ) Moreover, the perturbed drift and diffusion corrections to be inserted into (4.15) and (4.16) can also be written in a compact form with the auxiliary 6 8 function si in ( . ) as µ σ σ σ σ ˜i( ) = ˜iKisi + iKis˜i , η σ ν σ ˜i( ) = Rii iKis˜ , i (6.10) η ν ν ˜i( ) = Rii ˜iKisi.

σ σ σ Here s˜i is obtained by replacing in si by ˜ . The expressions in (6.9) and (6.10) are inserted into the formulas in (4.14)-(4.15) and give the ﬁnal form of the drift coefﬁcients as

ρ σ2K2 F σ − ij i i ai = iKisi , 2 6 11 K ν ( . ) ai = Rii iKisi,

F σ σ σ σ − ρ σ σ 2 F ν a˜i ( ) = ˜iKisi + iKis˜i ii i ˜iKi , a˜i ( ) = 0, K σ ν σ K ν ν 6 12 a˜i ( ) = Rii iKis˜i , a˜i ( ) = Rii ˜iKisi. ( . )

The diffusion coefﬁcients are already speciﬁed completely in terms of σ, ρ, ν, r and R in Subsection 4.3.1 so we do not specify those here.

6.2.2 The explicit dual functions In order to ﬁnd the ﬁrst and second order dual functions explicitly we ϵ ∆ need to differentiate the function gp,q(F (tN)) up to second order and then use the recursive expressions for the dual functions (4.27) and (4.28) in Subsection 4.3.2. 38 6 Sensitivities

ϵ ∆ The ﬁrst derivative of gp,q(F (tN)) is ( )′ ϵ ∆ ′ (1) Np,q ∂ ϕϵ ϕ 6 13 igp,q(F (tN)) =Qp(Mp,q)i + Qp Mp,q ϵ,i , ( . ) Mp,q i

(since Qp does not depend on Fi(Tp)) and the second order derivative is ( )′ ϵ ∆ ′′ ′ (1) Np,q ∂ ϕϵ ϕ ikgp,q(F (tN)) =Qp(Mp,q)ik + Qp(Mp,q)i ϵ,k + Mp,q k ( )′ ′ ϕ(1) Np,q + Qp(Mp,q)k ϵ,i Mp,q i ( )′( )′ (6.14) ϕ(2) Np,q Np,q + Qp Mp,q ϵ,ik Mp,q i Mp,q k ( )′′ ϕ(1) Np,q + Qp Mp,q ϵ,i , Mp,q ik ( ) Np,q where ϕϵ and all its derivatives are evaluated at − κ . The deriva- Mp,q tives are explicitly given by

q ′ exp(F ) (M ) = − ∑ B (T )δ2 i , p,q i k p + δ ( ) k=i 1 exp Fi ′′ ′ 1 − δ exp(F ) ( ) =( ) i Mp,q ii Mp,q i δ , 1 + exp(Fi) n 2 ′′ δ exp(F ) exp(F ) (M ) = ∑ i k δB (T ), p,q ik ( + δ ( ))( + δ ( )) k p k=max(i,k) 1 exp Fi 1 exp Fk 6 15 δ exp(F )B (T ) ( . ) ( )′ = i k p Np,q i δ , 1 + exp(Fi) ′′ ′ 1 − δ exp(F ) ( ) =( ) i Np,q ii Np,q i δ , 1 + exp(Fi) 2 ′′ δ exp(F ) exp(F ) ( ) = − i k ( ) Np,q ik δ δ Bk Tp , (1 + exp(Fi))(1 + exp(Fk))

( )′ ′ ′ N (N ) N (M ) p,q = p,q i − p,q p,q i 2 , Mp,q i Mp,q Mp,q ( )′′ ′′ ′ ′ N (N ) 2Np,q(Mp,q) (Mp,q) p,q = p,q ik + i j − (6.16) 3 Mp,q ik Mp,q Mp,q ′ ′ ′ ′ ′′ (Np,q)i(Mp,q)k + (Np,q)k(Mp,q)i + Np,q(Mp,q)ik 2 , Mp,q 6.3 Calculation of sensitivities 39

for p + 1 ≤ i, k ≤ q and i ̸= k. The ﬁrst and second order derivatives of ϕϵ are ( ) ( ) 1 ϕ 1 = − 5x7 + 21x5ϵ2 − 35x3ϵ4 + 35xϵ6 + 16ϵ7 , ϵ,i 32ϵ7 ( ) (6.17) ( ) 1 ϕ 2 = − 35(x2 − ϵ2)3 . ϵ,ik 32ϵ7

Now all the derivatives are specified and can be used to find the first and second order dual functions from the recursive formulas (4.27) and (4.28). This is done in the simulations and we omit the details.

