Counterparty Credit Risk Simulation of a Plain Vanilla IRS using
One-Factor Short-Rate Models
Author: BSc. E.J. van Viersen Student number: 10267980
October 25, 2013
Master thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Operations Research and Management
at the Faculty of Economics and Business
Dr. H.J. van der Sluis
Acknowledgments
Before the mathematics commence, I’d like to thank a few people. I would like to thank Dimitar Mechev for his help and time during the first half year of my time at NIBC. I really liked our sparring sessions on some concepts, and his dedication to guiding me throughout the internship. I would like to thank Eloy Cosijn for giving me the chance to explore this interesting subject during an internship which ultimately has led to my first job. I would like to thank Dr. van der Sluis, for the support he has shown me and his extensive knowledge of OR which he allowed me to call upon. I would like to thank Prof. Boswijk. With his knowledge of financial mathematics he was able and willing to provide comments and ideas which helped me very much. Last but not least, I would like to thank Junsheng Huang for our fun time together in the ‘’internship-room” and for our collaboration which allowed both of us to compare our models to a benchmark.
Table of Contents I. Introduction 3 II. Terminology 4 2.1 Financial Markets 4 2.1.1 Derivatives 4 2.1.2 Interest rates 4
2.1.3 Zero coupon Bond 5
2.1.4 Term structure 5
2.1.5 Forward rates 6
2.1.6 Interest Rate Swap 7
2.1.7 Interest Rate Cap 8
2.1.8 Counterparty Credit Risk 9
2.2 Risk-Neutral Measure and CVA 10
2.3 Interest rate models 12
2.3.1 Short rate models 12
2.3.2 LIBOR Market Model 13
2.4 Calibration 13
III. Short rate models 14
3.1 One-Factor Vasicek 14
3.1.1 Discretization 15
3.1.2 Calibration to current Term Structure 16
3.2 One-Factor Hull-White 19
3.2.1 Calibration to Caps 20
IV. Exposure simulation 23
4.1. Comparing 1FV to LMM 25
4.2 Comparing 1FHW to LMM 29
V. Variance Reduction 31
VI. Conclusion 34
VII. Further Research 34
I. Introduction
Counterparty credit risk is the credit risk to each party of an Over-The-Counter (OTC) derivative that the counterparty will not fulfill its contractual obligations (Gregory, 2010, p.9). The financial crisis that started in 2007 and that we are still experiencing has shown that financial risk management underestimated counterparty credit risk. Financial institutions such as Bear Sterns, Lehman Brothers and Fannie Mae all had OTC derivatives on aggregate notional values of hundreds of billions of dollars at bankruptcy; contracts that their counterparties lost to a large extent. The total OTC market had a notional outstanding of $601 trillion at December 31, 2010 and has been the largest derivative market for a long time. Assessing and managing counterparty credit risk has therefore become an important aspect for the financial system as a whole.
Mathematical tools are being used increasingly in the estimation of counterparty credit risk of OTC derivatives. For interest related OTC derivatives, such as interest rate swaps, interest rate models can be used to describe the evolution of the interest rate or related concepts such as forward rates and consequently the price of the OTC derivatives. The most well-known classes of interest rate models are short-rate models and the LIBOR market model. The former class tries to describe the future evolution of interest rates by modeling the short-rate, whereas the latter class models forward rates. The LIBOR market model is becoming increasingly popular as a benchmark model in the financial sector. On the other hand, (one factor) short rate models such as the one-factor Vasicek and one-factor Hull-White have been in use for a long time and additionally lead to explicit analytical formulas for derivatives traded in high volume such as Interest Rate Swaps and -Caps. The ability of these relatively simple short rate models to model counterparty risk for certain products is an additional rationale for the use of such models. Therefore, I would like to answer the following question:
How well do one-factor Short-Rate models perform in the assessment of counterparty credit risk of a plain vanilla IRS compared to the LIBOR market model1?
First the terminology needed to understand the setting is introduced. Subsequently a number of well-known short-rate models will be introduced and calibrated. Ultimately, a certain risk profile (the expected exposure) computed using the short-rate models discussed will be compared using the LIBOR market model as a benchmark.
1
During my internship at NIBC bank N.V., Junsheng Huang wrote his thesis on the LIBOR Market Model as part of the Stochastics and Financial Mathematics Master (University of Amsterdam). All curves for the LIBOR Market Model in this thesis are due to his work.
II. Terminology 2.1 Financial markets 2.1.1 Derivatives
A financial derivative has a value that is derived from one or more underlying factors. These factors can be assets such as commodities or stocks but also more fundamental market factors such as interest rates or foreign exchange rates. Derivatives, whose value depends on interest rates, are naturally called interest rate derivatives. This paper considers only plain vanilla (e.g. fixed-for-floating swap in the same currency) interest rate derivatives. Over-The-Counter (OTC) derivatives are derivatives traded between two counterparties without the involvement of an intermediary exchange such as AEX or NASDAQ.
2.1.2 Interest rates
Let be the value of a bank account at time described by the following differential equation (Brigo & Mercurio, 2006, p.2)::
, 0 1
which implies:
Here is called the instantaneous rate, or short-rate. The short rate can be thought of as the interest rate at which an entity can borrow money for an infinitesimal small period of time. The short rate will play a central role in this paper, since the models that will be introduced are aimed at modeling the evolution of the short rate over time.
Given the expression for the bank account, what is the value at time of one unit of currency available at time ? Assuming a deterministic short-rate , units of currency at time 0 will accrue to a value of at time . The following relationship must hence hold for :
1
Therefore at time zero, which can be computed explicitly since is deterministic. At time , is worth , which is exactly the value of one unit of currency at time as seen from time . The quantity , is known as the discount factor , between time and . If is stochastic, the discount factor becomes stochastic as well and can be
calculated by the following expectation with respect to the risk-neutral measure (explained in subchapter 2.2):
,
In this paper, the short rate is always assumed to be stochastic.
In the context of interest rates, the type of compounding is of importance. Compounding means that interest is added to the principal. On the added interest, again interest is earned. The simply-compounded spot interest rate , is the constant rate at which an investment of , at time is worth one unit of currency at maturity if accruing occurs proportionally to the investment time, and is given by:
, 1
,
1
, 1
In this paper, , denotes the length of the time interval , expressed as a fraction of a year. A well-known example of a simply discounted spot interest rate is the London Interbank Offered Rate (LIBOR).
