Implementation of Multiple Interest Rates
Modelling by Three-Dimensional Trinomial
Interest Rate Tree
Cheuk Lam Karen Lo
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics
Author: Cheuk Lam Karen Lo
Student nr: 11109807
Email: karen.locheuklam@gmail.com
Date: 28th October, 2016
Supervisor: Prof. Dr. Daniel H. Linders Second reader: . . .
Statement of Originality
This document is written by Student Cheuk Lam Karen Lo who declares to take full respon-sibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
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Abstract
Using multiple interest rates for discounting becomes an upcoming topic among researchers and practitioners since the public is no longer confident in discounting by the London Interbank Borrowing (LIBOR) rate. This thesis researches on the multi-curve modelling by interest tree which is pro-posed by Hull and White. The construction of the interest rate tree and its pricing approach are explained step by step. A comparison between single curve pricing and multi-curve pricing is illustrated. Multi-curve modelling shall be further investigated for adopting the current low or even negative interest rate environment.
List of Tables 4
1 Introduction 2
2 Discounting in Quantitative Finance 5
2.1 Main Principal of discounting . . . 5
2.1.1 Time Value of Money . . . 5
2.1.2 Risk-Free Interest Rate . . . 7
2.1.3 Risk Neutral World . . . 7
2.2 Bond Valuation . . . 9
2.2.1 Concept of Bonds . . . 9
2.2.2 Yield to Maturity . . . 9
2.2.3 Arrow Debreu Price . . . 11
2.2.4 Bond Valuation with Stochastic Calculus Approach . . . 12
2.3 Derivatives Valuation. . . 13
2.3.1 Options . . . 13
2.3.2 Interest Rate Swap . . . 15
3 Interest Rate 18 3.1 LIBOR - The Single-Curve World. . . 18
3.2 OIS - Better Risk-Free Proxy . . . 20
3.3 LIBOR-OIS Spread - Indicator of Credit Risk . . . 21
4 Interest Rate Model and Hull-White Interest Rate Tree 25 4.1 Interest Rate Model . . . 25
4.1.1 Equilibrium Model . . . 26
4.1.2 No-Arbitrage Model . . . 28
4.1.3 Comparison of Equilibrium Model and No-Arbitrage Model . . . 29 2
3
4.2 The Hull-White Model . . . 30
4.2.1 Parameters . . . 30
4.2.2 Tree Building . . . 31
5 Multi-Curve Modelling by Hull and White 39 5.1 Methodology . . . 39
5.2 Procedure of multi-curve modelling . . . 41
5.2.1 OIS tree building . . . 41
5.2.2 Computation 12-month forward OIS rate . . . 43
5.2.3 LIBOR-OIS spread tree building . . . 47
5.2.4 Three-dimensional tree building. . . 48
6 Pricing Derivative by Three-Dimensional Tree 55 6.1 Single Curve Approach. . . 55
6.2 Mutli Curve Approach . . . 56
6.3 Comparison . . . 61
6.3.1 Initial value of the bond option . . . 62
6.3.2 Interest rate. . . 63
7 Conclusion 68
A Probabilities of the three dimensional tree 70
B Adjustment parameters of spread tree 76
2.1 Interest rate swap cash flows . . . 16
4.1 Interest rate on the tree in Figure 4.1 . . . 31
4.2 Value of nodes on the tree in Figure 4.1 . . . 32
4.3 ∆r and probabilities of nodes on the tree in figure 4.3 . . . 35
4.4 Initial term structure and bond price for 4.4 . . . 37
4.5 Completion of second stage for the tree in Figure 4.3 . . . 38
5.1 Initial term structure of OIS zero rate, 12-month forward LIBOR . . . 41
5.2 OIS tree with θ(t) = 0 . . . 42
5.3 Bond price of the OIS zero rates . . . 42
5.4 Completion of second stage for the OIS tree . . . 43
5.5 Completion of second stage for the OIS tree up to i = 3 . . . 45
5.6 Completion of second stage for the OIS tree up to i = 4 . . . 45
5.7 12-month forward OIS rates, Fi,j from the OIS tree . . . 47
5.8 Value of node (i, j) on the spread tree . . . 47
5.9 Probabilities of node (2, 2, 1) at the three dimensional tree . . . 49
5.10 Adjustments of probabilities for positive correlation of the OIS and the spread 50 5.11 Adjusted probabilities on node (2, 2, 1) . . . 50
5.12 Arrow-Debreu prices of the three-dimensional tree at i = 1. . . 51
5.13 Arrow-Debreu prices of the three-dimensional tree at i = 2. . . 51
5.14 Arrow-Debreu prices of the three-dimensional tree at i = 3. . . 52
5.15 Adjustment parameter βi for the spread tree . . . 54
5.16 Adjusted spread tree . . . 54
6.1 12-month forward LIBOR tree . . . 55
6.2 Bond prices and call bond options value . . . 57
6.3 Bond prices and put bond options value . . . 57 4
5
6.4 Interest rate and bond price of three dimensional tree at i = 3. . . 58
6.5 Interest rate of three dimensional tree at i = 2 . . . 58
6.6 Interest rate of three dimensional tree at i = 1 . . . 58
6.7 Call bond option values at time i = 3 on three dimensional tree . . . 59
6.8 Put bond option values at time i = 3 on three dimensional tree . . . 59
6.9 Call bond option values at time i = 2 on three dimensional tree . . . 59
6.10 Put bond option values at time i = 3 on three dimensional tree . . . 60
6.11 Adjusted probabilities and the destination on node (2, 2, 1) . . . 60
6.12 Call bond option values at time i = 2 on three dimensional tree . . . 61
6.13 Put bond option values at time i = 2 on three dimensional tree . . . 61
6.14 The initial option values of the two approaches . . . 62
6.15 The highest and lowest interest rates in the single curve and multi curve approach 62 1 Initial term structure of the spread.. . . 76
2 Arrow-Debreu price for the spread tree. . . 76
3 Adjustment parameters . . . 76
Chapter 1
Introduction
Before the financial crisis in 2008, the classical approach to valuate financial instrument is by discounting its expected cash flows with the single yield curve. Discounting is the foundation of valuating different financial products. The most common traded product which is valuated by discounting is the zero coupon bond. The buyer only receives the face value of the bond at the maturity date and the present value of this bond is simply the face value discounted by an interest rate which most of the time is considered as risk free. The risk-free interest rate is a theoretical rate of return which means there is no risk of financial loss on the investment. One of the popular choices of risk-free rate is London Interbank Borrowing rate (LIBOR rate). It is a zero curve based on borrowing rates with different tenors 1 between banks. Note that the LIBOR rate is not necessarily used if there is no borrowing transaction occurring. The single curve model is designed to satisfy the no-arbitrage relationship. This is an important, yet strong restriction, because it allows the hedging of forward-rate agreements in terms of ”risk-free” rate. Nevertheless the no-arbitrage relationship does not always hold in reality. The credit crisis began around mid-2007, banks’ credit worthiness fell drastically. Skepticism thus grew towards lending between all financial institutions. Banks reported a higher rate for interbank lending or borrowing as the default probability2 was expected to be higher during the crisis. Hence, LIBOR rate can be seen as an estimation of the health of the financial sys-tem, especially in banking sectors. Whereas if banks are confident and expect that borrowers will be able to pay back their debts, the banks would report a low rate of lending. In other words, LIBOR rate would be low when there is no crisis and when the economy is stable. It was then questioned if LIBOR was still a good risk-free proxy for pricing derivatives, as a result of its significant increase mid-2007 (during the credit crisis) and this growth seemed to show the credit risk incorporated in LIBOR. Hence, the market started looking at an alter-native risk free proxy, the overnight swap index (OIS) rate. OIS rate is the fixed rate in the overnight index swap and it represents the assumed Federal funds rates. It is updated and controlled by the Federal Reserve. Hence, the LIBOR rate has to be considered as a reference rate because the OIS is a real rate used in transactions between financial institutions and central banks more frequently. One of the signals which called out the reconsideration of the reliability of LIBOR curve discounting is the growth in the LIBOR and Overnight Index
1
Tenors are the remaining of time for the loan payment.
2Probability of the borrowing party are not paying back his or her debt
Swap (OIS) spread. This spread is the difference between the LIBOR curve and OIS curve. Before 2008, the market did not pay so much attention to the LIBOR-OIS spread, because it was always in the minimal level, this implied the difference between the LIBOR rate and OIS rate was very small. However, it became larger when the crisis hit the market. The LIBOR rate rose sharply and widened the difference of LIBOR and OIS. Then, the public started to pay extra attention to the LIBOR-OIS spread and recognize it as a new indicator for credit risk in banking sector3. Thus, when the gap between LIBOR rate and OIS rate is becoming larger, this could be a sign that banking institutions are more likely to default on their debt i.e. high credit risk.
After the crisis, market participants grew more concerned over the accuracy of the single curve discounting with LIBOR rate and the implications of the widened LIBOR-OIS spread. An increasing number of parties chose to use OIS rate for valuating derivatives instead. Additionally, OIS has been seen as a better proxy for risk-free rate than LIBOR rate since the financial crisis4. Taking OIS rate as the risk free rate for valuation means that when traders come up with the price of a derivative, they used the term structure of OIS for discounting the future cash flows of the derivative.
The difference between a price of a derivative and its value is that, the value is market-fair, which means it is derived by discounting the expected cash flows of the derivative at risk-free rates. A price of a derivative is dependent on other factors such as the demand and the supply of the market or the interests of the parties which involved in the contract, it can be more subjective. The common valuation method is discounting cash flows. For pricing, academics have developed different models such as the Black-Schole model. Yet, the basis of valuating different types of derivatives are all starting from discounting the cash flows under the risk neutral setting. Nevertheless, the discounted cash flows of some interest rate deriva-tives are related to the underlying interest rates such as interest rate swap with LIBOR rate or LIBOR rate futures and so on. The public concerned that discounting the cash flows by only the LIBOR rate is not accurate enough especially their spread has increased during the crisis. So multi-curve framework for the risk-free rate have become the new tool as a part of derivative pricing. Multi-curve modelling refers to the observation that there is more than one interest rate which can be used to determine the risk-free curve for discounting. Multiple papers from Kijima et al. (2009, (19)), Henrard (2014, (8) ), Moreni and Pallavicini(2010, (22)) and Bormetti et. al. (2015) (2) have illustrated the complexity of the framework and the impact of multi curve discounting to valuations for various financial measures such as credit or liquidity. Combining several interest rate in order to construct a proper risk-free curve s more complicated than just taking the average. The papers from Henrard, Hull and White have shown that constructing the multi-curve framework also depends on, for example, the derivatives pricing method and the initial term structure. Furthermore, the techniques of multi-curve modelling is not as developed as the single-curve modelling. There might be some impact on direct derivatives valuation by incorrect computation or wrong assumptions. Thus, compared to the ordinary approach with single curve, multi-curve modelling seems to be more complex and difficult to implement.
3
LIBOR rate reflects how banks perceive the probability of borrowers default.
4
The implementation of multi-curve modelling in this thesis is based on the paper of Hull and White (2015)(16). They extended the trinomial tree method for one-factor interest rate model to the three-dimensional trinomial tree for OIS rate and LIBOR rate. The focus of this thesis is to illustrate the impact of the multi-curve modelling on valuation of bond options. This three dimensional tree approach is the most suitable framework compared to the multi-curve framework by Henrard or Moreni as the three dimensional tree is extended from the trino-mial interest rate tree because it is a simple and straight forward approach for calculating the initial value of an option.
The main goal of this thesis is to implement the interest rate tree in both single curve and multi curve environment and then use these trees to price bond options, the results are com-pared and analyzed. This thesis will first introduce the core concept of pricing derivatives, discounting in mathematical finance. As well as the key component of constructing the dis-count factor, interest rates, will be discussed. The common choice for single curve pricing, LIBOR rate and its alternative choice, OIS will be introduced in chapter 3. Furthermore, several short rate models are discussed and the reason of why the Hull and White model is selected among them is illustrated. A detailed explanation of building the three dimensional interest rate tree will be shown in chapter 5. After that, chapter 6 explains how these trees are used for pricing the bond options. The comparison of the results are made also in that chapter. Moreover, the analysis of the differences between construction of the single interest rate tree and multi interest rate tree is elaborated. Finally, the last chapter will give a con-clusion about whether or not multi-curve modelling of interest rate can produce the risk-free rate and what should be the further development point in terms of the current economy.
Discounting in Quantitative Finance
The foundation of pricing of financial instruments comes from the idea of discounting. Its core principle is using risk-free interest rate to discount the cash flows in the future in order to obtain the net present value of an instrument. Yet, the technique of discounting has been evolving for acclimating various financial instruments in the market. This chapter first discusses the main concept of discounting and then illustrates how it is developed into pricing different financial products such as bonds, options and interest rate swap.
2.1
Main Principal of discounting
2.1.1 Time Value of MoneyIn the financial world, it is known that the value of money, for instance, e100 today is not equal to e100 in five year. This is a basic concept of the time value of money. It is a same amount of money, but why is value of it in the future not equal to its today’s value? This is because the value of the money is compounded by the interest rate, therefore the value of the money increases when the interest rate is positive and decreases when the interest rate is negative. This rate is not consisting any risk, which implies it is certain for the value of this e100 in five years.
One might ask that, in which situation the interest rate rate will imply the risk consisting in the value of the money? For instance, when a certain amount of money is borrowed to a party from a bank, the bank first evaluates the chances of that party being unable to pay back the loan, this is as known as the default probability. When the party has higher default probability, the loan to the party consists of higher credit risk. Therefore, the bank demands a higher rate of interest.
In the 100 euro example, assumed that it is deposited in a bank account and the effective annual interest rate for deposit is 3%, then in five year it becomes 116 euro. The compounding method is as followed, denote F V as the future value, r as the effective annual interest rate, n as the number of time period between today to a certain time in the future and P V as the
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present value:
F V = P V (1 + r)n (2.1)
Discounting is the opposite of compounding. Its focus is not on how much the amount of money or a financial product worth in the future anymore, but it is, given the future value, how much it worth now. For instance, John is promised that he will receive e100 in 5 years. Even though he does not need to do anything to earn this amount of cash, he is still interested in how much is the present value of this futuree100. John can get his answer by simply discounting thise100 with the annual interest rate with the formula.
F V
(1 + r)n = P V (2.2)
There are some financial products guaranteed multiple cash flow to the buyers, such as coupon bonds or annuity products from insurance companies. When calculating the present value of this kind of instruments with multiple payments, discounting their cash flows is needed. For example, given N cash flows for N periods.
The present value of these N future cash flows is:
P V = N X t=0 Ct (1 + r)t (2.3)
Above has illustrated the basic discounting concept in discrete time. Another circumstance which is used very often in academics literature for quantitative finance is the continuously discounting, which will be introduced later in this part.
In the example above, it is assumed that r is the effective annual interest rate. It indicates the actual amount of interest that will be earned at the end of each year. In practice, banks also quote interest rates in terms of an annual percentage rate. It indicates the amount of simple interest earned in one year, in other words, the amount of interest that will be earned without the effect of compounding. Therefore, if an annual percentage rate (AP R) is given and the effective annual interest rate (EAR) is computed by:
1 + EAR = 1 + AP R k
k
(2.4) Where k represents the number of compounding in a year. Therefore, if now assuming r = 3% is the annual percentage rate and with monthly compounding, k = 12, the effective annual rate is:
When the term ”continuously compounding”, is used, meaning that the amount of money is compounded every instant with the interest rate, so k = ∞. By setting the limit to ∞, the effective annual rate can be obtained as:
lim k→∞ 1 + r k k = ek
Thus, the continuous discounting for single payment can be written as:
P V = F V e−t (2.5)
For discounting multiple payments continuously:
P V = N X
t=0
Cte−tr (2.6)
2.1.2 Risk-Free Interest Rate
When traders or practitioners want to obtain the present value of the cash flows, they want this present value to be ”risk-free”. Therefore, the discount rate has to be ”risk-free”. But what is a risk-free rate? Why do they need to have the risk-free cash flows? Risk-free interest rate is a rate of return for an investment which consists of no risk in any financial loss. How-ever, risk-free interest rate is a theoretical concept since in reality every investment consists of risk. Even for the treasury bills issued by the U.S. government, there is still a chance that they will be unable to pay back the treasury bill holders, yet it is only a very low probability. Therefore, the yields of the zero-coupon bonds from the governments with good credit rating are sometimes regards as the proxy of risk-free interest rate because it is the safest investment and the chance of investors losing their money in such bonds are minimal. There are more details about bonds and yield in the next section.
Risk-free rate is the starting point for banks and financial institutions to calculate the cost of capital. For instance, when a bank processes a loan application from a company. The bank starts with the risk-free rate as the interest. Then, as it takes other risks, such as inflation risk, default risk and currency risk into account and the interest of the loan offered to the company becomes higher than the risk-free interest rate. Additionally, risk-free interest rate is also a starting point for a lot of financial pricing theory, such as Capital Asset Pricing Model and Black-Scholes Formula, which are used by traders to valuate financial products. Thus, risk-free rate is not only for a proxy for comparing the underlying risk of a financial product but it is also a key input of a lot of financial theories development. Hence, it is obvious that why risk-free interest rate is very essential.
2.1.3 Risk Neutral World
In the previous part, the interest r is assumed to be a constant. However r is a function of time in reality. Later in the thesis, the interest rate models are illustrated. Before we go into these model, an crucial principle about risk neutral valuation is introduced.
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The risk neutral setting is used very frequently in derivative valuation. When a derivative is being valuated under the risk neutral setting, this implies that the investors do not increase the required expected return of an investment to compensate for the increased risk. In other words, the risk preference of the investors is not relevant in the pricing process. A world in which risk preferences of investors are not important is called a risk neutral world, which is absolutely different from reality. Since investors would expect to have higher return if they are taking more risks in the real world.
One might ask that, why risk-neutral setting is still being used even though it does not match the real world? For some financial instruments, for instance, options, investor’s risk preference is not important for the valuation of the options. This is because even if the investors are more risk-averse and they do not prefer invest their money in some instrument with high un-certainty, such as stocks, it drives the stock price declined, nevertheless, the formulas relating the options price and stock price remain unchanged. Later in this chapter, option valuation by binomial tree will be illustrate. The interested reader can have a more clear view of why the option price is not affected by the investor’s preference of risk.
The interest rate models are also the major focus of this thesis. But what is the connection between the risk-neutral valuation and the interest rate model? The risk-neutral measure in interest rate is as known as the Q-measure. The most general form of an interest rate model under a Q-measure is:
dr = µ(t, r)dt + σ(t, r)dz (2.7)
where r is the short rate, t represents the time period and σ is the volatility of the short rate. dz is the stochastic process, it represents the uncertainty.
Under the real world measure, as known as the P-measure:
dr = (µ(r, t) − λ(t, r...)σ(t, r))dt + σ(t, r)dz (2.8) In the P-measure, the market price of risk λ is taken into account1. Market price of risk can also be an indicator of investor’s risk-preference. If investors do not prefer the exposure of interest rate risk, then it is very likely that they will not invest in bonds since they are highly correlated with interest rate. Thus, the bond price will go up, as the calibration of short rates is based on the current bond price, plus bond prices and interest rate move in opposite directions (this will be explained more in detail in the next section). As a result, the interest rate from the P-measure is lower than the one from the Q-measure. This implies, λ is negative and the drift becomes positive.
Nevertheless, the interest rate model which is implemented in this thesis is under the risk-neutral measure. The interest rate model under the Q-measure is used to develop an interest rate scenario. This scenario constructs a yield curve and it will be used to discount the cash flows of a financial instrument in order to price it. Thus, as risk-neutral approach can ensure the valuation is free of arbitrage, it is more preferable for the interest rate model than the real-world approach.
2.2
Bond Valuation
2.2.1 Concept of BondsBonds belong to the category of fixed-income security, this implies bonds give fixed payment stream to the bonds’ buyers in a regular time interval within the date of maturity. Bonds can be issued by different parties. For instance, there are government bonds and corporate bonds. In general, a bond can be seen as a loan to the bond issuer from the bond holders. Bonds consist of three main components: face value, maturity date and coupon rate. Coupon rate is the promised interest rate. Some bond issuers offer to pay the coupon annually, some also offer to pay semi-annually, quarterly or even monthly. Face value of a bond is the notional amount for computing the interest or coupon payments. Maturity date is the time that the bond issuer pays back the face value to the bond holder. For instance, a bond, which has face value 1000 euro with 10% semiannual coupon payments and will be mature on 31st of August 2020, will paye1000 × 10%/2 = e50 every 6 months and the face value plus the last coupon payment will be made at the end of August in 2020. There are bonds pay no coupon, so their coupon rate is zero. They are called the zero-coupon bonds (ZCB). Hence, the ZCB holders receive the face value of the bond when it matures. With the logic of time value of money mentioned earlier in this chapter, it is obvious that the initial price of a ZCB is less than its face value.
2.2.2 Yield to Maturity
The other important terminology about bonds is the yield to maturity. The yield to maturity of a bond is the rate that sets the present value of the promised payments of a bond and its current market price equal to each other. For instance, there is a zero coupon bond with price e9000 and face value e10000 and its maturity date is in one year then the yield to maturity (Y T M ) can be computed as following:
9000 = 10000 1 + Y T M1
Hence, Y T M ≈ 11%. The yield to maturity of the zero-coupon bonds are also called the spot interest rate or the zero rate. The yield to maturity of n-year ZCB is computed by:
Y T Mn=
face value bond price
1/n − 1
When the yields to different maturities of zero-coupon bonds is obtained, the yield curve of a zero coupon bond can be constructed.