6.3 Calculation of sensitivities We can now calculate the sensitivities in (5.11) with respect to the perturbations σ˜ and ν˜ of the European swaption under the SABR/LIBOR market model. We combine the above results with the result established in Section 5.2 and ﬁnd in our case that the sensitivities are

∆ ∆ ∆ ε ∆,M ,M ,M ,M ∗ 2 (∂θ u )(x,(0,0)) = u (x) + E + E + E + O((∆ ) ), a p a a,d a,d,s a,s N 6 18 ∆ ∆ ∆ ( . ) ε ∆,M ,M ,M ,M ∗ 2 (∂θ )( ( )) = ( ) + + + + ((∆ ) ) b up x, 0,0 ub x Eb,d Eb,d,s Eb,s O N , where

M N−1 ∆ ∆ ∆t u ,M(x) = ∑ ∑ a˜ (t , X (t , w ))ψ (t ,w ) k , a i k k m i k m M m=1 k=0 6 19 − [ ] ( . ) M N 1 ∗ ∆ ∆,M ˜∗ ˜ ψ(1) tk ub (x) = ∑ ∑ [bb + bb ]ij(tk) ij (tk,wm) . m=1 k=0 2M

Using the exact form of the drift and diffusion coefﬁcients in Subsection 6.2.1 and the exact dual functions from the recursive formulas together with the derivatives in Subsection 6.2.2 we ﬁnd the sensitivities for an European swaption using equation (6.19). To summarize, one can calculate the sensitivities with respect to perturbations in the parameters σ and ν using equation (6.19). Since the leading order error terms are computable in a posteriori form one can control the error in the estimations and make sure that the total discrete errors are within a given tolerance. In this article we focus on simulating the sensitivities only.

7 Simulations

This chapter begins with a description of how the SABR/LIBOR market model is calibrated. We also present the market data used to accomplish this calibration. Then we continue with an explanation of how the sensitivities were simulated and which perturbations that were considered.

7.1 Calibration data In the simulations for calibrating the SABR/LIBOR market model we have used market data from 15 June 2011 with 6 month intervals between matu- 8 rity times. The data includes values of volatilities for caplets for different strike prices and maturity times and can be found in Appendix A.1. Note that the forward rates and maturity times in Table 1.1 in Appendix A.1 are multiplied with 0.01.

7.2 Calibration of the SABR/LIBOR Market Model To calibrate the SABR/LIBOR market model (4.11) we need to specify the volatility σ, the volatility of the volatility ν, the correlation matrices ρ, r and R and the initial values F(0) and K(0) so that the SABR/LIBOR market model replicates forward interest rates and volatilities of the market. In order to accomplish this we first use market data to find the initial SABR σTi νTi volatility 0 , the SABR volatility of volatility of the forward rates and the correlations Rii. All these values are needed in order to find the parameters in (4.11) that we are looking for. Applying the method of grid-search to the special case (β = 1), ap- 3 4 3 10 νTi σTi proximation in Subsection . . of [ ], we can find , 0 and Rii. The approximation is given by ( ) { [ ] } σTi νTi − 2 z Rii 2 3R bσ(K) = σTi · · 1 + 0 + ii (νTi )2 T , (7.1) 0 χ(z) 4 24 where ( ) νTi f z = ln , σTi K 0 ( √ ) (7.2) − 2 − χ 1 2Riiz + z + z Rii (z) = ln − . 1 Rii

νTi σTi The optimal , 0 and Rii are found by repeating the calculation of the root mean square, RMS, of the difference between the known volatilities

8 An interest rate caplet is a derivative in which the buyer receives payments at the end of each period if the interest rate exceeds the agreed strike price. 42 7 Simulations from the market data and the estimated volatilities by equation (7.1) for νTi σTi different values of , 0 and Rii and then choosing the ones who gives the smallest RMS. σTi νTi When 0 , and Rii are speciﬁed we move on with calculating the volatility σ and the volatility of the volatility ν. In Section 4.6 - 4.8 of [10] the authors argue that the volatility σ and the volatility of volatility ν to be 9 time homogeneous . Shortly, this choice is a result from the assumption that these parameters do not depend on the calendrical time, only on the residuals between maturity times. Hence, we let σ ( ) = ( − ) i t g Ti t , 7 3 ν − ( . ) i(t) = h(Ti t), for some functions g and h. Following Chapter 5 in [10] we note that the functions g and h, the correlations Rii and the initial values of the volatilities K(0), can be deduced from the SABR parameters corresponding to a collection of caplets. The function g is, according to some arguments from Section 2.2 and 5.2 in [10], chosen to be of the form − − − − 7 4 g(Ti t) = (a + b(Ti t)) exp( c(Ti t)) + d ( . ) and minimizes the squared discrepancies √ ( ) ∫ n 2 1 Ti σTi − − 2 7 5 ∑ 0 gˆ(Ti) , where gˆ(Ti) = (g(Ti t)) dt. ( . ) i=1 Ti 0

