Hedging Interest Rate Derivatives (Evidence from Swaptions) in a Negative Interest Rate Environment: : A comparative analysis of Lognormal and Normal Model

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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-721 23 VÄSTERÅS, SWEDEN

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Master thesis in mathematics / applied mathematics Date:

2017-01-19 Project name:

Hedging Interest Rate Derivatives (Evidence from Swaptions) in a Negative Interest Rate En-vironment: 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

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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.

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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 scholar-ship 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.

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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 20161central 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:}

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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 below2. 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 2016.

1.2 Motivation for Negative Interest Rate Policies

There are different motivations for implementing negative interest rate policy. Bech and Malk-hozov (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 pre-vent 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 13 below summarizes the rationale behind implementation of negative rate policy by different banks in Europe.

2_{The figure is extracted from https://www.bloomberg.com/quicktake/negative-interest-rates}

3_{The table is extracted from Jackson (2015). In the table; bp stands for basis points and DKK is the ISO code}

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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.

1.3.1 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.

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The need to change models

Black’s model has been used as the standard model in the market to price interest rate de-rivatives. 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 neg-ative 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).

1.3.2 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

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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 risk4. 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 x nx . (1.2)

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 eRnthat is an amount P invested for n years at rate R grows to

PeRn. (1.3)

If we suppose that Rcis a rate of interest with continuous compounding and Rxis the equivalent

rate with compounding x times per annum, equating equation (1.2) to equation (1.3) we obtain P(eRcn_{) =}_P₁₊Rx

x nx

. (1.4)

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 Rc=xln 1+Rx x . (1.5)

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and

R_x=x(eRcx − 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}

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invested today at time T0. We use bootstrap method to calculate the spot rate.

A_T =A_T₀(1+r_s)T, (1.7)

with the present value of AT is given by

A_T = 1

(1+r_s)TAT0. (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 RA and RB are the zero rates for maturities T1and T2, respectively, and RF is the forward

interest rate for the period of time between T1and T2, then

R_F = RBT2− RAT1 T₂− T1 . (1.9) If we write equation (1.9) as RF =RB+ (RB− RA) T₁ T₂− T₁ (1.10)

and take limits as T2approaches T1, 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".

RF =R+T

∂ R

∂ T, (1.11)

where R is the zero rate for a maturity of T .

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

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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.

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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 σ2 such as X∼ N(µ , σ2)if it has the density function of

f(x) = _p 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 2πe

−x2

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The cumulative distribution function of the standard normal distribution is given by the equa-tion below, see Kijima (2013)

Φ(x) = x Z −∞ 1 p (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 , σ2) is described with density function

f(x) = √ 1 2πσ xe

−(log x−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 , σ2)and u=log x, 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.

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Figure 2.2: Lognormal Distribution.

2.2.1 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 1997. 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

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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=µ Sdt+σ SdWt. (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[S₀erTΦ(d₁)− KΦ(d₂)], (2.6) P=e−rT[KΦ(−d₂)− S₀erTΦ(−d₁)], (2.7) where d₁= ln[S0/K] + (r+σ 2_/2₎₍_T₎ σ √ T , (2.8) d₂= ln[S0/K] + (r− σ 2_/2₎₍_T₎ σ √ T =d1− σ √ T. (2.9)

Φ(x) is a cumulative probability distribution function for a standardized normal distribution which is defined in Equation (2.3). S0 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 op-tions 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 carry1in 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 F0such as

F(0) =F₀,

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dF_t=σBFtdWt, (2.10)

σB is the normal volatility and Wt 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.

C=P(0, T)[F0Φ(d1)− KΦ(d2)], (2.11) P=P(0, T)[KΦ(−d2)− F0Φ(−d1)], (2.12) where d₁=ln[F0/K] + (σ 2_/2₎₍_T₎ σ √ T , (2.13) d₂=ln[F0/K] + (σ 2 B/2)(T) σB √ T =d1− σB √ T. (2.14)

F0 is the forward price of the asset at time 0, σB 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.

2.2.2 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

dSt =µ Stdt+σ S_tβdWt, (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_{6 β 6 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.

Schroder (1989) presents the price of a European call option in the CEV model as follows:

Ct(St, T − t) =St 1 − ∞

∑

n=1 g(n+1+γ , ˜Kt) n

∑

m=1 g(m, ˜Ft) ! (2.16) −Ke−r(T−t) ∞

∑

n=1 g(n+γ , ˜Ft) n

∑

m=1 g(m, ˜K_t),

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where γ=₂₍_1−β1 ₎ and g(p, x) = xp−1_Γ₍_pe₎−x is the density function of the Gamma distribution. For the forward price of a stock Ft = _B₍S_t,Tt ₎ we have:

˜ F_t= F 2(1−β) t 2X(t)(1 − β)2, K˜t= K2(1−β) 2X(t)(1 − β)2, X(t) =σ 2Z T t e2r(1−β)udu

is the scaled expiry of an option.

