MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Transcription

1 MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS Hedging Interest Rate Derivatives (Evidence from Swaptions) in a Negative Interest Rate Environment: A comparative analysis of Lognormal and Normal Model by Shadrack Lutembeka Masterarbete i matematik / tillämpad matematik DIVISION OF APPLIED MATHEMATICS MÄLARDALEN UNIVERSITY SE VÄSTERÅS, SWEDEN

2 Master thesis in mathematics / applied mathematics Date: Project name: Hedging Interest Rate Derivatives (Evidence from Swaptions) in a Negative Interest Rate Environment: A comparative analysis of Lognormal and Normal Model Author: Shadrack Lutembeka Supervisor(s): Jan Röman and Richard Bonner Reviewer: Anatoliy Malyarenko Examiner: Linus Carlsson Comprising: 30 ECTS credits

3 Abstract This thesis is about hedging interest rate derivatives in a negative interest rate environment. The main focus is on doing a comparative analysis on how risk varies between Lognormal and Normal models. This because Lognormal models do not work in the negative interest rate since they do not allow negative values, hence there is a need of using Normal models. The use of different models will yield identical price but different hedges. In order to study this we looked at the case of Swaptions and Swaps as an example of interest rate derivatives. To study risk in these two models we employed the method of risk matrices to measure and report risk. We created various risk matrices for both Black model and Normal Black model which included the price matrices, Delta and Vega matrices to study how Swaptions and Swaps with different maturities are sensitive to changes in different parameters. We also plotted how Delta and Vega vary between the two models.

4 Acknowledgements First and foremost I would like to thank God for always taking care of me. Secondly I would like to convey my special thanks to the Swedish Institute (SI) for awarding me a full scholarship to study the masters program Financial Engineering at Mäladalen University. Thirdly i would like to convey my sincere thanks to my supervisor Jan Röman who invested alot of his time and efforts to ensure that this thesis becomes a success, am truly grateful. I wouldn t also forget my other supervisor, Richard Bonner for his continuous and prompt guidance during the writing of this thesis. To my dear parents who have been so supportive since day one that I set foot at the nursery school. To my siblings; Lilian, Meshack, Godfrey and Gladness for their continuous support throughout this journey. Last but not least are my dear friends and family, Polite Mpofu, Erick Momamnyi, James Okemwa, Oliver Grace, Kakta Mpofu and Mahalet Haile Selassie for always being there throughout this journey. 1

5 Contents 1 Introduction Negative Interest Rates Environment Motivation for Negative Interest Rate Policies Motivation and Problem Formulation Motivation Problem Formulation Understanding Interest Rates Overview and Outline Lognormal Model versus Normal Model Pricing Models Log Normal Models Black s model The Constant Elasticity of Variance (CEV) Model The Stochastic Alpha Beta Rho (SABR) Model Normal Models Bachelier s Model Normal SABR Model Hedging Parameters Greeks in Black-Scholes Greeks in Other Models Black Model Normal Model

6 3.2.3 SABR Model Hedging Strategies Risk Matrices Interest Rate Derivatives Derivatives Interest Rate Derivatives Forward rate agreement (FRA) Caps Floors Bond Options Interest Rate Swap Swaption Implementation Bootstrapping a Swap Curve Premium and Risk Measures Calculations Plotting Delta and Vega in Black and Normal Black Model Risk Matrices for Swaptions and Swaps at Different Maturities Price Sensitivity Delta and Vega Sensitivity Conclusion 51 7 Notes on fulfillment of Thesis objectives 52 Bibliography 54 A More mathematics 56 A.1 Singular Pertubation Technique A.2 Solution to Black Model B APPENDIX B 59 B.1 Extract of VBA Program Codes

7 B.2 Risk Matrices for Swaptions and Swaps with Different Maturities

8 Chapter 1 Introduction 1.1 Negative Interest Rates Environment In a normal world, one would expect that a lender to receive from a borrower a rate on the amount borrowed. It is also expected that when one deposits money in the bank, he would expect to get back some form of interest on his deposit. However, when we have a situation where lenders have to pay borrowers for lending from them or when depositors are charged for keeping their money with the bank instead of receiving an interest income, we have what we call "negative interest rate". One could argue that interest rates were modelled to be positive to compensate a lender for undertaking the risk of borrowing. Most of economic theory fact that nominal interest rates should have a zero lower bound. In 1995, Black (1995) stated explicitly in his paper that it is possible to have negative real interest rate but we cannot have the negative nominal short rate. After almost twenty years we question if Black s assumption was correct. In the current negative interest rate environment with around $ 13.5 trillion of negative-yielding bonds as reported by financial times in August central banks such as the European Central Bank have cut the deposit rate to below zero per cent. As a result, instead of paying interest to the banks or financial institutions that deposits their excess reserves to the central bank, the central bank taxes these deposits. As irrational as this concept may seem to be, the main idea behind it is to discourage the banks from parking the balances at the central bank, instead increase their lending or investments. However negative nominal interest rates is not a completely new concept. One could trace negative nominal interest rates back in the 19th century when "Gesell Tax" was introduced to overcome the zero-lower-bound on nominal interest rates Menner (2011). Similary in the 1970s the Swiss National Bank also experimented with negative rates to control capital inflows in a bid to prevent the Swiss Franc from appreciating. Looking with fresh eyes, in the past few years, we have witnessed the changes of interest rate environment. The global financial system 1 This information was retrieved from Financial Times website: 11e6-ae3f-77baadeb1c93 on

9 has been venturing further into the whole new world of negative interest rates. Between 2014 and 2016, five central banks namely, European Central Bank (ECB), Sveriges Riksbank (SR), Bank of Japan, Denmark National bank (DN) and the Swiss National Bank (SNB) decided to implement negative rate policy. As it can be seen from Figure 1.1 below 2. ECB was the first bank to decrease their interest rates to below zero in June 2014, since then the rate has been dividing deeper below zero. Figure 1.1: This figure shows the European Central bank s interest rates from 2008 until early Motivation for Negative Interest Rate Policies There are different motivations for implementing negative interest rate policy. Bech and Malkhozov (2016) mention different reasons for the implementation of the negative interest rate policy in Europe. One of the major reason to implement negative rate policy has been to boost the economy and to raise inflation which is currently below zero. The other reason is to prevent high rising of the currency. By lowering negative interest rates, investors are discouraged by banks from buying the local currency hence preventing its value from rising up. Table 1 3 below summarizes the rationale behind implementation of negative rate policy by different banks in Europe. 2 The figure is extracted from 3 The table is extracted from Jackson (2015). In the table; bp stands for basis points and DKK is the ISO code for Danish krone 6

10 1.3 Motivation and Problem Formulation In this section, we discuss the motivation behind carrying out this research and highlight why studying risk in the negative interest rate environment is important Motivation After having looked at the current negative interest environment in the world (especially in Europe), it is important to now look at how all this has an impact on the interest rate derivatives traded in the market. There is need to incorporate this new reality of negative interest rates in our models and in our volatility assumptions. 7

11 The need to change models Black s model has been used as the standard model in the market to price interest rate derivatives. As will be discussed later, the key feature of this model is that it assumes that the forward rates are lognormally distributed. This assumption allows the Black model to work only with positive values, the Black model valuation formula is constructed in such a way that it rejects any negative values. It is then clear that in negative interest rate environment where we have negative values we cannot use the standard models like Black model to price and hedge interest rate derivatives. In order to have working models in the negative interest rate environment, we need to adapt to either normal distribution models or the shifted lognormal normal models. The need to change volatility When looking at the Black model which has been the standard model used to price interest rate derivatives, Hagan et al. (2002) noticed an interesting fact in the model s formula. He noticed that in the Black model formula, one can easily observe all parameters except for one parameter which is volatility. This makes volatility a key parameter in the Black formula. As a matter of fact it is has been standard practice for brokers to offer quotes on interest rate derivatives in the form of Black volatility. It is called Black volatility since the Black model is used to derive such volatility. It is also called implied volatility. Quoting a price of a derivative in volatilty eliminates the effects of non-volatility parameters such as its strike, maturity, yield curve and tenor. But just like it was the case for the Black model, Black volatilities also do not work in negative interest rate environment. Frankena (2016) observed that log-normal volatility tends to experience variations or jump drastically when interest rates are approaching zero or negative values, on the other hand the Normal volatility are relatively stable. As an alternative to the use of Black volatility, Antonov et al. (2015) suggested 3 options: 1. Quoting the option prices in dollar value, 2. Using normal volatility suitable for all negative strikes 3. Using the shifted, lognormal, volatility. It is expected that the use of either log-normal model or normal model will yield identical prices. However this is not the case with the hedges or risk measures, the use of different models yield different hedges, see Henrard (2005) and Rebonato (2004) Problem Formulation The above two needs motivate our focus of trying to incorporate negative interest rates in models and volatility. We are interested in seeing how risk for interest rate derivatives varies 8

12 between the Black model and the Normal model. In order to study this we are going to look at the case of swaptions and swaps as an example of interest rate derivatives. To study risk in these two models we will employ the method of risk matrices to measure and report risk. We will also compute and compare the premium(price) and risk measures for this case the delta, gamma and vega for the two models and plot how delta and vega vary between the two models. 1.4 Understanding Interest Rates An interest rate is the amount of money that a lender is promised to be paid by the borrower to cover for the credit risk 4. Hull (2008) points out that the credit risk determines the amount of interest rate, such as the higher the credit risk, the higher the interest rate. Interest rates can be measured with different compounding frequency. The compounding frequency defines the units in which an interest rate is measured. An interest rate can be compounded annually, semiannually, quarterly, monthly, weekly, or even daily. If ssume that an amount P is invested for n years at an interest rate of R per annum and if the rate is compounded once per annum, then the terminal value of the investment is given by P(1 + R) n. (1.1) If the rate is compounded x times per annum, the terminal value of the investment is ( P 1 + R ) nx. (1.2) x Interest rates can also be compounded continuously, a continuous compounding is when we have the limit compounding frequency x approaching infinity. We often use continuously compounded interest when pricing derivatives. This is also the measure of interest rate that will be used throughout this thesis. When we compound a sum of money at a continuously compounded rate R for n years, we multiply it by e Rn that is an amount P invested for n years at rate R grows to Pe Rn. (1.3) If we suppose that R c is a rate of interest with continuous compounding and R x is the equivalent rate with compounding x times per annum, equating equation (1.2) to equation (1.3) we obtain ( P(e Rcn ) = P 1 + R ) x nx. (1.4) x Equation (1.4) can be deduced to equations (1.5) and (1.6) that can be used to convert a rate with a compounding frequency of x times per annum to a continuously compounded rate and vice versa ( R c = xln 1 + R x x ). (1.5) 4 The risk that the borrower of funds, wont pay the interest and principal to the lender as promised 9