σTi In other words, the function g is chosen to match the expectation 0 as σTi closely as possible. In general, it is impossible to match 0 perfectly. To compensate for this ﬂaw, we let the initial volatility be deﬁned by

σTi 0 7 6 Ki(0) = . ( . ) gˆ(Ti)

So, if the model is well calibrated the initial volatility Ki(0) should be very close to one. Continuing in the same way for the function h, following Section 5.3 in [10] we choose − α β − −γ − δ 7 7 h(Ti t) = ( + (Ti t)) exp( (Ti t)) + ( . ) that minimizes the squared discrepancies ( √ ) ∫ 2 n K (0) Ti d ∑ νTi − i ( )2( ( ))2 7 8 T 2 g t h t tdt , ( . ) σ i 0 i=1 0 Ti 9 Homogeneous means uniform in composition or character. 7.3 Simulating the sensitivities 43 where √ ∫ d 1 t h(t) = (h(s))2ds. (7.9) t 0 The initial values F(0) are found by inserting the forward rates used by the market at some particular date into Fi(t) = log( fi(t)), i.e. equation (4.9) in Section 4.3. Finally we can use the SABR correlations between the forward rate fi(t) and its own volatility Ki(t), namely Rii found from the market data above, to specify the three correlation matrices. In [10] the correlation matrices ρ and r are specified for a normal market day, so for simplicity we will use these matrices but expand them to the right dimension. In order to expand them we try to find a function that describes the change in the matrix elements between maturity times and then add new maturity times into the function to obtain more matrix elements. For the correlation matrix R we already know the diagonal elements, Rii, but all the other elements 10 are chosen, just like in [ ], to be equal to the mean of all Rii = 0.48. This choice will effect wether the super-correlation matrix of ρ, r and R is positively definite or not, which is explained in Chapter 9.

7.3 Simulating the sensitivities In the simulations of sensitivities under the SABR/LIBOR market model we use the result from the calibration. We also need to specify some parameters and which perturbation we are interested in. The parameters are presented in Table 7.1. We also need to choose an appropriate value of ϵ. In [6] it is shown ϵ that the error in approximating the swaption price up with up decreases as ϵ2 when ϵ goes to zero, suggesting that a very small value of ϵ should be used. On the other hand, they also show that the sensitivities initially increase as ϵ is decreased and then saturate at some level. The conclusion − in [6] is that we should choose ϵ < 10 1 but not too small so we choose − ϵ = 10 2.

Parameter Value Description NperI 10, 20, 40, 80 Number of time steps per maturity time interval M 105 Number of trajectories κ 0.04 Swap rate Tp 5 When we enter the swap (years) T 10 End time for the swap (years)

Table 7.1: The chosen parameters used in the sensitivity simulations. 44 7 Simulations

7.3.1 Perturbations We consider three different perturbations of the volatility σ and the volatility of volatility ν. The ﬁrst perturbation is called high and the second is called low, which means that we perturb the value of a and α in the functions g and h by ±0.3. The third and ﬁnal perturbation is called right because we perturb the function g and h so that the maximum value is moved to the right. This is done by changing the g and h function as − − − − − g˜R(Ti t) = (a 0.7 + b(Ti t)) exp(( c + 0.09)(Ti t)) + d, (7.10) − α β − −γ − δ h˜ R(Ti t) = ( + 0.07 + (Ti t)) exp(( + 0.15)(Ti t)) + .

In Figure 7.1 and 7.2 we present the high, low and right perturbations of the g and h functions together with the normal functions found from the calibration.

Figure 7.1: The normal and perturbed g functions that estimates σ and σ˜ . The considered perturbations of g are used when simulating the sensitivities of the European swaption. 7.3 Simulating the sensitivities 45

Figure 7.2: The normal and perturbed h functions that estimates ν and ν˜. The considered perturbations of h are used when simulating the sensitivities of the European swaption.

7.3.2 Algorithm The algorithm for simulating the sensitivity with respect to one of the above perturbations can be described by the following steps. 1. Start the Monte Carlo simulations and loop over trajectories. 2. Using Euler approximation on the SABR/LIBOR market model, (4.11), µ η to find i, i, Fi and Ki. 3 ϵ . Define the swaption price formula gp,q and the involving functions in (3.16). 4 ϵ . Define the derivatives of gp,q and the involving functions according to (6.13)-(6.17). 5. Determine the derivatives of the drift coefficients µ and η using (4.10) and (6.5)-(6.6). 6. Determine the dual functions using the recursive formulas (4.26) and (4.28). 7 . Determine the drift and diffusion coefficients ai, a˜i, bi and b˜i as in (4.14)-(4.16) and (6.11)-(6.12). 46 7 Simulations

8. Calculate the sensitivities using equation (6.19), where we sum over time steps and then take the mean over all Monte Carlo simulations.

