Arbitrage-Free Interpolation in the LIBOR Market Model

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MSc Stochastics and Financial Mathematics

MSc Econometrics, track Financial Econometrics

Master Thesis

Arbitrage-Free Interpolation in the

LIBOR Market Model

Author: Supervisors:

T.E. ter Bogt

dr. A Khedher

prof. dr. ir. M.H. Vellekoop

Examination date: Company supervisors:

October 17, 2018

M. Michielon

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Statement of Originality This document is written by Student Tjeerd Egbert ter Bogt who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business and the Faculty of Science are responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

This thesis is concerned with the interpolation of forward rates in the displaced LIBOR Market Model. The displaced LIBOR Market Model produces a discrete set of forward rates, which does not necessarily contain all required forward rates. We propose a new method, which can be used to infer non-modelled forward rates from modelled forward rates. In contrast to existing methods, our method is both arbitrage-free and results in natural volatilities for interpolated forward rates. It is an extension of an interpolation method suggested by Werpachowski (2010), which is defined in the non-displaced LIBOR Market Model, and is incompatible with negative interest rates. The new method is the first arbitrage-free interpolation method that produces natural volatilities for interpolated forward rates in the displaced LIBOR Market Model. This interpolation method can be used in a negative interest rate environment. We demonstrate that the proposed interpolation method may be applied to improve valuation of interest rate derivatives and estimate CVA (Credit Valuation Adjustment) more accurately.

Title: Arbitrage-Free Interpolation in the LIBOR Market Model Author: T.E. ter Bogt, bertterbogt@gmail.com, 11273887 Supervisors: dr. A. Khedher, prof. dr. ir. M.H. Vellekoop Company supervisors: M. Michielon, dr. R. Pietersz

Second Examiners: prof. dr. H.P. Boswijk, prof. dr. P.J.C. Spreij Examination date: October 17, 2018

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

Amsterdam School of Economics University of Amsterdam

Roetersstraat 11, 1018 WB Amsterdam http://ase.uva.nl

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Acknowledgements

This thesis concludes the master’s degree in Stochastics and Financial Mathematics and the master’s degree in Econometrics, both at the University of Amsterdam. Conducting research and writing the thesis have been challenging, yet enjoyable experiences. In six months, I have learned much about both the theory and practice of quantitative finance.

I would like to express my gratitude to Dr. Asma Khedher and Dr. Michel Vellekoop for their supervision and encouragement. Moreover, I would like to thank Dr. Raoul Pietersz and Matteo Michielon for their many comments and remarks, for proposing the research topic, for their help with debugging my code and, along with all their colleagues, for creating an interest-ing and enjoyable workinterest-ing environment. I would like to thank Dr. Peter Spreij and Dr. Peter Boswijk for acting as a second examiner for the thesis. Finally, I would like to thank my family and friends for their support, and in particular Eline Palstra, also for her extensive proofreading.

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Introduction

Interest rates are ubiquitous throughout the entire financial system. Mortgages, deposit ac-counts, pensions and insurances are examples of very familiar financial products that crucially depend on the interest rate. For financial companies that provide such products, market changes in the relevant interest rates can dramatically impact their balance sheets and profits. Yet, also non-financial companies are affected by changes in interest rates, if only because they often fund their business with debt, which requires interest rate payments. For this reason, there exists a very large market for financial contracts that attempt to reduce the exposure of companies to changes in the relevant interest rates. Examples of these contracts are interest rate swaps, caps and floors. Typically, these so-called interest rate derivatives are sold by banks. One method to determine the correct price of an interest rate derivative is to simulate the future movements of the interest rates until the end of the financial contract. The evolution of interest rates is also highly important for pension funds and insurance companies, whose liabilities typically have long maturities and are therefore sensitive to changing interest rates. Since interest rates are often directly linked to both the assets and the liabilities of financial companies, they are a main driver of risk, and therefore interest rate risk is under much scrutiny. Especially with the recent influx of regulation aimed at creating a more stable financial system, risk management applications of interest rate modelling have become increasingly important.

Given the practical relevance of interest rate modelling, it should be unsurprising that there exists a large mathematical literature suggesting various different methods of stochastically modelling the interest rate. Prime examples are Vasicek (1977), Hull and White (1990) and Heath et al. (1992), who introduce different approaches of modelling the interest rate curve. These models are very well-known and all come with their own advantages and disadvantages, making them more or less useful in particular applications. Another methodology was introduced by Miltersen et al. (1997), Brace et al. (1997) and Jamshidian (1997), the so-called LIBOR Market Model. Here, a finite number of forward interest rates are modelled using the log-normal distribution. The name ‘market’ model is chosen as it differs from previous methods in that market observable quantities are modelled directly. Moreover, it allows application of Black’s formula (Black, 1976) which is often used in banks dealing in interest rate options. An advantage of the LIBOR Mar-ket Model is that it can be easily calibrated to the marMar-ket for interest rate derivatives such as caps and floors using Black’s formula. LIBOR, the London Interbank Offered Rate, is the most important class of forward interest rates in the financial system, and many other interest rates follow movements of the LIBOR rates, which is why the LIBOR Market Model is named after this class of interest rates. However, the model has no direct relationship with LIBOR rates, and in principle every type of forward rates could be modelled. It should have no effect on the usefulness of the model if LIBOR rates cease to be the dominant interest rate in the future. The LIBOR Market Model will be the focus of this thesis.

Of course, the LIBOR Market Model is not without its own disadvantages. For example, in its log-normal form, it is unable to cope with negative interest rates. Negative interest rates were long believed to be impossible, but in the aftermath of the European debt crisis, bonds of several northern European countries were valued at a price that implied a negative interest rate. Several Euro-denominated reference interest rates such as EURIBOR and Eonia have been negative for several years. A solution for this is the displaced LIBOR Market Model (see Brigo and Mercurio, 2001). This extension was originally introduced to allow for a volatility skew (the phenomenon

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that out-of-the-money options on the interest rate typically have a relatively higher price than similar in-the-money options on the interest rate if one uses the Black’s formula), yet also enables the model to cope with negative interest rates. In this thesis, we construct the displaced LIBOR Market Model, using the backward induction construction proposed by Musiela and Rutkowski (1997). This is a generalisation in two dimensions of the paper by Musiela and Rutkowski (1997). The displaced forward rates are modelled, rather than the original forward rates, which means the model can be applied with negative interest rates. Modelling the displaced forward rates also allows us to include a volatility skew in the LIBOR Market Model, which is not possible in the non-displaced LIBOR Market Model (theoretically, yet there is not much control over the skew in practice). Second, the elements of the multidimensional Brownian motion are allowed to be correlated. In fact, we employ a known method to construct an arbitrage-free LIBOR Market Model to a generalised setting and obtain a well-known model, which is suggested in Brigo and Mercurio (2001).

However, interest rates depend on the time period they apply to. The displaced LIBOR Market Model only models a finite number of forward interest rates. Sometimes we might need a forward interest rate that does not correspond to one of the modelled forward rates (if it applies to a different time period). In those cases, we need to interpolate the modelled forward rates appro-priately. The first requirement of any interpolation method is that the resulting model (after applying the interpolation method) is arbitrage-free. Only if the interpolation method results in an arbitrage-free model, we can use the model to price interest rate derivatives. This criterion is relatively well-understood and several methods exist that satisfy it, such as Schlögl (2002), Bev-eridge and Joshi (2009) and Werpachowski (2010). A second important aspect is the volatility of interpolated forward rates. Most interpolation methods in the literature result in interpolated forward rates with unnaturally low volatility. When volatility of forward rates (with a constant accrual period) is considered as a function of the start of the accrual period of the forward rate, we observe the phenomenon of (implied) volatility dips (for example, see Figure 2.4). Volatility of interpolated forward rates can be much lower than the volatility of the modelled forward rates, even though they have almost the same accrual period. This is problematic when interpolated forward rates are used to value an interest rate derivative: the value of a non-linear interest rate derivative depends crucially on the volatility of the underlying interest rate(s). The interpolation methods by Schlögl (2002) and by Beveridge and Joshi (2009) do not satisfy this requirement, whereas Werpachowski (2010) focuses on this difficulty. However, Werpachowski (2010) uses the original formulation of the log-normal, non-displaced LIBOR Market Model, which does not allow for negative interest rates. Because of this, the interpolation method by Werpachowski (2010) is not directly applicable in our case, as the ability to cope with negative interest rates is crucial in the current market environment. This leads us to the following research question:

How can we interpolate forward rates in a displaced LIBOR Market Model to ensure that in-terpolated forward rates are arbitrage-free and have a smooth volatility structure?