13 and R x = x(e R c x 1). (1.6) Interest rates can be defined in many ways depending on the situation and the market. For example to a repo trader we have the simple rate, to an option trader we have compounding rate, while for a bond trader we have yield-to-maturity, see Röman (2015). Let us give a brief description of several types of interest rates with a focus on those that will feature prominently in this thesis. Treasury rates Treasury rates can be defined as the rates that an investor earns on the instruments known as "Treasury bills" and "Treasury bonds". The government use these instruments to borrow its own currency. For example Swedish Treasury rates are the rates at which the Sweden government borrows in Swedish Kronors. Since it is assumed that the government will not default, these rates are sometimes refered to as risk free interest rate. Risk free rate Risk free rate is the rate earned by taking a risk-less position. This rate is normally used to discount projected or expected cash-flows to a present value. Various literature has used various rates e.g. Treasury rates, Swap or OIS rate and LIBOR rate as a proxy for the risk-free rate. Before 2007 LIBOR rates were used by financial institutions as one riskfree rate, however after the financial crisis they turned to overnight indexed swap (OIS) 5. Röman (2015) points out that the correct rate to use depends on what instrument is being valued, the counterparty and the agreements made. Zero coupon rate The zero coupon rate is the yield to maturity on a zero coupon bond(a bond that pays no coupon). We can use bootstrap method to obtain this rate from coupon bonds. The zero coupon rates are used for the discounting the future payments and can also be used to calculate the risk by shifting the Zero coupon curve. Spot rate The spot rate (short rate) can be defined as the theoretical profit given by a zero coupon bond. This rate is used to calculate the amount that will be obtained at time T (in the future) if A is 5 The reason to stop using LIBOR rates is because banks became very reluctant to lend to each other during the crisis and LIBOR rates soared, see Hull (2008) 10

14 invested today at time T 0. We use bootstrap method to calculate the spot rate. with the present value of A T is given by A T = A T0 (1 + r s ) T, (1.7) A T = 1 (1 + r s ) T A T 0. (1.8) Forward rate Forward rate is the rate which is referred to by the zero rate for future period. To protect investor s position if the future rates will be different from today s forward rates, "forward rate agreement (FRA)" is used. This type of interest rate derivative will be covered in Chapter 4. If R A and R B are the zero rates for maturities T 1 and T 2, respectively, and R F is the forward interest rate for the period of time between T 1 and T 2, then If we write equation (1.9) as R F = R BT 2 R A T 1 T 2 T 1. (1.9) T 1 R F = R B + (R B R A ) (1.10) T 2 T 1 and take limits as T 2 approaches T 1, and let the common value of the two be T, we can obtain a forward rate that is applicable to a very short future time period that begins at time T, known as an "instantaneous forward rate". where R is the zero rate for a maturity of T. R F = R + T R T, (1.11) Swap rate A swap rate is the fixed interest rate that is used to price a Swap to a zero value. As we will see in Chapter 4, a Swap is contract between two parties to exchange interest rate cash flows. Sometimes swap rates can are used as risk free interest rate. LIBOR rate LIBOR (London Interbank Offer Rate) is the rate of interest at which a bank is prepared to deposit money with other banks in the Eurocurrency market for maturities ranging from 11

15 overnight to one year, see Röman (2015) and Hull (2008). Banks use this rate as a benchmark when lending to one another. This rate is calculated and published by Thomson Reuters on behalf of the British Bankers Association (BBA) at around 11:45 AM each day (London time). LIBOR rate is calculated for 6 major currencies and is often used as the reference rate for floating-rate loans, especially swap contracts in the domestic and international financial markets. STIBOR rate STIBOR (the Stockholm Interbank Offer Rate) is the rate that the Swedish banks can borrow from each other s at different maturities, see Röman (2015). This rate is used to assess how the market views the risk of lending between banks. STIBOR rates are compared by the interest rate on government securities with the same maturity, to see which risk premium imposed on bank loans. STIBOR rate is calculated and published by Swedish bank association called Bankforeningen at around 11:00 AM each day (Swedish time). Furthermore STIBOR is used as a floating-rate in swap contracts in the swedish market. 1.5 Overview and Outline The thesis is organized as follows: This chapter has briefly reviewed the negative interest rate environment and provided motivation for our research question. The next chapter discusses the normal and lognormal models. Chapter 3 juxtaposes the hedging parameters used in hedging risk while Chapter 4 discusses interest rate derivatives with a focus on swaps and swaptions. Chapter 5 discusses the implementation while Chapter 6 gives the conclusion of the thesis. Finally Chapter 7 presents the notes on fulfillment of the thesis objectives. 12

16 Chapter 2 Lognormal Model versus Normal Model In this chapter we will discuss several models which are used in the pricing and hedging of financial derivatives. 2.1 Pricing Models Pricing models fall in two main categories namely; Lognormal Models and Normal Models. While the Normal models allow for the negative interest rates since their distribution is normal, Lognormal models can only allow positive interest rates. There are many models which fall under these two main categories, however we will limit our discussion to only the Black model, the CEV model and the SABR model as examples of log normal models. As for the examples of normal models we will consider the Bachelier model and the normal SABR model. This chapter will briefly present the valuation formulas of the respective models, further derivations of the models will be presented in the appendix section. But before we begin our discussion on the lognormal and normal models we first briefly illustrate the concept of normal and lognormal distribution. Normal (Gaussian) Distribution Random normal variable X, is normally distributed with mean µ and variance σ 2 such as X N(µ,σ 2 ) if it has the density function of f (x) = 1 (2πσ) e (x µ)2 2σ 2. (2.1) When µ = 0 and σ = 1 that is N(0,1) we have what we call a Standard Normal Distribution with density function φ(x) = 1 e x2 2. (2.2) 2π 13

17 The cumulative distribution function of the standard normal distribution is given by the equation below, see Kijima (2013) Φ(x) = x 1 (2π) e t2 2 dt. (2.3) Figure 2.1: Standard Normal Distribution. Log Normal Distribution The log-normal distribution, see Walck (2007) which is sometimes denoted as Λ(µ,σ 2 ) is described with density function f (x) = 1 2πσx e (logx µ)2 2σ 2, (2.4) where the variable x > 0, and parameters µ and σ > 0 are all real numbers. If u is distributed as N(µ,σ 2 ) and u = logx, then x is distributed according to the log-normal distribution. Figure 2.2 illustrates log-normal distribution for the basic form with µ = 0 and σ = 1. A variable that has a log normal distribution can take any value between zero and infinity. The other property which distinguishes log normal distribution from normal distribution is the fact that log normal distribution is skewed such that the mean, median, and mode are all different. 2.2 Log Normal Models In this section we are going to discuss the Log-normal-like models which include: The Black Model, CEV Model and SABR Model. 14

18 Figure 2.2: Lognormal Distribution Black s model Fisher Black s motive to come up with the Black model in 1976 was to extend on his original work famously known as the Black-Scholes model. In his paper Black (1976), Black modified the Black Scholes model in such a way that it could be used to value European call or put options on futures contracts. To enhance our understanding on the Black model it is important to first look at the Black-Scholes model. Black-Scholes model One of the greatest findings in finance, the Black Scholes Model was developed by Fischer Black, Myron Scholes, and Robert Merton in the 1970 s. This great achievement was later recognized by a Nobel prize for economics which was given to Robert Merton and Myron Scholes in Hull (2008) points out that Black and Scholes used the capital asset pricing model to determine a relationship between the market s required return on the option to the required return on the stock. Since its development in the 1970 s, the model has been a centre in pricing and hedging derivatives. In their paper Black and Scholes (1973) "The Pricing of Options and Corporate Liabilities". Black and Scholes made the following key assumptions for the model: 1. The stock price follows a random walk such as they may likely move in any direction at any given time 2. Stock returns are normally distributed, hence constant volatility 3. Frictionless market exists, i.e.,no transactions cost in buying or selling stocks or option. 4. The stock doesn t pay any dividends or other payments. 5. The option can only be exercised upon expiration, i.e.,european option 6. Constant and known interest rate r 15

19 They came up with a formula for pricing European call and put options. In this formula they assumed that the stock price S follows geometric Brownian motion with expected return µ and volatility σ ds t = µsd t + σsdw t. (2.5) The Black-Scholes formulas, see Hull (2008), Black and Scholes (1973) and Röman (2015) for the pricing European Calls C and Puts P for non dividend paying stocks at time 0 are given by; C = e rt [S 0 e rt Φ(d 1 ) KΦ(d 2 )], (2.6) where P = e rt [KΦ( d 2 ) S 0 e rt Φ( d 1 )], (2.7) d 1 = ln[s 0/K] + (r + σ 2 /2)(T ) σ, (2.8) T d 2 = ln[s 0/K] + (r σ 2 /2)(T ) σ = d 1 σ T. (2.9) T Φ(x) is a cumulative probability distribution function for a standardized normal distribution which is defined in Equation (2.3). S 0 is stock price at time zero, K is strike price, r is a continuously compounded risk free rate, T is time to expiry of an option and σ is the stock price volatility. Many studies have shown that, the assumption of constant volatility makes Black-Scholes model inadequate in pricing and hedging options. This is because the assumption of constant Implied volatility shows a dependence on the volatility smile (i.e. option strike and maturity). Thus the constant volatility method, which assumes that the volatility is constant for all the options on the same underlying, can lead to a significant model specification error, see Coleman et al. (2003). Black model is similar to the Black-Scholes model except for the adjustment made on the drift term and on dependence in time of the volatility. In Black model, Black attempts to address the problem of negative cost of carry 1 in the option pricing model by using forward prices F instead of spot prices S like in Black Scholes model. The use of spot prices versus the use of forward prices (discounted futures price) is the key difference between these two models. Hull (2008) points out the use of forward prices is what makes the use of Black model advantageous. This is because the forward prices used in Black model incorporates the market s estimate of convenience yield or income, hence there is no need to estimate this income. Pricing European Option Black s formula, is derived from the assumption that the forward prices F(t) are lognormally distributed about today s forward price F 0 such as F(0) = F 0, 1 Is a situation where an investor is losing money since the cost of holding a security exceeds the yield earned. 16

20 df t = σ B F t dw t, (2.10) σ B is the normal volatility and W t is a Brownian motion. The formula gives the price for a European call and put option of maturity T on a futures contract with strike price K and delivery date T are given by. where C = P(0,T )[F 0 Φ(d 1 ) KΦ(d 2 )], (2.11) P = P(0,T )[KΦ( d 2 ) F 0 Φ( d 1 )], (2.12) d 1 = ln[f 0/K] + (σ 2 /2)(T ) σ, (2.13) T d 2 = ln[f 0/K] + (σ 2 B /2)(T ) σ B T = d 1 σ B T. (2.14) F 0 is the forward price of the asset at time 0, σ B is the quoted volatility of the option and P(0,T ) is today s discount factor to the maturity date. In Appendix A.2 we show the solution to the black model The Constant Elasticity of Variance (CEV) Model CEV model which provides a basis of the SABR model covered in the next subsection was introduced by Cox and Ross (1976). The model is given by ds t = µs t dt + σs β t dw t, (2.15) where the drift µ is constant, α 0 and β 0 are real constant parameters. The key parameter in this model is the elasticity factor β since it controls the relationship between volatility and price in this model. In the interest rate market β range between 0 β 1. When β = 0, Equation (2.15) reduces to Bachelier model which will be covered in Section (2.3). When β = 1 we obtain the Black model in the previous subsection. Pricing European Option Schroder (1989) presents the price of a European call option in the CEV model as follows: ) C t (S t,t t) = S t (1 n=1 g(n γ, K t ) Ke r(t t) g(n + γ, F t ) n=1 n m=1 n m=1 g(m, K t ), g(m, F t ) (2.16) 17

21 where γ = 1 2(1 β ) and g(p,x) = xp 1 e x Γ(p) the forward price of a stock F t = S t B(t,T ) is the density function of the Gamma distribution. For we have: F 2(1 β ) t F t = 2X(t)(1 β ) 2, K K t = 2(1 β ) T 2X(t)(1 β ) 2, X(t) = σ 2 e 2r(1 β )u du is the scaled expiry of an option. t The Stochastic Alpha Beta Rho (SABR) Model The SABR model is derived by Hagan et al. (2002) is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. According to Hagan, this model came as an extension to CEV model which fails to calculate accurate hedges and to predict the dynamics of the Black model implied volatility accurately. The model has 4 main parameters which make the name of the model. These parameters include; 1. ρ is a correlation parameter which defines how the market moves in sync with the volatility dynamics, controls the skewness of the distribution. 2. β is a skewness parameter and it controls the relationship between the forward price and the at-the-money volatility. 3. α is a "volatility-like" parameter that cannot be observed from the market, it determines the at-the-money (ATM) forward volatility. 4. ν is a parameter, volatility of volatility, determines the skew. Under the martingale measure P, the forward rate F and its volatility α are assumed to obey the following equations: df t = α t F β t dw t, (2.17) with the initial condition: dα t = να t dz t, (2.18) F 0 = F, α 0 = α, where F is the forward price, the constant parameters β,ν satisfy the conditions 0 β 1,α 0 and the correlation between the two processes W t and Z t with correlation coefficient 1 < ρ < 1 is given by: dw t dz t = ρdt. (2.19) 18

22 Pricing European Option Assume that the volatility parameters α t and ν defined in Equation 2.17 and 2.18 are very small. At time t, F(t) = f, α(0) = α the value of the European call option at date t is given by: V (t, f,α) = E[(F te K) + F t = f,α(t) = α], (2.20) t e is the exercise time. The price of a European call option then becomes, see Zhang (2011) V (t, f,α) = [ f K] + [ f K] 2 π x 2 2τe e q (q) 3 2 dq, (2.21) where q = x2 2τ. Implied Volatility in SABR model Hagan et al. (2002) used singular perturbation techniques to obtain the black implied volatility. In Appendix A.2 we will show how singular perbutation technique can be used to analyze models and find an explicit expressions for the values of European options. The Black implied volatility σ B (F,K) is given by the formula below, see Hagan et al. (2002) and Röman (2015). α ( z ) σ B (F,K) = { } (2.22) (FK) (1 β )/2 (1 β ) (lnf/k) 2 (1 β ) (lnf/k)4 X(z) where and { [ (1 β ) α 2 (FK) 1 β ρβνα (FK) 2 3ρ2 + ν 2]} T, (1 β )/2 24 z = ν α (FK)(1 β )/2 ln(f/k), (2.23) { 1 2ρz + z x(z) = ln 2 + z ρ 1 ρ }, (2.24) when K = F, equation (2.22) gives at-the money (ATM) implied volatility σ AT M = σ B (F,F) = 2.3 Normal Models α { [ (1 β ) 2 F (1 β ) F 2 2β + 1 ρβνα 2 3ρ2 + ν 2]} T. (2.25) 4 F (1 β ) 24 In this section we are going to discuss the Normal-like models which include: The Bachelier or Normal Black Model and The Normal SABR Model. α 2 19

23 2.3.1 Bachelier s Model This model was first introduced by Bachelier (1900). The Bachelier model assumes normal distribution for the asset price, which makes it possible for the model to take in negative interest rates. This model is also known as Normal Black model. Pricing European Option The derivation of Bachelier formula assumes the forward rate F t follows the SDE df t = σ N dw t, (2.26) where σ N is normal volatility. The formula prices the European call and put option as follows: and where C = P(0,T )[(F K)Φ(d) + σ N T φ(d)], (2.27) P = P(0,T )[(K F)φ( d) + σ N T Φ(d)], (2.28) d = F K σ N T, (2.29) K is strike price and P(0,T ) is a zero coupon bond used to discount Normal SABR Model The SABR model can be extended to accommodate the negative interest rates. The easy way to incorporate negative rates in SABR model is by introducing a shift parameter s in the original SABR model presented by Equation (2.17) df t = (F t + s) β α t dw t, (2.30) where s is a deterministic positive shift which moves the lower bound on F t from 0 to s, This type of the model is called Shifted SABR model. The major drawback of this model is that it requires the shift parameter s to be changed whenever the rates change either to positive or more negative, which in turn leads to jumps into the model prices, Greeks, and the risk. Instead Antonov et al. (2015) suggests the use of Free Boundary SABR model that can handle negative rates.in his paper he derives an exact solution for zero correlation and approximation for non-zero correlation When we adjust the SDE in Equation (2.17) we obtain a model that allows for negative rates for β values ranging 0 β < 1/2 and a free bondary i.e. with no shift parameter s df t = F t β α t dw t, (2.31) 20

24 The normal implied volatility σ N (F,K) is given by the formula below, see Hagan et al. (2002). σ N (F,K) = α(fk) β / (lnf/k)2 + (1 β ) (lnf/k) 2 + { [ β (2 β ) α 2 (FK) 1 β (1 β )4 ρβνα (FK) 1920 (lnf/k)4 (1 β. ) (lnf/k)4 ( z ) X(z) 2 3ρ2 + ν 2]} T, (1 β )/2 24 z and x(z) are defined as in Equations (2.23) and (2.24) respectively. (2.32) 21

25 Chapter 3 Hedging Parameters Risk and hedge are two inseparable common terms in finance. When one discusses the concept risk, he will most certainly address hedging as well. A risk can generally be defined as a degree of uncertainty associated with an investment. On the other hand the term hedge simply means reduction of risk by exploiting correlations between various risky investments. Managing risk is a common problem that an option traders face in the OTC market encounters. For example to counterbalance risk in his portfolio, an option seller can buy an option which is similar to the option that he sold on the exchange. However, this is only possible if the option in question happens to be similar to the option traded on the exchange, contrary to that, a seller is faced with great difficulties in hedging his risk. To manage such risk we need risk measures or hedging parameters known as the Greeks or Greeks letters. Greeks are used to quantify risk in the portfolio. In this chapter we will present the Greeks and how they are used in the pricing and hedging of options. 3.1 Greeks in Black-Scholes Greek letters are vital tools in risk management which measures the sensitivity of the price of a derivative such as an option to a small change in a given underlying parameter, see Hull (2008), Wilmott (2007) and Röman (2015). Each Greek letter measures the risk in a different dimension in an option position. The main goal of the trader is to manage the Greeks so as to achieve the desired exposure or risk in a portfolio. Greeks are normally computed by trading software s to manage the risk in instruments and in portfolios. There are different Greek letters in mathematical finance, however our discussion in this section will be limited to Delta, Gamma, Vega, Rho and Theta as presented in Black Scholes model. Later we will also look at how the Greeks are presented in other models. 22

26 Delta The Delta of an option is defined as the sensitivity of the option or portfolio to the underlying. Frankena (2016) points out that depending on the market standards of the specific product, the underlying rate can either be a spot rate or a forward rate. Delta is a measure of a rate of change in value V of an option with respect to the asset S such as = V S. Delta is not constant, it always change over time. An investor can use delta to periodically determine a number of units of stock that needs to be held for each option sold in order to create a riskless portfolio. The Black-Scholes of a European call option on a non-dividend-paying stock is positive and the formula gives a Delta of a long position 1 in one call option such as: (c) = Φ(d 1 ), (3.1) On the other hand, the Delta of a put option is negative and is given by: (p) = Φ(d 1 ) 1, (3.2) Where; Φ(d 1 ) is defined in Equation (2.6). for call and put options are in the interval [0, 1] [-1, 0] respectively. If = 0 we have out-of-the-money options and if = 1 we have in-the-money options. If = 1/2 the options are at-the-money. As an option gets further in-the-money, the option s Delta increases and decreases when an option gets further out-ofthe-money. Gamma The Gamma Γ of an option is defined as the sensitivity of the Delta to the underlying. It measures the rate of change of Delta with respect to the underlying S such as Γ = S. Wilmott (2007) goes further to define Gamma Γ as a measure of how much or how often a position must be rehedged in order to maintain a delta-neutral position. Delta changes slowly if the absolute value of gamma is small. However as the absolute value of Gamma increases and becomes large, Delta becomes highly sensitive and to maintain the neutrality of the portfolio then we need frequent re-balancing. Gamma value tends to increase with time to maturity. Figure below shows the variation of Gamma with time to maturity from the top for atthe-money (ATM), out-of-the-money (OTM), and in-the-money(itm) options. Gamma value is large for the at-the-money options and decreases for both in-and out-of-the-money options as time to maturity decreases. Gamma Γ is always positive for for both call and put options. The Black-Scholes Γ of a European call or put option on a non-dividend-paying stock is given by: Γ = φ(d 1) Sσ T, (3.3) Where φ(d 1 ) is defined in Equation (2.8) 1 On the other hand the Delta for a short position in one call option is given by Φ(d 1 ), see Hull (2008) 2 The figure is extracted from Röman (2015) 23

27 Figure 3.1: This figure shows the variation of Gamma with time to maturity. The green line represents at-the-money (ATM) options, the black line represents out-of-the-money (OTM) options, and the red line represent in-the-money(itm) options. Vega The Vega υ of an option measures the rate of change of an option price V with respect to the implied volatility σ of the underlying asset. Vega is defined by υ = V σ, it simply measures the sensitivity of the price of an option to changes in volatility. This sensitivity depends on the absolute value of Vega, the higher the value of Vega, the more sensitive is the option s value to volatility changes and vice versa. An increase in volatility will increase the prices of all the options on an asset, and vice versa is true. The impact of volatility changes is greater for at-the-money options than it is for the in- or out-of-the-money options. The Black-Scholes υ of a European call or put option on a non-dividend-paying stock is given by: υ = S T φ(d 1 ), (3.4) Vega υ of a long position is always positive. Theta When the option is purchased, the one parameter that will definitely change is time, the amount on the time value remaining on an option starts to decrease while other things remain the same. Theta Θ of an option is a measure of the rate of change of the price of an option V with respect to time to maturity T. Theta is defined by Θ = V T. Θ always takes a negative sign, because as time to maturity decreases with all else being the same, the option tends to become less valuable. Figure 3.2 below which is extracted from Röman (2015) shows the variation of theta with time to maturity from the top for out-of-the-money (OTM), in-the-money(itm) and at-the-money (ATM) options. Theta value is large and negative for at-the-money options and gets lower for both in-and out-of-the-money options as time to maturity decreases. The 24

28 Figure 3.2: This figure shows the variation of theta with time to maturity. From the top, the red line represent out-of-the-money (OTM) options, the green line represents in-the-money(itm) options and the blue line represents at-the-money (ATM) options. Black-Scholes Θ of a European call on a non-dividend-paying stock is given by: Θ(c) = Sφ(d 1)σ 2 T rke rt Φ(d 2 ), (3.5) On the other hand Θ for a European put option is given by: Θ(p) = Sφ(d 1)σ 2 T + rke rt Φ( d 2 ), (3.6) where, all the parameters are defined in Equations (2.2), (2.6) and (2.7). Rho The Rho ρ of an option is a measure of the rate of change of price of an option V with respect to the interest rate r. It simply measures the sensitivity of the price of an option when only interest rate changes while all else remains constant. Rho is always positive and is defined by ρ = V r. The Rho value for options is maximum when in-the-money due to arbitrage activity with such options. As risk free interest rate increases, the option value also increases. The Black-Scholes ρ of a European call on a non-dividend-paying stock is given by: On the other hand ρ for a European put option is given by: ρ(c) = KTe rt Φ(d 2 ), (3.7) ρ(p) = KTe rt Φ( d 2 ). (3.8) 25

29 3.2 Greeks in Other Models In this section we will take a quick look at the way Greeks of an option are presented under other models apart from Black-Scholes model. We will present Greeks under Black model, Normal model and SABR model. We will limit our discussion to three major Greeks namely; delta, gamma Γ and vega Black Model Delta Delta for a call option is given by (c) = V F. (3.9) We substitute the value of the option from Equation (2.11) in Equation (3.9) We then apply differentiation we obtain (c) = [ ] (FΦ(d 1 ) KΦ(d 2 ), F = Φ(d 1 ) + FΦ (d 1 ) d 1 F KΦ (d 2 ) d 2 F = Φ(d 1 ) + d 1 F (Fφ(d 1) Kφ(d 2 ) (c) = Φ(d 1 ). (3.10) By the same analogy we obtain the Delta for a put option (p) to be (p) = Φ(d 1 ) 1. (3.11) Gamma Gamma for a European call option is defined by Γ(c) = (c) F. (3.12) We can substitute the delta for the call option from Equation (3.10) in Equation (3.12) to obtain Gamma Γ for the call option which is also similar to put option Γ(c) = Γ(p) = Φ(d 1) F = φ(d 1 ) d 1 F = φ(d 1) Fσ T. (3.13) 26

30 Vega Vega is defined by υ = V σ. (3.14) For a call option, we substitute the value of the option from Equation (2.11) in Equation (3.14) υ = [ ] (FΦ(d 1 ) KΦ(d 2 ) σ and we use differentiation to obtain Vega υ for the call option which is also similar to put option υ(c) = υ(p) = F T φ(d 1 ). (3.15) Normal Model Delta Delta for a call option is given by (c) = V F. (3.16) For a call option, we substitute the value of the option from Equation (2.28) in Equation (3.16) (c) = [ (F K)Φ(d) + σ ] T φ(d) F We then follow the same procedures we used in computing the Delta for the Black model to obtain delta (c) (c) = Φ(d). (3.17) Using the same analogy we obtain Delta (p) for a put option to be (p) = Φ(d) 1. (3.18) Gamma Gamma for a European call option is defined by Γ(c) = (c) F. (3.19) We can substitute Delta for the call option from Equation (3.17) in Equation (3.19) to obtain Gamma Γ for the call option which is also similar to put option Γ(c) = Γ(p) = φ(d) Φ(d) = F σ T. (3.20) 27

31 Vega Vega is defined by υ = V σ. (3.21) For a call option, we substitute the value of the option from equation (2.28) in equation (3.21) υ = [ (F K)Φ(d) + σ ] T φ(d) σ Using the same procedures as the ones we performed in computing the Vega of the Black model, we obtain a Vega υ for the call option which is also similar to put option υ(c) = υ(p) = T φ(d). (3.22) SABR Model Delta Assume the underlying forward rate F changes by F, then the SABR volatility α will also change on average by δ F α. This is because the two parameters are correlated. The Delta risk can be calculated from the following scenario: F F + F, α α + δ F α. (3.23) According to Bartlett (2006), to calculate the δ F α, we write the SABR dynamics in terms of independent Brownian motions W t and Z t, and obtain The change in the option value is given by Delta risk is then given by V = δ F α = ρν F β. (3.24) [ B F + B ( σ σ F + σ ρν )] α F β F. (3.25) = B F + B ( σ σ F + σ ρν ) α F β. (3.26) 28

32 Vega In this case we assume the SABR volatility α changes by α, because of this change then the underlying forward rate F also changes on average by δ α F. Vega risk can be calculated from the following scenario: F F + δ α F, α α + α. (3.27) Using the same procedures as for the delta risk we obtain δ α F to be, (See Bartlett (2006)). The change in the option value is given by Vega υ risk is then given by 3.3 Hedging Strategies δ α F = ρfβ ν α, (3.28) V = B ( σ σ α + σ ρf β ) α, (3.29) F ν υ = B ( σ σ α + σ ρf β ). (3.30) F ν The most common Greek letters that are use for hedging purposes are delta, gamma Γ and vega υ. In this section we will briefly discuss how these Greek letters are used for hedging 3 and how hedging is done in the practice. Delta Hedging Delta hedging can be defined as the use of Delta to eliminate all risk using an option and the underlying asset. Wilmott (2007) defines delta hedging as holding one of the option and short a quantity of the underlying. Delta hedging aims at keeping the investor s position to zero as close as possible to zero, a position known as delta neutral. Hull (2008) points out that for a European call option, delta hedging for a short position requires one to retain a long position of Φ(d 1 ) for each option sold. Likewise delta hedging for a long position requires retaining a short position of Φ(d 1 ) shares for each option purchased. The case is different for a European put option. In put option a long position is hedged with a long position in the underlying stock and short position is hedged with a short position in the underlying stock. Delta hedging is an example of dynamic hedging, in order to remain in a delta neutral position, a trader is required to do continual monitoring and periodic adjustments a process called rehedging or rebalancing the portifolio. Rehedging or rebalancing the portfolio is achieved by a 3 The discussion is based from Hull (2008) and Wilmott (2007) 29

33 sale or purchase of an underlying asset, a strategy which can be expensive due to transactions costs incurred on trade. However delta hedging strategy has been criticized not to be a perfect strategy and not to be able to perfectly hedge away the risk in the underlying. This is because of the high costs associated with this strategy and for a failure to have an accurate model for the underlying due to the risks which are always associated with the model. Gamma Hedging To encounter the problems of large costs and inaccurate model of the underlying which are common with the delta hedging strategy we can employ gamma-neutral strategy. Gamma hedging simply refers to hedging against changes in the hedge ratio. Wilmott (2007) defines this strategy as buying or selling more options, not just the underlying. Gamma hedging is used to reduce the size of each rehedge and/or to increase the time between rehedges. Gamma hedging is said to be more precise as it eliminates the effect of insensitivity of a delta hedged portfolio. If Γ p is the Gamma of the portfolio and Γ is the Gamma of a traded option, a position of Γ p /Γ in the traded option makes the portfolio vega neutral. For a portfolio to remain gamma neutral after a period of time, one needs to keep adjusting the position in the traded option to be Γ p /Γ. Vega Hedging Volatility of an underlying asset is a key parameter in determining the value of the contract. Vega hedging is used when a trader wants his portfolio to be insensitive to volatility. If υ p is the Gamma of the portfolio and υ is the Gamma of a traded option, a position of υ p /υ in the traded option makes the portfolio vega neutral. For a portfolio to be gamma and vega neutral, a hedger must use atleast two traded derivatives dependent on the underlying asset. For example an option will have zero Delta and Vega if hedged with both the underlying and another option. Hedging in Practice When it comes to the the actual practice of managing portfolio risk, portfolio rehedging is not normally done continuously. This is because the transaction costs involved in frequent rehedging or rebalancing are relatively expensive. So what the traders do is to analyze the individual risks in a portfolio in line with the risk limits set. Here is where Greeks are used to quantify various aspects of portfolio risk. If the trader finds out the risk to be acceptable no action is taken. If these risks exceed the limits then rehedging or rebalancing is carried out as discussed above. 30

34 3.3.1 Risk Matrices In the real world risk managers use different methods to measure and control the risk. One of the common methods employed is known as "Risk Matrices". Risk Matrices are used to measure the size of risk, control and report risks, see Röman (2015). He further states that "risk matrix is an outcome analysis of a scenario in which two risk factors are stressed at different intensities". The 2 factors which are stressed in these method are; the price of the underlying asset i.e. Delta and Gamma risk and the volatility i.e. vega risk. Figure 3.3: This figure shows an example of a risk matrix, it shows gains or losses when the 2 fractors; the volatilities and underlying prices are increased and decreased at different intervals. The matrix above in Figure 3.3 shows gains or losses when the 2 factors; the volatilities and underlying prices are increased or fluctuated within an interval of +/-5 percent and +/-0.5 percent respectively. In this thesis we will employ risk matrices to study how risk varies when stressed at different intensities. We decided to use risk matrices since it offers a very clear and comprehensible method to measure and report risk. Another reason for employing risk matrices is the fact that this thesis will only focus on 3 types of risks, which are Delta, Vega and Gamma. As pointed out earlier risks matrices measures these types of risk. 31

35 Chapter 4 Interest Rate Derivatives This chapter explores derivatives, specifically interest rate derivatives in order to give detailed background to the reader. Risk managers tend to use risk matrices to set limits for the level of the maximum acceptable loss. 4.1 Derivatives Definition 4.1. A derivative is a contract between a buyer and a seller entered into today regarding a transaction to be fulfilled at a future point in time. The value of a derivative normally derive from the price of an underlying asset which can be stocks, currencies, interest rates, indexes or commodities. This section will cover the derivatives whose underlying asset is an interest rate. A derivative can be used for a number of purposes including; taking position on the underlying asset, transferring or hedging risk, arbitrage between markets, and speculation. Parties involved in the derivative contract can trade specific financial risks embodied in the contract such as interest rate, credit, currency, equity and commodity price risk to other entities who want to manage these risks without trading in a primary asset or commodity. This can be done either by trading the contract itself, such as with options, or by creating a new contract which incorporates risk characteristics that nullifies those of the existing contract. Weber (2009) in his book gives a brief history of the derivatives markets. He points out that derivatives can be traced back to the origin of contracts for future delivery of commodities which originated in Mesopotamia and spread to the Roman world. This is evidenced by the fact that the first ever derivative contract was a contract of future delivery of commodities that were often combined with a loan. Upon the innovation of securities, derivatives started to be used in the security markets in Italy and the Low Countries during the Renaissance. The first derivatives on securities were written in the Low Countries in the sixteenth century. In the mid eighteenth century, derivative trading on securities went further to spread from Amsterdam to England, France and Germany. Today derivative market is one of the largest 32

36 market in the world with a notional amount of outstanding contracts amounting to $553 trillion at end June 2015 in the global OTC derivatives markets, see BIS (2016) 4.2 Interest Rate Derivatives Definition 4.2. An interest rate derivative is a contract to exchange payments based on different rates over a specified period of time. The level of interest rates determine the payoffs of these instruments, see Hull (2008). These instruments are mostly used for hedging and speculation against changes in interest rates. Interest rate derivatives are the most traded instruments in the global OTC derivative market today. According to BIS (2016), by end-june 2015, the interest rate derivatives contracts totalled $435 trillion which is 79 per cent of the market with swaps having the largest share amounting to $320 trillion. The most popular types of interest rate derivatives traded in the OTC market include: Forward Rate Agreements (FRA), Bond Options, Caps and Floors, Interest Rate Swaps and Swap Options. However, while we will define all these types of interest rate derivatives, we will put more emphasis on Swap and Swap Options as our thesis centers on them Forward rate agreement (FRA) Definition 4.3. A forward rate agreement (FRA) "is an over-the-counter agreement between two parties designed to ensure that a certain interest rate will apply to a prescribed principal over some specified period in the future", see Hull (2008). Hull (2008) points out that in this contract the assumption used is that the borrowing or lending would normally be done at LIBOR. FRAs are normally used to hedge future interest rate exposure. To elaborate FRAs, he considers a forward rate agreement where party A is agreeing to lend money to party B for the period of time between T 1 and T 2, and defines: R K is rate of interest agreed to in the FRA R F is the forward LIBOR interest rate for the period between times T 1 and T 2 calculated today R M is the he actual LIBOR interest rate observed in the market at time T 1 for the period between times T 1 and T 2 L is the principal underlying the contract R K, R F, R M are all assumed to be measured with a compounding frequency reflecting the length of the period to which they apply. From FRA we obtain the cashflows for Party A and 33

37 Party B at time T 2 to be given by Equation (4.1) and Equation (4.2) respectively. L(R K R M )(T 1 T 2 ), (4.1) L(R M R K )(T 1 T 2 ). (4.2) We can then interpret FRA as an agreement where party A will receive interest on the principal between T 1 and T 2 at the fixed rate of R K and pay interest at the realized LIBOR rate of R M. On the other hand party B will pay interest on the principal between T 1 and T 2 at the fixed rate of R K and receive interest at R M. To compute the payoff for Party A and Party B, we have to discount from time T 2 to T 1 since FRAs are settled at time T 1 rather than T 2. For Party A the payoff at time T 1 is given by for Party B, the payoff at time T 1 is given by L(R K R M )(T 1 T 2 ), (4.3) 1 + R M (T 1 T 2 ) L(R M R K )(T 1 T 2 ). (4.4) 1 + R M (T 1 T 2 ) Valuation of FRAs To value FRAs, we need to calculate the payoff assuming that R M = R F and then discount this payoff at the risk-free rate. Now if we consider 2 FRAs, where; 1. Promises that the LIBOR forward rate R F will be received on a principal of L between times T 1 and T 2 2. Promises that R K will be received on a principal of L between times T 1 and T 2 The present value of the difference between the interest payments of these 2 contracts is given by; L(R K R F )(T 2 T 1 )e R 2T 2, (4.5) where R 2 is the continuously compounded riskless zero rate for a maturity T 2. From equation 4.5 above we can compute the values of our FRAs. The value of our FRAs are given by; V FRA1 = 0, (4.6) V FRA2 = L(R K R F )(T 2 T 1 )e R 2T 2, (4.7) V FRA3 = L(R F R K )(T 2 T 1 )e R 2T 2. (4.8) The value V FRA1 of the first FRA when R F is received in Equation (4.6) is zero because we normally set R K = R F when the FRA is first initiated. Equation (4.7) gives the value of V FRA2 when R K is received and Equation (4.8) gives the value of V FRA3 when R K is paid. 34

38 4.2.2 Caps Definition 4.4. An interest rate cap is a series of caplets with a predetermined interest rate strike, expected to expire on the date when a floating rate is fixed again. In this contract, the purchaser of a cap has the right to exercise the option, but pays a premium to compensate the seller s risk. On the other hand, the seller agrees to compensate the buyer for the amount by which the floating rate exceeded the strike price during the contract period. Figure 4.1 below describes the payout from a Cap. From Figure 4.1 above, what happens is Figure 4.1: The payout from a Cap when the floating rate exceeds the strike-rate on each interest date, the current reference or floating rate is compared with the strike price. If the floating rate is lower than the strike price, no payment takes place. If the floating rate exceeds the strike price, the seller pays the difference. It ensures the holder of the cap that interest rate costs for his liabilities will be restricted to the agreed level. Interest rate caps are used by borrowers to hedge against the risk of paying very high interest rates on funds borrowed on a floating interest rate basis. Valuation of Caps When we have an cap with a total life of T, a principal of L, a cap rate of R K, reset dates t 1, t 2,...,t n. When T is defined as T =t n+1 and R k defined as the LIBOR interest rate for the period between time t k and t k+1 observed at time t k. The payoff of such a cap at time t k+1 (k=1,2,..,n) is given by Lδ k max(r k R K,0), (4.9) where δ k = t k+1 t k. The standard market model gives the value of the caplet as Lδ k P(0,t k+1 )[F k Φ(d 1 ) R K Φ(d 2 )], (4.10) where d 1 = ln(f k/r K ) + δ 2 k t k/2 δ k tk, (4.11) 35

39 d 2 = ln(f k/r K ) δ 2 k t k/2 δ k tk = d 1 δ k tk. (4.12) Φ(x) in Equation (2.3), F k is the forward interest rate at time zero for the period between time t k and t k+1, δ k is the volatility of this forward interest rate and P(0,t k+1 ) is the discount factor Floors Definition 4.5. An interest rate floor is a series of floorlets with a predetermined interest rate strike, which will expire on the date when a floating rate will be fixed again. In this contract, the buyer of a floor has the right to exercise the option. On the other hand, in exchange of premium, the seller agrees to compensate the purchaser if the floating rate is below the strike price during the contract period. Figure 4.2 below illustrates the payout of a floor. From the figure above, on each interest date, the reference or floating rate is compared Figure 4.2: The payout from a Floor when the floating rate falls below the strike-rate with the strike price, if the floating rate is higher than the strike price, no payment takes place. If the floating rate is lower than the strike price, then the seller pays the difference. Floor is similar to a Cap except that it is designed to hedge against the downside risk. Interest rate floors are used by lenders to hedge against the risk of receiving very low interest rates on funds lent on a floating interest rate basis. Valuation of Floors When we define a floor like the cap in the above section, the payoff of this floor at time t k+1 (k=1,2,..,n) will be given by The standard market model also gives the value of the floorlet as all the variables are described in Equation (4.10). Lδ k max(r K R k,0). (4.13) Lδ k P(0,t k+1 )[R K Φ( d 2 ) F k Φ( d 1 )], (4.14) 36

40 4.2.4 Bond Options Definition 4.6. A bond option is an option to buy or sell a bond at particular date and price. In order to make bond options more appealing to either the issuer or the purchasers, bond options are frequently embedded in bonds upon being issued. Examples of embedded bond options include a collable bond and a puttable bond. A callable bond is the one where an issuer can buy back the issued bond in the future at a predetermined price. In such contract the holder of the bond has sold a call option to the issuer in exchange of the predetermined price (which is the strike or call price.) However such bonds have what is known as a "lock-out period", which restricts an issuer from calling the bond on the first few years of their life. On the other hand a puttable bond is the one where the holder of the bond can ask for an early redemption at a predetermined price. In such contract the holder of the bond has purchased a put option on the bond as well as the bond itself, see Hull (2008). Valuation of Bond Options In this case we are only going to consider the case of a European bond options. By setting F 0 = F B, in Equation 4.11 and Equation 4.12 we can use Black model to obtain the value of the bond option as follows; where C = P(0,T )[F B Φ(d 1 ) KΦ(d 2 )], (4.15) P = P(0,T )[KΦ( d 2 ) F B Φ( d 1 )], (4.16) d 1 = ln[f 0/K] + (σ 2 /2)(T ) σ, (4.17) T d 2 = ln[f B/K] + (σ 2 B /2)(T ) σ B T = d 1 σ B T, (4.18) K is the strike price 1 of the bond option, T its time to maturity and F B is the forward bond price that can be calculated as F B = B 0 I P(0,T ), (4.19) where B 0 is the bond price at time zero, I is the present value of the coupons that will be paid during the life of the option Interest Rate Swap Definition 4.7. An interest rate swap is a contract where two parties agree to exchange streams of interest rate cash flows from a predetermined fixed rate for some specific period, based on a specified notional principal. 1 The strike price here should be the cash strike price also referred to as dirty price 37

41 When investor A makes a fixed rate payments on the specified notional principal to investor B, investor B in return makes floating rate payment to investor A for the same period of time. The vice versa is true. Traders use swaps for many different purposes including; hedging interestrate risk, changing a liability either from floating-rate loan to a fixed loan or otherwise. It can also be used to change the nature of an asset either from an asset earning a fixed rate interest into an asset earning a floating rate of interest or otherwise. The most common floating rate used in a swap agreement is the LIBOR, however in Sweden the floating rate used is the STIBOR. These are arleady discussed in Chapter one. Relationship between Swaps and Bonds As explained earlier a swap has two sides which are the fixed-rate side and the floating rate side. The value of all fixed interest rate payments at time t in a swap can be expressed as a sum of zero-coupon bonds such as r f ix N i=1p(t,t i ), (4.20) where r f ix is the fixed rate of interest, N is the number of payments one at each T i On the other hand the floating side of the swap has value 1 P(t,T N ). (4.21) Valuation of Interest Rate Swaps We can view the value of a swap as an equivalent to a portfolio of two bonds, a fixed-rate bond and a floating-rate bond. From a floating-rate payer view, a swap is regarded as a long position in fixed-rate bond and a short position in floating-rate bond such as; V swap = B f ix B f l. (4.22) From a fixed-rate payer this relationship can be expressed as a long position in floating-rate bond and a short position in fixed-rate bond such as; V swap = B f l B f ix, (4.23) where B f ix is the fixed-rate bond and B f l is the floating-rate bond. If we define B f ix as in Equation (4.20) and B f l as in Equation (4.21), we have the value of the swap as V swap = r f ix N i=1p(t,t i ) 1 + P(t,T N ). (4.24) Using Equation (4.19), we can obtain the quoted Swap rate as r f ix = 1 P(t,T N). (4.25) P(t,T i ) N i=1 38

42 Bootstrapping with Swaps Definition 4.8. Bootstrapping is a procedure or method used to calculate the zero coupon curve from the given market data. With this technique, we strip the bonds to create virtual zero coupon bonds of the coupons and the principal. Since we do not always have zero coupon bonds offered in the market, bootstrapping method is used to fill in the missing figures in order to derive the zero coupon curve. The bootstrap method uses interpolation to determine the spot rates for zero coupon securities with various maturities. It is important to cover the concept of bootstrapping with swaps since we will use this concept when we are doing our implementation. Swaps can be used to determine the yield curve, see Röman (2015). The zero coupon rate can be derived from the par Swap rate by means of bootstrapping. Given r f ix (T i ) for many maturities T i we can use the formula in Equation (4.25) to calculate the prices of zero-coupon bonds and thus the yield curve. At the first point on the discount-factor curve we use formula in Equation (4.25) to obtain the zero coupon bond as follows r f ix (T 1 ) = 1 P(t,T 1), (4.26) P(t,T 1 ) P(t,T 1 ) = After finding the first k discount factors the k + 1 is found from r f ix (T 1 ). (4.27) then P(t,T k ) = 1 + r f ix (T i ) k i=1 P(t,T i ), (4.28) 1 r f ix (T k+1 ) k P(t,T i ) i=1 P(t,T k+1 ) = 1 + r f ix (T k+1 ). (4.29) Procedures to Bootstrapping a Swap curve When bootstrapping a zero-coupon curve we use liquid instruments, the procedures are as follows. 1. Cash Deposit Rates: The first part of the yield curve is built using the cash deposits quoted from the Swedish market with maturities on over-night rate (O/N), a tomorrownext rate (T/N), one week, one, two and three month. We use the formula in Equation (4.27) to start calculating the discount factor by using O/N, such as D O/N = r par O/N. d O/N 360, (4.30) 39

43 we then obtain the zero rate as Z O/N = 100. ln(d O/N). (4.31) d O/N 365 Zero rates are normally given as continuous compounding, Act/365. After O/N we turn to T/N and then we proceed with the money-market instruments (1W,1M,2M and 3M) using the same formulas we calculate the discount factor and the zero rate for each of them. 2. Forward Rate Agreement (FRA): We then proceeded with the Short Future, here we use OMX STIBOR Forward Rate Agreements with maturities on IMM days. The IMM futures contracts are are available for the months March, June, September and December. Here we need to use a stub rate since these are forward contracts quoted in forward rates. The stub rate which can be found using linear interpolation will have maturity which is the same date as the start date of the first FRA contract. The stub discount factor 2 is given by ( D Stub = exp Z Stub. T Stub 365 then the discount factor of the FRA rate can be obtained from D i FRA = D i 1 FRA 1 + r i FRA. di FRA 360 where D 0 FRA = D Stub. We can then obtain our zero rate as ), (4.32), (4.33) ZFRA(T i ) = 100. ln(di FRA ). (4.34) dfra i 365 In most markets the FRA contracts are commonly quoted in clean price, however in the Swedish market all instruments are quoted in yield. If FRA contracts are quoted in price, the discount factor is given by D FRA = D Stub (t) 1 + ( 100 PFRA 100 ), (4.35) where P FRA is the quoted price of the FRA contract and 91 are the days between the two IMM dates. 3. Swap Rates: When we move away from the spot date we either run out of the futures contract or the futures contract become unsuitable due to lack of liquidity. Therefore to generate the yield curve we need to use the next most liquid instrument, which is the swap rates. We use Equation (4.25) to obtain swap rates. For the years where we are missing the swap rate we use linear extrapolation to compute the zero rate and then we proceed to calculate the discount factors using similar formulas as in cash deposits and FRAs. 2 It is important to note that whenever we calculate the discount factor we use 360 days in a year while we use 365 days for the other calculations 40

44 4.2.6 Swaption Definition 4.9. A swaption is an option to enter into a swap, which gives the holder the right to buy or sell an underlying asset on the expiry date at a fixed strike price. A swaption has combined features of both a swap and an option. When the holder has the right but not an obligation to enter into a Swap at a given date, paying the fixed rate in exchange of floating rate, we have what is called a payer swaption. On the other hand a receiver swaption gives the holder the right to enter into a swap, receiving fixed rate in exchange of paying a floating rate. Like stock options, swaptions can either be European or American. European swaption can only be exercised on the expiration date. On the other hand American swaption can only be exercised on cash flow dates unlike American stock options which can be exercised at any time during the life of the option. A special American swaption that can only be exercised at reset dates is called a Bermudan swaption. Valuation of European Swaption The model used to value a European option on a swap assumes that the underlying swap rate at the maturity of the option is lognormal. Consider a swaption where the holder has the right to pay a rate s K and receive LIBOR on a swap that will last n years starting in T years. For this case we assume that there are m payments per year under the swap and that the L is the notional principal. Now suppose that the swap rate for an n-year swap starting at time T is s T. By comparing the cashflows on a swap where the fixed rate is s T, to the cash flows on a swap where the fixed rate is s K, we obtain the payoff from a swaption to consists of cashflows equal to L m max(s T s K,0). The cashflows are received m times per year for the n years of the life of the swap. If we assume that the swap payment dates are T 1, T 2,..,T mn measured in years from today, It is then approximately true that T i = T + m i. Each cash flow represents the payoff from a call option on s T with strike price s K. The standard market model gives the value of the payer swaption V PS by summing up all individual call options where V PS = mn L i=1 m P(0,T i)[s 0 Φ(d 1 ) s K Φ(d 2 )], (4.36) d 1 = ln(s 0/s K ) + σ 2 T /2 σ, (4.37) T d 2 = ln(s 0/s K ) σ 2 T /2 σ T = d 1 σ T, (4.38) 41

45 mn P(0,T i ) is the discount term and Φ(x) as defined in Equation (2.3). i=1 By introducing A, we can reduce Equation (4.37) into V PS = LA[s 0 Φ(d 1 ) s K Φ(d 2 )]. (4.39) A is the value of the contract that pays 1/m at times T i and is defined as A = 1 m mn i=1 by analogy, for the European receiver swaption, obtaining a payoff of P(0,T i ), (4.40) L m max(s K s T,0), (4.41) The standard market model gives the value of the swaption V RS as V RS = LA[s K Φ( d 2 ) s 0 Φ( d 1 )], (4.42) with A defined as in equation (4.41) and d 1, d 2 as in Equation (4.38) and Equation (4.39) Hedge Parameters in European Swaption In the previous chapter we covered in detail various hedging parameters. Since we will later use our data to compute Greeks for swaption in different models, it is important to take a look at how these Greeks can be presented with respect to swaptions. Delta From the previous discussion we know that is the rate of change in value V of an option with respect to the price of the underlying S. For the case of swaption our underlying is the fixed swap rate s K. Delta for the payer swaption PS is given by on the other hand, Delta for the receiver swaption RS is given by all parameters are defined as in equation (4.43). PS = LAΦ(d 1 ), (4.43) RS = LAΦ(d 1 ) 1, (4.44) 42

46 Gamma Gamma Γ for the swaption is similar for both payer and receiver swaption through differentiation we obtain: Γ PS = Γ RS = LAφ(d 1) s K σ T, (4.45) φ(x) as in Equation (2.2) and other parameters as in Equation (4.43). Vega We also use differentiation to obtain Vega υ for the swaption which is similar for both payer and receiver swaption. υ PS = υ RS = LAs K T φ(d1 ), (4.46) parameters here are defined as in Equation (4.46) 43

47 Chapter 5 Implementation In this chapter we put into practical all the theory assembled so far. We present the procedures we followed to perform these and the corresponding results. The tests will be performed by the help of Excel VBA programming language Bootstrapping a Swap Curve The first thing we did was to bootstrap a zero-coupon curve which required us to use liquid instruments as can be seen in Figure 5.1 below. To bootstrap we followed the procedures discussed in the previous chapter. We first used the cash deposits quoted from the Swedish market with maturities on over-night rate (O/N), a tomorrow-next rate (T/N), one week, one, two and three month. We then proceeded with the Short Future were we used OMX STIBOR Forward Rate Agreements with maturities on IMM days instead of shorter Swaps because FRA are more liquid. Finally we used Swap rates from SEK STIBOR A 3M from year 4 to year 30. Figure 5.1: This figure shows liquid instruments; Cash deposits, Short future and the Swap rates which were used for bootstrapping the Swap curve. 1 This VBA codes were programmed originally by Jan Roman 44

48 From bootstrapping we obtained the following curves Figure 5.2: This figure shows three different curves obtained as a result of bootstrapping namely: Zero Rate Curve, Forward Rate Curve and Discount Rate Curve 5.2 Premium and Risk Measures Calculations We computed the premium (prices) and risk measures, i.e. Delta, Gamma and Vega of the "at the money (ATM) receiver swaption of different maturities. In this section we present the results of our computations and compare to see how premium and risk measures vary between the Normal model and the Black model. From Figure 5.3 and Figure 5.4 below we can see that for both models we obtained the same premium or price i.e for a 10y x 10y ATM receiver swaption and for 5y x 10y ATM receiver swaption. On the other hand this is not the case for the risk measures. From Figure 5.3 and Figure 5.4 we can also see that the value of Delta, Gamma and Vega vary form one model to the other. We also noted that Delta varies more significantly between the models than vega and gamma. These results coincide with several authors such as Henrard (2005) and Rebonato (2004) who concluded that the use of either log-normal or normal model will yield identical prices but different risk measues. Another interesting thing observed was higher values for delta and vega for a receiver swaption in the black model compared to the normal model. On the other hand normal model has higher gamma values compared to the black model 45

49 Figure 5.3: Premium and Risk Measures of a 10y x 10y ATM receiver swaption Figure 5.4: Premium and Risk Measures of a 5y x 10y ATM receiver swaption Plotting Delta and Vega in Black and Normal Black Model We also went further to analyse and plot how risk measures i.e. Delta and Vega vary between the Black model and the Normal Black model. To make this analysis we considered 3 different scenarios as seen in Figure 5.5. The first scenario was when the forward rate was set to be equal to the strike price (also known as ATM). The second scenario was when the forward rate was set to be less than the strike price (also known as OTM). The third scenario was when the forward rate was set to be greater than the strike price (also known as ITM). Figure 5.5: Different Scenarios Figure 5.6 below demonstrates the first scenario. When the Black volatility increases, we notice a sharp decline of the Black vega curve while the curve of the Normal vega remains constant. For the Delta, the Black delta curve increases while the curve of the Normal delta remains constant. 46

50 Figure 5.6: Black Vega(Delta) Vs Normal Vega(Delta) when F = K Figure 5.7 below demonstrates the second scenario. When the Black volatility increases, the curves of the Black vega and Normal vega move in opposite directions. For the Delta, the Black delta curve tends to increase. Figure 5.7: Black Vega(Delta) Vs Normal Vega(Delta) when F < K Figure 5.8 below demonstrates the third scenario. When the Black volatility increases, we notice that the curve of the Black vega tends to decrease while that of the Normal vega remains constant. For Delta, we notice that the two curves are moving in the opposite direction. 47

51 Figure 5.8: Black Vega(Delta) Vs Normal Vega(Delta) when F > K 5.3 Risk Matrices for Swaptions and Swaps at Different Maturities In the real world risk managers or traders use risk matrices to measure, control and report risks. Risk matrices are used to show different scenarios such as the best or worst case scenario that a trader can be exposed to. In this thesis we used the same concept of risk matrices to study how risk varies between the Black Model and the Normal Black Model. However instead of studying an entire portfolio of interest rate derivatives we only looked at the case of swaptions and swaps with different maturities. We created various risk matrices for both Black model and Normal Black model which included the price matrices, Delta and Vega matrices to study how swaptions and swaps with different maturities are sensitive to changes in different parameters Price Sensitivity Here we analyze the price sensitivity for both the Black model and the Normal model when the Black and Normal volatility is varied or changed with respect to forward rates. With the change of volatility, the prices also changes. For Black model we start with 30 percent for black volatility while for Normal model we start with the corresponding normal volatility of 1.15 percent.thereafter to see the sensitivity we varied the volatility up and down by a certain percentage. From Figure 5.9 below we notice that for both models every time we increase our volatility from our starting point were we have zeros we obtain more profit. For the Black model we have a maximum profit of while for the Normal model we have a maximum profit of On the other hand when we decrease our volatility from our starting point we obtain less profit or loss. For the Black model the maximum loss is while for the Normal model we have a maximum loss of

52 Figure 5.9: Black Vs Normal Price Matrix of a 10y x 10y receiver swaption When we compute the price matrices of a receiver swaption with different maturities, see Appendix B.2, we observe a similar pattern for both models. Every time we increase our volatility from our starting point were we have zeros we obtain more profit. On the other hand when we decrease our volatility from our starting point we obtain less profit or loss Delta and Vega Sensitivity At this point we analyze the sensitivity in the change of Delta and Vega for both the Black model and the Normal model when the Black and Normal volatility is varied or changed with respect to strike rates. We use the same procedure as the one used when we calculated the price matrices. In this section we only present the results of a receiver swaption with maturity 10y x 10y. Other matrices with different maturities can be found in the appendix Figure 5.10: Black Vega vs Black Delta Matrix of a 10y x 10y receiver swaption 49

53 Figure 5.10 above shows the Black delta vs Black vega matrices of a 10y x 10y receiver swaption. The 2 matrices also show how the changes in Delta and Vega in a Black model move in opposite directions. For the delta matrix, at ATM when you increase the Black volatility we tend to have positive delta changes while the decrease of Black volatility results into negative delta changes. On the other hand for the vega matrix when normal volatility is increased at ATM we tend to have negative vega changes while the decrease in normal volatility results into positive delta change. Figure 5.11: Normal Vega vs Normal Delta Matrix of a 10y x 10y receiver swaption Figure 5.11 above shows the Normal delta vs Normal vega matrices of a 10y x 10y receiver swaption. The 2 matrices also show how the changes in Delta and Vega in a Normal model move in opposite directions. For the delta matrix, at ATM when you increase or decrease the normal volatility we tend to have zero delta changes. On the other hand for the vega matrix, at ATM when normal volatility is increased or decreased we also obtain the same results as in delta matrix. 50

54 Chapter 6 Conclusion In this paper we have showed how risk varies between the Black model and the Normal Black model. We employed risk matrices method to study how risk varies between these two models. We also computed and compared premium(price) delta, gamma and vega for the two models and plotted how delta and vega differs between these 2 models. In Chapter 5 we have presented the implementation of the emperical test for the swaptions. Throughout this chapter we saw how risk vary between the Black model and the Normal model. We saw both from Figure 5.3 and Figure 5.4 that both models yield identical prices but different risk measures (hedges). The curves in Figures 5.6, 5.7 and 5.8 further plots the difference between vega and delta in these two model. Delta and Vega seemed to have higher values in the Black model than in the Normal models. On the other hand Gamma seemed to have higher values in the Normal model than in Black model. To this end it is our proposition that for risk managers concerned about pricing interest rate derivatives, that a rich literature exists which conclude that with the current change in interest rate environment it is vital to move from Black model to either Normal model or to Shifted Black model. When you compare the two alternatives, Normal model appears to be a better choice for modelling and measuring risk compared to the Shifted Black model. This is because with a Normal model you can easily obtain an analytic expressions for the valuation formula and its hedges. Furthermore one doesn t need to introduce an additional (shift) parameter like in Shifted Black model. However as it has been shown in our results, when one changes from one model to the other, hedges or risk metrics also change. Hence risk managers need to be careful when hedging their portfolios. Since our thesis was limited to only swaptions and swaps as an example of interest rate derivatives, further studies in this topic may include looking at other interest derivatives such as caps and floors. Since this thesis was focused on Black model and Normal model as examples of lognormal and normal model, another area for further research would be to look at other types models e.g. SABR model, CEV model which we discussed earlier in Chapter 2 of this thesis. 51

55 Chapter 7 Notes on fulfillment of Thesis objectives This section provides an analysis on how the requirements of the Swedish National Agency for Higher Education to master theses have been fulfilled in this thesis. Objective 1: Knowledge and understanding in the major field The author started this thesis by introducing to the readers the concept of negative interest rate environment, sighting specific examples of different central banks which have adopted the negative interest rates and the rationale behind their move. The author then goes further to introduce interest rate derivatives and how they perform in the current market. In this thesis the author also discusses log-normal and normal models in use today and how these models are used in hedging risk. Objective 2: Deep methodological knowledge In this thesis the author used tools from financial mathematics to formulate a mathematical problem from a financial problem. The author shows a logical framework within which the problem is solved. Sections before the model implementation in chapter 5 described the theory that required for hedging swaptions in a negative interest rate environment. Objective 3: Critical thinking and systematic integration of knoledge Here the author had to refer to different relevant sources which included; different papers in the field of financial mathematics and economics, different text books, other students thesis and lecture notes. The author went further to use primary data from one of the trading system at Swedbank, Murex, Mx3 to implement his work. Objective 4: Ability to critically, independently and creatively formulate issues with a plan to carry out advanced tasks In order to accomplish this thesis within a required time frame, the author had to make a project plan which specified specific time line for each phase of the project. In the implementation chapter (chapter 5), the author uses the help of Excel VBA programming language as a practical application of the theory advanced in the thesis. Objective 5: Knowledge presentation in both national and international contexts 52

56 In this thesis, the author was keen to present his work using the language that could be understood by any reader with a financial mathematics background. In the presentation of this thesis (June 4 th 2015) the author was able to logically argue his conclusion and answered questions from the audience. Furthermore, the author provided an investment recommendation as well as potential further work in the field. Objective 6: Ability to make judgement considering the relevant scientific, social and ethical aspects As stated earlier, it is clear that the topic on pricing interest rate derivatives under negative interest rate environment is quit new and hot. Few research has been done in this area so far. The author believes that this work will be helpful in adding knowledge in this new area and trigger interests for further research. Finally to ensure ethics aspect, the author was keen give due credit to the ideas, work, research and findings of other authors from different sources he used in this thesis. 53

57 Bibliography Antonov, A., Konikov, M. and Spector, M. (2015). The Free Boundary SABR: Natural Extension to Negative Rates, Available at SSRN Bachelier, L. (1900). Théorie de la spéculation, Gauthier-Villars. Bartlett, B. (2006). Hedging under SABR Model, Wilmott magazine 4: 2 4. Bech, M. L. and Malkhozov, A. (2016). How have central Banks Implemented Negative Policy Rates?, BIS Quarterly Review March pp BIS (2016). Statistical release OTC derivatives statistics at end-december 2015, BIS Quarterly Review pp Black, F. (1976). 3(1): The Pricing of Commodity Contracts, Journal of Financial Economics Black, F. (1995). Interest Rates as Options, The Journal of Finance 50(5): Black, F. and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities, The Journal of Political Economy pp Coleman, T. F., Kim, Y., Li, Y. and Verma, A. (2003). Dynamic Hedging in A Volatile Market, Technical report, Cornell University. Cox, J. C. and Ross, S. A. (1976). The Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics 3(1-2): Frankena, L. H. (2016). Pricing and Hedging Options in A Negative Interest Rate Environment, PhD Thesis, PhD thesis, TU Delft, Delft University of Technology. Hagan, P. S., Kumar, D., Lesniewski, A. S. and Woodward, D. E. (2002). Managing Smile Risk, The Best of Wilmott pp Hagan, P. S. and Woodward, D. E. (1999). Equivalent Black volatilities, Applied Mathematical Finance 6(3): Henrard, M. (2005). Swaptions: 1 Price, 10 Deltas, and /2Gammas, Technical report, Derivatives Group, Banking Department, Bank for International Settlements, Basel Switzerland. 54

58 Hull, J. C. (2008). Options, Futures and Other Derivatives, Seventh edn, Pearson Prentice Hall. Jackson, H. (2015). The International Experience with Negative Policy Rates, Bank of Canada Staff Discussion Paper ( ): Kijima, M. (2013). Stochastic Processes with Applications to Finance, Second edn, CRC Press. Menner, M. (2011). "Gesell Tax" and Efficiency of Monetary Exchange, Working Papers. Serie AD, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie). URL: Rebonato, R. (2004). Volatility and Correlation, Second edn, John Wiley & Sons. Röman, J. (2015). Lecture Notes for Analytical Finance II, Mälardalen University, Sweden. Schroder, M. (1989). Computing The Constant Elasticity of Variance Option Pricing Formula, The Journal of Finance 44(1): Walck, C. (2007). Hand-Book on Statistical Distributions for Experimentalists, Technical report, University of Stockholm. Weber, E. J. (2009). A Short History of Derivative Security Markets, Vinzenz Bronzins Option Pricing Models, Springer, pp Wilmott, P. (2007). Paul Wilmott Introduces Quantitative Finance, John Wiley & Sons. Zhang, N. (2011). Properties of The SABR Model, Technical report, Working Paper, Department of Mathematics, Uppasala University. Available at divaportal. org/smash/get/diva2: /FULLTEXT01. 55

59 Appendix A More mathematics A.1 Singular Pertubation Technique In this section of the appendix we will show how singular perbutation technique can be used to analyze models and find an explicit expressions for the values of European option. This is based on Hagan and Woodward (1999). Consider a European call expiring at date t i and its to be settled at date t j, with strike K. We also let F(t) be a stochastic process for the forward price at date t. Assuming under the forward measure we have df = α t AFdW, (A.1) under this measure the value of the option at date t is given by V (t,f t ), where V (t, f ) is given by the expected value V (t, f ) = P(t,t j )E[(F ti K) + F t = f ], (A.2) where P(t,t j ) is the discount factor to the settlement date t j at date t. When we strip the discount factor from V we obtain We can define Q(t, f ) as V (t, f ) = P(t,t j )Q(t, f ). Q(t, f ) = E[(F ti K) + F t = f ]. The expectation is over the probability distribution generated by the process df t = α t AFdW t, (A.3) (A.4) (A.5) therefore Q(t, f ) for t < t i satisfies the backward Kolmogorov equation. (Here we skipped several steps to arrive to the equation below) att = t i we have a final condition Q t α2 t A 2 f Q f f = 0, Q = E[ f K] +. (A.6) (A.7) 56

60 A.2 Solution to Black Model The derivation is based from Frankena (2016). When we define transformation Y t = logf t on a log-normal SDE, according to Ito s formula we obtain: dy t = 1 F t df t Ft 2 (df t ) 2 (A.8) = σdw t 1 2 σ 2 dt Y t = Y 0 + σw t σ 2 t /2. (A.9) This implies that Y t N (Y 0 σ 2 t /2,σ 2 t ) and that F t is log-normally distributed: F t = exp(y t ) = F 0 exp(σw t σ 2 t ). (A.10) The forward F t is a martingale under Black s symbol, since E[F 0 F t ] = exp((log(f 0 ) σ 2 t/2) + (σ 2 t)/2) = F 0. (A.11) Assuming a deterministic zero coupon bond P(0,t), the value V c (0) of a European-type call option at time 0 under Black s model is given by = ( Φ [ 1 Φ V c (0) P(0,t) = E[F t K,0 F 0 ] = E[F t K F t > K]P(F t > K) + 0 P(F t < K) ( logk (Y0 σ 2 )] [ (( t/2) σ exp Y 0 σ 2 ) t + σ 2 ) t t 2 2 ) ( /Φ logk (Y 0 σ 2 t/2) σ t logk (Y 0 σ 2 t/2) σ t ( σ t = exp(y 0 )Φ 2 logk Y 0 σ t where d 1 σ t 2 + log(f 0/K) σ and d t 2 d 1 σ t. σ t ) ( KΦ σ t 2 logk Y 0 σ t ) K ) ] (A.12) = F 0 Φ(d 1 ) KΦ(d 2 ). (A.13) By the pull-call parity the value V p (0) of a European-type put option at time 0 under Black s model is V p (0) = V c (0) P(0,t)(F 0 K) (A.14) 57

61 = P(0,t)[F 0 (Φ(d 1 ) 1) K(Φ(d 2 ) 1] = P(0,t)[K(Φ( d 2 ) F 0 Φ( d 1 )]. 58

62 Appendix B APPENDIX B B.1 Extract of VBA Program Codes Below we present a sample of codes that was used to compute the Black normal model ## Base function for the normal Black model (C = Call, P = Put options) ## For all functions below, the following is used. ## The SwapRate, StrikeRate, r and vol is given in %. ## The SwapTenor and SwaptionMaturity are given in years. ## F is the frequency, i.e., the number of cash-flows per year ## N (the face value, notional) is not used (use this outside this function) ## Calculating Annuity Function Annuity(SwapRate As Double, Tenor As Double, F As Double) As Double Annuity = (1-1 / Pow(1 + SwapRate / F, F * Tenor)) / SwapRate End Function Function BlackNormalC(SwapRate As Double, StrikeRate As Double, maturity as Double, r As Double, vol As Double) As Double Dim d1 As Double, d2 As Double, nd1 As Double d1 = (SwapRate - StrikeRate) / (vol * Sqr(maturity)) d2 = vol * Sqr(maturity / (2 * )) * Exp(-d1 * d1 / 2) nd1 = CND(d1) BlackNormalC = Exp(-r * maturity) * ((SwapRate - StrikeRate) * nd1 + d2) End Function Function BlackNormalP(SwapRate As Double, StrikeRate As Double, 59

63 maturity As_ Double, r As Double, vol As Double) As Double Dim d1 As Double, d2 As Double,_ nd1 As Double d1 = (SwapRate - StrikeRate) / (vol * Sqr(maturity)) d2 = vol * Sqr(maturity / (2 * )) * Exp(-d1 * d1 / 2) 33 nd1 = CND(-d1) BlackNormalP = Exp(-r * maturity) * ((StrikeRate - SwapRate) * nd1 + d2) End Function

64 B.2 Risk Matrices for Swaptions and Swaps with Different Maturities The Price Matrix for tenor (5,10) Figure B.1: Black Vs Normal Price Matrix for tenor (5,10) Risk Measures for tenor (5,10) Figure B.2: Black Vega vs Black Delta Matrix for tenor (5,10) 61

65 Figure B.3: Normal Vega vs Normal Delta Matrix for tenor (5,10) Figure B.4: Black Vs Normal Price Matrix for tenor (3,10) Figure B.5: Black Vega vs Black Delta Matrix for tenor (3,10) 62

66 Figure B.6: Normal Vega vs Normal Delta Matrix for tenor (3,10) Figure B.7: Black Vs Normal Price Matrix for tenor (3,5) Figure B.8: Black Vega vs Black Delta Matrix for tenor (3,5) 63

67 Figure B.9: Normal Vega vs Normal Delta Matrix for tenor (3,5) Figure B.10: Black Vs Normal Price Matrix for tenor (3,3) Figure B.11: Black Vega vs Black Delta Matrix for tenor (3,3) 64

68 Figure B.12: Normal Vega vs Normal Delta Matrix for tenor (3,3) Figure B.13: Black Vs Normal Price Matrix for tenor (2,10) Figure B.14: Black Vega vs Black Delta Matrix for tenor (2,10) 65

69 Figure B.15: Normal Vega vs Normal Delta Matrix for tenor (2,10) Figure B.16: Black Vs Normal Price Matrix for tenor (2,5) Figure B.17: Black Vega vs Black Delta Matrix for tenor (2,5) 66

70 Figure B.18: Normal Vega vs Normal Delta Matrix for tenor (2,5) Figure B.19: Black Vs Normal Price Matrix for tenor (1,5) Figure B.20: Black Vega vs Black Delta Matrix for tenor (1,5) 67

71 Figure B.21: Normal Vega vs Normal Delta Matrix for tenor (1,5) Figure B.22: Black Vs Normal Price Matrix for tenor (1,10) Figure B.23: Black Vega vs Black Delta Matrix for tenor (1,10) 68

72 Figure B.24: Normal Vega vs Normal Delta Matrix for tenor (1,10) Figure B.25: Black Vs Normal Price Matrix of a 5y x 5y receiver swaption Figure B.26: Black Vega vs Black Delta Matrix of a 5y x 5y receiver swaption 69

73 Figure B.27: Normal Vega vs Normal Delta Matrix of a 5y x 5y receiver swaption 70