9. Plot result and calculate conﬁdence intervals.

All these steps follow the theory previously described. We continue with a short explanation of how to simulate the error terms. As mentioned before we will focus on simulating the sensitivities only and leave the error term approximation. However, in the next section one can read about how to implement the error simulations.

7.4 Simulating the error terms The difference between simulating the sensitivities and the error terms is that we need to determine a lot more involved derivatives and functions. The algorithm is almost the same but each step involves a lot more computations. Instead of just finding the dual function ϕ and the first derivative ϕ(1) we also need the second and third derivative of the dual function, ϕ(2) and ϕ(3). To determine these we also need higher derivatives of all the ϵ involving functions in the option pricing formula and on gp,q itself. One new calculation that we do not need in the sensitivity simulations is to find the Euler approximations and dual functions of the extended system. This involves new derivatives of the drift and diffusion coefficients and is described in detail in Section 5.1. To summarize, simulating the error terms is a quite computational task but follows the same algorithm as when simulating the sensitivities. 47

8 Results

In this chapter we present the result from the calibration of the SABR/LIBOR market model and the sensitivity simulations.

8.1 Calibration results When calibrating the SABR/LIBOR market model we want to find the optimal values of the volatility σ, the volatility of the volatility ν, the correlation matrices ρ, r and R and the initial values F(0) and K(0) that fits the market data. νTi σTi Before we could find these parameters we needed to find , 0 and Rii by using the method of grid-search. The optimal values of these parameters were find using a limited interval of values for each parameter. − νTi σTi The intervals are Rii = [ 0.7,0.6], = [0,1] and 0 = [0,1] and are all divided into 70 subintervals in the grid-search. To find these intervals we started with a large interval and performed the grid-search a number of times in order to distinguish shorter and shorter intervals until we finally were satisfied with the results. The result from this grid-search is shown in Table 8.1. These values should give an approximation of the volatilities for different strike prices, the volatility smile. We present four graphs, one for each of the first four interest rates, of how accurate the approximations are to the volatility smile in Figure 8.1.

0 4550 0 41731 0 4927 0 4550 0 4927 0 4927 Rii = [- . - . - . - . - . - . -0.5115 9 -0.473 -0.4362 -0.4550 -0.4739 -0.4550 -0.4739 -0.4927 -0.4927 -0.5115 -0.5115 -0.5304 -0.5304] ν 0 7826 0 6087 0 4348 0 5072 0 5217 0 5217 Ti = [ . - . . . . . 0.5217 0.5217 0.5362 0.5217 0.5217 0.5217 0.5217 0.5217 0.5072 0.5072 0.5072 0.5217 0.5217] σ0 = [0.1739 0.2319 0.3043 0.3043 0.3333 0.3188 0.3333 0.3043 0.2899 0.2899 0.2899 0.2754 0.2754 0.2754 0.2754 0.2754 0.2754 0.2754 0.2754]

νTi σTi Table 8.1: The calibrated values of Rii, and 0 found from the market data using the method of grid-search. Each value represent the parameter for each maturity time Ti.

Using the result in Table 8.1 we ﬁnd the functions g and h to be

( − ) = ( + ( − )) (− ( − )) − g Ti t 0.2778 1.0482 Ti t exp 0.4133 Ti t 0.6167, 8 1 − − − − − ( . ) h(Ti t) = (0.0786 0.1(Ti t)) exp( 0.7(Ti t)) + 0.5286. 48 8 Results

Figure 8.1: The estimated and known volatilities for each strike price for the maturity 1 3 times T1 = 2 , T2 = 1, T3 = 2 and T4 = 2 for the market data.

σ ν These functions should approximate i and i, respectively, and in Fig- ure 8.2 - 8.3 these approximations are visualized. On a first look one would think that the approximations of g and h are not so good, but in fact they only have a RMS value of 0.0023 and 0.14 respectively, compared to the − volatility and volatility of volatility values in order 10 1. The reason that we can not make the functions g and h fit the curves even better is the form of the functions defined in [10], see (7.4) and (7.7), together with the fluctuations occurred in the forward rates. But the approximations are in fact good even if the graphs may be a little misleading since we get values of the initial volatility very close to one. This is namely an indication of a successful calibration. The initial values F0 and K0 is shown in Table 8.2 below. 8.1 Calibration results 49

Figure 8.2: The rings represents the ”known” volatilities found from the market data and the curve is the g function that estimate theses volatilities for each maturity time.