2.2.3 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 volat-ility 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 measureP, the forward rate F and its volatility α are assumed to obey the following equations:

dFt=αtFtβdWt, (2.17)

dαt=ν αtdZt, (2.18)

with the initial condition:

F₀=F, α0=α ,

where F is the forward price, the constant parameters β ,ν satisfy the conditions 0_{6 β 6} 1, α_{> 0 and the correlation between the two processes W}t and Zt with correlation coefficient

−1 < ρ < 1 is given by:

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Assume that the volatility parameters αt 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[(Fte− K)

+_|F

t = f, α(t) =α], (2.20)

t_eis 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 Z ∞ x2 2τe e−q (q)32 dq, (2.21) where q= x_2τ2.

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 σB(F, K)is given by the formula below, see Hagan et al. (2002) and Röman (2015).

σB(F, K) = α (FK)(1−β)/2n₁₊(1−β)2 24 (ln F/K)2+ (1−β)4 1920 (ln F/K)4 o z X(z) (2.22) ∗n1+h(1 − β) 2 24 α2 (FK)1−β + 1 4 ρ β ν α (FK)(1−β)/2+ 2 − 3ρ2 24 ν 2io_T_, where z= ν α(FK) (1−β)/2_ln₍_F_/K₎_, _(2.23) and x(z) =ln np1 − 2ρz+z2+z− ρ 1 − ρ o , (2.24)

when K=F, equation (2.22) gives at-the money (ATM) implied volatility σAT M =σB(F, F) = α F(1−β) n 1+h(1 − β) 2 24 α2 F2−2β + 1 4 ρ β ν α F(1−β) + 2 − 3ρ2 24 ν 2io_T_{. (2.25)}

2.3 Normal Models

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.

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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.

The derivation of Bachelier formula assumes the forward rate Ft follows the SDE

dF_t =σNdWt, (2.26)

where σN is normal volatility. The formula prices the European call and put option as follows:

C=P(0, T)[(F− K)Φ(d) +σN √ T φ(d)], (2.27) and P=P(0, T)[(K− F)φ(−d) +σN √ TΦ(d)], (2.28) where d= F− K σN √ T, (2.29)

Kis strike price and P(0, T)is a zero coupon bond used to discount.

2.3.2 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

αtdWt, (2.30)

where s is a deterministic positive shift which moves the lower bound on Ft 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_{6 β < 1/2 and a free bondary i.e. with no shift parameter s}

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The normal implied volatility σN(F, K)is given by the formula below, see Hagan et al. (2002). σN(F, K) =α(FK)β/2. 1+₂₄1(ln F/K)2+(1−β₁₉₂₀)4(ln F/K)4 1+(1−β₂₄)2(ln F/K)2₊(1−β)4 1920 (ln F/K)4 . z X(z) (2.32) ∗n1+h−β(2 − β) 2 24 α2 (FK)1−β + 1 4 ρ β ν α (FK)(1−β)/2+ 2 − 3ρ2 24 ν 2io_T , zand x(z)are defined as in Equations (2.23) and (2.24) respectively.

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Chapter 3 Hedging Parameters

Riskand 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.

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Delta

The Delta ∆ of an option is defined as the sensitivity of the option or portfolio to the un-derlying. 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 position1in one call option such as:

∆(c) =Φ(d1), (3.1)

On the other hand, the Delta of a put option is negative and is given by:

∆(p) =Φ(d1)− 1, (3.2)

Where; Φ(d1) 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-of-the-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 GammaΓ 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 3.12 below shows the variation of Gamma with time to maturity from the top for at-the-money (ATM), out-of-at-the-money (OTM), and in-at-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. GammaΓ is always positive for for both call and put options. The Black-ScholesΓ of a European call or put option on a non-dividend-paying stock is given by:

Γ= φ(d1)

Sσ√T, (3.3)

Where φ(d₁)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)}

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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(d₁), (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. ThetaΘ 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 byΘ=−∂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

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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-ScholesΘ of a European call on a non-dividend-paying stock is given by: Θ(c) =−Sφ(d1)σ

2√T − rKe

−rT_Φ₍_d

2), (3.5)

On the other handΘ for a European put option is given by: Θ(p) =−Sφ(d1)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:

ρ(c) =KTe−rTΦ(d2), (3.7)

On the other hand ρ for a European put option is given by:

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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