Analogously to the simply-compounded spot interest rate is the continuously-compounded interest rate , , which is the constant rate at which an investment of , at time accrues continuously to yield one unit of currency at maturity . One can express , as follows:
, ln ,
,
2.1.3 Zero-coupon bond
A zero-coupon bond with maturity is a contract that at maturity guarantees the payment of one unit of currency. The value of the contract at time is denoted by , . Intuitively , 0 for
→ ∞ and , 1. If the short-rate is deterministic we have , = , , i. e. the price of the bond at time equals the corresponding discount factor. If the short-rate is stochastic the bond is equal to the following expectation (Brigo & Mercurio, 2006, p.6)::
,
(2.1) We express the continuously-compounded interest rate , in terms of a zero-coupon bond:
, ln ,
, (2.2)
The continuously-compounded interest rate , is therefore consistent with the zero-coupon-bond prices in the following sense:
, , , 1
In the same manner, we express the simply-compounded spot-rate , in terms of a zero-coupon bond: , 1 , 1 , 1 2.1.4 Term structure
A term structure shows the continuously-compounded interest rate at some fixed future time as a function of maturity and is given by : → , . Figure 1 shows an example of a term structure for a maximum maturity of twelve years for 0, that is : → 0, , ∈ 0,12 :
Figure 1: Term structure
2.1.5 Forward rates
A forward rate ; , is the future yield on a zero-coupon bond for the period [ , at time . Forward rates are determined by three time instances: the time when the forward rate is determined , its expiry (the beginning of the specific time period) and its maturity (the end of the specific time period). Forward rates can be calculated using the term structure (Brigo & Mercurio, 2006, p.12)::
0 2 4 6 8 10 12 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Years Yi e ld
; , 1 , ,
1 , ,
, 1
Forward rates for any can additionally be expressed in terms of zero-coupon bonds:
; , 1
,
,
, 1 (2.3)
2.1.6 Interest Rate Swap
An Interest Rate Swap ( ) is an OTC derivative that gives the buyer the possibility to lock in the interest rate at a given fixed rate K. Starting at time , for given instances in the predetermined set of times , … , the seller pays amounts according to the floating leg:
,
Where , is the spot rate for the period , , whereas the buyer pays amounts according to the fixed leg:
The time between two payments and is known as the tenor of the swap. Typically, an has a three or six month tenor. In this paper we will always consider an from the perspective of the receiver of fixed ( who pays the floating leg and calculate the discounted payoff as follows (Brigo & Mercurio, 2006, p.14):
, , , , , ,
(2.4)
Expression (2.4) can be rewritten in terms of zero coupon bond prices and forward rates as follows:
, , , , , ; ,
(2.5) can be used for example in the LIBOR market model context. Expression (2.4) can also be rewritten fully in terms of zero coupon bond prices, which shall be used in the context of short-rate models:
, , , , , , , (2.6)
Usually the fixed rate is chosen such that at inception of the contract, the value of the is zero. Using (2.6), can be determined in terms of zero coupon bond prices:
t, , , , , , , 0
dividing both sides of the equation by
, , , 0
rearranging for yields:
, ,
∑ , (2.7)
2.1.7 Interest Rate Cap
An Interest Rate Cap (henceforth known as Cap) is a contract which entitles the owner to receive payments similar to those in a payer (a swap from the perspective of the payer of fixed), but only if they have positive value. The discounted payoff is hence given by (Brigo & Mercurio, 2006, p.16):
0, , , , 0, ,
(2.8)
Where N denotes the nominal, , the floating rate (e.g. LIBOR), , the discount factor for period 1 and the fixed rate. A cap allows one to prevent losses from (large) movements in interest rates. For example, suppose a company has to make payments on a loan at times paying floating rate resetting at times , … , . The company fears that for some the interest rate will increase heavily. In order to lock the rate at a maximum , the company considers entering a
Cap. If , , a cash flow of , results from the Cap. The company hence pays at most the fixed rate:
min ,
2.1.8 Counterparty Credit Risk
Counterparty credit risk (CCR) is the risk that a counterparty in a derivatives transaction such as an OTC derivative will default prior to the expiration of the derivative and will therefore not make the current and future payments as stated in the contract. CCR risk is similar to lending risk or traditional credit risk, of which the latter is defined as the risk of the borrowing party not being able to make the contractual payments or repay the (full) amount lent due to insolvency. CCR differs however from traditional credit risk for the following reasons (Gregory, 2010, p.9):
The value of the derivative in the future is uncertain. As an example let us reconsider an from subsection 2.1.6 for which counterparty pays the floating leg and counterparty pays the fixed leg. The value of the at time depends on the rates of the floating leg. These rates change throughout time and it is not known what the value of the will be at the time of a counterparty default beforehand. This future value is highly uncertain and could be either positive or negative.
CCR is typically bilateral which means that both counterparties have risk to each other. If from the perspective of counterparty ( ) the value of the derivative at time is negative, counterparty ( runs CCR on counterparty ( .
As discussed in subsection 2.1.6, the fixed rate is usually chosen such that the initial value of the is zero. After the inception of the swap, the term structure changes (and hence the rates of the floating leg) but the fixed rate does not. This causes the value of the swap to become either negative or positive. The mark-to-market with respect to a counterparty can be thought of as the present discounted value of all the payments one is expecting to receive minus those one is obliged to make. The therefore is a measure of the CCR of a swap. If we consider one single , this means that is equal to the value of the swap. Close to the expiration of the swap the number of remaining payments reduces to zero, which causes the to reduce to zero as well. The next figure shows a scenario of the (as a percentage of the notional) from the inception of a swap until the expiration (ten years) for an :
MtM scenario for an IRS
We can see that the can be either positive or negative, and is zero at both the inception and expiration of the swap. From now on we shall define exposure as max , 0 ; exposure is defined as non-negative.
2.2 Risk-Neutral Measure and CVA2
In order to compute the at time of a single , we need to compute the value of the swap at time as explained in subsection 2.1.8. In order to value a derivative in general, one could compute its expected future payoff under the real-world measure (statistics) and consequently adjust for the risk it bears. A risky asset would have a lower current value than the expected payoff; since investors generally need to be rewarded for this risk (i.e. investors are risk averse). Under a certain condition (absence of arbitrage) one can incorporate the risk right away, transforming the real-world measure to the risk-neutral measure . To compute the value of any derivative at time , one then computes the discounted (at the risk free rate) expectation of the payoff at the future time with respect to
:
,
2
For the purpose of this paper, some concepts will be introduced informally. For a more formal discussion of the subjects, the reader is referred to Brigo & Mercurio, 2006.
0 20 40 60 80 100 120 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 Months Mt M
Where , is the discount factor at the risk-free rate (money market interest rate) for the time period , .
For risk management purposes, parameters of models are typically estimated using historical data (i.e. using real-world distribution). This is because one would like scenarios to be realistic in terms of their statistical properties such as variance and mean. The valuations in this paper will not be used for scenario generation however but for the computation of the value of the CCR of an . The fair value of an OTC derivative such as an should reflect the default probability of the counterparty. If either counterparty is more likely to default, the other counterparty should be rewarded for this fact. Credit Value Adjustment (CVA) is the difference between the value of a derivative assuming the counterparty default probability is zero and the value given the actual default probability
distribution : → 0,1 ∀ ∈ 0, (Gregory, 2010, p.194): 1
Here, is the expected exposure under the risk neutral measure and is the recovery rate. The expected exposure is defined as the expectation of max 0, , i.e. the expectation of the positive part of the values (Gregory, 2010, p.37). It is possible to separately hedge the part of the exposure as illustrated by the following example:
1
10 11 2 4
∗ 8 7
The total exposure (the exposure of the at time at time is 10 of which the is hedged (e.g. with a Credit Default Swap) leaving an exposure of 8 at risk. If because of a movement in the interest rate at time 1 the total increases to 11, the (hedged exposure) could increase to 4. The exposure at risk ∗ actually decreases to 7 because of the hedged . Hedging hence leads to risk mitigation.
Risk neutral simulation
For the risk management of counterparty risk, future scenario simulation (e.g. short rate path
simulation) should be done using the real-world measure , while revaluation at future dates should be done under the risk neutral measure . approaches however are solely based on market data and need to produce consistent prices among counterparties which allow for successful hedging (e.g. using a credit default swap). In order to achieve consistency in pricing, both the future scenario
simulation and future evaluation should be performed under the risk neutral measure . Real world calibration allows for more freedom than risk neutral calibration. With real world calibration for example, a counterparty decides for himself what time period he considers most suitable for calibration. If the data underlying the calibration differs between two risk counterparties, the two counterparties possibly arrive at different future scenarios, and hence at different values of the , which essentially is a function of future . If both counterparties however use risk neutral
calibration, the ‘’ market knows best’’ principle assures that both counterparties use consistent data and arrive at the same future scenarios, estimates and . The following figure from (Gregory, 2010, p.166) illustrates this principle:
Risk management simulation versus CVA pricing. Here P denotes the real world measure, while Q denotes the risk neutral measure.
2.3 Interest rate models
Interest rate models are used to model the evolution of the interest rate (short- rate models) or related concepts such as forward rates (e.g. LIBOR market model). The former class models the evolution of the short-rate over time. This approach is convenient due to the fact that fundamental quantities such as rates and bonds (and therefore financial products as well) are readily defined as functions of the short rate. A drawback of these models is that the short rate it models is not observable in the market. Therefore, in calibration procedures it should be either approximated by a short term interest rate such as the overnight rate or should be calibrated additionally to the other parameters. Lately the LIBOR market model (LMM) is becoming increasingly popular, since the forward rates it models are directly observable in the market. Market observed volatilities lead directly to implied volatilities in the LMM avoiding the need for additional numerical fitting procedures.
2.3.1 Short-rate models
rate models model the continuously compounded spot rate, discussed in subsection 2.1.2. Short-rate models model the dynamics of the short Short-rate as a stochastic differential equation (SDE) of the following form (Brigo & Mercurio, 2006, p.52):
Where denotes a Brownian motion under a specified measure (in this paper the risk neutral measure will be used). With μ we denote the drift and with σ the volatility (a measure for the variation with respect to the mean) of the process. As an example, the one-factor Vasicek (1FV) model is given by the following equation:
, 0
Here , and , (i.e. a constant volatility over time is assumed). The above model is called a one-factor model since there is one stochastic factor . Analogously, models with stochastic factors are called -factor models.
2.3.2 LIBOR Market Model
Rather than modeling the short rate, the LIBOR market model models forward rates for a set of time periods ′ , , … , , which have the advantage of being directly observable in the market. The assumption of the model is that forward rates are log-normally distributed. More
specifically, for a future time period , ∈ ′ the forward rate , , at time is modeled as the following stochastic differential equation (Brigo & Mercurio, 2006, p.200):
, , , , , (2.9)
Where , is the volatility of the forward rate for the time period , , and again denotes a
Brownian motion. The volatilities vector , : ∈ 0, … , 1 is calibrated (a concept which is explained in subchapter 2.4) to market observable Cap prices for the set .
2.4 Calibration
Once a certain interest rate model is chosen, its parameters need to be estimated. For the computation of the risk neutral expected exposure as discussed in subsection 2.2, estimates for the
parameters should be market implied (i.e. risk neutral calibration). Estimating parameters on market data means that the parameters of the model are chosen such that market observed entities (e.g. Cap prices) and entities as produced by the model differ as little as possible. For example, one could try to find , and for the 1FV model such that current market observed term structure is fitted as well as possible.
III. Short rate models
This chapter serves as an introduction to the models that will be tested against the LIBOR market model benchmark. First, the most basic short rate model is introduced: the one-factor Vasicek (1FV) model. Secondly, an extended version of the 1FV is considered: the Hull-White model.
3.1 One-Factor Vasicek
Under the risk neutral measure, the one-factor Vasicek with constant coefficients is described by the following SDE (Brigo & Mercurio, 2006, p.58):
, 0 1 (3.1)
where κ, θ and σ are assumed to be constants. The parameter σ is a scaling factor that determines the amplitude of the process; higher values for σ imply more randomness whereas taking the limit σ→0 means that the process becomes deterministic. The parameter is the long term average or mean of the process. This makes sense because for the drift is negative whereas for the drift becomes positive so that on average tends to . In addition, κ determines the speed of reversion; how fast trajectories will return to the neighborhood of . If (3.1) is integrated then for every :
1
Implying that is , distributed with
1
and
2 1
If we consider the limit → ∞, is , ) distributed: in the long run the process’s variance is finite and its mean is equal to . Because of the normal distribution, negative short-rates can occur especially when the model is calibrated to low market observed interest rates. These negative short-rates can result in simulated term structures with negative values (consequently negative spot-short-rates). Because negative interest rates were an anomaly in financial markets before the current financial
crises, this fact has traditionally been thought of as a reason to disqualify use of the model. Sweden, Denmark, Switzerland and Japan all have or have had negative interest rates and it is likely that more countries will introduce negative interest rates due to the ongoing financial crises (on 02-05-2013 the European Central Bank hinted at the introduction of negative interest rates). In conclusion, the generation of negative interest rates by the 1FV model is no longer a reason to not use the model. For the purposes of this paper, I will not explain the monetary considerations behind negative interest rates.
If one would like to exclude the possibility of negative interest rates, the Exponential-Vasicek (EV) model is often considered. The EV model however does not imply explicit formulas for pure discount bonds (see Brigo pp 71), which makes it unsuitable for this paper. Another issue is that the 1FV model endogenously produces the current term structure, which may be quite different from the actual observed market structure. Motivated by this notion the Hull-White model is considered in subchapter 3.2, which takes the currently observed market structure as an exogenous input.
One-factor models in general assume the correlation between interest rates of different maturities to be perfect, i.e. 0, , 0, 1 ∀ , 0. This assumption is not realistic, long and short term interest rates are generally not perfectly correlated. This means that the entire term structure moves in the same direction given a shock in its initial point 0 .