F0 = [-3.5526 -3.4991 -3.4594 -3.4335 -3.4150 -3.3928 -3.3690 -3.3437 -3.3186 -3.3088 -3.3042 -3.2780 -3.2752 -3.2981 -3.2855 -3.2702 -3.2749 -3.2893 -3.3102] K0 = [1.0060 0.9805 1.0559 0.9528 0.9995 0.9514 1.0149 0.9606 0.9563 1.0017 1.0454 1.0278 1.0513 1.0609 1.0568 1.0408 1.0163 0.9865 0.9543]

Table 8.2: From the calibration we ﬁnd these values of the initial interest rates and the initial volatilities. Each value of F0 and K0 is found for one speciﬁc maturity time Ti.

Finally, we consider the correlation matrices. The correlation matrices for the market data are presented graphically in Figure 8.4, 8.5 and 8.6. In Figure 8.7 we also present the result of the super-correlation matrix after we have adjusted it so that it is positively deﬁnite. This adjustment is explained further in Chapter 9. 50 8 Results

νTi σTi Figure 8.3: The rings represents the values of ( 0 Ti)/K0 for each maturity time, found from the market data. The curve is the estimation to those ”known” values and represents how accurate the h function is to the known volatilities of volatilities for the market data. 8.1 Calibration results 51

Figure 8.4: The correlation matrix ρ, i.e. the forward rate/forward rate correlations.

Figure 8.5: Plot of the correlation matrix r, i.e. the volatility/volatility correlations. 52 8 Results

Figure 8.6: Plot of the correlation matrix R, i.e. the forward rate/volatility correlations.

Figure 8.7: The positive deﬁnite super-correlation matrix P. 8.2 Simulation results 53

8.2 Simulation results In this section we present the result from simulating the sensitivities with the new method when considering the high, low and right perturbation of the volatility σ and the volatility of volatility ν. We start with the sensitivities when perturbing σ. In Figure 8.8 we can see the two sensitivity terms ε ε (∂θ ) 1 (∂θ ) 2 a(σ) up , denoted sensitivity and b(σ) up , denoted sensitivity for the high perturbation of σ. Together, these terms give rise to the total sensitiv- σ ∂ ε ity of the price when perturbing the volatility , denoted as ( θσ up). This total sensitivity is found in Figure 8.9 for the high perturbation of σ. The same results for the other perturbations of σ is found in Figures 8.10 - 8.13 and the corresponding results for the perturbations of ν can be seen in Figures 8.14 - 8.19. In all the graphs we also presents the 95% conﬁdence 10 interval of the results. Since perturbations in σ affects both the drift and diffusion coefﬁcient it is more interesting to analyze the total sensitivity than sensitivity 1 and sensitivity 2 separately but we still present these results so one can see the convergence of the two sensitivities.

Figure 8.8: The sensitivity with respect to the drift and diffusion coefficient when we consider the high perturbation of σ. Sensitivity 1 is with respect to the drift coefficient and sensitivity 2 is with respect to the diffusion coefficient.

10 The upper conﬁdence limit is denoted as UCL and the lower conﬁdence limit as LCL. 54 8 Results

Figure 8.9: The total sensitivity of the swaption price when we consider the high perturbation of σ.

In Tables 8.3 - 8.5 we present the result for all the simulations when we have used 80 time steps and 105 trajectories, with respect to the high, low and right perturbations.

Sensitivity LCL Value UCL ε (∂θ ) 0 0584 0 0574 0 0564 a(σ) up - . - . - . ε (∂θ ) 0 0371 0 0579 0 0786 b(σ) up . . . ∂ ε 0 02073 0 00047 0 02166 ( θσ up) - . . . ε (∂θ ) 0 00162 0 00158 0 00154 a(ν) up - . - . - . ε (∂θ ) 0 1344 0 1435 0 1526 b(ν) up . . . ∂ ε 0 1328 0 1419 0 1510 ( θν up) . . .

Table 8.3: The sensitivities for the high perturbations with 95% conﬁdence intervals from the simulations based on the new method and SABR/LIBOR market model. 8.2 Simulation results 55

Sensitivity LCL Value UCL ε (∂θ ) 0 0322 0 0317 0 0312 a(σ) up - . - . - . ε (∂θ ) 0 0347 0 0353 0 0359 b(σ) up . . . ∂ ε 0 0029 0 0036 0 0043 ( θσ up) . . . ε (∂θ ) 0 00139 0 00135 0 00131 a(ν) up - . - . - . ε (∂θ ) 0 1154 0 1232 0 1311 b(ν) up . . . ∂ ε 0 1141 0 1219 0 1297 ( θν up) . . .