The definition of smooth is non-standard and is introduced in Section 2.1. For now, it suffices to consider smooth to mean that no dips occur in (implied) volatility. As the main contribution of this thesis, we propose a forward rate interpolation method that is both arbitrage-free and leads to a smooth volatility structure. This method can be applied in the displaced LIBOR Market Model. It could be considered as an extension of the method by Werpachowski (2010) to the displaced LIBOR Market Model. Previous literature on interpolation methods in the (displaced) LIBOR Market Model paid much attention to lower bounds on interpolated interest rates. An important property of the displaced LIBOR Market Model is that modelled forward rates have a lower bound, equal to (minus) the so-called displacement parameter. Using the method introduced in this thesis, it turns out that the lower bound for interpolated interest

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rates is different (smaller) than minus the displacement parameter, yet the differences are small in practice. Schlögl (2002) introduces a class of interpolation methods called ‘short bond’ meth-ods, and we find that the new interpolation method fits into this class. The interpolation method results in smooth volatility of interpolated interest rates, and we demonstrate how this results in more accurate prices of caps and floors. Moreover, we adapt an alternative interpolation method that is used by practitioners, leading to an arbitrage-free and volatility-preserving interpolation method. We use similarities to the new interpolation method proposed in this thesis to adapt this alternative method.

Another application of the (displaced) LIBOR Market Model (together with an interpolation method) is the computation of Credit Valuation Adjustment (CVA). CVA is roughly defined as the adjustment to the price of a derivative contract, when counterparty credit risk is incorpo-rated in the valuation procedure. Especially after the financial crisis, CVA (and other types of adjustments, ‘XVA’) has become very influential in the financial sector. We demonstrate that using the new interpolation method leads to better estimation of CVA for interest rate swaps (and by implication, other interest rate derivatives) when the LIBOR Market Model with in-terpolated interest rates is used in the calculation of CVA. Again, this result stems from the fact that interpolated interest rates have a more natural volatility using the new interpolation method. The main advantage is that the number of modelled forward rates can be drastically reduced. This speeds up the calculation of CVA, which is computationally intensive, without loss of accuracy.

This thesis is organised as follows. Chapter 1 introduces the displaced LIBOR Market Model. We derive the displaced LIBOR Market Model using the method of backward induction proposed by Musiela and Rutkowski (1997). The rest of Chapter 1 is dedicated to standard results in the LIBOR Market Model: we derive the dynamics of forward rates under different measures and the famous Black’s formula for caps and floors, mainly based on Filipović (2009). Chapter 2 focuses on forward rate interpolation. We define a set of criteria for a successful interpolation method. The most important criteria are absence of arbitrage and smooth volatilities of interpolated for-ward rates. Then, we provide some general theory of forfor-ward rate interpolation in the LIBOR Market Model, following Schlögl (2002). We review several existing interpolation methods for forward rates and conclude that none of the existing methods allow for arbitrage-free interpola-tion in the displaced LIBOR Market Model while also providing a smooth volatility structure. Chapter 3 answers our research question by introducing a new methodology for forward rate interpolation in the displaced LIBOR Market Model. We demonstrate how the construction of the displaced LIBOR Market Model can be changed, so that the new interpolation method nat-urally fits into the framework described in Chapter 1, and derive a lower bound for interpolated interest rates. Finally, we adapt another interpolation method we introduced in Chapter 2, lead-ing to better interpolation properties. Chapter 4 applies the new interpolation method to the computation of CVA for an interest rate swap. We introduce CVA and demonstrate that the new method leads to better estimations of CVA for interest rate derivatives when the LIBOR Market Model is used to model interest rate derivatives depending on interpolated rates. A discussion concludes this thesis.

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1 The LIBOR Market Model

In this chapter we formally introduce the LIBOR Market Model, also referred to as the BGM model (after the authors of the paper Brace et al. (1997)) in the literature. This mathematical model for forwards rates will be used throughout the rest of this thesis, where the focus of the latter chapters is on interpolation issues. We start with a short overview of interest rate models to provide some context. The earliest stochastic models of the interest rate modelled the short rate. The short rate is a mathematical variable denoting the interest rate which currently applies over an infinitesimally short period of time. Given some mathematical assumptions, the entire interest rate curve can be derived from a model of the short rate. The pioneering paper of this approach is by Vasicek (1977). A disadvantage of the short rate model by Vasicek (1977) is that it is not possible to pick the parameters of the model so that the current initial interest rate curve is reproduced by the model (the same holds true for many related models). Essentially, the models are not flexible enough. Ho and Lee (1986) and Hull and White (1990) introduce extensions of the model by Vasicek (1977) that allow us to impose an arbitrary initial interest rate curve on the short rate model. A more fundamental problem is the difficulty of modelling the complex dynamics of the entire interest rate curve with one underlying quantity (the short rate). Another class of models was introduced by Heath et al. (1992). Instead of modelling the short rate, they propose to model the instantaneous forward rates, which are interest rates that apply over some infinitesimally short period of time in the future. These models can provide richer dynamics of the interest rate curve. Still, these models attempt to model a rather abstract mathematical construct, which is not directly visible in the financial markets (in practice, inter-est rates accrue over periods which are longer than an infinitesimally short period of time).

We consider the LIBOR Market Model (Brace et al., 1997), which we shortly discussed in the introduction. It is an arbitrage-free model for forward rates. The famous Black formula (a for-mula often used as an approximation) is analytically true for call and put options on the interest rate (called caps and floors respectively) when using the LIBOR Market Model. Black’s formula for caps and floors is inspired by the option pricing formula for equity call and put options de-rived by Black and Scholes (1973), and assumes that the underlying financial product (in this case the relevant forward rate) is log-normally distributed. The validity of Black’s formula al-lows practitioners to determine the parameters of the LIBOR Market Model from the observed market quotes for financial products such as caps and floors (the procedure for determining cor-rect parameter values of a mathematical model from market data is called calibration in general).

One aspect of the log-normal distribution is that it can only produce positive values. Hence, the standard LIBOR Market Model can only use positive interest rates as input (initial values) and will only produce positive interest rates. Interestingly, this used to be perceived as a positive aspect, as negative interest rates were assumed to be unrealistic. Yet, in the current financial climate, where negative interest rates are observed in the market (especially near the short end of the curve), this is inconvenient. A practical solution to this problem is to displace the for-ward (LIBOR) rates. Instead of assuming that the forfor-ward interest rates follow a log-normal distribution, we assume that the sum of a forward interest rate and a (positive) constant follows a log-normal distribution. Here, we pick the constant so that all initial interest rates observed in the market become positive after adding the constant. In the LIBOR Market Model, the uncertainty in the evolution over time of forward rates is modelled using a (multidimensional) Brownian motion. We use this Brownian motion as a proxy for new market information, which

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can lead to interest rate changes. Since we want to allow new market information to have a different impact on different forward rates (forward rates which have a different start or end date for the accrual period), it is worthwhile to consider a Brownian motion that is not perfectly correlated between its elements as the source of uncertainty in the LIBOR Market Model. To incorporate these two properties (i.e. the ability to cope with negative interest rates and to allow for a different impact of market news on different forward rates), a slightly more general version of the LIBOR Market Model will be constructed in this chapter, using the backward induction method of Musiela and Rutkowski (1997). Another reason for using the more general model is that the interpolation method by Beveridge and Joshi (2009) is defined in this more general model. Apart from the theoretical appeal in demonstrating that a displaced LIBOR Market Model with a correlated Brownian motion exists mathematically speaking, deriving it explicitly allows us to relate the different mathematical objects constructively, reducing the dependence on more general theorems and ‘tricks’ in further derivations and making this thesis more self-contained.

Furthermore, the dynamics of the different forward rates under a unifying measure are derived, which is both of theoretical and practical (for simulation purposes) interest, and we show that the constructed model is arbitrage-free. It is important that the model is arbitrage-free, as then we can use it to compute risk-neutral prices (prices that take the relevant market risk premia into account). This leads to several general pricing formulas for many derivatives. These formu-las may be used to derive an explicit pricing formula for caps and floors in the LIBOR Market Model. This explicit formula is exactly the well-known Black’s formula. We end the chapter with a summary and explain the connection with the further chapters. As a note, this chapter relies heavily on general theory about stochastic integration and some well-known results such as Girsanov’s theorem, Novikov’s condition and Itô’s lemma. We refer the reader unfamiliar with these topics to Karatzas and Shreve (1998).