The yield to maturity is crucial for valuating a bond. If the coupon (CP N ), face value (F V ), maturity date (it will be mature in N years) and the yield to maturity are given. The present value of a coupon bond, P , can be computed by:
P = CP N × 1 Y T M 1 − 1 (1 + Y T M )N + F V (1 + Y T M )N
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However, the yields of a bonds are not constant in practice. For instance, in Figure 2.1, the solid line represents the average of the AAA-rating euro government bonds yields to the maturity of 30 years, and the dashed line illustrates all euro area central government bonds (included AAA-rating government bond) yields up to maturity of 30 years. This figure clearly illustrates that the AAA-rating government bonds have lower yields when it is compared to the yield curve will all euro government bonds. Moreover, the difference of the yields becomes larger in the long-term. These two phenomenons can be explained by the default probabilities and the compensation of the risks in the long-term bonds or the government bonds with credit ratings lower than AAA. First, when the horizon of the maturity is larger, the higher of the default probability is. As the credit rating is deteriorated over the maturity horizon (Collin and Solnik, 2001 (3)). Moreover, bonds with longer maturities consist of higher interest rate risk. This can be observed in the duration2 of the bond. When duration is large, this implies the bonds are sensitive to the movement of the interest rate. Long-term bonds always have larger duration than short-term bonds, therefore the duration illustrates that they also con-sist of more interest rate risk. Thus, it can explain why the long term yield rates are higher than the short term yield rates as the higher yield (higher return) is for compensating the risk.
AAA-credit rating government bonds have lower yield is because the default risk of these the AAA- credit rating governments is lower. The good credit rating or low credit risk implies that the default probability of these bond issuers is low and they will very likely pay back the debt. Even though there are a little credit risk among the good-rating bonds, it is still very low, or at least lower than those bonds issued by the parties with worse credit rating. There-fore, it is more certain about the bonds holders’ payoff. Hence, the yields of these AAA-credit rating government bonds do not need to be high for compensating the underlying risk.
Figure 2.1: Euro Government Bond Yield Curves
Source:European Central Bank: http://www.ecb.europa.eu/stats/money/yc/html/index.en.html 2
Duration is a sensitivity measure of how the interest rate movement affected the present value of the bonds.
During the European sovereign debt crisis, several countries such as Greece, Portugal, Ireland, Cyprus and Spain faced the severe challenges in their economies. This crisis began in 2009, these countries were unable to repay or refinance their government debt, or bail out their banks without the assistance of third-party financial institutions such as the European Central Bank (ECB), the International Monetary Fund (IMF). The global financial crisis in 2007 to 2008, the great recession during 2008-2012 3 and the unstable real estate markets were the main causes of the European sovereign debt crisis. Figure 2.2 illustrated the 10-year Greek government bond yields from 2008 to 2016. Back to 2009, the new government in Greece unveiled that the previous government under-reported its budget deficit. It caused the confidence of investors to Greek government bonds sharply eroded. This lead to the significant increase in their bond yields, which is shown in Figure2.2during year 2011 and 2012. This crisis changed the investors’ expectation toward government bonds. Moreover, they no longer consider the zero rate of the governments’ zero coupon bonds as risk-free interest rate.
Figure 2.2: Greek 10 year-Government Bonds yield from 2008 to 2016
2.2.3 Arrow Debreu Price
Arrow-Debreu price is a tool which is crucial in the pricing approach by the Hull-White inter-est rate tree. Arrow-Debreu price is the value of an Arrow-Debreu security, which pays one unit when a specified state is reached and zero otherwise (Tavella, 2003 (24)). This framework is developed by Kenneth Arrow and Gerard Debreu. It is used very often as the foundation of a lot of economics idea. The Arrow-Debreu model is one of the most general models of competitive economy and is an important part of general equilibrium theory of economy.
Suppose a portfolio of Arrow-Debreu securities is purchased, denote that Fj,n(T ) is the amount of the j-th Arrow-Debreu security in the portfolio which has pay at time T if the j−th state is reached. The payoff of the n-th asset in state j will be matched with the Arrow-Debreu price. Let πs to be the value of the s-the Arrow-Debreu security in Equation (2.9), matching
3
The great recession is the largest downtown in global economy after the great depression in the US after 1920s
12
the present value of the n−th asset, Vn(0) and the present value of this portfolio, following equation is derived: Vn(0) = s=1 X s=S Fs,n(T )πs, n = 1, ..., N (2.9)
Where S is the total number of states.
If the equation above is not satisfied, it implies that there are arbitrage in the market. If the portfolio of Arrow-Debreu securities are more valuable than the asset, one can short sell the portfolio and buy the asset, or vice versa when the asset is more valuable. In either way, there is riskless profit in the transaction.
The value of the Arrow-Debreu securities πs can be used to determine the price of other securities whose payoffs are known. However, it is only feasible if the number of the states and the number of the independent assets are the same, i.e. S = N . Once this is satisfied, Equation (2.9) can be rewritten and used for determining πs:
s=1 X
s=S
Fs,n(T )πs= Vn(0), where n = 1, ..., N and S = N (2.10)
In this thesis, Arrow-Debreu prices are used in the computation of the adjustment parameters in the interest rate tree. The adjustment parameter is to fit the initial term structure to the interest rate tree and using the Arrow-Debreu price helps to facilitate the calculation. The state assumed is a security pays 1 unit if the rate on the particular node of the interest rate tree is reached and it pays zero otherwise. Further explanation of how to use the Arrow-Debreu price to find the adjustment parameter is discussed in the later chapter.
2.2.4 Bond Valuation with Stochastic Calculus Approach
There are a lot of examples from textbook used constant interest rate for illustrating the basic concepts of discounting cash flows. It is an easy approach to demonstrate the main idea of the valuation method in financial theory. In the previous sections, the interest rate for the bond valuation is assumed to be deterministic. However, stochastic interest rate is considered more often in quantitative finance. In fact, the stochastic approach of modelling the values of interest rate derivatives or bonds with optionality, such as callable bonds, is more appropriate. Since the interest rate in the future is indeed uncertain, it is important to take the stochastic factors into account when the payoff of the financial instruments are related to the interest rates. Under the risk-neutral world, the price of a zero coupon bond can be derived by this partial differentiation equation (PDE)4:
1 2σ(r) 2∂2P ∂r2 + [a(r) + σ(r) + ϕ(r, t)] ∂P ∂r + ∂P ∂t − rP = 0 (2.11)
Where σ(r) is the volatility of the interest rate, a(r) is the mean reversion rate and .. And the solution of the PDE is the price of any zero coupon bond:
P [t, T, r(t)] = EtQ[e−R(t,T )] (2.12)
where P [t, T, r(t)] represents the price of the bonds which will be mature in T periods and with the interest rates r(t) at time t. EtQ is the expected value under the risk neutral set-ting. R(t, T ) is the stochastic variable representing the interest rate. The concept related to the interest rate models and stochastic calculus approach for bond valuation is discussed in chapter 5.
2.3
Derivatives Valuation
2.3.1 OptionsOption is one of the major derivatives in the financial market. It is a contract that pro-vides the option holder a right to buy or sell a certain financial instrument in a pre-agreed price within or on a certain date. European option only allows option holders to buy or sell the underlying financial instrument on the expiration date. American option allows option holders to buy or sell before or on the expiration date. Options can be applied in a lot of different financial instruments, the most common option is the option on stock price. There are also different type of options which gives the right to the option holder to buy or sell the bond at a certain price on a certain date. One of the main types of bond with optionality is callable bonds, which allow the bond issuers to buy back the bonds from the bond hold-ers at a certain price. Even though option is a derivative contract but it is only a right but not an obligation, option holders can choose to not exercise the options if that is not profitable.
Many academics in the past have developed different valuation models for options. For option with stock index or currencies, the popular option pricing approach is using binomial tree. This is a diagram technique which illustrates the possibilities of the underlying instrument’s movement. It is assumed the stock price follows random walk5. At every time step, the diagram shows two paths for each node, either goes up or down. Each path carries certain probability of the stock price’s movement. When the size of the time step on the binomial tree becomes small enough, this model is the same as the famous pricing model, the Black-Scholes-Merton model which is commonly used to price different types of financial derivatives.
The pricing approach chosen in this thesis is similar to the binomial tree technique, which is found by Hull and White and called the trinomial tree. The detail of building and using the trinomial tree is illustrated in Chapter 4 and 5. In this part, the focus is on the ordinary valuation of options by binomial tree
5Random walk theory suggests that the changes in stock price have a same distribution and they are
independent to each other. In other words, the historical stock prices cannot be used to predict the movement of the stock prices in the future.
14
Option priced by binomial tree
If one would like to valuate an option, the first step is to know the possible payoff the option gives. The price of a option depends on the following characteristics: strike price, current stock price, maturity date of the option, the volatility of the stock price, the risk free interest rate and the dividends that are expected to be paid.
The maturity date of an option is known as the expiration date, it represents when the option contract will be expired. The call option holder can buy (or sell if it is a put option) the stock at the strike price. It takes a main role to determine whether the option is profitable. When the strike price of a call option is lower than the underlying stock price on the maturity date, then the option holder can acquire the difference between the stock price and the strike price. Otherwise, it is assumed that the option holder do not exercise the option hence his payoff is zero. The payoff of a option can be represented in this formula:
Payoff of an European call option = max(ST − K, 0) Payoff of an European put option = max(K − ST, 0)
(2.13) Where ST is the stock price on the maturity date of the option and K is the strike price. The binomial tree of stock price has the following formulas, assume the size of each time step is ∆t. During each time step, it either moves up to u times the initial stock price or moves down to d times the initial stock price. σ is the volatility of the stock price and p be the upward branch on the tree, (1 − p for the downward branch). r here represents the risk-free rate. The size of the stock price going up and down are determined by these formulas6:
u = eσ √ ∆t d = e−σ √ ∆t p = e r∆t− d u − d (2.14)
The value of the option on stock price is denoted by f . A two-step tree is used to illustrate how an option is priced under a binomial tree. Suppose the initial stock price is S0. ∆t, p, u, d and r are defined same as above. fu and fd represented the option value after one up and down movement on the tree. Similarly, fuudenotes the option value after two up movements. Assumed the option is expired in 2∆t, therefore, the option values in the last step of the tree are known by Equation (2.13). Figure 2.3 illustrates the binomial tree of pricing a option with two time stpe. The values of fu and fd can be found by:
fu = e−r∆t[pfuu+ (1 − p)fud] fd= e−r∆t[pfud+ (1 − p)fdd]
(2.15) Similarly, the initial value of the option can be found by calculating backward with the option values in the first time step:
f = e−r∆t[pfu+ (1 − p)fd] (2.16)
6
The value of u and d are determined by matching the volatility of the stock price, read chapter 12.7 of Hull (14) for details.
Figure 2.3: Stock and option prices in two-step tree
The option price is equal to the expected payoff of the option in a risk-neutral world dis-counted by the risk-free interest rate which is assumed to be the same in every step. How-ever, when the trinomial interest rate tree is used for price the bond option7, the discount rate is the rate on the node. The detail of pricing the bond option is shown later in this thesis.
2.3.2 Interest Rate Swap
Interest rate swap is another kind of common derivatives in the financial world. Swap is an over-the-counter agreement, which means this type of instrument is traded in some context other than a formal exchange such as New York Stock Exchange. The swap agreement in-cluded the specified dates which the cash flows are to be paid, as well as the way of calculating these cash flows. The calculation of the cash flows usually depends on a market variable. The popular choices of this variable are interest rates and exchange rates.
The most common swap contract is called a ”plain vanilla” interest rate swap. In this con-tract, party A makes an agreement with party B which party A pays cash flows equal to interest at a predetermined fixed rate on a notional principal for a specified time period to party B, thus party A is the fixed-rate payer. In the same contract, party B agrees to pay party A cash flows to the floating interest rates, such as LIBOR or OIS rates, on the same amount of notional principal, so party B is the floating-rate payer. Swap is a useful hedging tool. For instance, investment with a large amount of fixed income securities, such as loans, government bonds or corporate bonds are exposed to interest rate risk. Issuers use interest rate swap to match their funding and risk exposure requirement while still tailor the bonds to match the investors’ demand (Fabozzi, 1998 (5)).
Even though swap is a very useful tool for hedging risk in an investment, its initial value is difficult to calculate. Since the payoff of a swap is the difference between the amounts paying
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by both parties. Assume that party A is paying the fixed rate 5% on notional 100 million euro and receiving the floating rate of 12-month LIBOR on the same amount of notional principal for 3 years. The agreement also stated that the payments are made every year and the contract starts on the 31st of January, 2017. The net cash flow will be paid are shown in Table2.1. Party A therefore by the end of the swap contract, they pay party B 0.95 million which is the net cash flow of the whole swap contract.
A swap contract has zero initial value, however its value changes over time, depends on the underlying variables. Similar to other derivatives, swap is a zero sum instrument, when one party has positive cash flow, another party has negative. Valuation of a swap can be done in terms of forward rate agreement (FRA).
Date 12-month Floating Fixed Net
LIBOR cash flows (receiving) cash flows (paying) cash flows
31st January, 2017 4.050% -
-31st January, 2018 4.500% +4.05 -5 -0.95
31st January, 2019 5.500% +4.50 -5 -0.5
31st January, 2020 - +5.50 -5 +0.5
Table 2.1: Interest rate swap cash flows
Forward Rate Agreement
A forward rate agreement (FRA) is an over-the-counter agreement used for ensuring that a certain interest rate will apply to either borrowing or lending a certain principal during a predetermined period. The assumption of this agreement is that the borrowing or lending would normally be done at the LIBOR rate. Later in this thesis, the FRA is used for the assumption of calibrating parameters in the three-dimensional interest rate tree.
The valuation of a forward rate agreement will be illustrated by an example. Consider com-pany X agree to lend money to comcom-pany Y for the period between time t1 and t2, moreover, they enter a forward rate agreement. Denote that RK as the interest rate agreed in the FRA, RF as the forward LIBOR rate8 for the period between t1 and t2 calculated at present time. RM represents the actual LIBOR rate observed in the market at time t1 for the period between t1 to t2. And finally L as the notional principal stated in the contract. For the usual market practice of FRAs, all the rates are measured with a compounding frequency which is the length of the time period, for instance if t2− t1 = 0.5 then the rates are expressed as semiannually compounding.
If companies did not enter the FRA, company X earns RM of the loan. After they entered the FRA, company X earns RK. Thus, entering the FRA gives them an extra earning of RK− RM, of course it can also be the case that they lose money with that when the interest
8Forward LIBOR rates are the interest rate implied by current LIBOR rates for periods of time in the
future. Denote R1, R2as the LIBOR rate at t1and t2, forward LIBOR rate between t1and t2can be calculated
rate pre-specified in the contract is lower than the observed LIBOR rate. The extra interest company X may earn at time t2 is expressed as:
L(RK− RM)(t2− t1) (2.17)
Similarly for company Y at time t2, the extra cash flow might be (if RM > RK):
L(RM − RK)(t2− t1) (2.18)
In most of the cases, RK is set equal to RF when the FRA starts, so that its initial value is zero. It is equal to zero because a large financial institution can keep the forward rate for the future time period as the same at no cost. They can ensure that they earn the forward rate for the time period between, for instance, year 3 and year 4 by borrowing a certain amount of money for 3 years and investing it for 4 years. Similarly, they can ensure themselves paying the forward rate by borrowing a certain amount of money for 4 years and investing it for 3 years.
The valuation of a FRA can be done as follow. Consider there are two FRAs, one stated that the LIBOR forward rate RF will be received and one stated that the pre-specified rate RK will be received. Both on rates are pay on the notional principal L between times t1 and t2. These FRAs are exactly the same, except the payments of the interest rate which is paid at time t2. Similar to Equation (2.17), the payment difference of the two FRAs is L(RK− RF)(t2− t1) and the present value is:
L(RK− RF)(t2− t1)e−R2t2 Where R2 is the risk-free zero rate for maturity of t2.
As when one FRA pays RF and receives RK and the other FRA pays RK and receives RF, the present value of the FRA which receiving RK is:
VF RA= L(RK− RF)(t2− t1)e−R2t2 (2.19) The FRA which pays the RK is:
VF RA= L(RF − RK)(t2− t1)e−R2t2 (2.20) By comparing Equation (2.17) and (2.19) and Equation (2.18) and (2.20), one can find the value of a FRA by discounting the difference of the interest payment by the risk-free rate under the assumption of RM = RF.
Chapter 3
Interest Rate
There are different kinds of interest rates curve in the market, they have different purposes and reflect particular economic or financial conditions in different countries or sectors. For example, given a currency Euro, there are Euro government bonds yield curve, Euro swap curve etc. A term structure of interest rate is an essential information for pricing as it is used for discounting the present value of a financial instrument. However, risk-free interest rate is the primary assumption in many valuation models such as the capital asset pricing model or the Black-Scholes model. Yet, which rates should be used to apply the financial pricing theories with this assumption? Traditionally, LIBOR rate is always used for the risk free rate proxy for derivative dealers. But after the credit crisis, using LIBOR curves for discounting has been questioned. Thus, the public has paid more attention to OIS discounting and multi-curve modelling.In this chapter, these two interest rates, LIBOR and OIS, as well as their difference, the LIBOR-OIS spread will be introduced.
3.1
LIBOR - The Single-Curve World
LIBOR rate is tailored to indicate the interest rate at which banks are prepared to make a large amount of deposit with other banks. It is calculated by the Intercontinental Ex-change(ICE). Every day, the member banks of the British Bankers’ Association (BBA) have to respond to the survey: ”At which rate could you borrow funds, were you to do so by asking for and then accepting inter-bank offers in a reasonable market size just prior to 11 am?”. The BBA selects the highest four and lowest four responses, and take the average of the remaining ten responses and then yielding a 23% trimmed mean1. The average is then published by the mass media corporation Thomson Reuters every day at 11:30am. The member banks in the BBA must have the credit rating AA or above. Because the deposits from bank A to bank B can be seen as a loan that bank B borrows from bank A. Therefore these banks must have a good credit rating to satisfy certain credit worthiness criteria in order to receive the deposits. LIBOR rate is quoted in 5 major currencies for 7 maturities from overnight up to twelve months. An 1-month LIBOR is the rate of one month deposits are offered (Hull, 2012(14)). Treasury or government bonds yields curves are referred to risk-free curve in many of the
1When the lowest 23% and the highest 23% are discarded.
textbook cases. However practitioners uses to choose LIBOR rate as a risk-free proxy instead of government bond yield in practice. The reason why they do not use the government bonds yield curve as risk free rate is that the low yield rates of these government bonds is made artificially. It is caused by the regulatory requirements to financial institutions that they must purchase a minimum amount of the government fixed-income securities. And, for the case in the United States, the treasury instruments are given a favorable tax regime compared to other fixed-income securities. As a result, the prices of the government bonds are pushed up by the increasing demand and the yields go down.
As mentioned above, LIBOR rate is a short-term borrowing rate of the AA financial insti-tutions. For instance, if a financial institution with AA rating borrows a fund at 6-month LIBOR rate, there is a small chance that the financial institution might default. There-fore, some academies suggested that LIBOR still carries some credit risk. Collin and Solnik (2001,(3)) modelled the default risk embedded in the term structure of swap in their paper. They verified that the positive spread between LIBOR and swap2 proves that LIBOR con-sisting of more credit risk than the swap contract. They also found that the positive spread due to not only swap carries less credit risk than LIBOR, but also the deterioration of the credit quality of the issuer over the horizon of the loan.
The public only reviewed the reliability of LIBOR as a risk-free rate for pricing after the crisis occurred. The financial crisis started in mid-2007, with the decline of the credit wor-thiness among financial institutions. Banks were doubtful if the borrowers (other banks or financial institutions in this case) can pay back their debts, therefore they set the interbank rate high, so that the borrowers find borrowing funds at such high rates less appealing. In 2012, the traders from several sector-leading banks were charged of manipulating the LI-BOR rates. The scandal arose when it was found that banks were inflating or deflating their rates deliberately so as to profit from trades, or to give the impression that they were more creditworthy than they actually were. Because LIBOR is used in the derivative markets, es-pecially in the U.S., therefore manipulating LIBOR rates implied manipulating the derivative markets which is certainly against the law. Large Banks such as Barclays banks, Union Bank of Switzerland (UBS) and Royal Bank of Scotland (RBS) involved in this scandal. The settle-ment is up to 3.5 billions dollars recorded in 2014. An article from a former trader of Morgan Stanley, Douglas Keenan, was published by Financial Times in July of 2012, which stated the manipulation of LIBOR started in 1991 (Keenan, 2012 (18)). As LIBOR rate is very relevant to the financial contracts underpinning trillions of pound or euro, this scandal entirely ripped the credit market apart. The public was shocked by the story of how the banks use LIBOR to make profits and by how weak the regulation system was toward determining the interest rate which affects the global financial system. Thus in the aftermath of the scandal, the regulatory around the world is facing the challenge of reforming LIBOR in the way to ensure the creditability and trust for both the financial market and the consumers. Importantly, main goal of the reform of LIBOR is to ensure the market works well and the consumers can have fair trades. Reforming LIBOR is not a simple task, let alone the trades involved trillions of pounds are attached to it, the regulatory also has to decide for the underlying issues such as if the LIBOR should be a rate reflecting the credit risk or a risk-free rate in the vein of
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the overnight index swap rate, or if LIBOR should remain an uncollateralized rate or reflect collateralized lending3. Several approaches are suggested for reforming the LIBOR in order to improve the financial system, however, all of them are having drawbacks4.
3.2
OIS - Better Risk-Free Proxy
The introduction of overnight index swap started from 1990. It is used for hedging the movement in the overnight rates. It is an interest rate swap (IRS) where a fixed rate for a period is exchanged for the geometric average of the overnight rates during that period. Overnight rates is the interest rate that large financial institutions use to borrow or lend from one another in the overnight market. The period of an OIS is short, if a bank borrows funds at the overnight rate for a period, then its effective interest rate is the geometric average of the overnight rates of each business day during the entire period. For example, the bank is borrowing the fund for n days, so the interest of this fund will be charged on top of the principal is the geometric average, let rn be the overnight rate of the n-th business day of that week and the effective interest rate, ref f is :
ref f = (r1+ r2+ ... + rn)1/n
ref f will be the amount of interest paid by the borrower bank. Similarly for lending at overnight rate for a period, the bank which lends will earn the geometric average of the overnight rates. Therefore, an OIS allows these overnight lending or borrowing at the overnight rates to be swapped for making these transactions at a fixed rate. For instance, suppose the fixed rate of an 1-month OIS is 2%, and the geometric average of the daily overnight rates during the same period is 1.5%. If the loan has notional of 100 million euro, then the net cash flow is (0.02 − 0.015) × 1/12 ×e100million.
These overnight transactions are made because at the end of the day, banks need to satisfy the liquidity needs, which is a requirement for banks that they must have certain amount of cash or liquid securities holdings5 in order to repay the liabilities with short term maturities. For example, some large clients want to transfer their cash deposit from bank A to bank B, after bank A made these transactions, it might have a shortage of fund (a surplus of fund for bank B) which implies the liquidity requirement is not satisfied. Then the banks with shortage of fund may borrow the excess reserve from the banks with surplus of fund at the overnight rates. Bank A can also enter an OIS, so it can borrow at a fixed rate. For example, Bank A enters an OIS and has the transactions as following:
1. Borrow 100 million dollars in the overnight market for 1 months, rolling the loan forward each night.
2. Lend the 100 million dollars for 1 month at LIBOR to other bank.
3
Collateralized lending means the borrowers are obligated to offer the lends their property to secure to loan.
4
See Hou and Skeie, 2014 (7) for more discussion.
3. Use an OIS to exchange the overnight borrowings for the fixed rate borrowing with LIBOR.
So after one month, bank A receives the fixed LIBOR 1-month rate and pays the geometric average of the overnight rate in that month. As it is an interest rate swap, hence there is no exchange of the notional. The amount bank A is paying or earning at the maturity is only the net difference in the LIBOR 1-month rate and the OIS floating rate, which is the geometric average of the daily overnight rates in that month.
The overnight rate is often targeted by the central bank to influence monetary policy. It is popular among the major currencies. For example, Federal Funds rates are the overnight rates for the US dollar and Euro Overnight Indexed Average (Eonia) is the rate for unsecured overnight inter banks lending in Euro. The overnight rate is a good indicator of the health of country’s economy and the banking system. Since if liquidity in the market increases, this implies it is easier to borrow, hence the overnight rate decreases and vice versa. Therefore, when the market is stable, the central banks are more freely lending out money as they have confidence to other financial institutions paying back their borrowing, thus the overnight rates are low.
The OIS rate is considered to be less risky. First of all, central banks are involving in the OIS in a lot of cases. Therefore, the default risk, liquidity risk and counter-party risk are seen as minimal in OIS since central banks in general are more reliable. However, the counter-parties of the other side of the swap do still carry a little credit risk and it depends on the possibili-ties of them defaulting. Thus, an adjustment to the fixed rate is needed. This adjustment is computed from various factors such as the slope of the term structure, the default probability by the counter-party, the volatility of interest rate, the maturity of the swap and if the trans-action is collateralized (Hull and While, 2012 (13)). If both parties have equally high ratings with the flat term structure and the transaction is at-the-money, the size of the adjustment is very small because the probability of defaulting in this case is also small. Thus, it is shown that OIS swap rate is a good proxy for a longer-term risk free rate.
3.3
LIBOR-OIS Spread - Indicator of Credit Risk
LIBOR-OIS spread is the difference between the LIBOR and the OIS rates. LIBOR is the rate with higher level as it incorporates the risk premium for credit risk and liquidity risk6. Hou and Skeie explained the theoretical components of the LIBOR rates:
LIBOR =overnight risk free rate over the term + term premium + bank term credit risk + term liquidity risk
+ term risk premium
Overnight risk free rate is a hypothetical overnight interest rate at which a riskless institution could expect to borrow over loan period of LIBOR. For instance, the geometric average of the
22
overnight risk free rate for 1 month is the basis component for the LIBOR 1 month quote. Term premium is the intertemporal rate of substitution for the term of the loan which implies the rate of foregoing the loan for the current term and make the transaction in the future. The idea of the component credit risk, is mentioned earlier in this chapter, this is because the financial institutions have some chances of defaulting. Term liquidity risk compensates for maturity risk caused by the lender when they tie up the funds for a longer period of time. This could include market illiquidity for interbank funds that may increase the lender’s rollover refinancing costs. The last term incorporates the compensation for the risk of the differences between the realization of the previous terms and their expected amount. Hou and Skeie decomposed LIBOR in this way in order to explain and analyse its movement during the crisis. Nevertheless, it provided some insight of what kind of risks LIBOR consist of. The reason of why the credit risk and liquidity risk incorporated in LIBOR is larger than those in the OIS rates is because the lending and borrowing parties involving in using LIBOR are mostly financial institutions but central banks, which have even less default risk, are the participants in the OIS.
A three-month LIBOR- OIS spread exhibits the difference of the credit risks of a three-month loan and a daily-refreshed loan to a bank which has acceptable credit quality. As OIS rate has been seen as the accurate measure of investors’ expectation of the effective federal funds rate over the period of the swap plus the liquidity in the market and LIBOR is now being consid-ering involving credit risk. Hence, the gap between these two interest rates are reflecting the credit risk. Especially the LIBOR-OIS spread increased significantly when the credit crisis emerged, now the public LIBOR-OIS spread is an important measure of credit risk and the liquidity in the money market. Before 2008, the LIBOR-OIS spread was generally assumed to be constant or deterministic and this spread always stayed at the minimal level. When the crisis hit the financial market, this gap rose significantly. It increased from about 10 basis points to 62 basis points for the average 12-months spread. However, the term structure of spread for the short term remained flat.
The Figures3.1 illustrated the LIBOR (USD) and OIS rate for 1 month and 12 month from middle of 2011 to middle of 2016, as well as the Euribor and Eonia (Euro Overnight Indexed Average) for the same terms and same period. Figure 3.1a and 3.1c illustrated that the 1 month interbank borrowing rate (-ibor) and overnight indexed swap rate in US dollars and euro. It shows the opposite movements of LIBOR and OIS in the end of 2011 to beginning of 2012 in the U.S and a sharp decline in European interbank borrowing rate (Euribor) and Eonia. The increase in LIBOR and the higher yield of the one month Euribor and Eonia can be explained by the concern of the European sovereign debt crisis,the effects were even more significant in FIgure3.1band3.1dwhich show the rates for longer term. The rates in Europe remained low after the middle of 2012 and even become negative in middle of 2015. The low interest rates environment around the globe in the recent years is the symptom of chal-lenges in the world economy. The low short term rates are for stimulating the consumption, the low long term rates are for rebalancing and restructuring the saving and long term in-vestment, especially for the pensions and insurances sector (European Central Bank, 2016 (4). In the Figure 3.3, it displayed the spread of -ibor rates and OIS rates in both short term and long term for the period middle of 2011 to middle of 2016. In general, the spread of the
(a) LIBOR and OIS for 1 month (b) LIBOR and OIS for 12 month
(c) Euribor and Eonia for 1 month (d) Euibor and Eonia for 12 month Figure 3.1: Historical interbank borrowing rates and overnight indexed swap rate (2011-07 to 2016-06)
Source:http://www.macrotrends.net/
short term rates is quite small and the long term rates are observed to be larger. Especially during the Euro crisis, the LIBOR-OIS spread and Euribor-Eonia spread even rose to about 100 and 160 basis points respectively, plus the spread for the short term rates also increased significantly. The difference in Euribor and Eonia for the short term and long term remains rather stable after the end of 2012, about 40 b.p. . The long term LIBOR reduced slowly after end of 2012, but it remained about 40 b.p. difference between short term and long term spread until 2015. The sharp increase can be explained by global economy outlook announced by the Federal Reserve in the last quarter of 2016, which was not as positive as the public expected to be and possibly the EU referendum in the United Kingdom affected the financial institutions’ prospective of the future economy, especially in the financial market in London. The figures show the connection between the crisis and the movements of the interest rates, together with the movements of their spread. These figures can verify some of the assumption about the LIBOR consisting of more credit risks. Since under the unstable economy, the LI-BOR or Euribor did increase, this can support the assumption of the banks rise the interbank borrowing rates since they are reluctant to lend or they do not have enough confidence for the creditworthiness of each other. The larger spreads in the 12-month LIBOR rate and OIS rate, as well as in the 12-month Euribor and Eonia, can also validate the concept of loans
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(a) LIBOR-OIS spread for 1 month and 12 month (b) Euribor-Eonia spread for 1 month and 12 month Figure 3.2: Historical spread of interbank borrowing rates and overnight indexed swap rate
(2011-07 to 2016-06)
Source:http://www.macrotrends.net/
with longer horizon carry more default risk, therefore the rates are higher.
Nevertheless, credit risk is not a main risk factor for the rates themselves. First, for the OIS rate, since under the overnight indexed swap the cash flow is the difference between two interest rate streams, not the principal. Thus, there is no risk of losing the principal but the risk of the counter-parties not able to obligated for the swap contracts. Yet, this risk is rather low as the parties in the overnight indexed swap often are central banks. Plus, for the LIBOR rate, when the economy is expected to be normal, credit risk is not reflected in LIBOR either because the financial institutions are less likely to default when the economy is stable. The major reason of why the public see LIBOR is no longer risk free and LIBOR - OIS spread as an indicator of the credit risk is that financial institutions are reluctant to lend during the bad time in the market as they concerns the credit worthiness of the counter party more. It indeed can be observed the rose of the LIBOR-OIS spread during the crisis. Therefore, it implies the financial sector is at critical point when the gap of LIBOR and OIS is widened.
Interest Rate Model and
Hull-White Interest Rate Tree
This chapter provides some insight about what an interest rate model is and why Hull and White select their own model for the multi-curve modelling. Moreover, the interest rate tree from Hull and White is introduced in this chapter as the interest rate tree is the tool of constructing the basis of the multi-curve framework in their paper. Hull and White had done a lot of research regard to their interest rate model as well as the interest rate tree (10; 12; 13; 17; 16). They proposed that a three-dimensional interest rate tree can model the interest rates and price derivatives in a multi-curve environment. The first two papers (10; 11) they published which illustrated the details of the interest rate tree construction. The interest rate tree built in this chapter is based on the procedure provided in those two papers from Hull and White.
4.1
Interest Rate Model
When the interest rate term structure is modelled, it is more realistic to model its behaviour than forecast its future movement because interest rates are seen as random variables. There-fore, it is impractical to forecast the interest rate movement.The short rate rt at time t is the rate that applied to an infinitely short time period at time t. It is sometimes called the instantaneous short rate. The values of bonds, options and derivatives depends on the behavior of the short rates. Academics investigated the process followed by the short rates and developed different interest rate models (as known as the short rates models). Hull and White proposed their own interest rate model for the basis of the multi-curve modelling. It is assumed that the OIS and the LIBOR-OIS spread follow such process. It is an extension of the first one-factor no-arbitrage model1, Ho-Lee model. To explain why the Hull-White model is a proper choice for multi-curve modelling, one should first focus on understanding the two types of interest rate models, equilibrium model and no-arbitrage model. The reasons of using the Hull-White interest rate model are disclosed. Before the interest rate models are
1One-factor model means there is only one source of uncertainty involve in the short rate process.
26
illustrated, some background of short rate r is clarified.
The expected value of an interest rate derivatives with payoff fT at time T is ˆ
E[e−¯r(T −t)fT] (4.1)
Where ¯r is the average of short rate r from the starting time t to the ending time T and ˆE denotes the expected value under the risk-neutral measure.
The price of a zero-coupon bond at time t which will mature at time T is denoted as P (t, T ), it pays 1 unit when the maturity date comes. From Equation (4.1),
P (t, T ) = ˆE[e−¯r(T −t)] (4.2)
If the price of the zero-coupon bond is continuously compounded by interest rate R(t, T ) between time t and T , it becomes,
P (t, T ) = e−R(t,T )(T −t) (4.3)
And, the interest rate R(t, T ) can be derived,
R(t, T ) = − 1
T − tln P (t, T ) Finally, substitute Equation (4.2),
R(t, T ) = − 1
T − tln ˆE[e
−¯r(T −t)] (4.4)
Equation4.4 can determine the term structure of the interest rate at any given time by the value of the r at that time and the short rate process. To model the yield curve, academics thus proposed different interest rate model to define the process of r.
4.1.1 Equilibrium Model
There are two kinds of short rate model used by academics, which are the equilibrium model and the no-arbitrage model. Equilibrium models start with the assumption of economic variables and then derive a process for the short rate. Then, they explore what the short rate process implies about the bond price. An equilibrium model process described by an Ito’s process has the form:
dr = m(r)dt + s(r)dz (4.5)
Where m(r) is the instantaneous drift, s(r) is the instantaneous volatility. They are both assumed to be functions of the short rate r. dz is the stochastic term of the short rate model where z is assumed to follow a Wiener process. This means z follow a normal distribution, z ∼ N (0, t) at time t2. Here are some examples of equilibrium model:
2
More about information about Wiener process and the short rate model can be found in Chapter 13 of Hull (14).
1. The Rendleman and Bratter Model (RB), dr = µrdt + σrdz
• µ, σ are constants, this implies that the short rate r follows geometric Brownian mo-tion.
2. The Vasicek Model, dr = a(b − r)dt + σdz • a, b and σ are constants.
• r is pulled to a level b at rate a, so it has mean reversion feature. • σdz is a normally distributed stochastic term
3. The Cox, Ingersoll, and Ross Model (CIR), dr = a(b − r)dt + σ√rdz • It also has mean reversion features, same as the Vasicek model.
• The standard deviation of the process of r is proportional to √r, this implies when r increases or decreases, the standard deviation σ√r follows the same movement.
In these examples of equilibrium model, the Vasieck and the CIR model are used more often than the Rendleman and Bratter model. This is because the RB model is not mean-reverting and this feature is important when interest rates are modelled. Mean reverting of interest rates means that when r is high, the mean-reverting drift will tend to be negative and pulls the interest rate down, when r is low, the drift will tend to be positive and brings the interest rate up. This movement of interest rates can actually be observed in the economy. When the interest rates are high, the economy tends to slow down and borrowing money is not appealing since borrowers need to pay more for the interest rate, thus the demand of fund becomes low. Subsequently, the rate will decrease. When the rate is low, people tend to borrow at a low interest rate and the demand of fund thus becomes high, and the rates tend to rise (Mankiw, 2007 (20)).
The importance of mean reversion has been illustrated, next we look at how the models with mean-reverting feature determine the price of a zero coupon bond that pays 1 euro at the maturity time, T . Vasicek model can be used to computed the price of the zero-coupon bond:
P (t, T ) = A(t, T )e−B(t,T )r(t) (4.6)
Where r(t) in this Equation is the short rate value and time t. B(t, T ) and A(t, T ) are defined as:
B(t, T ) = 1 − e −a(T −t)
a (4.7)
and,
A(t, T ) = exp (B(t, T ) − T + t)(a 2b − 1 2σ2) a2 − σ2B(t, T )2 4a (4.8)