Table 8.4: The sensitivities for the low perturbations with 95% conﬁdence intervals from the simulations based on the new method and SABR/LIBOR market model.

Sensitivity LCL Value UCL ε (∂θ ) 0 0385 0 0379 0 0373 a(σ) up - . - . - . ε (∂θ ) 0 0383 0 0391 0 0399 b(σ) up . . . ∂ ε 0 0002 0 0012 0 0022 ( θσ up) . . . ε (∂θ ) 0 001514 0 001476 0 001438 a(ν) up - . - . - . ε (∂θ ) 0 1183 0 1321 0 1459 b(ν) up . . . ∂ ε 0 1168 0 1306 0 1445 ( θν up) . . .

Table 8.5: The sensitivities for the right perturbations with 95% conﬁdence intervals from the simulations based on the new method and SABR/LIBOR market model. 56 8 Results

Figure 8.10: The sensitivity with respect to the drift and diffusion coefficient when we consider the low perturbation of σ. Sensitivity 1 is with respect to the drift coefficient and sensitivity 2 is with respect to the diffusion coefficient.

Figure 8.11: The total sensitivity of the swaption price when we consider the low perturbation of σ. 8.2 Simulation results 57

Figure 8.12: The sensitivity with respect to the drift and diffusion coefficient when we consider the right perturbation of σ. Sensitivity 1 is with respect to the drift coefficient and sensitivity 2 is with respect to the diffusion coefficient.

Figure 8.13: The total sensitivity of the swaption price when we consider the right perturbation of σ. 58 8 Results

Figure 8.14: The sensitivity with respect to the drift and diffusion coefficient when we consider the high perturbation of ν. Sensitivity 1 is with respect to the drift coefficient and sensitivity 2 is with respect to the diffusion coefficient.

Figure 8.15: The total sensitivity of the swaption price when we consider the high perturbation of ν. 8.2 Simulation results 59

Figure 8.16: The sensitivity with respect to the drift and diffusion coefficient when we consider the low perturbation of ν. Sensitivity 1 is with respect to the drift coefficient and sensitivity 2 is with respect to the diffusion coefficient.

Figure 8.17: The total sensitivity of the swaption price when we consider the low perturbation of ν. 60 8 Results

Figure 8.18: The sensitivity with respect to the drift and diffusion coefficient when we consider the right perturbation of ν. Sensitivity 1 is with respect to the drift coefficient and sensitivity 2 is with respect to the diffusion coefficient.

Figure 8.19: The total sensitivity of the swaption price when we consider the right perturbation of ν. 61

9 Discussion

In this chapter we discuss the method used to ﬁnd the sensitivities and the results from the simulations. We start by discussing the calibration technique and continue with the sensitivity simulations.

9.1 The Calibration When calibrating the SABR/LIBOR market model to the market we used data from June 15 2011, when the market was quite low. Even thugh the market was low we decided to follow some of the assumptions made in [10] when calibrating the SABR/LIBOR market model, where they both considered normal and excited data. When we compare the result in our calibration to the results in [10] we can right away distinguish some differences. The differences can mostly be explained by fluctuations in our market data. When the data was constructed from caplets data with 1 year intervals between the maturity times and forward rates it occurred some fluctuations in the forward rates. For 6 month intervals these fluctuations were minor but if one would consider a market data with even shorter intervals between the maturity times the fluctuations would become larger. These fluctuations makes it more difficult to calibrate the data and especially to approximate the g and h func- σ ν tions to the values of 0 and Ti . In order to get as good approximations as we did for the 6 month data we had to limit the possible values of the ν σ parameters in the grid-search of Rii, Ti and 0. Hence, if one would like to make the same simulations for say 3 month data or even closer maturity times this limitation is something to keep in mind. In Figure 8.1 we observe the first step in the calibration and can see ν σ that the approximations of Rii, Ti and 0 give a very good approximation of the volatility smile, which is expected. These values are used to specify the parameters in the SABR/LIBOR market model, which also provided good results since values of K0 are very close to 1 for all forward rates. As mentioned in Section 8.1 the g and h functions are in fact good ap- σ ν 8 2 8 3 proximations of i(t) and i(t) respectively, even if Figure . and . may indicate differently. One can of course approximate even better functions σ ν to i(t) and i(t) but then one would have to assume a different form of the functions. Hence, this is also something to consider if one would like to calibrate the SABR/LIBOR market model to a data set with shorter intervals between the forward rates and maturity times. One thing that we did not choose as they did in [10] was the value of β. The choice of β = 1 was made because it is preferable when the forward rates are either very low or very high and in contrast β = 0.5 is used when we have normal market conditions. Of course, this choice could affect the result since it changes the SABR/LIBOR market model 62 9 Discussion and then indirectly the sensitivities but the conclusion about the method, if it is a good method or not, should not be affected by the choice of β since in theory it should work for all choices of β between 0 and 1.