3.2.1 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) ∆(c) = ∂

∂ F h

(FΦ(d₁)− KΦ(d₂)i, We then apply differentiation we obtain

=Φ(d₁) +FΦ0(d₁)∂ d1 ∂ F − KΦ 0₍_d 2) ∂ d2 ∂ F =Φ(d₁) +∂ d1 ∂ F (Fφ( d₁)− Kφ(d₂) ∆(c) =Φ(d₁). (3.10)

By the same analogy we obtain the Delta for a put option∆(p)to be

∆(p) =Φ(d₁)− 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 GammaΓ for the call option which is also similar to put option

Γ(c) =Γ(p) = ∂Φ(d1) ∂ F =φ( d₁)∂ d1 ∂ F = φ(d1) Fσ√T. (3.13)

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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 = ∂

∂ σ h

(FΦ(d₁)− KΦ(d₂)i

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(d1). (3.15)

3.2.2 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 h (F− K)Φ(d) +σ √ T φ(d)i

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 GammaΓ for the call option which is also similar to put option

Γ(c) =Γ(p) = ∂ ∂ FΦ( d) = φ(d) σ √ T. (3.20)

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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 = ∂ ∂ σ h (F− K)Φ(d) +σ √ T φ(d)i

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)

3.2.3 SABR Model

Delta

Assume the underlying forward rate F changes by∆F, then the SABR volatility α will also change on average by δFα . This is because the two parameters are correlated. The Delta risk can be calculated from the following scenario:

F → F+∆F,

α → α+δFα . (3.23)

According to Bartlett (2006), to calculate the δFα , we write the SABR dynamics in terms of

independent Brownian motions Wtand Zt, and obtain

δFα =

ρ ν

Fβ. (3.24)

The change in the option value is given by ∆V =h∂ B ∂ F + ∂ B ∂ σ _{∂ σ} ∂ F + ∂ σ ∂ α ρ ν Fβi∆F. (3.25)

Delta∆ risk is then given by

∆= ∂ B ∂ F + ∂ B ∂ σ _{∂ σ} ∂ F + ∂ σ ∂ α ρ ν Fβ . (3.26)

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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 δαF. Vega risk can be calculated from

the following scenario:

F→ F+δαF,

α → α+∆α. (3.27)

Using the same procedures as for the delta risk we obtain δαF to be, (See Bartlett (2006)).

δαF =

ρ Fβ

ν ∆α, (3.28)

The change in the option value is given by ∆V = ∂ B ∂ σ _{∂ σ} ∂ α + ∂ σ ∂ F ρ Fβ ν ∆α, (3.29)

Vega υ risk is then given by

υ = ∂ B ∂ σ _{∂ σ} ∂ α + ∂ σ ∂ F ρ Fβ ν . (3.30)

3.3 Hedging Strategies

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 hedging3 and how hedging is done in the practice.

Delta Hedging

Delta hedgingcan 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 ofΦ(d₁)for each option sold. Likewise delta hedging for a long position requires retaining a short position ofΦ(d₁)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 re-hedging or rebalancing the portifolio. Rere-hedging or rebalancing the portfolio is achieved by a

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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 Γp is the Gamma of the portfolio and Γ is the Gamma of a traded option, a

position of −Γp/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 −Γp/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 υp

is the Gamma of the portfolio and υ is the Gamma of a traded option, a position of −υp/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.

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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.

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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 de-rivatives 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, ar-bitrage 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

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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 differ-ent 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. In-terest 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, In-terest Rate Swaps and Swap Options. However, while we will define all these types of inIn-terest rate derivatives, we will put more emphasis on Swap and Swap Options as our thesis centers on them.

4.2.1 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 T1and T2, 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 T1and T2calculated

today

R_M is the he actual LIBOR interest rate observed in the market at time T1for the period

between times T1and T2

Lis the principal underlying the contract

R_K, RF, RM are all assumed to be measured with a compounding frequency reflecting the

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Party B at time T2to be given by Equation (4.1) and Equation (4.2) respectively.

L(R_K− RM)(T1− T2), (4.1)

L(R_M− RK)(T1− T2). (4.2)

We can then interpret FRA as an agreement where party A will receive interest on the principal between T1 and T2at the fixed rate of RK and pay interest at the realized LIBOR rate of RM.

On the other hand party B will pay interest on the principal between T1and T2at the fixed rate

of RK and receive interest at RM.