3.1.1 Discretization
For modeling purposes, the discrete version of (3.1) is given by the following process under the risk neutral measure:
1 1 √ (3.2)
The discretization scheme used to obtain (3.2) from (3.1) is known as Euler discretization. Euler discretization makes use of the left point rule, which is used to approximate the value of an integral in case the value of the function is known in the left point (Glasserman, 2003). The left point rule simply
states that . Starting from (3.1):
which implies:
now the left point rule is used to compute:
is distributed as √Δ ε where ε is N(0,1) distributed leading to the following expression:
√
Thus expression (3.2) can be recognized.
3.1.2 Calibration to current Term Structure
For the calibration of the 1FV model, the parameters , and are chosen such that the model implied term structure : → 0, fits the market observed term structure : → 0, as well as possible. To this end, we first compute zero-coupon bond prices which can then be used to compute yields.
Under the risk neutral measure, zero-coupon bond prices in the 1FV are given by the following expression (Brigo & Mercurio, 2006, p.59):
, exp , , with , exp 2 , 4 , and , 1 1
Important to note here is that , the short-rate at time , is part of the expression for the zero-coupon bond. We therefore also obtain an estimate for the short-rate. Using equation (2.2) we obtain
, ln ,
Where denotes the yield implied by the 1FV. If we now consider a set of times , … , and consider 0 (we are calibrating to the current market observed term structure) we minimize the following expression:
, , 0, 0,
Where , , denotes the residual sum of squares given , and . We hence solve a nonlinear least-squares problem. To solve this minimization problem, we use the function ‘lsqnonlin’ MATLAB which solves nonlinear least-squares problems and uses the following syntax:
x = lsqnonlin(fun,x0,lb,ub) where
‘x’ is the vector of parameters for which the problem should be minimized, thus x ( , , ‘fun’ is the function to be minimized : ∑ 0, 0,
‘x0’ is the initial guess of the parameters , and ‘lb’ is a lower bound for the estimates , and ‘ub’ is an upper bound for the estimates , and
Estimation
We obtain the continuously compounded term structure from Bloomberg on 24-01-2013 for a granularity of three months up until a final maturity of 120 years. That is, we calibrate using the continuously compounded interest rates for the set : 1, … ,120 with k in years. The curves on Bloomberg are bootstrapped from the prices of a set of interest rate swaps which have for example a three month tenor. In a financial context, bootstrapping refers to a method for obtaining the term structure from the prices of a set of market observed figures such as swap prices. This method shall not be explained in detail for the purposes of this paper. The minimization scheme leads to the following estimates of the parameters: 0.0284, .0145, 0.079 and a residual of , , 1.2409 022. Figure 3 shows the market observed term structure in blue, and the model implied term structure in red for a maturity up to 30 years. In this example, the 1FV is able to fit the observed market structure quite well 3.
3
Figure 3: Model implied and market observed term structure
As noted in subchapter 3.1, the 1FV model generates non-positive short-rates with high probability if it is calibrated under adverse market conditions. For example, for 1/12 (one month) and using as a proxy for 0 the three month rate at 24-01-2013 (. 00251 we obtain the following distribution of
: 1 12 ~ 1 12 , 1 12 . 00251 ∗ . .0284 1 . , . 0145 2 ∗ .079 1 . .00267,0.00656)
The distribution has a large part of the mass over the negative values of ; the probability of 0 is 48%.
The calibrated model can now be used to simulate short rate paths. Figure 4 shows the 50th percentile in blue, the 95th percentile in red and the 5th percentile in black for a simulation of 10000 short rate
paths and a period of 50 years . We observe that the positive extreme scenario continues to have reasonable values. In addition, the 50th tends to 0.0284, which is the long term average of the
model. In the beginning of the simulation, we can see that a large amount of negative paths are generated, which corresponds to the high probability we obtained for negative interest rates after one month. Due to the mean reversion property of the model this probability declines and stabilizes, which can be seen from the 5th percentile which increases at first and stabilizes at circa 300 months.
0 5 10 15 20 25 30 0 0.005 0.01 0.015 0.02 0.025 Years M a rk et obs er v ed y iel d c u rv e ( b lu e) , V a s ic e k i m pl ie d y iel d c u rv e ( red )
Figure 4: Simulated mean, 95th and 5th percentile for 1FV
Fitting limitations
The 1FV model produces the current observed term structure as an output, which might lead to a poor fitting of the actual observed market term structure. That is, the calibrated rates → 0,
ln 0, / may not necessarily match those observed in the market regardless of how , and are chosen. In fact the 1FV model can only model the following three shapes (Puhle, 2008, p.51):
For ,the term structure is monotonically increasing (upward). For , the term structure is monotonically decreasing
(downward).
For , the term structure is humped.
Upward Downward Humped
3.2 One-factor Hull-White
Hull and White (1990) were the first to propose a model which takes the current observed term structure as an input. The potential poor fitting of current term structure by the 1FV model can be reduced by introducing a time dependent level of the mean to the model (Brigo & Mercurio, 2006, p.72): , 0 (3.3) 0 100 200 300 400 500 600 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Months 1F V pe rc e n ti le s 0 20 40 60 80 100 120 0.1 0.101 0.102 0.103 0.104 0.105 0.106 0.107 0.108 0.109 Upward 0 20 40 60 80 100 120 0.396 0.397 0.398 0.399 0.4 Downward 0 20 40 60 80 100 120 0.2994 0.2995 0.2996 0.2997 0.2998 0.2999 0.3 0.3001 0.3002 0.3003 Humped
Model (3.3) is known as either the extended Vasicek model or the one-factor Hull-White (1FHW) model. Contrary to the 1FV model, the mean is no longer a constant but a function of time. The mean reversion parameter and volatility parameter however are still constant. The function is chosen as to exactly fit the current observed term structure and is given by the following expression:
0,
0,
2 1
Here corresponds to the (market observed) instantaneous forward rate at time 0 for the maturity which can be expressed in terms of a market observable zero-coupon bond:
0, ln 0,
Similarly to the 1FV model, we can use Euler discretization to obtain the discrete version of the 1FHW model which is given by:
1 √ (3.4)
The parameters in (3.3) need to be calibrated first in order to simulate paths or to price products. At time , the short rate is distributed as:
~ 1 ,
2 1
Since follows again a normal distribution, negative short rates are obtained with non-zero probability.