9.2 The Sensitivities One problem that occurred when we started simulating the sensitivities was that the super-correlation matrix, P was not positive definite. This matrix needs to be positive definite in order to construct the correlation 10 increments dzi and dwi in the SABR/LIBOR market model. In [ ] a solution to this problem is presented but in our case we could not use the suggested solution since our matrix was too small. Instead we simply defined the three negative values of P as zero. This is of course not an optimal solution but we were running out of time and could at the time not come up with another solution. The number of negative eigenvalues in the correlation matrix P depends on how the correlations in ρ, r and R are defined. If one have more 11 values in the data such that P have greater than or equal to M number of positive eigenvalues, then a positive definite matrix can be constructed following the strategy suggested in Subsection 7.2.2 in [10]. But for our P matrix this was not the case and we had to come up with another solution. This simple solution could have affected the values of the forward rates and volatilities found from the SABR/LIBOR market model but we could not see any such influences when comparing them to the original values. We also checked that this simplification did not affect the correlation con- ρ dition, i.e. that the unit vectors of and r, defined as bi and ci satisfies · bi ci = Rii, which it did not. So our conclusion is that this correction of the super-correlation matrix P has not affected the results significantly. A consequence of the fact that we had to adjust the super-correlation matrix was that we had to use new correlation matrices of ρ, r and R in the formulas of b, b˜ and the derivatives of µ, η and c. These changes made it difficult to analyze the sensitivities with respect to perturbations in the correlation parameters, which we had planned to simulate. The problem is that if we perturb the correlation ρ it will affect all of the new correlation matrices and make it difficult for us to distinguish how much each correlation parameter affects the sensitivity. Hence, in order to perform a sensitivity simulation with respect to perturbations in the correlation parameters we need to analyze how much each new correlation matrix is affected by the adjustment and how this can be interpreted into the perturbed drift and diffusion coefficients a˜ and b˜, respectively. Another issue was how we were going to verify that the new method actually works, since we do not know the correct values of the sensitivities.

11 M is in [10] the dimension of the Brownian motion. 9.2 The Sensitivities 63

We can not draw any conclusions based only on the numbers in the result, of course the 95% confidence interval gives an interpretation of how much the results spread but if the result itself is wrong this confidence interval does not give any useful information. Hence, to analyze the result and come to a conclusion about the new method of calculating sensitivities we have chosen to plot the results against increasing number of time steps. If the method works, the sensitivities should converge as we increase the number of time steps since the calculations then becomes more accurate. These graphs for the considered perturbations are shown in Section 8.2 and they all show how the sensitivities converge, even if the confidence levels are a little odd. Another way of determining if the method works or not would have been to calculate the sensitivities using another method, for example the finite difference approximation. But this would take a bit more time, which we did not have, so instead we simulated the sensitivities with respect to increasing number of time steps. We end this discussion with some thoughts about the numerical results. In Tables 8.3 - 8.5 we can see that all perturbations of σ give a lower sensitivity than the perturbations of ν. This means that the price of the Eu- ropean swaption is more sensitive to changes in the volatility of volatility ε 8 3 8 5 ∂θ then in the volatility itself. We can also se from Tables . - . that ( b up), the sensitivity with respect to the diffusion coefficient, is always larger ∂ ε than ( θa up), the sensitivity with respect to the drift coefficient. Hence, the sensitivity with respect to the diffusion coefficient contributes the most to the total sensitivity of the price. Another observation is that the sensitivity with respect to the drift coefficient is always negative, which means that the price of the swaption decreases when the volatility σ and the volatility of volatility ν increase on the market. ε ∂θ We can also see that the statistical error is larger for ( b up) than for ∂ ε ( θa up). The unexpected result is that the statistical error often increases in the third simulation, i.e. when we consider 40 number of time steps. This can bee seen by the widening of the confidence interval in the figures in Section 8.2. We do not have a reasonable explanation for this behavior since the only difference between the simulations is number of time steps. Most likely, this is a bug in the code or a round off error in the program and should not be seen as a default of the method since the sensitivity still continues to converge.

10 Conclusion

Based on the results and the discussion above we have come to the conclusion that the new method of calculating sensitivities based on a stochastic volatility market model is a good choice when considering a complex financial derivative. Since the method is quite computational it is recom- mended to use the method when considering a small number of securities with respect to a large number of parameters. The advantages of this method is that the result is more market-driven since we use a stochastic volatility market model which better predicts the volatility smile than earlier used deterministic volatility market models. For further research we suggest that an analysis of how perturbations of the correlation parameters affects the price and how one can implement this using the new method. The first step in this analysis is to find out how a perturbation in for example ρ affects the new correlation matrices after the adjustment making the super-correlation matrix positively definite. When this is clear one can continue with the implementation and simulate the sensitivities with respect to correlation perturbations and analyze the results. Another suggestion connected to this is to analyze how the adjustment of the super-correlation matrix can be done more correctly. Since the strategy explained in [10] only can be used when the number of positive eigenvalues is greater than the dimension of the Brownian motion, it would be a good idea to find another strategy that can be used when this condition is not fulfilled. Finally we can recommend the interested reader to further test the new method on other complex derivatives and also try to estimate the time discretization error using the new method and the a posteriori approach. 66

11 References

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A Appendix

For interested readers we presents some useful material in the following appendix.

A.1 Market data The market data from 15 June 2011 is given in Table 1.1 below. Note that the forward rate and strike prices are multiplied by 0.01. 70 AAppendix 10 2820 2840 2861 2957 3071 3015 3014 2833 2880 2810 2740 2690 2640 2595 2550 2510 2470 2445 2420 ...... 01 7 . 2820 0 2840 0 2864 0 2985 0 3098 0 3017 0 3009 0 2950 0 2880 0 2810 0 2740 0 2690 0 2640 0 2595 0 2550 0 2510 0 2470 0 2445 0 2420 0 0 ...... 6 2550 0 2666 0 2802 0 2923 0 3061 0 2968 0 2979 0 2905 0 2840 0 2775 0 2710 0 2665 0 2620 0 2570 0 2520 0 2485 0 2450 0 2425 0 2400 0 ...... 2170 0 2440 0 2787 0 2884 0 3051 0 2947 0 2963 0 2880 0 2830 0 2770 0 2710 0 2670 0 2630 0 2585 0 2540 0 2505 0 2470 0 2445 0 2420 0 ...... 5 5 . 2100 0 2394 0 2790 0 2889 0 3066 0 2971 0 2998 0 2885 0 2840 0 2790 0 2740 0 2700 0 2660 0 2620 0 2580 0 2545 0 2510 0 2485 0 2460 0 ...... 2050 0 2381 0 2828 0 2918 0 3115 0 2983 0 3043 0 2905 0 2880 0 2835 0 2790 0 2755 0 2720 0 2680 0 2640 0 2610 0 2580 0 2555 0 2530 0 ...... 5 4 4 . 2000 0 2407 0 2908 0 2977 0 3208 0 3076 0 3168 0 2955 0 2940 0 2905 0 2870 0 2840 0 2810 0 2775 0 2740 0 2715 0 2690 0 2665 0 2640 0 ...... K 1920 0 2462 0 3019 0 3081 0 3371 0 3196 0 3301 0 3038 0 3022 0 3010 0 2990 0 2965 0 2940 0 2915 0 2890 0 2865 0 2840 0 2815 0 2790 0 ...... Note that the forward rate and strike prices are multiplied by K 5 3 3 . 1930 0 2541 0 3158 0 3259 0 3586 0 3420 0 3558 0 3235 0 3240 0 3225 0 3210 0 3190 0 3170 0 3145 0 3120 0 3095 0 3070 0 3050 0 3030 0 ...... 2350 0 2858 0 3461 0 3611 0 3926 0 3771 0 3882 0 3642 0 3658 0 3625 0 3600 0 3575 0 3550 0 3520 0 3490 0 3465 0 3440 0 3415 0 3390 0 ...... 5 2 2 . 2830 0 3299 0 3896 0 4097 0 4398 0 4282 0 4392 0 4200 0 4190 0 4155 0 4120 0 4090 0 4060 0 4025 0 3990 0 3960 0 3930 0 3900 0 3870 0 ...... and different maturity times f 4000 0 4232 0 4518 0 4814 0 5123 0 5040 0 5121 0 4970 0 4950 0 4910 0 4870 0 4830 0 4790 0 4750 0 4710 0 4675 0 4640 0 4610 0 4580 0 ...... month data used in the calibration of the SABR/LIBOR market model and sensitivity simulations. The data includes the market 5 1 1 6 . 0 6380 0 6042 0 5633 0 6087 0 6435 0 6359 0 6450 0 6305 0 6270 0 6215 0 6160 0 6110 0 6060 0 6015 0 5970 0 5925 0 5880 0 5845 0 5810 0 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The : 1 . · 1 f 01 . 8650 0225 1450 2275 2875 3615 4425 5305 6205 6560 6730 7705 7810 6955 7420 8000 7820 7280 6510 ...... 0 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Table volatilities for each forward rate