Abstract

This thesis focuses on calculation of CVA for Bermudan swaptions. Two-factor interest rate model is calibrated to market data and used for the simulation of future interest rate paths. We find that the model is able to fit well to market data. The default probabilities for each counterparty are directly extracted from CDS market quotes. In order to calculate CVA of a Bermudan swaption, Matlab code has been built. We assume no correlation between the default times of the counterparties.

Chapter 1

Introduction

One of the major causes of the financial crisis from 2008 was the underestimation of the credit risk importance. Before this point in time, the standard practice of pricing derivatives contracts was to mark them to the market without taking the counterparty credit risk into account. Future cash flows were simply discounted back using the LIBOR curve and the subsequent values were considered for default-free.

Nowadays, financial institutions recognize the importance of the possibility of futures losses due to counterparty default when pricing derivative contracts. Moreover, regulators impose further restrictions on the specifics related to derivatives trading. As a result, the Counterparty Value Adjustment (CVA) becomes a key risk measure. It gives the price difference between a default-free contract and the true value of the contract when the probability of counterparty’s default is taken into account.

There exist two types of CVA ‘ unilateral and bilateral. One differentiates from the other by the fundamental assumption of whether only one or both counterparties may default. The question which gives better approximation of the credit risk price is still debatable. Moreover, the complexity of calculating CVA is also due to the challenging nature of its components. It depends on the distribution of future exposures and the counterparties’ probability to default during the life of the trade. Both quantities are subject to the choice of model for their computation, require simulation of market factors’ values and assumption of interdependency between those factors.

Many interest rate models have been used in the past for the simulation of future exposures. However, in the current global economic situation, where negative interest rates are no longer a myth, some of the models are considered market inconsistent. Brigo and Mercurio (2006) introduce the Linear Gaussian Two Factor model which overcomes this modelling disadvantage by being able to produce negative values for the yield curve evolution. We calibrate the model to At-the-money European swaption prices.

Another important step of computing CVA is the choice of credit modelling approach. Producing default probability curve which incorporates the possibility of random shocks is replicated by the times of Poisson process jumps. As those jumps are exogenous, introducing model dependency on market factors is challenging and has not been considered in this thesis.

This thesis is organized as follows. In Chapter 2 we define counterparty credit risk and credit value adjustment and discuss the component of the CVA. Chapter 3 is devoted to the interest rates modelling, the nature of the Linear Guassian two factor model and its application for mark to market pricing of interest rate contracts. In Chapter 4 we review the literature on credit modelling frameworks and discuss the choice of intensity function for producing default probability curves. Chapter 6 closes with the numerical results for the CVA of Bermudan swaptions and compares the measure among different counterparties.

Chapter 2

Counterparty Credit Risk and Credit Value Adjustment

2.1 Introduction

Counterparty Credit Risk (CCR) is the risk that a counterparty defaults before the last transaction’s payment. A loss occurs if at the default time the contract has a positive value for the other counterparty. For many years, contracts over-the-counter have been priced without consideration of the credit risk. However, after the financial crisis from 2008, it has become clear that managing counterparty credit risk cannot be neglected anymore. Without a proper understanding and management of counterparty risk, any further measure of derivatives markets’ growth and development is compromised.

Credit Valuation Adjustment (CVA) is by definition the market price of the counterparty credit risk. Thus, it measures the difference between the risky and the default-free price of the derivative contract. Under the unilateral framework, it is assumed that the counterparty which performs the CVA analysis is default-free. Both counterparties require premium for the risk they are bearing, therefore, they could not agree on the credit risk price under the unilateral framework. Under the bilateral CVA approach, both the bank and the counterparty could default.

Under both unilateral and bilateral frameworks, pricing counterparty credit risk is not easy, especially when the target is an exotic derivative contract or a portfolio of exotic products. When a simple contract is present, such as a coupon paying bond, one should simply account for the possibility of default when calculating the discounted value of the cash flows. All the cash flows have one direction and in a case of default, they are fully risky. In comparison, payments on a swap contract are made in both directions and if a default occurs, the future cash flows are not considered fully risky due to the partial cancellation with the payments made in the opposite direction. Furthermore, the credit risk on the swap is relatively smaller than the bond. At the same time, complexity arises from pricing this risky part of the swap future payments as they depend on a range of market factors.

In this Chapter we review the basic concepts and notation of the counterparty credit risk, expected exposure and CVA. Further, we discuss the approaches for calculating CVA. We follow the notation provided by Jon Gregory (2010).

Since the main components of the CVA are the expected exposure and default probabilities, this section builds better understanding of the nature of the CVA by evaluating in detail and careful consideration of every step taken for its calculation. We also consider the different types of CVA and explain in detail the CVA components used for its calculation.

2.2 Unilateral approach

Under the assumption that only one of the counterparties could default, we could assess the credit risk by calculating Unilateral CVA. However, this scenario is unrealistic as all the participants carry a certain probability of default.

We adopt the notation provided by Jon Gergory (2010). V(t,T) and V ??(t,T) denote the risk-free and the risky value of the contract at valuation time t for maturity at T, respectively. Let ?? be the time when the counterparty defaults and s corresponds to a point in the future such that t<sT} V(t,T)

Where 1_((‘)) is the indicator function such that it returns one if the default has occurred prior to the expiration of the contract, and zero otherwise. In other words, the equation states that the risk-free and the risky value of the contract are equal.

Counterparty does default before the contract has matured

In this case the total payoff would be a sum of the cash flows paid before default

1_{??< T} V(t,T) and the default payoff at time ?? 1_{‘??T} (‘??V(??,T)’^-+’V(??,T)’^+) Where ?? is the recovery fraction, assumed to be known, independent and constant, and x^-=min'(x,0) and y^+=max'(y,0). By putting the payoffs together, the expression showing the difference between the risky and the default-free value of the contract is derived V ??(t,T)=V(t,T)-E^Q ((1-??)1_{‘??T} ‘V(??,T)’^+ ) Where Q is risk neutral measure and E^Q denotes risk-neutral expectation. Therefore, CVA could be approximated by taking the risk-neutral expectation of the discounted loss. CVA=E^Q ((1-??)1_{‘??T} ‘V(??,T)’^+ ) It is crucial to note that to adopt a CVA formula for the numerical implementation we assume independency between exposure and counterparty credit quality. The opposite would present the concept of right and wrong way risk to the equation. Wrong way risk is observed when worsening of the credit condition of the company leads to higher exposure. With the right way risk the correlation between exposure and credit quality is positive and they worsen or improve together. However, this dependency assumption is important for equity and commodity trades, but appears to be rather negligible when related to FX and fixed income derivatives as the majority of banks’ counterparty credit risk comes from interest rate contracts. Further, we assume no wrong way risk and deterministic discount factors and survival probabilities. Under this approach, Gregory (2010) derives the formula for unilateral CVA. Following the notation from his book, let (1-?? ??) be the recovery fraction of the value of the contract at time of default, this is, the Loss Given Default which is assumed to be known, constant and independent measure. Then, we denote B(t_j) the risk-free discount factor at time t_j as all the future possible losses need to be discounted back at the time of pricing the credit risk. EE(t_j) is the term which represent the expected exposure measure for any future date t_j. At last, q(t_(j-1), t_j) is the default probability measure between the dates t_(j-1) and t_j. For the purpose of calculating default probabilities, we introduce different credit models in Chapter 4 and explain the need of bootstrapping the term structure of credit-default swaps (CDS). Definition: Unilateral CVA CVA'(1-?? ??)’_(j=1)^m”B(t_j )EE(t_j )q(t_(j-1),t_j)’ Where we have m periods such that t_0=t,t_1,’,t_m=T and T is the last payment date of the contract or the longest maturity in a portfolio. 2.3 Bilateral approach Before the financial crisis from 2008, there were counterparties considered to be riskless, for instance, banks had high credit ratings and tight credit spreads. Thus the most widely used credit risk measure was the unilateral CVA. After the collapse of Lehman Brothers, the idea of default-free counterparty has become no longer applicable. Basic for the case of bilateral CVA is the fundamental assumption that both the bank (B) and the counterparty (C) may default. In order to elaborate on this point, we first assume that both parties cannot default at the same time and that there is no wrong way risk. The two parties have also agreed on the price of the credit risk and entered the trade. In addition to the notation for unilateral CVA, we introduce ‘first-to-default time’ denoted by ??^1=min'(??_B,??_C). Up to maturity T, we might observe one of the following payoffs depending on the particular default event. Neither the bank nor the counterparty default before T: 1_{??^1>T} V(t,T)

In other words, there is no difference between the value of the risky and the risk free position.

If the counterparty defaults first and before time T, the Loss Given Default is expressed as

1_{??^1’T} 1_{??^1=??_c } (??_C ‘V(??^1,T)’^++’V(??^1,T)’^-)

The additional term to the Unilateral CVA formulation is derived from the case when the bank itself is first to default and the event occurs before time T. Suppose in that case the bank owes to the counterparty, then they pay a recovery fraction of the value owed (negative market position). If the opposite is true, then they will receive the full amount from the counterparty (positive market position). Consequently, the payoff is the opposite of the one in the case of the counterparty’s default:

1_{??^1’T} 1_{??^1=??_B } (??_B ‘V(??^1,T)’^-+’V(??^1,T)’^+)

If either the bank or the counterparty default, then all the payments prior to the default time must be settled:

1_{??^1’T} V(‘t,’??^1 )

By putting all the payoffs together and simplify, we obtain the formula for Bilateral CVA:

BCVA(t,T)=E^Q [1_{??^1’T} 1_{??^1=??_c } (1-??_C ) ‘V(??^1,T)’^+

+ 1_{??^1’T} 1_{??^1=??_B } (‘1-‘??_B)’V(??^1,T)’^-]=

‘ E’^Q [1_{??^1’T} 1_{??^1=??_c } (1-??_C ) ‘V(??^1,T)’^+

– 1_{??^1’T} 1_{??^1=??_B } (‘1-‘??_B)'(-V(??^1,T))’^+]

The first term in this equation is CVA of the counterparty while the second term is exactly the opposite known as Debt Value Adjustment (DVA). It corresponds to the fact that if the bank defaults, it will make a profit if the market position value is negative i.e. will only repay a fraction of the quantity owed.

Definition: Bilateral CVA

CVA(t,T)=’ E’^Q [1_{??^1’T} 1_{??^1=??_c } (1-??_C ) ‘V(??^1,T)’^+ ]

DVA(t,T)=’ E’^Q [1_{??^1’T} 1_{??^1=??_B } (‘1-‘??_B)(-V(??^1,T))^+ ]

BCVA(t,T)=CVA(t,T)-DVA(t,T)

Under the above formulation, we might obtain negative CVA value if the value of the DVA is larger than the CVA. Consequently, the value of the risky portfolio must be larger than its value under the risk free probability measure which could be a result of manipulation where the bank attached value to its own default. As it appears that the second term in the equation is subject to great uncertainty, using bilateral CVA should be cautious. Due to the opportunity for moral hazard imposed by the presence of the DVA term, both media and regulators proposed DVA to be excluded from the calculation of the credit value adjustment. On the other hand, it is of fundamental importance that both parties in the trade agree upon the value of the credit risk and excluding the credit risk measure on one of them leads to another problem. The debate on this issue is growing and so worthy of consideration in this thesis. We chose to adopt the unilateral approach for the purpose of this thesis.

2.4 CVA components

CVA measure depends on three main components ‘ recovery rate, expected exposure and probability to default. Usually, the recovery rate is a fixed fraction of the realized loss in case of default. As this component is usually quoted by the market, we assume it is a fixed number. The other two components are more complex and depend on a variety of market factors. In the following section we discuss in detail their nature, approximation and importance to evaluate CVA.

2.4.1 Contract-level exposure

If the bank has a position in a single derivative contract with a counterparty which defaults, we assume that the bank will enter another contract with a different counterparty to keep its market position. In that case, the realized loss is the replacement cost when the default occurs. Thus, the bank can take one of the following actions:

If the contract value is negative, the bank closes out the position and pays to the counterparty the contract’s current market value. Simultaneously, it opens a position on a similar contract with the other counterparty. The price of the contracts cancels out without a cost for the bank.

If the contract value is positive to the bank when the default occurs, it closes the position with the defaulting counterparty but receives nothing from it. For the same reason as before, it enters into a similar contract. The loss for the bank in this case is the price of the new contract.

Following, the contract-level exposure could be expressed as

E_i (t)=max'{V_i (t),0}

where the contract-level exposure equals the maximum value of the contract i at time t, or zero. Taking into account that the contract between the two counterparties could be either an asset or liability, follows that the nature of the counterparty credit risk is bilateral.

2.4.2 Counterparty-level exposure

If we assume that there are multiple trades with the defaulting counterparty and the risk is not mitigated in any way, the total exposure is simply the sum of all individual exposures

E_i (t)=’_i”max'{V_i (t),0}’

However, in this case netting agreements could be introduced which would result in a significantly lower credit exposure. By definition, a netting agreement is a contract for offsetting the positive with negative positions in the portfolio at the default time. In that case, the exposure would be defined by the maximum between zero and the sum of all individual contract exposures

E_i (t)=max'{‘_i”V_i (t) ‘,0}

In a certain portfolio, the bank may have a number of netting agreements with the same counterparty. For simplicity, we do not consider netting in this thesis but rather analyse the case when the counterparty risk is not moderated.

2.4.3 Modelling Credit Exposure

Calculating credit exposure is essential for the bank to price and hedge counterparty credit risk and to estimate economic and regulatory capital. The purpose of this section is to present a universal framework for obtaining the exposure distribution for any future date. The process contains the following steps:

Scenario Generation

The first step is simulating possible future values of the risk factor. For the purpose of pricing interest rate exotic derivatives, we need different scenarios for the evolution of the interest rates. We can choose between two ways of generating possible values of the risk factor. The first one generates a trajectory which the risk factor follows from the initial time t=0 to the last simulation date t=T. Similar to Pykhtin and Zhu (2010), we refer to this method as Path-Dependent Simulation (PDS). Alternatively, the futures values could be generated by a direct jump from the initial value to the simulation date, called ‘Direct Jump to the Simulation date’ (DJS). The dynamics of the interest rate under the Two Factor Gaussian model is presented in Chapter 3.

Simulation of the scenarios could be performed under both risk-neutral and real probability measure. By the fundamental theorem of asset pricing, the absence of arbitrage opportunities is equivalent to existence of risk neutral measure . Hence, we adopt the risk neutral approach to calibrate the drifts and match the volatility parameters to the market implied volatilities of the interest rate derivatives considered.

Instrument Valuation

When the possible future values of the interest rates are generated, a valuation of the exotic derivative contract for each simulation date is performed. In our case we calculate the prices of Bermudan Swaptions, presented in Chapter 3.

The complexity with pricing Bermudan swaptions or other path-dependent derivatives for a certain set of scenarios is imposed by the uncertainty of whether the instrument has been exercised or barrier has been hit at each of the future simulation dates. This imposes a challenge for the DJS approach as the simulated jumps do not contain information for each other. Lomibao and Zhu (2005) provide a solution to the issue by introducing the concept of ‘conditional valuation’ which incorporates the events that could have occurred in the past into the marked-to-market model. The framework calculates the mean of the future marked-to-market values for all the paths for a certain simulation. In other words, the value of the contract is assumed to be equal to the expectation of the marked-to-market value at any future date, conditional on the information available between today and the simulation date. In terms of the PDS approach we solve

V_PDS (t_k,{r(t_j )}_(j=1)^k )=E[V_MTM (t_k,{r(t)}_(t’t_k ) )|{r(t_j )=r_j }_(j=1)^k]

where r is the underlying factor, in our case ‘ the interest rate.

For DJS implementation we only have the simulation date and the value of the price factor:

V_DJS (t_k,{r(t_k )})=E[V_MTM (t_k,{r(t)}_(t’t_k ) )|{r(t_k )=r_k }]

Lomibao and Zhu (2005) have also shown that both approximations provide results consistent with the market values of barrier, average and interest rate options.

Portfolio Aggregation

Depending on the nature of the portfolio, the total exposure is calculated by taking into account the presence of any collateral, margin or netting agreements.

Following, the future exposure is a product of all factors noted above. However, there are two main effects that determine the exposure over time ‘ so called ‘diffusion -‘ and ‘amortization’ effect. Diffusion effect increases the exposure with time while amortization effect reduces the exposure as the time passes because it decreases the number of cash flows exposed to default.

2.4.3 Default Probabilities

The default probability reflects the degree of likelihood that counterparty will not be able to make necessary scheduled payment. Calculating default probabilities is performed via bootstrapping the term structure of the credit-default swaps (CDS). This point is developed in Chapter 4 and a credit model has been proposed.

2.5 Computational complexity

This thesis focuses on calculating CVA in the absence of wrong way risk – we assume independency between the three components recovery rate, default and exposure.

In order to price CVA we need both the current value of the contract and the distribution of its possible future values. For the purpose of analysing CVA of interest rate exotic derivatives, we need a model which generates the risk factor, in this case, the whole yield curve. Those models usually require high computational power and are also time demanding.

Most of those derivatives do not have a close ‘form pricing formula and the valuation process requires implementation of a numerical technique as Monte Carlo, for example. It is used to generate future scenarios of the risk factor for a number of simulation dates. However, the number of simulated scenarios is limited to a few thousands and the dates considered are daily, weekly, monthly or even yearly.

Another difficulty in the calculation of the CVA is imposed by the complexity of the portfolio subject to valuation, as it might contain collateral, margin or netting agreements. To be accurate, CVA must account for the effect of those features on the portfolio value.

As the topic of this thesis requires pricing of exotic products, we adopt Longstaff-Schwartz method of Monte Carlo simulation due to its ability to price contracts with specific clauses like early exercise optionality.

Chapter 3

Interest Rate Models

3.1 Introduction

3.1.1 Choice of model

So far we have examined the importance of the credit model as a requirement for obtaining the CVA measure. However, for the purpose of calculating the CVA of any traded interest rate product, a model which provides the dynamics of the term structure process is necessary as it allows pricing derivatives analytically.

Literature provides a huge scope of choice between interest rate models, which differ in the underlying process for the fundamental quantity modelled and the number factors assumed to characterise the yield curve.

Very promising and widely applicable are the so called interested rate market models, more specifically, the LIBOR and the Swap Market Models. It must be taken into account that as a result of the recent events in the financial industry, LIBOR is no-longer considered to be a risk-free rate and the applicability of those models has become questionable. As an advantage, they both provide closed form formulas for swaption pricing and terminal correlations, leading to a computational efficiency. However, both models perform Monte Carlo pricing and do not produce a recombining lattice for the short rates, then pricing path-dependent derivatives with an option for early exercise (like American- and Bermudan-style) becomes problematic.

Heath, Jarrow and Morton (1992) framework specifies the evolution of the forward rates through their instantaneous volatility structures. The problem with this method comes from the restricted number of volatility structures that imply recombining short rate trees. The complication for the evaluation of the exotic derivatives is that a particular volatility structure has to be considered.

In comparison to the HJM framework, with one-factor models, the volatility of the short rates itself does not characterize the model. One-factor short-rate models like Vasicek (1977), Cox, Ingersoll and Ross (1985), Dothan (1978), Hull and White (1990), have the advantage of relatively easy implementation and computational simplicity. However, those models base the dependence of the derivatives payoff only on one single rate (for instance, six-month rate) and assume perfect correlation between the rates. As the correlation between near rates is relatively high, those models are still practical to an extent. In this theses however, we aim to achieve higher computational accuracy by implementing a model which considers more precisely the relevance of the correlations.

The logical question that follows is how many factors should be used for the purpose of pricing exotic products. Research papers on the multi factor models provide an answer in terms of numerically-efficient implementation and capability of fitting the market data. Most satisfactory option is the Linear Gaussian Two Factor model (G2++). In this case, the instantaneous short rate process is presented in terms of two correlated Guassian processes and a deterministic function, chosen such that the model provides an exact fit to the market data. The Gaussian distribution gives an opportunity for obtaining explicit formulas for wide range of vanilla products, which is used for relatively easy and tractable evaluation of the prices of any payoff. Moreover, the model captures the volatility smile in the market and once the model has been calibrated to plain vanilla market prices, it could be used for pricing path-dependent exotic products.

There exist a close relationship between the two-factor Gaussian and the two ‘factor Hull-White (1990) model, which provides equivalence between both approaches. For the purpose of this thesis, the two-additive factor model is more suitable as it provides less complication regarding formulation and implementation in practice.

In a summary, the most desired model is the one which provides an accurate fit to the initial term structure and the option for building recombining trees for the evolution of the instantaneous spot rates to lie in the base of pricing path dependent products with an option for early exercise. Moreover, we require analytical tractability and computational-efficiency. The most suitable model in this case is the two-additive-factor Gaussian model.

Having this decided, some drawbacks of the model have to be taken into account. Its most problematic feature is that the model allows for negative interest rates. However, considering the current economic conditions when the negative rates are no longer impossible, this feature increases the model’s applicability in practice.

3.1.2 Notation

The notation and the concept applied in this project are introduced below and follow Brigo and Mercurio (2007) Chapter ‘

First of all, we define the short rate as it is the building base of the model. B(t) is the value of the Money Market Account and represents a riskless investment such that

B(t)=’_0^t”r_s ds’

With dynamics:

dB(t)=r_t B(t)dt, B(0)=1

In words, investing a unit at time 0 returns the amount B(t), where r_t is the instantaneous spot rate.

We denote by P(t,T) the fundamental coordinate which characterised the yield curve, that is, the price of a discount bond at time t, with maturity time T.

P(t,T)= E_t [exp(-‘_t^T”r_s ds’)]

Therefore, as the price of the bond is known at time t, the whole evolution of the curve is characterised by the quantity r.

At last, we denote the continuously compounded instantaneous forward rate at time t for maturity T by f(t,T). The forward rate can be also seen as the rate of the forward contract as it approaches its expiry date

f(t,T)= -‘/’T ln”P(t,T)’

3.2 The Linear Gaussian Two Factor model (G2++)

Under the risk-adjusted measure Q, Brigo and Mercurio (2006) provide the dynamics of the short rate process such as

r(t)=x(t)+y(t)+??(t), r(0)=r_0

dx(t)=-ax(t)dt+??dW_1 (t), x(0)=0

dy(t)=-by(t)dt+??dW_2 (t), y(0)=0

Where t’0; r_0,a,b,??,?? are positive constants; ??(t) is a deterministic function and ??(t)’ [0,T^* ] and T^* is the time horizon and ??(0)= r_0. W_1 and W_2 are two-dimensional Brownian motions with correlation parameter ?? such that:

dW_1 (t)dW_2 (t)=??dt

By simple integration of the processes above, we obtain

r(t)=x(s) e^(-a(t-s) )+y(s) e^(-b(t-s) )+’??_s^t”e^(-a(t-u) ) dW_1 (u)’+’??_s^t”e^(-b(t-u) ) dW_2 (u)’+??(t)

3.2.1 Pricing a zero coupon bond in G2++

Zero-coupon bond price is required for the process of calculating CVA of any product as it supplies the model with the necessary discount factors.

The formula for the bond price at time t which matures at time T in this framework is given by:

P(t,T)=A(t,T) exp'{-B(a,t,T)x(t)-B(b,t,T)y(t)}

Where:

A(t,T)=(P^M (0,T))/(P^M (0,t)) exp'{1/2 [V(t,T)-V(0,T)+V(0,t)]}

B(z,t,T)=(1-e^(-z(T-t)))/z

And the variance of a bond is represented by:

V(t,T)=??^2/a^2 [T-t+2/a e^(-a(T-t) )-1/2a e^(-2a(T-t) )-3/2a]+??^2/b^2 [T-t+2/b e^(-b(T-t) )-1/2b e^(-2b(T-t) )-3/2b]+2?? ‘?/ab [T-t+(e^(-a(T-t) )-1)/a+(e^(-b(T-t) )-1)/b-(e^(-(a+b)(T-t))-1)/(a+b)]

The calibration of the model has to be performed to a class of very liquid financial instruments, giving it the flexibility to adopt the market term structure at any point of time. Hence, we focus on the pricing of the highly traded interest rate caps and floors, and swaptions.

3.2.2 Pricing European Swaption

For the purpose of calibrating the model, we choose to use European swaptions prices because this approach is more suitable for the particular products whose price we evaluate later, this is, the Bermudan interest rate options. In this case we want to fit the model around the volatilities of the most liquid underlying swaptions.

The arbitrage-free price at time t=0 of European swaption in G2++ model, following the notation provided by Brigo and Mercurio (2006), is given by:

ES(0,T,T,N,X,??)=N??P(0,T)’_(-‘)^”’e^(-1/2 ((x-??_x)/??_x )^2 )/(??_x ‘2??) [??(-??h_1 (x))-‘_(i=1)^n”??_i (x)e^(k_i (x) ) ??(-??h_2 (x))’]dx’

Where:

h_1 (x)=(y ??-??_y)/(??_y ‘(1-??_xy^2 ))-(??_xy (x-??_x))/(??_x ‘(1-??_xy^2 ))

h_2 (x)=h_1 (x)+B(b,T,t_i)??_y ‘(1-??_xy^2 )

??_i (x)=c_i A(T,t_i)e^(-B(a,T,t_i )x )

k_i (x)=-B(b,T,t_i)[??_y-1/2 (1-??_xy^2 ) ??_y^2 B(b,T,t_i )+??_xy ??_y (x-??_x)/??_x ]

y ??=y ??(x) is the unique solution of the equation:

‘_(i=1)^n”c_i A(T,t_i ) e^(-B(a,T,t_i )x-B(b,T,t_i ) y ?? )=1’

??_x=-(??^2/a^2 +?? ‘?/ab)[1-e^(-aT) ]+??^2/’2a’^2 [1-e^(-2aT) ]+?? ‘?/(b(a+b)) [1-e^(-(a+b)T) ]

??_y=-(??^2/b^2 +?? ‘?/ab)[1-e^(-bT) ]+??^2/’2b’^2 [1-e^(-2bT) ]+?? ‘?/(a(a+b)) [1-e^(-(a+b)T) ]

??_x=’??((1-e^(-2aT))/2a)

??_y=’??((1-e^(-2bT))/2b)

??_xy=”/((a+b)??_x ??_y ) [1-e^(-(a+b)T) ]

Also ??=1 for a payer -, and ??=-1 for a receiver swaption. X is the strike of the swaption with maturity T and nominal value N. The holder could enter a swap at time t_0=T, with payment times T'{t_1,’ t_n }, and t_1>T.

??_i denotes the year fraction from t_(i-1) to t_i, i=1,’,n and c_i=X??_i for i=1,’,n-1 and c_n=1+X??_n.

3.2.3 Pricing Bermudan swaption

As the topic of this thesis requires pricing exotic interest rate derivatives, we choose value Bermudan swaptions for the set of simulated dates.

Due to the dependence of the pricing process on the opportunity of early exercise, most exotic options, Bermudan swaption included, do not provide a closed form solution for the price. Consequently, a numerical technique shall be used to construct the exposure distribution through time. Cesari et al. (2010) proposes the implementation to be performed via Monte Carlo technique. Then the computation of the Bermudan Swaption prices is numerically performed according to Longstaff and Schwartz’s (2000) approach for a set of Monte Carlo simulations (LSMC) which provides the exercise behaviour with swap rates basic function. The algorithm’s specifics presented in this thesis follow the notation provided by Brigo and Mercurio (2006).

Definition (Bermudan Swaption): ‘A (payer) Bermudan swaption is a swaption characterised by three dates T_k<T_h0|V_t’L)’

Further expansions of these models try to incorporate stochastic interest rates and stochastic barrier (for example, Longstaff and Schwartz, 1995). Moreover, some modify the asset value dynamics to follow exponential of non-Brownian Levy process. For example, Zhou (1996, 2001) introduces log-normally distributed jumps; Lipton (2002) consider downward jumps which not only hit but also cross the barrier; Kou and Wang (2003, 2004) adopts fluctuation identities for the purpose of pricing path dependent options on assets, where the underlying is driven by exponentially distributed Poissonian jumps; Madan and Schoutens (2008) state that the jumps considered must be negative in any case.

An important part of the structural models application is their calibration to market data so accurate estimates of default probabilities could be obtained. For this purpose, the approach is calibration to credit default swap term structure as CDS in their nature are financial instruments designed to provide securitization against default. However, the firm-value models dependence on information from the company’s balance sheet is an obstacle for the process of calibration as such information is available at most four times per year. Jarrow and Protter (2004) justify that nonlinear filtering problems are observed with structural credit risk models due to incomplete data on the current firm’s assets and liabilities value.

Eom, Helwedge and Huang (2004) empirical tests of corporate bond pricing gives an insight of the structural models relevance to the real world. Starting from the Merton’s and considering some of its extensions up to date. The outcome suggests a significant room for improvement as none of the models shows the ability to fit exactly a given term structure of spreads. For instance, Merton’s model predicts too low spreads compared with the observable in the bond market. Other models tend to underpredict the spread on relatively safer bonds or overpredict the credit risk of risky bonds and firms with high leverage.

O’Kane and Turnbull (2003) also stress on another limitation of the firm-value models, crucial for the purpose of this thesis, is that they cannot be easily modified to price exotic interest rate contracts.

4.2.2 Reduced form approach

The other approach for calculating default probabilities is known as reduced form as it models directly the probability that a default is inaccessible event, independent of market quantities. Under this framework, through derivatives pricing models default probabilities can be extracted directly from market quotes. They also prove high flexibility in terms of fitting a given term structure of wide variety of credit derivatives, including contracts with exotic features.

Jarrow and Turnbull (1995) provides one of the most commonly applied reduced form approach for credit risk modelling where the default is assumed to follow a Poisson process with constant intensity.

Later, Duffie and Singleton (1999) as well as White and Hull (2000) introduce similar models. One of their main advantages is the easy mathematical tractability. Jarrow and Protter (2004) argue that those models do not seem to contain all the information about the credit event which from theoretical point of view is more consistent with the real world.

Dependence on market variables can be introduced by assuming a stochastic process for the default intensity parameter dynamics. Such models are referred to as doubly Poisson processes, introduced later in this thesis.

Consequently, reduced form models have simultaneously the advantage and disadvantage of flexibility in their functional form. They have been proven to fit easily the term structure of credit spreads, although a result could be strong in-sample fitting properties and weak out-sample predicting ability.

Due to the lack of exact theoretical characterization of the default intensity function, there is not many empirical studies which test the reduced form models. Duffee (1999) provides an evidence of high instability of parameters when a square-root process for the intensity function has been used. Accessing reduced models creditability is challenged by the methodology of using bonds data is used to fit the model but also to test the model’s performance. As a result, those models are usually calibrated to CDS quotes.

In this thesis, we focus on Jarrow and Turnbull (1995) which is the most widely adopted by the market method for calculating default probabilities. The probability of a default event, occuring at time ?? and characterized as the first jump of a Poisson continuous process, is defined by

Pr'[??<t+dt ‘| ‘??t]=??(t)dt In words, the probability that the counterparty survives to some point of time in the future, denoted by t+dt, conditional on no default event occurring before t, is equal to the product of the function ??(t), also known as hazard rate, and the time interval dt. Once we obtain the implied market values for the term ??(t)dt, we can calculate the survival probabilities equal 1- ??(t)dt. Then, the probability to receive the recovery rate R if a default event occurs, is equal to ??(t)dt and could be easily obtained. To proceed, an assumption for the nature of the hazard rate needs to be settled. Initially, we adopt the assumption of a deterministic hazard rate function. In other words, it does not depend on the yield curve and recovery rates. The validity of this assumption is discussed further in Chapter 4. For the purpose of calculating CVA, however, we need an expression for the continuous time survival probability as a simulation of different scenarios for multiple time periods is required. In that case the survival probability expression is extended to: Q(t_V,T)=exp'(-‘_(t_V)^T”??(s)ds’) Where t_V is the valuation date. 4.2.3 Comparison Arora, Bohn, Zhu (2006) are the pioneers in empirically testing and comparing the performance of structural and reduced form approaches by considering Merton’s, Vasicek-Kealhofer (VK) and White and Hull model of credit risk. For this purpose, they access the models on a broad cross-section data of credit default swaps and consequently, neither of the models is calibrated to CDS spread quotes. They conclude that the reduced form White and Hull model underperformes compared to Vasicek and Kealhofter, but significantly outperforms the simple Merton model. However, as mentioned above reduced form models are expected to perform best when calibrated to CDS spreads. This research cannot provide a significant guidance to the credibility of the models neither of those models represent the latest developments in the sector. Advantages and shortcomings of both approaches need to be accessed depending on the purpose of their application. For the aim of pricing interest rate exotic derivatives, we choose to implement reduced form models as it can perfectly first the term structure of the credit spreads and is easier to extend one to pricing exotic options. To keep it simple, we assume no correlation between the credit events and the yield curve. Calibration to CDS spreads 4.3.1 Valuing the premium leg Credit Default Swap is a contract which provides the buyer with a credit protection while the seller guarantees credit worthiness of the debt security. The premium leg is the sum of all defalt swap spread payments made before the maturity of the contract is default does not occur. Contrary, is the presence of a default event, it consists of the value of all the payments made prior to the credit event. The value of the premium leg when ignoring the premium accrued is given by: Premium Leg PV(t_V,t_N )=S(t_0,t_n ) ‘_(n=1)^N”(t_(n-1),t_n,B)Z(t_V,t_n )Q(t_V,t_n ) Where t_n represents the contractual payment dates for n=1,’,N, and t_N is the maturity date of the default swap. S(t_0,t_n ) denotes the CDS spread for time period from 0 to n; The term ‘(t_(n-1),t_n,B) represents the day count fraction between t_(n-1) and t_n with the appropriate discount base B; Z(t_V,t_n ) is the Libor discount factor for the valuation date to premium payment date n; Accordingly, Q(t_V,t_n ) is the risk-neutral survival probability for the same time period. The formula above does not consider that in a case of default, the protection buyer is obligated to pay a fraction of the premium that has been added since the previous payment date up to the time of default. Thus, the next step for including this effect is to calculate the expected premium arising in this time gap when considering the default probability and to calculate the probability weighted premium payment. The premium period is assumed to between dates t_(n-1) and t_n. We derive the survival probability from valuation time to each time point s within the premium period and the probability of default in the interval ds. This is given by Q(t_V,t_n )??(s)ds. Each premium accrued payments up to any s has to be estimated and discounted to the valuation date by Libor discount curve and sum all premium periods. We obtain the following equation for the premium accrued S(t_0,t_N ) ‘_(n=1)^N”_(t_(n-1))^(t_n)”(t_(n-1),s,B)Z(t_V,s)Q(t_V,s)??(s)ds O’Kane and Turnbull (2003) note that if the credit event occurs within two payment dates, the average premium that would have to be paid is half of the amount which would be paid at the end of the period. Hence, the above equation can be simplified to S(t_0,t_N )/2 ‘_(n=1)^N”'(t_(n-1),t_n,B)Z(t_V,t_n )(Q(t_V,t_(n-1) )-Q(t_V,t_n ))’ By definition, the value of the premium leg is the product of the CDS spread and the risk-adjusted present value of one basis point (RPV01). Thus, we have Premium Leg(t_V,t_N )= S(t_0,t_N )RPV01 and RPV01=’_(n=1)^N”(t_(n-1),t_n,B)Z(t_V,t_n )[Q(t_V,t_n )-1_PA/2 (Q(t_V,t_(n-1) )-Q(t_V,t_n ))] Where 1_PA is an indicator function which returns 1 is a premium accrued has been specified, and 0 otherwise. 4.3.2 Valuing the protection leg The protection leg is a fraction of the face value paid in the case of default. The timing of the default is crucial for approximating its present value. Thus, we introduce conditioning on each small interval from s to s+ds. O’Kane and Turnbull (2003) note that this time interval does not need to be less than a day as the assumption of intra-day occurrence if the default does not lead to a significant change in the results. If R is the recovery rate, the expected value of the protection leg, discounted back to the valuation time is given by: Protection Leg PV(t_V,t_N )=(1-R)’_(t_V)^(t_N)’Z(t_V,s)Q(t_V,s)??(s)ds The integral in the equation above imposes high computational complexity. In order to simplify the above equation, we adopt the assumption that the default could happen a finite number of times throughout the life of the contract. We denote this quantity by M such that m=1,’,M ?? t_N. Thus, Protection Leg PV(t_V,t_N )=(1-R)’_(m=1)^(M ?? t_N)”Z(t_V,t_m )(Q(t_V,t_(m-1) )-Q(t_V,t_m ))’ 4.3.3 Calculating the breakeven default swap spread Now we can compute the default probabilities as a function of the CDS spread. By definition, the breakeven spread requires equality between the present values of the premium and the protection leg of the default swap: Protection Leg PV=Premium Leg PV We set t_V=t_0 for new contract. Hence S(t_0,t_N )=((1-R)’_(m=1)^(M ?? t_N)”Z(t_V,t_m )(Q(t_V,t_(m-1) )-Q(t_V,t_m ))’)/RPV01 Obviously, given just one equation, we can calculate only one value for the survival probabilities. Here comes the use of the hazard rate function specified before. Through our choice of inhomogeneous Poisson process for the intensity, we derive all the survival probabilities needed by implementing the bootstrap procedure described previously. 4.4 Intensity models for credit risk 4.4.1 The Poisson Process It is the most simple jump Levy process existing. The distribution, ??, is Poisson and the process depends on the intensity parameter ?? only, characterised as follows: ??_Poisson (u; ??)=exp'(??(exp'(iu)-1)) Where i is the imaginary number such that i^2=-1,?? is the characteristic function of the distribution, or equivalently of a random number X such that, by definition, it is the Fourier-Stieltjes transform of the distribution F(x)=P(X’x). Note that an assumption of the probability measure, P, has not been made. At this stage, we do not need such specification as the Poisson process is approached from a pure theoretical point of view. Definition: The Poisson Process ‘A stochastic process N={N_t ,t’0} with intensity parameter ?? > 0 is a Poisson process if:

N_0=0;

The process has independent increments;

The process has stationary increments;

For s0. The corresponding default probability model is developed by Jarrow and Turnbull (1995). Under this framework, the expected time of default is represented by ??=1/?? and hence, the probability that no default appears between time 0 to time t is:

P_Surv^HP (t)=exp'(-??t)

Mathematically, the above expression gives the probability that a jump has not occurred prior to time t.

The inhomogeneous case (IHP)

The base of the inhomogeneous Poisson process is the assumption of time varying default intensity which is more realistic assumption than the homogenous case. Now, the survival probability for the time interval from 0 to t is given by:

P_Surv^IHP (t)=exp'(-‘_0^t’??(s)ds)

To consider the case of piecewise constant default intensity, let ??_t=K_j,T_(j-1)’t