3.2.1 Calibration to Caps
At time , the price for a zero-coupon bond in the Hull-White model is given by (Brigo & Mercurio, 2006, p.75):
, , , (3.5)
where
,
, 0, 0,
, , ,
Let us now consider a set of dates , … , of payments dates and a set of times
, … , where denotes the difference between payment date and settlement date . In addition, denotes the first reset time. The 1FHW gives the following analytical form for a Cap on a
notional with strike price (Brigo & Mercurio, 2006, p.76): , , , ∑ , ) 1 , (3.6) Where 1 2 , and 1 , 1 , 2
Setting 0, we see from (3.5) that we then set the bond prices for 0 equal to the observed market price 0, . We calculate these bond prices using the market term structure via the equation:
, , 0,
We obtain estimates for and by performing nonlinear least square regression and minimizing the following expression:
,
Where and is the price for Cap according to the 1FHW model and the market respectively
For the example calibration of and we first consider Cap prices as given by Bloomberg on 02-05-2013. We consider two Caps with final maturities 5,7 years, strike 1 and notional
1000000 and quarterly payment dates. Here we obtain the following estimates: .0069 and 0.0675 leading to a residual4 of
, 2.3470 022. The next table shows the market prices and 1FHW model implied prices for maturities 5,6,7,8,9 , as well as their difference as a percentage of the market price
5 6 7 8 9
Market price 11825 20482 31074 43768 57788
1FHW price 11721 20205 30678 43341 57567
Difference % .88 1.35 1.27 .975 .3846
The calibrated model can again be used to simulate short rate paths. Figure 6 shows for a simulation of 10000 short rate paths the mean in blue, the 95th percentile in red and the 5th percentile in green.
Additionally, the market observed term structure per 07-04-2013 is plotted in black. For a large number of simulated paths, is constructed such that on average, paths are ensured to follow the initial market observed term structure.
Figure 6: Simulated mean, 95th and 5th percentile for 1FHW
We see that the mean of the paths follows the initial market observed term structure quite well. Negative paths are generated again, but as discussed this fact no longer disqualifies use of the model due to the current market conditions.
4
For the MATLAB code, the reader is referred to the appendix
0 20 40 60 80 100 120 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 Months P e rc e n ti le s ( g re en ,r ed an d bl ue ), T e rm s tr u c tur e ( b la c k )
IV. Exposure simulation
We have now two short-rate models at our disposal and are able to calibrate them to market data. Either calibrated model can be used to simulate short-rate paths. For each of these short rate paths, we can simulate term structures at each point in time. Using the 1FHW model, the following figures illustrate this principle for 100 simulated short rate paths of which one path is highlighted and consequently three example term structures are shown5:
Simulated short-rate paths Three example term structures
We can then use these simulated term structures at each point in time to value an , or more specifically an receiver of fixed swap . If we generate a large number of short-rate paths (typically 10,000) and consider the average of all the simulated values at each point in time, we obtain a measure for the expected value of the . In this chapter we will use this technique (also known as the Monte Carlo method) to simulate the Expected Exposure (EE) of a plain vanilla . As defined in subsection 2.1.6, an is a contract that entails a number of future payments. The current discounted value of these payments or rather the differs from time to time depending on the development of the short rate. If we simulate for 0 36 (months) for 10,000 paths and compute the at each point in time for each path of an with a tenor of three months and an expiration of three years we obtain the following figure 6:
5
For the MATLAB code, the reader is referred to the appendix
6
For the MATLAB code, the reader is referred to the appendix
0 20 40 60 80 100 120 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Months r(t )
Figure 6: 1000 simulated MtM paths using 1FV
Both positive and negative values are generated, and all values lie on a path of values which all correspond to a particular simulated short rate path. At first we observe an increase in the values in absolute value because of the movement of and the large number of remaining payments. As the number of payments decreases to zero, so does the . The distribution of the
paths at time depends on the probability distribution of the underlying short rate paths at . As discussed in subchapter 2.2, the expected exposure of an is defined as max , 0 . If we first generate a large number of paths and consider the average of max , 0 we obtain a close approximation for this expectation. This principle is illustrated by the following figure (Gregory, 2010, p.24):
From simulated MtM paths to EE
Using the 1FV to simulate short rate paths, the following graph of → is obtained for an IRS with a maturity of three years and three month tenor:
0 5 10 15 20 25 30 35 40 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5x 10 5 Months Mt M
Figure 7: EE profile using 1FV
From now on, the shall be presented as a percentage of the underlying notional of the swap. The estimates produced using the 1FV model (or using any short rate model in general) show a ‘’sawteeth’’ profile which stems from the payments made during the lifetime of the swap. After each payment, the exposure (i.e. the reduces with the notional amount of that payment, while between payments the evolution of the underlying short rates (potentially) increases the exposure. It can be seen that each sawtooth spans a three month period; exactly the tenor of the swap under consideration. The LMM does not show this sawteeth profile due to the fact that future values are simulated simultaneously, and consequently interpolated to obtain the entire curve.
4.1 Comparing 1FV to LMM
If the LMM and the 1FV are calibrated on the same time instant (albeit on different market data), profiles for both models can be computed and compared. Using the bootstrapped market observed term structure obtained from Bloomberg on 17-04-2013 and 10-05-2013 for calibration of the 1FV model, the following graphs of are obtained for a 5 and 10-year with a three month tenor (the green and blue line correspond7 to the LMM and 1FV respectively):
7
The estimated parameters can be found in the MATLAB appendix
0 5 10 15 20 25 30 35 40 0 2000 4000 6000 8000 10000 12000 14000 Months EE
Figure 8: 5 year EE profiles by 1FV and LMM (17-04-2013)
Figure 9: 10 year EE profiles by 1FV and LMM (17-04-2013) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 Years EE L M M ( g re e n ), EE 1 F V ( b lu e ) 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 Years E E L M M (g re e n ), E E 1 F V (b lu e )
Results
For both swaps on both dates, we observe an overestimation of the predicted by the 1FV model compared to the LMM. The amortization profiles are very similar for both models, and the maxima of the curves of the 1FV model and LMM share their horizontal coordinate in all four examples. To analyze what causes the differences, we first compare the 3 month spot rates generated by both the LMM and 1FV model. The LMM models throughout time the set of forward rates for the set of
payment periods , , … , , given by:
0; , 1
,
0,
0, 1 : , ∈
Then using equation (2.5) one is able to compute the of an . The first forward rate modeled by the LMM is 0; 0,3 which is in fact the 3 month spot rate 0,3 . Since we obtain the entire bond term structure → 0, at each point in time for the 1FV model we can compute 0,3 with the 1FV model as well. Figure 12 shows the 95th , 50th and 5th percentile of 0,3 for both the
LMM and 1FV model as calibrated on 10-5-2013 in red and blue respectively.
Figure 10: 5 year EE profiles by 1FV and LMM (10-05-2013)
Figure 11: 10 year EE profiles by 1FV and LMM (10-05-2013) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 Years E E 1 F V (b lu e ), E E L M M (g re e n )
Figure 12: Simulated three month spot rate percentiles for the 1FV model and LMM (10-05-2013)
If we remember expression (2.4) we can see that this picture might be a first hint as to why the 1FV model overestimates the , or rather the underlying . We observe that the 3 month spot rates are generally speaking lower for the 1FV model than those generated by the LMM. In addition, we do not observe any negative 3 month spot rates for the LMM because it assumes forward rates to be distributed lognormal.
Lower values for the set of floating rates { , ∶ 1, … , result in higher values of the . Figure 12 shows us that the first payment corresponding to 0,3 will in general be higher in the 1FV model than in the LMM. We already know that negative short-rates are generated with non-zero probability in the 1FV model (see Figure 4). Could these negative short rates explain the negative spot rates, and do they consequently contribute to the overestimation? We reconsider the expression for 0,3 : 0,3 0; 0,3 11 4 0,0 0,3 1 4 1 0,3 1
The three month spot rate is only negative if 0,3 1. We reconsider the expression for the three
month zero bond price:
0,3 0,3 , with 0,3 exp 2 0,3 3 4 0,3 0 1 2 3 4 5 6 7 8 9 10 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 Years 3 m o n th s p ot r a te LM M ( red ), 1F V ( b lu e )
and
0,3 1 1 ∗
We first compute the value of 0,3 , which is . 999. This means that if , 1.001, 0,3 is larger than one. Since 0, we know that 0,3 0. This means that 0 0 in order for
, to be larger than one. We conclude that the negative 3 month spot rates stem from
negative short-rates. To investigate the overestimation due to negative short rates in general, figure 13 plots the for the 10 year swap on 10-05-2013 if we first take the non-negative transformation of the short rate which is given by max 0, :
Figure 13: 1FV EE profile for max(r(t),0)
The resulting figure shows that the 1FV model now underestimates the compared to the LMM. This is because one cannot simply take the nonnegative part of the values of ; the model is
calibrated such that these negative values are generated with a certain probability. The simulated paths and their distribution are therefore a result of the simulation which in turns depends on the calibration. If there was a calibration such that negative values do not occur, the values for , and κ would imply a different distribution of the paths and different term structures shapes. Figure 13 merely illustrates that the overestimation in graphs 8-11 must be due to the negative short rates generated by the 1FV since disregarding the negative values of leads to an underestimation of the profile compared to the LMM. A second reason for the overestimation might be the imperfect fit of the market term structure as discussed in 3.1.2. Figure 13 plots the implied term structure by the 1FV in green, and the market observed term structure in blue:
0 20 40 60 80 100 120 140 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Months E E 1F V f o r m a x (r, 0)
Figure 13: Market and 1FV implied term structure
Clearly, there is a large discrepancy between the two shapes even though a global minimum was achieved in the optimization scheme. This stems from the fact that the observed term structure is not of a shape that can be reproduced by the 1FV model (see subchapter 3.1). To this end, a two factor Vasicek model should improve the fit and could possibly decrease the overestimation. That is, the implied parameters from this better fit may decrease the number of negative paths generated. One factor models in general model correlations between interest rates for different maturities as being perfect, as discussed in subchapter 3.1. This means that a shift in the term structure is parallel, meaning that all rates move up/down simultaneously. Since the (and hence the ) of an depends on a large set of short- and long-term interest rates, it might be that this perfect correlation leads to an overall estimation of the . Again, multiple factor models could be used to test this hypothesis and might lead a decrease in the overestimation.
4.2 Comparing 1FHW to LMM
Analogously to subsection 4.1.2, we can compare the profiles generated by the 1FHW model and LMM. Both models are calibrated on the prices of market observed caps, first on 17-4-2013 and on 10-05-2013.
Figure 14: 5 year EE profiles by 1FHW and LMM (17-04-2013)
Figure 15: 10 year EE profiles by 1FHW and LMM (17-04-2013) 0 1 2 3 4 5 6 7 8 9 10 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Years T e rm S tr u c tur e m a rk e t ( g reen) , 1F V ( b lue) 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 Years E E 1F H W ( b lue) , E E LM M ( g reen) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 Years E E 1F H W ( b lue) , E E LM M ( g reen)
Figure 16: 5 year EE profiles by 1FHW and LMM (10-05-2013)
Figure 17: 10 year EE profiles by 1FHW and LMM (10-05-2013)
Similar to subsection 4.1.2, we obtain an overestimation of the compared to the LMM for all four scenarios. For the swaps under consideration, the overestimation of the by the 1FHW model is larger compared to the overestimation by the 1FV model. We again analyze the 5th, 50th and 95th percentile of the generated three month spot rates paths by comparing figures on 10-05-2013 for the 1FHW model (blue) and LMM (red):
Figure 18: Simulated three month spot rate percentiles for the 1FHW model and LMM (10-05-2013)
Figure 18 shows that similar to subsection 4.1.2, the percentiles are generally lower for the 1FHW model than for the LMM. Negative spot rates are again a qualitative difference between the models. Following the same logic as in subsection 4.1.2 we conclude that these negative spot rates contribute to a higher , that is a higher . The 50th percentile of the three month spot rates for the 1FHW
is lower than that of the 1FV model. This can be explained by the fact that the mean of the short rates generated by the 1FHW simulations follows the initial term structure. The fact that the 50th percentile
of the three months spot rates is lower could explain why the 1FHW model overestimates the to a
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Years E E LM M ( g re en) , E E 1F H W ( b lu e) 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Years E E L M M (g re e n ), E E 1 F H W ( b lu e ) 0 1 2 3 4 5 6 7 8 9 10 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 Years 3 m ont h s p o t r a te LM M ( re d ), 1F H W ( b lu e)
larger extent than the 1FV model. To again determine the overestimation due to negative short rates, we again plot the values for the transformation max , 0 for the 10 year on 10-05-2013:
Figure 19: 1FHW EE profile for max(r(t),0)
Since figure 19 shows that the 1FHW now would underestimate the compared to the LMM, we observe that the negative values for explain the overestimation. Figure 19 is again merely of illustrative use because of the fact that the short rate distribution is a result of the simulation, and that this distribution cannot be altered independently of the calibration.
The 1FHW assumes perfect correlation for interest rates of different maturities, since it is a one-factor model. A two factor Hull-White model could be used to test the overestimation that might stem from this assumption.
V. Variance Reduction
Simulated values as presented in figure 6 and their underlying short rate values are random, which affects the estimate obtained of the . Simulating for example only two interest rate paths potentially could lead to a highly unlikely estimate if one of the paths is “extreme”. The estimate
of the expected value based on simulated values , … , is in general calculated as follows (Lyuu, 2008):
∑
The law of large numbers states that the more paths are simulated, the closer our simulated average will approximate the true . Simulating more paths however is time consuming, especially for more exotic products than the plain vanilla swap we have been considering until now. Let us consider the variance of the estimate given by:
0 20 40 60 80 100 120 140 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Months E E 1 F H W f o r m a x (r, 0 )
2 , 4
If and are independent , 0, if however , 0 we
could reduce the variance of and obtain an estimate which is subject to less variance using the same number of paths. This principle generalizes to any value of ; if we can construct
, … , such that we can sort them in pairs of negatively correlated variables, we can reduce the variance obtained for the estimate . More specifically, we consider i.i.d. realizations
, … , and their antithetic variates , … , . If the elements of each pair , is negatively correlated, we obtain the desired structure. In the context of this paper, we let ; the simulated short rate at time instant . We recall the expression of the simulated interest rate in the 1FHW model:
1 √
Clearly, and are negatively correlated. We are now hence in a position to decrease the variance in the evolution of the short rate . This will result ultimately in a decrease of the variance of the estimate for the , which is a function of . Using the 1FHW model calibrated on 17-04-2013, the following figures shows estimates obtained from a total of 100 separate
simulations, using 100 runs per simulation.
Standard Monte Carlo Monte Carlo using Antithetic Variates
Although both figures show that estimates for 100 runs are in the neighborhood of the estimates obtained for 10,000 runs (see figure 15), clearly the antithetic variates method leads to a smaller range of outcomes (i.e. less variance). This means that for a low number of runs, using antithetic variates enhances the estimates obtained for the . A related additional advantage is the decrease in computational time. For a total of 10,000 paths, the computational time of calculating the
estimates using the standard Monte Carlo method is 229.244 seconds. Using the antithetic variates
0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 Months E E e s ti m a te s 0 20 40 60 80 100 120 0 1 2 3 4 5 6 Months E E e s ti m a te s
technique for 5,000 original paths, which ultimately yields 10,000 paths of which 5,000 are unique and 5,000 are antithetic variates, reduces the computational time to 109.993 seconds. For this example simulation, the computation time is hence reduced with more than 50%. If the circumstances demand that the estimate needs to be obtained as fast as possible (e.g. for trading purposes) this reduction is a clear reason to opt for the use of the antithetic variates technique. If the swap under considerations is more exotic, that is not single currency nor plain vanilla, it could be that the computational
complexity demands the use of the antithetic variates technique in order to run the simulation on the computer that is available.
VI. Conclusion
The LMM and short-rate models considered in this paper are different in nature. They describe the evolution of different subjects and do so under different assumptions (e.g. flat volatility of short-rates versus a volatility surface of log normally distributed forward rates). The approach I chose in this paper is not to compare the models, but rather the output of the models. I considered a specific measure, the for a five and ten year , to compare short-rate models with the LMM. It could be that under a different measure (for example the 95th percentile of the paths of a different
product) the comparison would imply different results.
Based on my results, both the 1FV model and the 1FHW model overestimate the of an compared to the LMM. The prevailing market conditions lead to a calibration of both short-rate models such that positive short rates are generated with negligible probability. These non-positive short-rates explain the overestimation, but cannot be altered independently because the paths are a result of the simulation which in turn depends on the calibration of the model. Given different market conditions (e.g. higher market observed term structures), calibration of both models leads to a lower probability of generating negative short-rates. It could be that the 1FHW model and 1FV model are then more consistent with the LMM.
Non-positive interest rates are likely to become more prevalent given the financial crises at hand; it therefore depends on the market conditions which model is preferable. In markets where non-positive interest rates do not occur and never have occurred, the use of the short-rate models considered in this paper could lead to an estimate of the that is unrealistically high. In markets (e.g. Japan) where non-positive interest-rates do occur and have occurred for some time, the use of the LMM could lead to an estimate of the that is unrealistically low.
In general, if the product under consideration is more exotic, or if the computation time needs to be minimized (e.g. for trading purposes), the use of variance reduction is a valuable technique.
VII. Further Research
The framework for EE estimation built for both short-rate models should be backtested. This means that the predictive power of the framework is tested against historical data. Multiple-factor models should be explored in order to exclude problems that stem from the use of one-factor models, such as a perfect correlation between short- and long run interest rates.
References
Brigo, D. & Mercurio, F. (2006). Interest Rate Models-Theory and Practice. New York, NY: Springer-Verlag Berlin Heidelberg
Gregory, J. (2010). Counterparty Credit Risk. West Sussex: John Wiley & Sons Ltd
Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. New York, NY: Springer-Verlag Berlin Heidelberg
Puhle, M. (2008), Bond Portfolio Optimization, New York, NY: Springer-Verlag Berlin Heidelberg
Appendix
Vasicek Calibration
function r=may1(x)
zr=0.2077/100; %proxy for short rate, e.g. 1 month interest rate k =[1/4:1/4:5]; %in years
%FOR CURVES, USE ICVS MENU IN BLOOMBERG
y=[0.2077 0.1979 0.2062 0.2168 0.2273 0.2418 0.2598 0.2776 0.3034 0.3325 0.3607 0.3881 0.4249 0.4616 0.4986 0.5336 0.5776 0.6219 0.666 0.7073]/100; B=(1/x(3))*(1-exp(-x(3).*k)); A=exp((x(1)-((x(2).^2)/(2*x(3)^2))).*((B-k))-((x(2).^2)/(4*x(3)).*B.^2)); P=A.*exp(-B*x(4)); yv=(-log(P)./k)
%yields according to vasicek plot(k,yv,k,y)
%for lsqnonlin% r=y-yv;
end
Hull-White Calibration
function CapValues = CC(k) % WORK UNDER ONE CURVE, i.e. EURO vs 3M or 6M
sigma=k(1); a=k(2); X = 0.01; %strike N = 1000000; %notional x = [1/4:1/4:10]; %in years y=[0.2118 0.19 0.1963 0.1922 0.201 0.2124 0.2256 0.239 0.2544 0.2698 0.2855 0.2996 0.3321 0.3635 0.3947 0.425 0.4625 0.4999 0.5375 0.5732 0.616 0.6591 0.7019 0.7422 0.7832 0.824 0.8646 0.9033 0.9444 0.9851 1.0264 1.0633 1.102 1.1412 1.1789 1.2138 1.2506 1.286 1.3209 1.3545] ;% market yields from Bloomberg
y = y/100; % make percentage into fraction CapValues1 = zeros(1,2);
for i=1:2
tau = 1/4; %quarterly t = tau:tau:5*i
r = interp1(x,y,t); %quarterly rates
P = exp(-r.*t); %corresponding zero bond prices for corresponding maturities
%STEP 2 --- from zero bonds to caplets to CAP
n = length(t)-1;
P0 = P(1:n); P1 = P(2:(n+1)); t0 = t(1:n);
h = log((P1.*(1+X*tau))./P0)./s + s./2; Caplet = P0.*normcdf(-h+s)-(1+X*tau).*P1.*normcdf(-h) %---for pricing---% %CapValues(i) = N*sum(Caplet) % marketprices = [11825 20482 31074 43768 57788] %---% %---for lsqnonlin---% Cap_Values(i) = N*sum(Caplet); CapValues1=Cap_Values CapValues=[11825 31074]-CapValues1; %---% end %d=[5 6 7 8 9] %plot(d,CapValues,'.r',d,marketprices,'.b') end Vasicek EE Simulation %VASICEK MtM% %James van Viersen%
%---% %(0) PARAMETERS SETUP%
N=1; %notional
x=[ 0.0202 0.0239 0.1777 0.0006]; i_p=3; %interval between payments in months T=1; %final maturity in years
T=T*12; %final maturity in months n_s=1000; %number of simulations
%---% %(1) SHORT RATE SIMULATION FOR T PERIODS AND n_s PATHS%
deltat=1/12;
r(i,1)=x(4)/100; %zero rate, use 3 month rate for j=2:T; r(i,j)=r(i,j-1)+x(3)*(x(1)-r(i,j-1))*deltat+x(2)*normrnd(0,1)*sqrt(deltat); end end % ---%
%(2) CALCULATE SWAP RATE AT t=0% k=1:T; k=k*(1/12); B=(1/x(3)).*(1-exp(-x(3).*k)); A=exp((x(1)-((x(2)^2)/(2*x(3)^2))).*((B-k))-((x(2)^2)/(4*x(3)).*B.^2)); P=A.*exp(-B*r(1,1))
for l=1:T/i_p %index for payments dates sr_1(l)=P(l*i_p);
sr_2=(i_p/12)*sum(sr_1);
K=(1-P(T))/(sr_2); %formule 1.25 pp 15 from Brigo, make sure this
expression agrees with the payment dates in the next section (3), i.e. all shifted on to left (-1)
end
for j=1:T %time from 0 to maturity
for i=1:n_s;
%in this part you put in the expression for bond prices, in this case
k=1:T+1; %months you want the yield curve to be constructed for at each time j for each path i
k=k*(1/12);
B=(1/x(3)).*(1-exp(-x(3).*k));
A=exp((x(1)-((x(2)^2)/(2*x(3)^2))).*((B-k))-((x(2)^2)/(4*x(3)).*B.^2)); P=A.*exp(-B*r(i,j));
if j<T-1 % j < last reset data
P=[0 P]; %indices move one place to the right, the column of zeros is added to make sure the amortising effect works
for l=1:T/i_p
MtM_1(l)=P(max(1,l*i_p-j+2)); % j is fixed in each iteration! all intermediate coupons, +2 (or extra +1) is from the column of zeros and the fact we start from alpha+1
end
MtM_2=sum(MtM_1);
MtM(i,j)=-N*1+N*P(max(1,T-j+2))+N*(i_p/12)*K*(MtM_2); % see pp 14 from Brigo, -N*1 because we consider alpha=0
end
if j>T-2 % j => last reset date-1 P=[0 P]; MtM(i,j)=-N*1+N*P(T-j+2); end if j>T-1 MtM(i,j)=0; end end end w=max(0,MtM); w1=mean(w) ; plot(100*w1'); % plot EE
Hull-White EE Simulation (including Antithetic Variates)
%general Hull-White MtM% %James van Viersen%
%(0 a) PARAMETERS SETUP% tic
%for q=1:100
x(2) = 0.0063 ; %sigma x(1) = 0.0386 ; %a
i_p=3; %interval between payments in months
T=10; %final maturity in years <-> relationship with part(0 b)where the market observed yield curve needs to be given for the
corresponding maturity
n_s=5000; %number of simulations
%(0 b) INSTANTANEOUS FORWARD RATE CALCULATIONS%
time =[1/4:1/4:10]; %in years
%FOR CURVES, USE ICVS MENU IN BLOOMBERG
market_yields=[ 0.2077 0.1979 0.2062 0.2168 0.2273 0.2418 0.2598 0.2776 0.3034 0.3325 0.3607 0.3881 0.4249 0.4616 0.4986 0.5336 0.5776 0.6219 0.666 0.7073 0.7538 0.8002 0.8462 0.8893 0.9347 0.9798 1.0245 1.0672 1.111 1.1553 1.1979 1.2375 1.2791 1.319 1.3585 1.3963 1.4336 1.4705 1.5074 1.5418] ; % zero rates from Bloomberg, 18-02-2013 , USE ICVS OPTION
market_yields=market_yields/100; interpolation_times=[1/12:1/12:T];
q=(find(time>T-1)); %according length of 'time' to maturity, q(1) is the index of the length
interpolated_rate=interp1(time(1:q(1)),market_yields(1:q(1)),interpolation_ times,'spline'); %interpolation for each month
Q=exp(-interpolated_rate.*interpolation_times);%interpolated bond prices for each month
R=log(Q);
z1=-12*diff(R); %instantaneous forward rate at time 0, rough first order estimate, dt=1/12! z1=[z1 z1(T*12-1)]; z2=diff(z1); z2=[z2 z2(T*12-1)]; T=T*12; %---% %(1a) BUILD FUNCTION THETA%
theta=z2+x(1)*z1 +((x(2)^2)/(2*x(1)))*(1-exp(-2*x(1).*interpolation_times));
% (1b)SHORT RATE SIMULATION FOR T PERIODS AND n_s PATHS% for i=1:n_s
deltat=1/12;
r(i,1)=.211/100; %zero rate, use proxy for j=2:T a(i,j)=normrnd(0,1); r(i,j)=r(i,j-1)+theta(j-1)*deltat-x(1)*r(i,j-1)*deltat+x(2)*sqrt(deltat)*a(i,j); end end for i=n_s/2+1:n_s;
% optional; Antithetic Variates
for j=2:T; r(i,j)=r(i,j-1)+theta(j-1)*deltat-x(1)*r(i,j-1)*deltat+x(2)*sqrt(deltat)*-a(i-(n_s/2),j); end
end
%---% %(2) CALCULATE SWAP RATE AT t=0%
for l=1:T/i_p ; %index for payments dates sr_1(l)=Q(l*i_p);
sr_2=(i_p/12)*sum(sr_1); end
K=(1-Q(T))/(sr_2); %Swap rate
%---%
% (3) MtM at each time j for each simulated path i%
for j=1:T %time from 0 to maturity
for i=1:n_s; if j<2 for l=1:T/i_p MtM_1(l)=Q(l*i_p-j+1); end MtM_2=sum(MtM_1); MtM(i,j)=-1+Q(T)+(i_p/12)*K*(MtM_2);
end
D=-12*(log(Q(1))); % f^M(0,0)
%in this part you put in the expression for bond prices, in this case
%for the Hull-White model
g=1:T; k=1:T; k=k*(1/12); B=(1/x(1))*(1-exp(-x(1).*(k))); A=(Q(g)).*exp(B*D-((x(2)^2)/(4*x(1)))*(1-exp(-2*x(1)*(k))).*B.^2); P=A.*exp(-B*r(i,j)); if j>1 %
P=[0 P]; %indices move one place to the right, the colum of zeros is added to make sure the amortising effect works
for l=2:T/i_p MtM_1(l)=P(max(1,l*i_p-j+2)); end MtM_2=sum(MtM_1); MtM(i,j)=-1+P(max(1,T-j+2))+(i_p/12)*K*(MtM_2); end if j>T-1 MtM(i,j)=0; end end end w=max(0,MtM); w1=mean(w) ; plot(100*w1'); % plot EE