1.1 Model construction

As mentioned above, we first construct the LIBOR Market Model we use in this thesis. This construction is based on the paper by Musiela and Rutkowski (1997) yet extended to a more gen-eral setting. Formally, the extension encompasses that the multi-dimensional Brownian motion driving the stochastic process is allowed to have correlated elements, and that the requirement of positive initial forward rates is relaxed to the requirement that the each initial forward rate is larger than a possibly negative constant. This generalisation allows the model to incorpo-rate negative interest incorpo-rates, and to achieve decorrelation, i.e. to specify the correlation between different forward rates in the model (for example, to match correlations observed in a financial market). To the best of my knowledge, this particular extension of the construction of the model by Musiela and Rutkowski (1997) was not described before. Yet, (special cases of) this exten-sion has often been used in academia as a starting point, for example to derive more elaborate models or to show how to calibrate a LIBOR Market Model when the objective is to price a particular type of financial product, but also by practitioners in financial institutions to price financial products or for risk management, especially after negative interest rates started to ap-pear after the financial crisis (see Brigo and Mercurio, 2001). Yet, there are other variations of the LIBOR Market Model in the literature which can produce negative interest rates. For example, Eberlein and Özkan (2005) generalise the Brownian motion driving the uncertainty in the LIBOR Market Model to a non-homogeneous Lévy process and proceed to model the ratio of bond prices. This is similar to our approach, yet we restrict the Lévy process to a Brownian motion (with correlated components). The model by Eberlein and Özkan (2005) may also lead to negative interest rates in general. However, if one is primarily interested in accommodating negative interest rates, the model of Eberlein and Özkan (2005) is a rather complicated solution

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Years 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 T0 T1 T2 T3 T4 T5 T6 T7 T8 L(t,1.5,1.75) L(t,1.25,1.5) L(t,1,1.25) Current time (t = 0.65)

Model start date Model end date

Schematic representation of LIBOR Market Model Structure

Figure 1.1: The represented model consists of 8 forward rates (N = 8). The first tenor date T₀ equals zero, whereas the final tenor date T8 equals two years. The forward rates in

the represented model have accrual periods of three months (δ = 0.25 years). The current model time is equal to 0.65 years in the representation. Only three out of the total of eight forward rates are included in the figure.

(at least, in its full generality). Moreover, the Brownian motion part of the Lévy process is still assumed to be a standard multidimensional Brownian motion with independent components, as opposed to correlated components.

We start by assuming there exists a sequence of tenor dates 0 =: T₀ < T1 < ... < TN < ∞,

where N ∈ N = {1, 2, ...}. These dates are such that Ti − Ti−1 = δ for all i ∈ {1, ..., N }, for

some δ ∈ R+. The dates T0, ..., TN −1 will correspond to the maturity dates of the modelled

forward rates. To ease notation, define q(t) as the unique integer such that T_q(t)−1 ≤ t < T_q(t), for t ∈ [0, T_N). For TN, q(TN) := TN. Then, we assume that there is a filtered probability space

(Ω, {Ft}t∈[0,TN], P

TN_{), where {F}

t}t∈[0,TN] satisfies the usual conditions (Karatzas and Shreve,

1998). The subscript T_N indicates its relation with the zero coupon bond maturing at time T_N, which will be explained later. Let { ˜WTN_(t)}

t∈[0,TN]denote a d-dimensional Wiener Process

(con-sisting of independent Brownian motions) and assume that {F_t}_t∈[0,T_N_]_{is the P}TN_{−augmentation}

of the filtration generated by { ˜WTN_(t)}

t∈[0,TN](i.e. for all t ∈ [0, TN], Ftis the smallest σ-algebra

that contains the natural σ-algebra F_tW˜TN and N , the collection of null sets of F_TW˜TN

N ). Let

P (t, T ) denote the price at time t of a zero-coupon bond (an asset that pays out 1 at maturity) maturing at time T , with t, T ∈ [0, TN], t ≤ T . Note that we use price and value interchangeably

in this thesis. Throughout this chapter, we assume that:

Assumption I. {P (t, T )}_{t∈[0,T ]} is a strictly positive, real-valued and adapted stochastic process for all T ∈ {T₀, T1, ..., TN}, with P (T, T ) = 1. Moreover, there exists a frictionless market for

every zero-coupon bond with maturity date Ti, i ∈ {0, ..., N }.

Now, we formally define simply-compounded forward rates as follows (S, T ∈ {T0, T1, ..., TN}, t ≤

S < T ): L(t, S, T ) := 1 T − S P (t, S) P (t, T )− 1 . (1.1)

For the maturity dates Ti ∈ {0, T1, ..., TN −1}, define L(t, Ti) := L(t, Ti, Ti+1) = L(t, Ti, Ti+ δ).

The forward rates defined in (1.1) may be interpreted as the interest rate one obtains by deciding at time t to invest money in the bank account between S and T . Indeed, by selling at time t one zero-coupon bond maturing at S and buying P (t,S)

P (t,T ) maturing at T , the cost at time t equals 0, at

time S one has to pay out one unit of currency and at time T one receives _{P (t,T )}P (t,S), which leads to the forward interest rate defined in (1.1). In Figure 1.1, we present a schematic representation of the model structure introduced so far. We also define the displaced forward rates as follows, for some β ∈ R.

˜

L(t, S, T ) := L(t, S, T ) + β. (1.2) As explained before, the deterministic parameter β is added to guarantee that initial displaced forward rates are positive, allowing us to assume displaced forward rates have a lognormal distri-bution. We will proceed to assume that the dynamics of forward rates are of the following form,

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with W denoting a d-dimensional Brownian motion and µ a (relative) drift:

dL(t, Ti) = L(t, Ti)µ(t)dt + L(t, Ti)λ(t, Ti)dW (t).

Here, λ(t, T_i) is referred to as the (relative, instantaneous) volatility of forward rate L(t, Ti), and

this function influences the variance of L(t, T_i). To ease notation when taking dot products, this function is defined as a horizontal d-dimensional vector function (this function produces a row vector for every pair (t, T_i) in its domain). Finally, to construct the LIBOR Market Model, we need the following assumptions for the rest of the chapter:

Assumption II. There exists a real-valued constant β such that ˜L(0, Ti) := L(0, Ti) + β > 0 for

all i ∈ {0, ..., N − 1}, with δβ < 1.

Assumption III. For every T_i, i ∈ {0, ..., N − 1}, there exists a known, bounded and determin-istic Rd-valued function λ(t, T_i), t ∈ [0, Ti], that represents the (relative, instantaneous) volatility

of ˜L(·, Ti).

Assumption IV. The forward rate dynamics are driven by a correlated, d-dimensional Brownian motion {WTN_(t)}

t∈[0,TN], where the constant instantaneous correlations (Karatzas and Shreve,

1998) between the i-th WTN

i and j-th WjTN elements of the Brownian motion, ρij :=

dhW_iTN,W_jTNit

dt

for i, j ∈ {1, ..., d}, t ∈ [0, TN], are such that the correlation matrix ρ is positive definite. Here,

hX, Y i := 1₄(hX + Y i − hX − Y i) for square integrable martingales X, Y and hXi denotes the quadratic variation process of square integrable martingale X.

Note that Assumption II is effectively a restriction on the prices of zero-coupon bonds, as it is equivalent to the fact that P (0, T_i) ≥ P (0, Ti+1) − βδP (0, Ti+1), for all i ∈ {0, ..., N − 1}. Hence,

the prices of zero-coupon bonds can only be slightly increasing as a function of maturity T (for a positive value of β) or have to be at least somewhat decreasing as a function of maturity T (for a negative value of β). For β = 0, the restriction is that the initial zero-coupon bonds are non-increasing. As the parameter β has been introduced to allow for negative initial interest rates, it is most often positive in practice. Define C as the Cholesky decomposition of ρ such that ρ = CC>, and define (for all i ∈ {1, ..., d})

WTN i (t) := d X j=1 CijW˜_jTN(t) for all t ∈ [0, TN].

As desired, we obtain that

dhWTN i , WjTNit= dh d X k=1 CikW˜_kTN, d X l=1 CjlW˜_lTNit= d X k=1 CikCjkdt = ρijdt.

We postulate now that

d ˜L(t, TN −1) = ˜L(t, TN −1)λ(t, TN −1)dWTN(t), t ∈ [0, TN −1],

with initial condition ˜L(0, TN −1) = 1_δ

P (0,TN −1) P (0,TN) − 1 + β. This is equivalent to ˜ L(t, TN −1) = 1 δ P (0, TN −1) P (0, TN) − 1 + β Et(λ(·, TN −1)•_WTN_), _{t ∈ [0, T}_{N −1}_],

where E denotes the Doléans exponential (Karatzas and Shreve, 1998). We define the bounded, progressive process

σTN −1,TN(t) :=

δ ˜L(t, TN −1)

δL(t, TN −1) + 1

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To see that this process is bounded, note that both the numerator and the denominator of the fraction in (1.3) are always positive by Assumption II and that the fraction is monotonically increasing between bounds 0 for ˜L(t, TN −1) → 0 and 1 for ˜L(t, TN −1) → ∞. Since σTN −1,TN is

bounded and T_N < ∞, we can apply Novikov’s condition (Karatzas and Shreve, 1998) to define an equivalent probability measure PTN −1 _{on F}

TN −1 by dP TN−1

dPTN = ETN −1(σTN −1,TN •W

TN_{). The}

process σ_T_{N −1}_,T_N(t), t ∈ [0, TN −1], is chosen so that

d P (t, TN −1) P (t, TN) = δdL(t, TN −1) = δd ˜L(t, TN −1) = δ ˜L(t, TN −1)λ(t, TN −1)dWTN(t) = P (t, TN −1) P (t, TN) σTN −1,TN(t)dW TN_(t), which is equivalent to P (t, TN −1) P (t, TN) = P (0, TN −1) P (0, TN) Et(σTN −1,TN •W TN_), _{t ∈ [0, T} N −1].

Moreover, by Girsanov’s theorem for correlated Brownian motions (Theorem A1 in the Ap-pendix), we have that

WTN −1_{(t) := W}TN_{(t) −} Z t 0 ρσTN −1,TN(s) >_ds, _{t ∈ [0, T} N −1] (1.4)

is a PTN −1_{-Brownian motion, where the instantaneous correlations are invariant under this}

mea-sure change, i.e. dhW_iTN −1, WTN −1

j it= ρijdt for i, j ∈ {1, ..., d}, t ∈ [0, TN −1]. Again, we postulate

that

d ˜L(t, TN −2) = ˜L(t, TN −2)λ(t, TN −2)dWTN −1(t), t ∈ [0, TN −2],

with initial condition ˜L(0, TN −2) = 1_δ

_{P (0,T}

N −2)

P (0,TN −1) − 1

+ β. We define the bounded progressive process

σTN −2,TN −1(t) :=

δ ˜L(t, TN −2)

δL(t, TN −2) + 1

λ(t, TN −2), t ∈ [0, TN −2]

and an equivalent probability measure PTN −2 _{∼ P}TN −1 _{on F}

TN −2 by

dP_TN−2

dP_TN−1 = ETN −2(σTN −2,TN −1•W

TN −1

). For the ratio of zero coupon bonds, it holds that

P (t, TN −2) P (t, TN −1) = P (0, TN −2) P (0, TN −1) E_t(σTN −2,TN −1•W TN −1_), _{t ∈ [0, T} N −2],

and applying Girsanov’s theorem we obtain a PTN −2-Brownian motion

WTN −2_{(t) := W}TN −1_{(t) −} Z t 0 ρσTN −2,TN −1(s) >_ds, _{t ∈ [0, T} N −2],

where the instantaneous correlation is given by dhWTN −2

i , W TN −2

j it = ρijdt for i, j ∈ {1, ..., d},

t ∈ [0, TN −2]. By continuing this procedure until we have constructed a probability measure PT1

with corresponding Brownian motion {WT1_(t)}

t∈[0,T1], we obtain a family of lognormal processes

( ˜L(t, Ti))t∈[0,Ti] under their measures P

Ti+1_{, i ∈ {0, ..., N − 1} (as we have restricted λ(·, T}

i) to

be deterministic functions). This allows us to construct a LIBOR Market Model with correlated Brownian motions and negative interest rates.

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Summarising, we have constructed the following model, for i ∈ {0, ..., N − 1}:

d ˜L(t, Ti) = ˜L(t, Ti)λ(t, Ti)dWTi+1(t), t ∈ [0, Ti],

with initial condition ˜L(0, Ti) =

1 δ P (0, Ti) P (0, Ti+1) − 1 + β. σTi,Ti+1(t) = δ ˜L(t, Ti) δL(t, Ti) + 1 λ(t, Ti), t ∈ [0, Ti] dPTi dPTi+1 = ETi σTi,Ti+1 •WTi+1 P (t, Ti) P (t, Ti+1) = P (0, Ti) P (0, Ti+1) E_t(σTi,Ti+1•W Ti+1_), _{t ∈ [0, T} i] WTi_{(t) = W}Ti+1_{(t) −} Z t 0 ρσTi,Ti+1(s) >_ds, _{t ∈ [0, T} i] (1.5a) (1.5b) (1.5c) (1.5d) (1.5e) (1.5f)

This thesis will use the following special case of the model constructed above, which is used but not formally constructed in Brigo and Mercurio (2001). Here, the dimension d of the Brownian motions {WTN_(t)}

t∈[0,TN] and { ˜W

TN_(t)}

t∈[0,TN] is taken equal to N , i.e. to the number of

mod-elled tenor dates. Then, it is assumed that all d-dimensional volatility functions λ(., T_i−1) are in fact scalar, such that for all i ∈ {1, ..., N },

λ(., Ti−1) =0 0 ... λi(., Ti−1) ... 0 0 ,

where the only non-zero entry is at the i-th position. Then, we obtain the following dynamics of the forward rate, where WTi

i denotes i-th element of the Brownian motion WTi:

d ˜L(t, Ti−1) = ˜L(t, Ti−1)λi(t, Ti−1)dW_iTi(t), t ∈ [0, Ti−1], i ∈ {1, ..., N }. (1.6)

In this model, the one-dimensional Brownian motions driving the different forward rate processes have instantaneous correlation dhWTi

i , W Ti

j it= ρijdt for i, j ∈ {1, ..., N }, t ∈ [0, Ti−1].

1.2 Forward rate dynamics under different measures and the

absence of arbitrage

We have seen that displaced forward rates are log-normal under the appropriate measures (see equation (1.6)). However, this is not convenient when simulating forward rates, as it requires a different simulation measure and a different Brownian motion for every forward rate. Instead, we would prefer to use a single simulation measure, which preserves absence of arbitrage (we have not yet shown the constructed model is arbitrage-free, but this is explained below). An obvious choice is PTN _{with numéraire P (·, T}

N), as the TN-bond is the only zero-coupon bond

that with a stochastic price process for the entire time period considered (until TN) and hence

does not require us to change measure at some time point in a simulation. Note that this section is entirely based on known results in the literature, which we extend to cover the more general framework with displaced forward rates and a correlated Brownian motion by slightly tweaking the statements and proofs. Before we derive the dynamics of the forward rates under this measure, note that equation (1.5e) implies the following result for P (t,Ti)

P (t,TN), which is shown

in Lemma 11.2 in Filipović (2009).

Lemma 1.1. Given the constructed model in equations (1.5a-1.5f), and in particular if ˜L has dynamics given by equation (1.6), the following equation holds for i ∈ {0, ..., N − 1} :

P (t, Ti) P (t, TN) = P (0, Ti) P (0, TN) E_t σTi,TN•W TN_, _{t ∈ [0, T} i], (1.7)

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where we define the bounded, progressive process σTi,TN(t) := N −1 X j=i∨q(t) σTj,Tj+1(t), t ∈ [0, TN),

with q(t) the unique integer such that T_q(t)−1 ≤ t < T_q(t), for t ∈ [0, T_N). Hence, P (t,Ti)

P (t,TN),

t ∈ [0, Ti], is a positive PTN-martingale.

Proof. By equations (1.5e) and (1.5f), for t ≤ T_i,

P (t, Ti) P (t, TN) = N −1 Y j=i P (t, Tj) P (t, Tj+1) = P (0, Ti) P (0, TN) N −1 Y j=i E_t(σTj,Tj+1•W Tj+1₎ = P (0, Ti) P (0, TN) exp   N −1 X j=i Z t 0 σTj,Tj+1(s)dW Tj+1₋1 2 Z t 0 σTj,Tj+1(s)ρσ > Tj,Tj+1(s)ds   = P (0, Ti) P (0, TN) exp N −1 X j=i Z t 0 σTj,Tj+1(s)  dWTM ₋ N −1 X k=j+1 ρσ_T>_k_,T_k+1(s)ds   −1 2 Z t 0 σTj,Tj+1(s)ρσ > Tj,Tj+1(s)ds !! = P (0, Ti) P (0, TN) exp   Z t 0 N −1 X j=i σTj,Tj+1(s)dW TM −1 2 Z t 0 N −1 X j=i  2 N −1 X k=j+1 σTj,Tj+1(s)ρσ > Tk,Tk+1(s)ds + σTj,Tj+1(s)ρσ > Tj,Tj+1(s)ds     = P (0, Ti) P (0, TN) exp Z t 0 σTi,TN(s)dW TN₋1 2 Z t 0 σTi,TN(s)ρσ > Ti,TN(s)ds = P (0, Ti) P (0, TN) E_t σTi,TN•W TN_,

which proves the first claim. That P (t,Ti)

P (t,TN) is a martingale follows from Novikov’s condition, given

the boundedness of σ_T_i_,T_N.

From Lemma 1.1, we can see that the P (·, T_N)-discounted Ti-bond price processes are martingales

for i ∈ {0, ..., N }, under PTN_{. We call the unit used to express the prices of the zero-coupon}

bonds the numéraire process. This can be the domestic currency, say Euro, yet we can also opt to define all prices in terms of the value of one of the assets in the bond market, as long as it is almost surely positive. We call PTN _{the T}

N-forward measure with numéraire process

P (t, TN), t ∈ [0, TN]. Since there exists a measure PTN (trivially equivalent to itself) such that

the price processes of all assets (the zero-coupon bonds with maturities T0, ..., TN) divided by the

numéraire process P (t, T_N), t ∈ [0, TN], are martingales under this measure, the LIBOR Market

Model we have constructed is arbitrage-free by the first theorem of asset pricing.

Is it possible to compute an arbitrage-free price for every contingent T -claim X (X is defined as an F_T-measurable random variable, T ∈ {T₀, ..., TN})? This depends on whether the claim is

attainable. A claim is attainable if there exists a self-financing trading strategy that replicates X (Shreve, 2004). If all claims are attainable, the market is dynamically complete. By the second theorem of asset pricing, an arbitrage-free market is complete if and only if martingale

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measures are unique, i.e. there exists only one measure per numéraire process that makes all asset prices martingales when divided by the numéraire process. Hence, if we are willing to assume that the martingale measure PTN _{is unique, the market is complete and every}

contin-gent claim is attainable. An arbitrage-free price process π(t) for any contincontin-gent T -claim X with EPTN |X| P (T,TN) < ∞ satisfies π(t) P (t, TN) = E PTN t X P (T, TN) , t ∈ [0, T ], where EP

t (Y ) is used as short-hand notation for EP[Y |Ft] for any random variable Y and measure

P. If the market is not complete, the arbitrage-free price process π is not unique and multiple arbitrage-free prices co-exist. We can also switch to one of the measures PTi_{, i ∈ {0, ..., N − 1},}

introduced in (1.5d), as we show in the following lemma, see Lemma 11.3 in Filipović (2009). Lemma 1.2. Let the model be constructed as in Section 1.1, i.e. (1.5a-1.5f) are true. For all i ∈ {0, ..., N }, for any FT-measurable contingent claim satisfying

EPTN |X| P (T,TN) < ∞, with T ∈ {T0, ..., Ti}, π(t) P (t, Ti) = E PTi t X P (T, Ti) , t ∈ [0, T ]. (1.8)

Moreover, if X ≡ 1, we obtain for all T ∈ {T₀, ..., TN}, s ≤ t ≤ Ti∧ T :

P (s, T ) P (s, Ti) = E PTi s P (t, T ) P (t, Ti) .

Proof. For i = N , there is nothing to prove. Let i ∈ {0, ..., N − 1}. By Bayes’ rule for conditional expectations, EPtTi X P (T, Ti) = E PTN t X P (T,Ti) dPTi dPTN EPtTN dPTi dPTN .

Equations (1.5d) and (1.5e) imply that

dPTi dPTN|Ft = N −1 Y j=i dPTj dPTj+1|Ft = N −1 Y j=i P (t, Tj) P (t, Tj+1) P (0, Tj+1) P (0, Tj) = P (0, TN) P (0, Ti) P (t, Ti) P (t, TN) . We obtain EPtTi X P (T, Ti) = E PTN t X P (T,TN) P (t,Ti) P (t,TN) . Hence, EPtTi X P (T, Ti) = π(t) P (t, Ti) .

Now, if X ≡ 1, the claim is proved by (using the martingale property of _{P (t,T}P (t,T )

N) under P TN₎ EPsTi P (t, T ) P (t, Ti) = E PTN s P (t,T ) P (t,Ti) dPTi dPTN EPsTN dPTi dPTN = EPsTN P (t,T ) P (t,TN) P (s,Ti) P (s,TN) = P (s, T ) P (s, Ti) .

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From Lemma 1.2, we conclude that the prices P (t, T ) of zero-coupon bonds maturing at T ∈ {T0, ..., TN} divided by P (t, Ti) are PTi-martingales for t ∈ [0, Ti∧ T ]. Hence, we call measure

PTi the Ti-forward measure for all i ∈ {0, ..., N }, with numéraire process P (t, Ti),t ∈ [0, Ti].

Specifying to dynamics of the displaced forward rates given by (1.6), the dynamics of the displaced forward rates under PTN _{(the terminal measure) are given in the following proposition, which}

was, to the best of my knowledge, first proven as Proposition 6.3.1 in Brigo and Mercurio (2001). However, the proof given in this thesis deviates from their approach and is based on Lemma 14.2.2 in Andersen and Piterbarg (2010). An important consequence is that the (displaced) forward rates are not log-normal, as the drift term depends on the other forward rates. In practice, simulation is necessary for the LIBOR Market Model.

Proposition 1.1. Let the model be constructed as in Section 1.1, i.e. (1.5a-1.5f) are true. In particular, assume the dynamics of displaced forward rates are given by (1.6). Under the terminal measure PTN_{, the dynamics of ˜}_{L(t, T}

i) are given by

d ˜L(t, Ti−1) = λi(t, Ti−1) ˜L(t, Ti−1)

 − N −1 X j=i δ ˜L(t, Tj)ρi,j+1λj+1(t, Tj) 1 + δL(t, Tj)  dt

+ λi(t, Ti−1) ˜L(t, Ti−1)dW_iTN(t), t ∈ [0, Ti−1].

(1.9)

Proof. Denote the i-th row of the matrix ρ by ρi, i ∈ {1, ..., d}. By (1.5f),

dWTN_{(t) = dW}TN −1_{(t) +} δ ˜L(t, TN −1) δL(t, TN −1) + 1 ρλ(t, TN −1)>dt = dWTN −2_{(t) +} δ ˜L(t, TN −1) δL(t, TN −1) + 1 ρλ(t, TN −1)>dt + δ ˜L(t, TN −2) δL(t, TN −2) + 1 ρλ(t, TN −2)>dt = dWTi_{(t) +} N −1 X j=i δ ˜L(t, Tj) 1 + δL(t, Tj) ρλ(t, Tj)>dt. (1.10)

For our specification of the volatility functions λ(·, Ti) (i.e. scalar volatility), (1.10) reduces to

dWTN_{(t) = dW}Ti_{(t) +} N −1 X j=i δ ˜L(t, Tj) 1 + δL(t, Tj) ρ>_j+1λj+1(t, Tj)dt.

Hence, for the element i of WTi_{, we have that}

dWTi i (t) = dWiTN(t) − N −1 X j=i δ ˜L(t, Tj)ρi,j+1λj+1(t, Tj) 1 + δL(t, Tj) dt. (1.11)

By plugging in (1.11) into (1.6), we obtain (1.9).

Spot LIBOR measure

An alternative popular measure is the risk-free measure, where the numéraire is a bank account with discretely-accrued interest (i.e. only at tenor dates T1, ..., TN of the model). This is a natural

numéraire in the context of the LIBOR Market Model, as it focuses on modelling a discrete number of simply-compounded forward interest rates. It is defined as the value of the asset constructed as follows. At time T0= 0, 1 is invested in zero-coupon bonds maturing at time T1(at

T0, the value of this asset is clearly one). Then, at time T1, all proceeds (_{P (T}1₀_,T₁₎ = 1+δL(T0, T0))

are reinvested (“rolled”) in the zero-coupon bond maturing at time T2(then the portfolio consists

of _{P (0,T}1

1)

1

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time T₂). This process is repeated until time T_{N −1}. It is clear that the value of the numéraire at time t can be given by N (0) = 1 and, for t > 0, by

N (t) = P (t, Ti+1) i Y n=0 1 P (Tn, Tn+1) = P (t, Ti+1) i Y n=0 (1 + δL(Tn, Tn)), t ∈ (Ti, Ti+1]. (1.12)

Now we present the following lemma, which can be found without proof as Lemma 11.4 in Filipović (2009).

Lemma 1.3. Let the model be constructed as in Section 1.1, i.e. (1.5a-1.5f) are true. For all t ∈ [0, TN −1],

EPtTN (N (TN)P (0, TN)) = Et σT0,TN•W

TN ,

where we define the bounded, progressive process

σTi,TN(t) :=

N −1

X

k=q(t)∨i

σTk,Tk+1(t), t ∈ [0, TN).

Proof. From (1.12), we have, for t ∈ [0, T_{N −1}],

EPtTN (N (TN)P (0, TN)) = EPtTN N −1 Y i=0 1 P (Ti, Ti+1) P (0, TN) ! = EPTN t N −1 Y i=0 P (Ti, Ti) P (Ti, Ti+1) P (0, TN) ! .

Now we use the formula (1.5e) to obtain

EPtTN (N (TN)P (0, TN)) = EPtTN N −1 Y i=0 E_T_i σTi,Ti+1•W Ti+1 = EPTN t exp N −1 X i=0 Z Ti 0 σTi,Ti+1(s)dW Ti+1₋1 2 Z Ti 0 σTi,Ti+1ρσ > Ti,Ti+1ds ! .

Applying formula (1.5f), we derive

EPtTN (N (TN)P (0, TN))) = EPTN t exp N −1 X i=0 Z T_i 0 σTi,Ti+1(s)  dWTN ₋ N −1 X j=i ρσ>_T_j_,T_j+1(s)ds   −1 2 Z T_i 0 σTi,Ti+1(s)ρσ > Ti,Ti+1(s)ds !! = EPTN t exp Z TN −1 0 σT0,TN(s)dW TN −1 2 N −1 X i=0 Z Ti 0  2 N −1 X j=i σTi,Ti+1(s)ρσ > Tj,Tj+1(s)ds + σTi,Ti+1(s)ρσ > Ti,Ti+1(s)ds   ! = EPTN t exp Z TN −1 0 σT0,TN(s)dW TN₋ 1 2 Z TN −1 0 σT0,TN(s)ρσ > T0,TN(s)ds = exp Z t 0 σT0,TN(s)dW TN ₋1 2 Z t 0 σT0,TN(s)ρσ > T0,TN(s)ds × EPtTN exp Z TN −1 t σT0,TN(s)dW TN ₋1 2 Z T_{N −1} t σT0,TN(s)ρσ > T0,TN(s)ds = Et σT0,TN •W TN_,

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where we used that σ_T₀_,T_N is a bounded process implying that E_t σT0,TN •WTN, t ∈ [0, TN −1],

is a PTN _{martingale in the final equality. Indeed, the claim is proved.}

It is clear that N (T_N)P (0, TN) > 0 and, from Lemma 1.3, we can see that EP0TN (N (TN)P (0, TN))

= E0 σT0,TN •W

TN = 1, so we can define an equivalent probability measure PN _{∼ P}TN _{on F}

TN

by defining the Radon-Nikodym derivative as dPN

dPT N = N (TN)P (0, TN). Because of Lemma 1.3,

we can apply Girsanov’s theorem and obtain a Brownian motion under the measure PN: WN(t) = WTN_{(t) −} Z t 0 ρσT0,TN(s) >_{ds, t ∈ [0, T} N −1]. (1.13)

The measure induced by (1.12) is known in the literature as the spot (LIBOR) measure. The market for the T_i-bonds remains arbitrage-free under this measure. The statement and proof given here are based on Lemma 11.5 in Filipović (2009), yet extended to continuous-time.

Proposition 1.2. Let the model be constructed as in Section 1.1, i.e. (1.5a-1.5f) are true. For all i ∈ {0, ..., N }, s ≤ t ≤ T_i ≤ T_N EP N s P (t, Ti) N (t) = P (s, Ti) N (s) . Proof. EP N s P (t, Ti) N (t) = E PTN s P (t,Ti) N (t) dP N dPT N EPsTN dPN dPT N = EPsTN P (t,Ti) N (t) N (TN)P (0, TN) EPTN s (N (TN)P (0, TN)) . (1.14)

Using (1.12) and (1.5e),

EPsTN (N (TN)) = EPsTN   N −1 Y j=0 P (Tj, Tj) P (Tj, Tj+1)  = q(s)−1 Y j=0 P (Tj, Tj) P (Tj, Tj+1)E PTN s   N −1 Y j=q(s) P (Tj, Tj) P (Tj, Tj+1)   = q(s)−1 Y j=0 P (Tj, Tj) P (Tj, Tj+1)E PTN s   P (0, T_q(s)) P (0, TN) N −1 Y j=q(s) E_T_j(σTj,Tj+1•W Tj+1₎   = q(s)−1 Y j=0 P (Tj, Tj) P (Tj, Tj+1) P (0, T_q(s)) P (0, TN) E PTN s ETN −1 σTq(s),TN •W TN = q(s)−1 Y j=0 P (Tj, Tj) P (Tj, Tj+1) P (0, T_q(s)) P (0, TN) E_sσTq(s),TN •W TN = N (s) P (s, TN) , (1.15)

where the third and fourth equality follow using the same steps as in the proof of Lemma 1.3. We then use the martingale property of the resulting stochastic exponential and Lemma 1.1.

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Finally, EPsTN P (t, T_i) N (t) N (TN) = EPTN s   P (t, Ti) Qq(t)−1 j=0 P (Tj,T1j+1)P (t, Tq(t)) N −1 Y j=0 1 P (Tj, Tj+1)   = EPTN s   P (t, Ti) P (t, T_q(t)) N −1 Y j=q(t) 1 P (Tj, Tj+1)   = EPTN s   P (t, Ti) P (t, T_q(t)) P (0, T_q(t)) P (0, TN) N −1 Y j=q(t) E_T_j σTj,Tj+1•W Tj+1   = EPTN s P (t, Ti) P (t, TN) P (t, TN) P (t, T_q(t)) P (0, T_q(t)) P (0, TN) E_T_{N −1}σTq(t),TN •W TN = EPTN s   P (t, Ti) P (t, TN) 1 P (0,Tq(t)) P (0,TN) Et σTq(t),TN •WTN P (0, T_q(t)) P (0, TN) E_T_{N −1}σTq(t),TN •W TN   = EPTN s   P (t, Ti) P (t, TN) EPtTNETN −1 σTq(t),TN •W TN Et σTq(t),TN •WTN  = EP TN s P (t, Ti) P (t, TN) = P (s, Ti) P (s, TN) . (1.16)

Again, we use the same idea as in the proof of Lemma 1.3 in the fourth equality, Lemma 1.1 for equality five and the fact that the stochastic exponentials are martingales under PTN _{for equality}

seven. Plugging results (1.15) and (1.16) into (1.14), the claim follows.

We conclude the spot measure is also an equivalent martingale measure. Of course, this could have been proven by relying on the paper by Geman et al. (1995), which has shown that in-troducing a new numéraire asset leads to an equivalent martingale measure in a more general setting, yet an explicit proof is also instructive. The fact that the spot measure is an equivalent martingale measure implies that any contingent T -claim X satisfying EPTN |X|

P (T,TN)

< ∞ can also be valued equivalently using the following formula, for T ∈ {T0, ..., TN}:

π(t) N (t)= E PN t X N (T ) . (1.17)

Now, we want to derive the dynamics of the forward rates under the PN-measure. The proof is based on Lemma 11.6 in Filipović (2009).

Proposition 1.3. Let the model be constructed as in Section 1.1, i.e. (1.5a-1.5f) are true. In particular, assume that the dynamics of displaced forward rates are given by (1.6).

Under the spot measure PN defined by dPN

dPT N = N (TN)P (0, TN)), the dynamics of ˜L(t, Ti) are

given by

d ˜L(t, Ti−1) = λi(t, Ti−1) ˜L(t, Ti−1)

  i−1 X j=q(t) δ ˜L(t, Tj)ρi,j+1λj+1(t, Tj) 1 + δL(t, Tj)  dt

+ λi(t, Ti−1) ˜L(t, Ti−1)dWiN(t), t ∈ [0, Ti−1].

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Proof. From Proposition 1.1, we know that, for t ∈ [0, T_i−1],

d ˜L(t, Ti−1) = λi(t, Ti−1) ˜L(t, Ti−1)

 − N −1 X j=i δ ˜L(t, Tj)ρi,j+1λj+1(t, Tj) 1 + δL(t, Tj)  dt +λi(t, Ti−1) ˜L(t, Ti−1)dWiTN(t). (1.19) From (1.13), dWN(t) = dWTN_{(t) − ρσ} T0,TN(t) >_{dt, we obtain} dWTN i (t) = dWiN(t) + N −1 X j=q(t) ρi,j+1σTj,Tj+1,j+1dt (1.20)

and hence by plugging (1.20) into (1.19) and using (1.5c), we obtain

d ˜L(t,Ti−1) = λi(t, Ti−1) ˜L(t, Ti−1)

 − N −1 X j=i δ ˜L(t, Tj)ρi,j+1λj+1(t, Tj) 1 + δL(t, Tj)  dt + λi(t, Ti−1) ˜L(t, Ti−1)   N −1 X j=q(t) δ ˜L(t, Tj)ρi,j+1λj+1(t, Tj) 1 + δL(t, Tj)  dt + λ_i(t, T_i−1) ˜L(t, T_i−1)dW_iN(t). (1.21)

The result follows from cancelling terms in (1.21).

1.3 Black’s formula

Black’s equation has been the standard method to price caps and floors (series of call and put options on interest rates) for many years. The formula can be derived analytically if forward rates are log-normal. Given the underlying assumption of the LIBOR Market Model that for-ward rates have a log-normal distribution, it should be no surprise that we can indeed derive Black’s formula for caps and floors in this setting. One of the main advantages of the LIBOR Market Model is the consistency with Black’s formula, and given that the formula will be used extensively later on, it is worthwhile to repeat its derivation over here.

First, define a caplet with reset date T , settlement date T + δ and strike κ as the financial contract that pays the holder δ(L(T, T, T + δ) − κ)+ at time T + δ. A cap is a series of caplets with different reset and settlement dates. Using Proposition 1.4, the price of a cap is easily computed, if all reset and settlement dates are tenor dates of the considered LIBOR Market Model. A floorlet corresponds to a put option, i.e. a floorlet with reset date T , settlement date T + δ and strike κ pays out δ(κ − L(T, T, T + δ))+. A floor consists of a series of floorlets. Black’s formula derived below is very well-known. The particular versions in equations (1.22) and (1.23) are based on page 198 in Filipović (2009), where the proof is left as an exercise.

Proposition 1.4. Let the model be constructed as in Section 1.1, i.e. (1.5a-1.5f) are true. The value at time t ≤ Ti of a caplet with reset date Ti, settlement date Ti+1 and strike κ is given by

πcaplet(t) = δP (t, Ti+1) ˜L(t, Ti)Φ(d1) − (κ + β)Φ(d2)

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where Φ denotes the standard normal cumulative distribution function and d1 = log _˜ L(t,Ti) κ+β + 1₂RTi t ||λ(s, Ti)||2ds q RTi t ||λ(s, Ti)||2ds d2 = d1− s Z T_i t ||λ(s, T_i)||2_ds.

Similarly, the price at time t ≤ T_i of a floorlet with reset date T_i, settlement date T_i+1 and strike κ is given by πfloorlet(t) = δP (t, Ti+1) (κ + β)Φ(−d2) − ˜L(t, Ti)Φ(−d1) . (1.23)

Proof. Clearly, the caplet with settlement date T_i+1 is F_T_i+1-measurable. By Lemma 1.2,

πcaplet(t) = P (t, Ti+1)EP Ti+1 t δ(L(Ti, Ti) − κ)+ P (Ti+1, Ti+1) = P (t, Ti+1)EP Ti+1 t δ(L(Ti, Ti) − κ)+ .

From equation (1.5a) and (1.5b), we can see that

˜ L(Ti, Ti) = ˜L(t, Ti) exp Z Ti t λ(s, Ti)dWTi+1(s) − 1 2 Z Ti t ||λ(s, T_i)||2ds .

Hence, conditional on F_t, ˜L(Ti, Ti) is lognormal with parameters µ = log( ˜L(t, Ti))

−1₂RTi

t ||λ(s, Ti)||2ds and σ2 =

RTi

t ||λ(s, Ti)||2ds. Denote the probability density function of such

a log-normal random variable X by f_X. Then,

EPtTi+1 ˜L(Ti, Ti)1{ ˜L(Ti,Ti)≥κ+β} = Z ∞ κ+β yfX(y)dy = exp µ + 1 2σ 2 Φ − log(κ + β) + µ + σ 2 σ = ˜L(t, Ti)Φ   logL(t,T˜ i) κ+β +1₂RTi t ||λ(s, Ti)||2ds q RT_i t ||λ(s, Ti)||2  ,

where we used the standard result that R∞

γ yfX(y)dy = exp µ +12σ2 Φ − log(γ)+µ+σ2 σ for a random variable X following a log-normal distribution with mean µ and variance σ2. Moreover,

EPtTi+1 (κ + β)1_{{ ˜}_L(T i,Ti)≥κ+β} = (κ + β)PTi+1 ˜_L(T i, Ti) ≥ κ + β|Ft = (κ + β)PTi+1   log( ˜L(Ti, Ti)) − log( ˜L(t, Ti)) + ₂1R_tTi||λ(s, Ti)||2ds q 1 2 RT_i t ||λ(s, Ti)||2ds ≥ log(κ + β) − log( ˜L(t, Ti)) + 1₂ RT_i t ||λ(s, Ti)||2ds q 1 2 RT_i t ||λ(s, Ti)||2ds |F_t   = (κ + β)PTi+1  Z <− log(κ + β) + log(L(t, Ti)) − 1 2 RT_i t ||λ(s, Ti)||2ds q 1 2 RT_i t ||λ(s, Ti)||2ds |F_t   = (κ + β)Φ   logL(t,T˜ i) κ+β −1₂RTi t ||λ(s, Ti)||2ds q RTi t ||λ(s, Ti)||2ds  ,

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where Z denotes an independent standard normal random variable. Hence,

πcaplet(t) = P (t, Ti+1)EPtTi+1 δ(L(Ti, Ti) − κ)+ = P (t, Ti+1)EPtTi+1

δ( ˜L(Ti, Ti) − (κ + β))+

= δP (t, Ti+1) (L(t, Ti)Φ(d1) − (κ + β)Φ(d2)) .

The value process of a floorlet can be derived along the same lines and will not be repeated.

The use of the formulas (1.22) and (1.23) for the price of caplets and floorlets in the LIBOR Market Model is twofold. First of all, it implies that there is no requirement to do simulations if one uses the LIBOR Market Model to price caps or floors corresponding to LIBOR tenor interest rates, as these simulations would eventually converge to the analytical prices derived above. Second, it is possible to observe a (market or simulated) price of a cap or floor, and then determine the constant, one-dimensional volatility function (λ(t, Ti) ≡ λi for all t ∈ [0, Ti],

λi ∈ R+) that leads to this price (if one picks a displacement parameter β that allows for this

price). The value of the obtained constant volatility function is referred to as implied volatility.

1.4 Summary

In this chapter we have formally constructed the LIBOR Market Model that is used in this thesis. The model allows for correlated Brownian motions driving the different forward rates, and also allows for negative interest rates. These two extensions are both relevant for modelling financial markets, as forward rates that are modelled using Brownian motions with independent elements have been found to be too rigid to accurately reflect market behaviour, and negative interest rates have also been observed over the last years. After constructing the model, we have shown how to obtain dynamics of the different forward rates under a single measure (either the terminal forward measure or the spot LIBOR measure), which is required for an efficient simulation of the model. It also results in arbitrage-free pricing formulas for T -contingent claims. These formu-las have been applied to explicitly derive the price of caps and floors in the LIBOR Market Model.

Yet, there are still many questions. First, how should the volatility functions λ(·, T_i) be de-fined in practice? This is a question of calibration, i.e. defining the parameters of the model so that the model reproduces market prices for some set of market instruments. The set of market instruments one uses also depends on which aim one has (for example, what kind of financial product one wishes to value). Other parameters to be calibrated are the correlation matrix of the Brownian motion, and it is also necessary to specify the displacement parameter β somehow. Second, how should forward rates be simulated according to the LIBOR Market Model? Simulation is a very common tool for valuation of financial products. Finally, there is a practical problem with some of the formulas derived in this chapter that has been neglected until now. For example, in equations (1.8), (1.12), (1.22) and (1.23), the term P (t, T_i) appears, where t ≤ T_i. Note that, P (t, T_i) = _1+(T 1

i−t)L(t,t,Ti). It should be clear that if t 6= Tj for some

j ∈ {0, ..., i}, there is no expression for this particular forward rate L(t, t, Ti). Therefore, it is

currently impossible to compute the price of a caplet if the evaluation date t of equation (1.22) does not match one of the tenor dates. This is not satisfactory, both from a theoretical and practical perspective. Of course, it is theoretically possible to add another tenor date to the model, yet there will always be some dates that are not tenor dates as long as one uses a finite set of tenor dates. Since one of the main advantages of the LIBOR Market Model is that the modelled quantity is market-observable and there are only a finite number of LIBOR rates in practice, directly modelling an infinite amount of rates as in Musiela and Rutkowski (1997) is not an appealing solution. This example motivates us to study methods of interpolating forward rates in the LIBOR Market Model, enabling us to compute P (t, T_i) for arbitrary t ∈ [0, Ti]. We

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2 Interpolation of modelled forward rates

As explained in the final section of the previous chapter, it is sometimes necessary to obtain an expression for an unmodelled forward rate L(t, S, T ), where either S or T are not equal to one of the tenor dates T_i. Equivalently, we require an expression for the arbitrary discount factor P (t, T ) for all T between t and T_N. The problem of obtaining the entire discount curve P (t, T ) from few observations P (t, T_i), i ∈ {0, ..., N }, is a classical and relatively well-understood problem in interest rate modelling. Many different approaches exist, for example parametric methods, where the basic principle is to fit a parametric function (in terms of time to maturity) on the known zero-coupon bond prices to obtain an approximation of all zero-coupon bond prices. An exam-ple of this approach is the method of Nelson and Siegel (1987), and its extension by Svensson (1994). Another example is approximation of the yield curve by piecewise polynomial functions, called splines, see Hagan and West (2006), which also provides a review of many other methods, as does James and Webber (2000). However, these methods are not applicable to the LIBOR Market Model. In general, only ratios of bond prices are known, not the bond prices themselves. Indeed, we only have an analytic expression for the prices of bonds maturing at tenor dates, only on a discrete set of tenor dates. On non-tenor dates, ratios of modelled bond prices are known (P (t,Ti)

P (t,Tj), 0 ≤ t ≤ Ti, Tj, TN) whereas absolute bond prices P (t, T ) are unknown for all

0 ≤ t ≤ T . Hence, any interpolation method that is based on interpolation of known zero-coupon bonds P (t, T_i) does not work here. Because of this anomaly, interpolation of forward rates (or equivalently zero-coupon bonds) in the LIBOR Market Model is relatively special and distinct from interpolation of zero-coupon bonds in other situations. Given that discount factors and (arbitrary) forward rates are essential in many applications of interest rate modelling, Werpa-chowski (2010) mentions that no implementation of the LIBOR Market Model can be considered complete unless a method to obtain arbitrary forward rates and discount factors is specified.

Note that we use the theoretical framework derived in Chapter 1 throughout this chapter. We first formally list a number of requirements for a successful interpolation method. Then, we intro-duce some general theory of interpolation in the LIBOR Market Model, review several methods in the literature and attempt to determine if they satisfy our criteria. The methods considered are those of Schlögl (2002), Beveridge and Joshi (2009), Piterbarg (2004) and Werpachowski (2010). For some interpolation methods, not all properties we require have been studied before. Therefore, some new (to the best of my knowledge), theoretical results have been derived for the interpolation method by Werpachowski (2010). These results allow us to embed the method proposed by Werpachowski (2010) in the theory of interpolation in the LIBOR Market Model derived by Schlögl (2002) and Beveridge and Joshi (2009). Finally, we consider another possible interpolation method. The focus of this chapter is the problem of non-smooth volatilities: many existing (arbitrage-free) interpolation methods suffer from the problem that the volatility of for-ward rates is relatively higher for modelled forfor-ward rates than for interpolated forfor-ward rates, leading to so-called ‘volatility dips’ (see Figure 2.4b as an example). One would intuitively expect the volatility of interpolated forward rates to be monotonically increasing or decreasing between two modelled forward rates. The volatility of the interpolated forward rates is important for the pricing of most derivatives (for example, for caps and floors). Using interpolated forward rates with lower volatility leads to misleading pricing of financial products depending on interpolated forward rates (even though the method is arbitrage-free). The goal of this chapter is to find a method for forward rate interpolation that does not suffer from this problem.

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Years 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 T0 T1 T2 T3 T4 T5 T7 T8 L(t,1.5,1.75) L(t,1.25,1.5) L(t,1,1.25) L(t,1.35,1.7) Current time (t = 0.65)

Model start date Model end date

Schematic representation of LIBOR Market Model Interpolation

Figure 2.1: Schematic representation of LIBOR Market Model interpolation. The goal here is to obtain an expression for L(t, 1.35, 1.7), by interpolating L(t, 1.25, 1.5) and L(t, 1.5, 1.75) or possibly some of the other modelled forward rates which are known.

2.1 Theory of interpolation in the LIBOR Market Model

In this chapter we extend the market for zero-coupon bonds with maturity dates T₀, ..., TN,

introduced in Chapter 1, to a market for zero-coupon bonds of every maturity T ∈ [0, T_N]. In Chapter 1 we made Assumptions I-IV. Again, we use these assumptions. We have seen that these assumptions allow us to construct the LIBOR Market Model. We will work throughout this chapter with the LIBOR Market Model (1.5a-1.5f), where we impose the additional structural assumptions leading to the dynamics of displaced forward rates given by (1.6). The goal of Chapter 2 is to find an expression for an arbitrary forward rate L(t, S, T ), t ≤ S ≤ T ≤ T_N in this setting, by specifying an interpolation scheme. In Figure 2.1 we have represented the problem schematically. The desired properties of an interpolation scheme are as follows:

(a) Absence of arbitrage: no arbitrage opportunities are introduced by extending the market from zero-coupon bonds with a tenor date as maturity date to all zero-coupon bonds with maturity date before the last tenor date.

(b) No additional forward rates need to be simulated besides L(·, T_i), i ∈ {0, ..., N − 1}.

(c) Interpolated rates with accrual period δ (i.e. L(t, T, T +δ) with t ≤ T ≤ T_N) are guaranteed to be larger than −β, the displacement parameter.

(d) The implied volatility of at-the-money caplets depending on L(T, T, T + δ), is a ‘smooth’ function of T for all T ∈ [0, T_{N −1}]. Here, smooth is defined as follows: between two tenor dates T_i and T_i+1, implied volatility of a L(T, T, T + δ)-caplet is continuous and approx-imately linear in T , for all i ∈ {0, ..., N − 2}. Moreover, at T_i (respectively T_i+1) the implied volatility of the L(T_i, Ti, Ti+1)-caplet (L(Ti+1, Ti+1, Ti+2)-caplet) equals its

the-oretical value according to Black’s formula, see Proposition 1.4, i.e. q 1 Ti RT_i 0 ||λ(s, Ti)||2 ( q 1 Ti+1 RT_i 0 ||λ(s, Ti+1)||2).

(e) Consistent interpolation: the interpolation scheme behaves as the identity function on modelled forward rates, i.e. modelled forward rates remain unchanged by applying the interpolation scheme.

The main reason for interpolating forward rates L(t, S, T ) is to be able to value derivatives de-pending on a forward rate that has reset date S or settlement date T that is not equal to one of the tenor dates T₀, T1, .., TN. It is not possible anymore to compute the arbitrage-free price as

an expectation of the numéraire-relative pay-off of the contract under an arbitrage-free measure if the interpolation method introduces arbitrage into the model. Hence, an interpolation method that is not arbitrage-free defeats its purpose, leading us to criterion (a). Theoretically, if any forward rate with reset date S or settlement date T is not modelled (i.e. S /∈ {T₀, ..., TN} or

T /∈ {T₀, ..., TN}), the problem of interpolation can be avoided by enlarging the set {T0, ..., TN}

Arbitrage-Free Interpolation in the LIBOR Market Model

MSc Stochastics and Financial Mathematics

MSc Econometrics, track Financial Econometrics

Master Thesis

Arbitrage-Free Interpolation in the

LIBOR Market Model

T.E. ter Bogt

dr. A Khedher

prof. dr. ir. M.H. Vellekoop

October 17, 2018

M. Michielon

Abstract

Acknowledgements

Contents

Introduction

1 The LIBOR Market Model

1.1 Model construction

1.2 Forward rate dynamics under different measures and the

absence of arbitrage

1.3 Black’s formula

1.4 Summary

2 Interpolation of modelled forward rates

2.1 Theory of interpolation in the LIBOR Market Model