To compute the payoff for Party A and Party B, we have to discount from time T2to T1since

FRAs are settled at time T1rather than T2. For Party A the payoff at time T1is given by

L(RK− RM)(T1− T2)

1+R_M(T₁− T2)

, (4.3)

for Party B, the payoff at time T1is given by

L(RM− RK)(T1− T2)

1+R_M(T₁− T2)

. (4.4)

Valuation of FRAs

To value FRAs, we need to calculate the payoff assuming that RM=RF and then discount this

payoff at the risk-free rate. Now if we consider 2 FRAs, where;

1. Promises that the LIBOR forward rate RF will be received on a principal of L between

times T1and T2

2. Promises that RK will be received on a principal of L between times T1and T2

The present value of the difference between the interest payments of these 2 contracts is given by;

L(R_K− RF)(T2− T1)e−R2T2, (4.5)

where R2 is the continuously compounded riskless zero rate for a maturity T2. From

equa-tion 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)(T₂− T₁)e−R2T2_, _(4.7)

V_FRA₃ =L(R_F− R_K)(T₂− T₁)e−R2T2_. _(4.8)

The value VFRA1 of the first FRA when RF is received in Equation (4.6) is zero because we

normally set RK =RF when the FRA is first initiated. Equation (4.7) gives the value of VFRA2

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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 RK, reset dates t1,

t2,....,tn. When T is defined as T =tn+1and Rkdefined as the LIBOR interest rate for the period

between time tkand tk+1observed at time tk. The payoff of such a cap at time tk+1(k=1,2,..,n)

is given by

Lδkmax(Rk− RK, 0), (4.9)

where δ_k=t_k+1− tk.

The standard market model gives the value of the caplet as

LδkP(0,tk+1)[FkΦ(d1)− RKΦ(d2)], (4.10) where d1= ln(F_k/RK) +δ_k2tk/2 δk √ t_k , (4.11)

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d2= ln(F_k/RK)− δ_k2tk/2 δk √ t_k =d1− δk √ t_k. (4.12)

Φ(x)in Equation (2.3), Fkis the forward interest rate at time zero for the period between time

t_kand tk+1, δkis the volatility of this forward interest rate and P(0,tk+1)is the discount factor.

4.2.3 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

Lδkmax(RK− Rk, 0). (4.13)

The standard market model also gives the value of the floorlet as

LδkP(0,tk+1)[RKΦ(−d2)− FkΦ(−d1)], (4.14)

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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 F₀=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;

C=P(0, T)[F_BΦ(d₁)− KΦ(d₂)], (4.15) P=P(0, T)[KΦ(−d2)− FBΦ(−d1)], (4.16) where d₁=ln[F0/K] + (σ 2_/2₎₍_T₎ σ √ T , (4.17) d₂= ln[FB/K] + (σ 2 B/2)(T) σB √ T =d1− σB √ T, (4.18)

K is the strike price1 of the bond option, T its time to maturity and FB is the forward bond

price that can be calculated as

FB=

B0− I

P(0, T), (4.19)

where B0is the bond price at time zero, I is the present value of the coupons that will be paid

during the life of the option.

4.2.5 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.

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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 interest-rate risk, changing a liability either from floating-interest-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=1

P(t, Ti), (4.20)

where rf ixis the fixed rate of interest, N is the number of payments one at each Ti

On the other hand the floating side of the swap has value

1 − P(t, TN). (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;

Vswap=Bf ix− Bf 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}− Bf ix, (4.23)

where Bf ix is the fixed-rate bond and Bf l is the floating-rate bond. If we define Bf ix as in

Equation (4.20) and Bf l as in Equation (4.21), we have the value of the swap as

Vswap=rf ix N

∑

i=1

P(t, Ti)− 1+P(t, TN). (4.24)

Using Equation (4.19), we can obtain the quoted Swap rate as r_{f ix}=1 − P(t, TN) N ∑ i=1 P(t, Ti) . (4.25)

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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 rf ix(Ti) for many

maturities Tiwe 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}(T₁) = 1 − P(t, T1) P(t, T1) , (4.26) P(t, T1) = 1 1+r_{f ix}(T₁). (4.27)

After finding the first k discount factors the k+1 is found from P(t, Tk) =1+rf ix(Ti) k

∑

i=1 P(t, Ti), (4.28) then P(t, Tk+1) = 1 − r_{f ix}(T_k+1) k ∑ i=1 P(t, Ti) 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 tomorrow-next 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 = 1

1+r_O/Npar .dO/N

360

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we then obtain the zero rate as

Z_O/N =−100.ln(DO/N)

dO/N

365

. (4.31)

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 fu-tures 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 factor2 is given by D_Stub=exp − Z_Stub.TStub 365 , (4.32)

then the discount factor of the FRA rate can be obtained from

Di_FRA= D i−1 FRA 1+r_FRAi .d i FRA 360 , (4.33)

where D0_FRA=D_Stub. We can then obtain our zero rate as Z_FRAi (T) =−100.ln(D i FRA) d_FRAi 365 . (4.34)

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= DStub(t) 1+100−PFRA 100 .₃₆₀91 , (4.35)

where PFRAis 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}

Hedging Interest Rate Derivatives (Evidence from Swaptions) in a Negative Interest Rate Environment: : A comparative analysis of Lognormal and Normal Model

MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS