Volatility- An investigation of the relationship between price- and yield volatility

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Volatility- An investigation of the relationship

between price- and yield volatility

Samia Nasir

Supervised by: Jan R¨oman

DIVISION OF APPLIED MATHEMATICS

M ¨ALARDALEN UNIVERSITY

SE-721 23 V ¨ASTER˚AS SWEDEN

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Master thesis in mathematics / applied mathematics

Date:

2020-09-29

Project name:

Volatility- An investigation of the relationship between price- and yield volatility

Author : Samia Nasir Supervisor(s): Jan R¨oman Co-Supervisor : Ying Ni Reviewer : Marko Dimitrov Examiner : Anatoliy Malyareko Comprising: 30 ECTS credits

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Abstract

This report investigates the relationship between the yield volatility and the price volatility in the Swedish market. The method given in our report can be used to analyze any market with appropriate data set.

We have used a time-series data of interest rate yield curves from Swedish government bonds. The curves are bootstrapped from these bills and bonds. The linear interpolation on these curves results in the nodes i.e. 1Y, 2Y, . . . , 10Y.

We also need prices for instruments. A good choice is to use the synthetic government bonds namely SE GVB 2Y, SE GVB 5Y, and SE GVB 10Y. They are issued every day with maturity 2, 5, and 10 years. We also use the time-series of these bonds. These bonds have a yearly coupon of 6%. We can get zero-coupon values of these bonds by stripping their coupons using the interest rate yield curves.

We have time-series data of zero-coupon prices with maturities 2, 5, and 10 years and time-series data of interest rates with the same tenors. We can use our data to calculate their respective volatilities to investigate how they are related to each other.

Keywords: Price Volatility, Yield Volatility, Python, Bonds, Zero-Coupon rate, COVID-19.

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Acknowledgements

All praise to Almighty Allah who gave me the ability and determination to complete this thesis.

I would like to express my special thanks and gratitude to my supervisor Jan R¨oman for introducing the topic and providing me the data. This thesis has given me a great chance to challenge my theoretical knowledge in practical work. The completion of this thesis could not have been possible without the guidance, patience, and help of my supervisor.

I would like to take this opportunity and pay a huge thanks to my family who has supported me during the thesis, especially in tough and stressful moments.

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List Of Symbols

σday Standard deviation

σ2 _Variance

σannual Annual volatility

d Number of trading days

F (t, T, r) Zero-coupon price given in forward rate σr Yield volatility

σp Price volatility

W (t) Brownian motion

M D = Dmod Modified Duration

P Price of bond

N Face value, principal amount of a bond P V Present value of the coupon

C Coupon rate r Interest rate ZCP Zero-coupon price D Macaulay Duration y YTM T Time to maturity X Random variable xt Observation t on variable x ¯

x Sample mean for variable x

n Number of observations in the sample pt Price of an asset at time t

Rt Logarithmic returns

p(t, T ) Zero-coupon price of bond at time t with ma-turity T

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K Strike Price

a Determines the strength of mean reversion in H-W model

σN Normal or Bachelier Volatility

σB Black (log-normal) Volatility

F SABR forward rate

f SABR forward rate starting value

Z(t) Brownian motion of the volatility in SABR τ Time to exercise date in years

σ SABR volatility

β Exponent for the forward rate

ρ Correlation between the Brownian motions in SABR

α starting value of SABR volatility ν SABR volatility of volatility

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List of Figures

1.1 The stock prices of Volvo, ABB and H&M . . . 2

2.1 Government bond zero-coupon rate on date 2019-12-30 . . . . 9

2.2 Government bond zero-coupon rate on date 2020-04-08 . . . . 10

2.3 YTM for Synthetic 2Y, 5Y and 10Y bonds . . . 13

2.4 Rates for Synthetic 2Y, 5Y and 10Y bonds . . . 15

4.1 Bond Prices for 2Y bond . . . 35

4.4 ZCP for 2Y, 5Y and 10Y bonds . . . 38

4.5 Plot of 2Y Log-Returns . . . 39

4.6 Plot of 2Y Simple Returns . . . 41

4.7 Normal and Log-Normal distribution of bond prices with ma-turity 2-years . . . 45

4.8 Normal and Log-Normal distribution of bond prices with ma-turity 5-years . . . 45

4.9 Normal and Log-Normal distribution of bond prices with ma-turity of 10-years . . . 46

4.10 Normal and Log-Normal distribution for ZCP of a bond with maturity 2-years . . . 48

4.11 Normal and Log-Normal Distribution for ZCP of a bond with maturity 5-years . . . 48

4.12 Normal and Log-Normal Distribution for ZCP of a bond with maturity 10-years . . . 49

4.13 Normal and Log-Normal distribution of bond rates with ma-turity of 2-years . . . 51

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List of Tables

4.1 Annual Price Volatility Using Log-Returns . . . 40

4.2 Annual Price Volatility Using Simple Returns . . . 41

4.3 Zero-Coupon Price Volatility Using Log-Returns . . . 42

4.4 Zero-Coupon Price Volatility Using Simple Returns . . . 42

4.5 Annual Yield Volatility . . . 42

4.6 Descriptive Statistics of 2Y, 5Y, and 10Y Bond . . . 44

4.7 Descriptive Statistics of 2Y, 5Y, and 10Y Zero-Coupon Bond . 47 4.8 Descriptive Statistics of 2Y, 5Y, and 10Y Rates . . . 50

4.9 Price Volatility Calculation . . . 53

4.10 Price Volatility Calculation . . . 54

4.11 Price Volatility Calculation Using Beta . . . 54

4.12 Optimal Beta Values for Positive Rates . . . 56

4.13 Optimal Beta Values for Negative Rates . . . 57

4.14 Calculating Price Volatility Using HW Model . . . 57

4.15 Calculating Normal Volatility From Log-Normal Volatility . . 58

4.16 Optimal Beta Values . . . 58

4.17 Calculating Normal Volatility From CEV Volatility . . . 59

4.18 Summary of Calculations . . . 59

B.1 Data of Bootstrapped Government Curves in SEK . . . 65

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Chapter 1 Introduction

The financial crisis in 2007-2008 has shaken the financial markets globally. These crises are considered as the worst tragedy since the Great Depression of the 1930s. According to data from the World Federation of Exchanges, at the end of 2007 the world equity market capitalization was more than $64 trillion and sharply declined in 2009 to stand at $49 trillion—a drop of 22%, which is equal to 25% of global Gross Domestic Product (GDP)1 _{for 2009}

[8]. The crisis was started in the US stock market and rapidly spread in the emerging markets. The crisis didn’t only affect the stock markets but also the employment rate, housing markets, and some banks and companies faced bankruptcy. The financial crisis affected the banks which lead to a decline of customer trust on the banks. The decrease in customer trust affected the return of banks and stock prices due to which the stocks became riskier to invest and it also increases the stock volatility.

The world is facing a new pandemic and we see another financial crisis right now. The COVID-19 pandemic is marked as a global crisis and the situation is different as compared to the 2008 crisis. This pandemic is not only hitting the market of goods and services but also trade, hotels, restaurants, shop-ping malls and transport. The social distancing and quarantine is making everything difficult whether its travelling or shopping and the demand for the goods is diminishing along the production. This crisis is different than the financial crisis as the real economy is directly affected and the supply and demand have been drastically disturbed. The business sector is in great distress and uncertainty whether the business will survive or not.

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Figure 1.1: The stock prices of Volvo, ABB and H&M

Figure 1.1 shows the stock prices of three Swedish companies i.e. Volvo, ABB and H&M 2. The graph represents the stock prices in the last seven months and we can see how drastically the stock prices went down in the month of March and April (which is the start of this pandemic). The drastic movements and fluctuations in the stock prices means that the stock market is facing high volatility.

The term volatility is a key concept in financial markets and the unexpec-ted increase and decrease in the volatility can lead to a financial crisis. The volatility is measured in terms of standard deviation and it is influenced by the risk. We can define volatility as a measure of the uncertainty of the re-turn realized on an asset [4]. In other words it is the fluctuations in the price of a security over a given time period. The term price volatility is used to describe the price fluctuations in a security. All securities in the market are not equally volatile and the level of volatility varies on the type of securities. The stocks are more volatile as compared to the bonds but on the other hand, they provide more returns [4]. The increase in volatility causes stock prices to fall whereas a decrease in volatility leads to an increase in stock prices.

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In other words, we can say if the price of an investment moves rapidly over a short period, it means it has high volatility. We can also say that market fluctuations increase with big price movements.

The fixed income securities are less volatile as compared to the stocks and they also provide a secure return to the investor. The fluctuations in the price of the fixed securities are common like any other security in the mar-ket. The volatility in term of the bond is the change in the bond price in return to a change in the interest rates. The main cause behind these price fluctuations in the fixed income securities more specifically the bonds is the change in the interest rates. When the interest rates move up the bond prices move down which means there is an inverse relationship between bond prices and interest rates. The relationship between the bonds and the interest rate occurs because the bonds have fixed coupons but the interest rate changes every day which make the investors pay more or less for the bonds.

1.1 Aim

The focus of this thesis is to study the relationship between interest rate volatility and price volatility. We need two sets of data for this study i.e interest rate yield curves from government bonds and zero-coupon prices. We have used a time-series of interest rate yield curves from Swedish government bonds. The curves are bootstrapped from these bills and bonds. The linear interpolation on these curves results in the nodes i.e.1Y, 2Y, ..., 10Y. We also need prices and a good choice is to use synthetic government bonds R2, R5 and R10. The bonds are issued every day with maturity 2,5 and 10 years. These bonds have a yearly coupon of 6%. So, we can easily get zero-coupon values of them by stripping their coupons using the curve discussed above. We have time-series of zero-coupon prices with maturities 2, 5, and 10 years and we also have time-series of interest rates with the same tenors. The aim of our thesis is to use our time-series (zero-coupon prices and interest rates) to calculate their respective volatilities and to investigate how they are related to each other.

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1.2 Purpose

The purpose of the thesis is to study interest rate instruments in the market and compare different models to check how we can convert between yield and price volatility. We want to check which formula seems to be most correct in the market and how the yield- and price volatility is related in the market.

1.3 Method

In this thesis we will cover some models i.e. Black model, Normal Black model, Hull-White and SABR model (with different beta values) to convert between the price and yield volatility.

1.4 Outline

The thesis consists of five chapters starting with the first section which is the introduction. It covers a brief background, aim, purpose, and method that will be used in the thesis. This section also covers the main research question and an approach that will help to acquire the answer. The second section is the theoretical review which focuses on the two main topics of the bond theory and aspects of volatility. This section gives an overview to the reader and provides better understanding about the topic. The third section covers the methodology used in the study. This section covers the brief information about the models and theories behind them. The next section covers the empirical analysis and discussion about the results that are drawn from the models. The last section focuses on the conclusion where we conclude all the sections and sum up the thesis.

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Chapter 2 Literature Review

This section covers the bond theory and the aspects of volatility. We will use the data of three synthetic bonds to perform calculations in the next sections therefore it is crucial to know important details and characteristics of bonds. The volatility is the main subject of the thesis hence we will define risk, standard deviation, and different types of volatility to explain this subject in detail.

2.1 Bond Theory

The bond is a fixed-income instrument that is issued by the government, companies, and many other types of institutions and corporations to raise the money or to pay the debt. It is a way for someone to participate in the lending to a company which means that a person who buys a bond is a partial lender. The bonds are debt and are issued for a period of more than one-year [5]. The bond is known as a fixed-income instrument because it gives a fixed amount of interest to the holder for a specific time or duration. The difference between the bank loan and a bond is that the bonds are traded in the secondary market where already own securities are traded [1]. When an investor invests in a bond they can face two different types of risk i.e. credit risk and the interest rate risk.

• Credit Risk: The risk that a loss will be experienced because of a default by the counter-party in a derivatives transaction [4]. Credit risk is also known as default risk in which the bond issuer might default and

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therefore not able to pay back the portion of interest and principal amount.

• Interest Rate Risk: The risk that the bond value will change due to the change in the interest rate. There is an inverse relationship between the bonds and interest rates. If the interest rate increase the bond’s price drops similarly if the interest rate falls the bond’s price increase. When a person buys a bond, they are lending money to the bond seller and the bond issuer, on the other hand, agrees to repay the face value of the loan at the specified time also, known as the maturity date. The fixed amount of interest that is paid to the holder is known as the coupon. A bond can also come without a coupon and in this case, the bond is called a zero-coupon bond. The price of a zero-coupon bond is lower as compared to the bond with coupons. The bonds have the following characteristics [7]:

• Principal Amount: The principal amount is also known as face value and it is the amount that the borrower promises to pay back to the bondholder at the time of maturity. It is the amount that the bond issuer agrees to pay at the maturity date.

• Issuer Price: The price at which the bond is sold initially (when someone is issuing the bond) in the financial market. In other words, it is the price at which the bonds are offered to the public.

• Coupon: The coupon represents the interest rate on a bond. It is the interest payments that the bondholder receives during the life of a bond. The coupon is usually paid annually or quarterly.

• Time to Maturity: The term time to maturity refers to the time between when you buy or sell the bond until the maturity date. It is the time until the final payment is repaid to the bondholder which is the face value of the bond.

• Yield - Yield to Maturity (YTM): It is the interest payment that the bondholder receives for a particular period of time. It is the effective rate of interest that is paid on a bond or note [5]. In simple words, the yield to maturity is calculated by dividing coupons with the principal amount. The yield tells about the buyer earning from holding a bond for a year.

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• Clean Price: The clean price of a bond corresponds to the price at which the transaction takes place without including accrued interest [7].

Clean = P − Coupon ·365 − d 365 ,

where P is the bond price and dis the number of days until the next coupon payment.

• Dirty Price: The dirty price is the price including accrued interest. It is the equal to the clean price and accrued interest.

2.1.1 Bond Pricing

The theoretical price of a bond is the present value of its future cash flows. It means the bond price is obtained by discounting the cash flows using the discount rate. The bonds can be traded at par, below par, or above par [7]. The coupon rate of the bond is compared to the interest rate to determine whether the bond is trading above or below par. A bond trading at par means it is traded at the face value and the investor will get the face value of the bond at maturity. The term below par means that the bonds are traded below its face value and at maturity, the bondholder receives par value that is higher than the bond purchase value. The above par means the bonds are traded above the face value and at maturity the bondholder receives less value. The price of a bond with no coupons can be calculated using the following formula:

P = N

(1 + r·d/360), (2.1)

where N is the face value of a bond and r is the market interest rate, and d/360 is the day-count convention (quoted in 30/360). The day-count is used for calculating the number of days between two dates. The price of a coupon bond is the sum of all coupon payments plus the face value and it is calculated as follows: P = N (1 + r(T ))T + n X i=1 C (1 + r(ti))ti , (2.2)

where P is the bond price, N is the face value, C is the cash flow or coupon (which can be annually, semi annually or quarterly), r is the market interest

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rate, T is the time to maturity and ti is the time for individual coupons. The

present value of the coupon can be find using the following formula:

P V = C

(1 + r(d))360d

, (2.3)

where C is the nominal amount, r is the rate (with d/360 as day-count) and d is the number of days to maturity. We can easily find the zero-coupon price by subtracting the present value of coupon from the bond price.

ZCP = P − P V. (2.4)

where P is the price of a bond and P V is the present value of the coupon given in Equation (2.3). Here is an example where we have a 3-years bond with a face value of 1000 and coupon rate of 5% per year. The market interest rate is also 5%. There will be three coupon payments of 50$ each. The present value of the bond is simply:

P = 50$ (1 + 5%)1 + 50$ (1 + 5%)2 + 1050$ (1 + 5%)3 = 1000$

Now the market interest rate changed to 6% and the coupon rate is still 5%.

P = 50$ (1 + 6%)1 + 50$ (1 + 6%)2 + 1050$ (1 + 6%)3 = 973$

The bond price is 973$ which means the bond is trading at a discount or in other words bond is trading below par value. Now the market interest rate changed to 4% and the coupon rate is still 5%.

P = 50$ (1 + 4%)1 + 50$ (1 + 4%)2 + 1050$ (1 + 4%)3 = 1027$

If the market interest rate is 4% and the coupon rate is 5%, it means that the bond would be trading at a premium (or above par value) and its price would be 1027$.

2.1.2 Yield Curve

Yield curves are used in the financial market to generate and discount cash flows from traded or quoted instruments. The yield curve is a graph that

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relates the current spot yields of any set of bonds to maturities [7]. The word bonds here means bills, notes and bonds which are defined by the same issuer. The yield curve can be constructed for corporate bonds, government bonds and mortgage bonds of the same risk and credit rating. The yield curve is a line plot with time to maturity on the x-axis and yields on the y-axis. The yield curve summarizes the connection between the bond maturities and interest rates.

The zero-coupon yield curve is the most important subclass among the yield curves. It is a plot of zero-coupon yields against the time to maturity. A zero-coupon yield curve defines a set of discount factors that are used to obtain the present value of future cash flows. This curve can be derived directly from the money market or a groups of coupon paying bonds using the technique called bootstrapping. In this technique we strip the bonds to create virtual zero-coupon bonds from the coupons and the principal amount.

Figure 2.1: Government bond zero-coupon rate on date 2019-12-30 The yield curve helps to forecast the phases of the business cycle [5]. A yield curve can be upward sloping, downward sloping or flat [4]. Figure 2.1 show an upward sloping yield curve which starts with a low-interest rate for lower

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maturity bonds and then increases for the bonds with higher maturity. In simple words, it means long-term bonds have a higher yield as compared to short-term bonds with the same credit rating. The downward-sloping yield curve also knows as the inverted yield curve appears when the short-term interest rates are higher. This yield curve is uncommon and it forecast the economic recession. In the case of a flat yield curve, the difference between the short-term and long-term term rates is minimal. This yield curve appears between the transition of normal and inverted curves.

Figure 2.2: Government bond zero-coupon rate on date 2020-04-08

2.1.3 Duration

The duration is a common gauge that is used to measure the price sensitivity of a fixed income asset [5]. There are different types of duration that are discussed in this section.

• Macaulay Duration: The Macaulay duration measures the price sensit-ivity of an interest rate instrument with respect to an absolute change in yield to maturity [7]. It is a measure of the average life of a bond. It

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is easy to show that the duration of the zero-coupon bond is the same as its time to maturity because the zero-coupon bond has no coupon payments. The drawback of Macaulay duration is that it is defined in terms of the bond’s own ytm and it does not provide information on how the change in the zero-coupon yield curve will affect the bond price. The duration D is defined as:

D = n P i=1 ti·Ci (1+y)t_i + tn·N (1+y)t n P ,

where P is the current price or the quoted price of a bond, y is the bond yield to maturity, N is the principal amount, C is the coupon rate multiplied with the principal amount and n is the number of years to maturity.

• Modified Duration: The modified duration M D also written as Dmod

determines the percentage change in bond price with respect to an absolute change in the yield. This type of duration is used when the bond is a function of yield [7]. It is defined as

M D = −1 P ∂P ∂y = D 1 + y_n,

where n is the number of cash flows per year, y is the yield to maturity, and D is the Macaulay duration. The duration indicates how much risk on bond an investor faces from the change in interest rate. A bond with higher duration will have a lower coupon rate along with more term to maturity and more volatility. In other words we can say long duration means high risk on bonds.

• Dollar Duration: The Dollar duration DV 01 measures the change in bond price (in $) with respect to change in the market interest rate [7]. The DV 01 is short form of dollar value of a 01. It is defined as the derivative of price with respect to yield. The formula of Dollar duration is given below:

D$ = DV 01 = −

∂P (y) ∂y ,

where P in the formula is the price of a fixed income instrument (such as bond) and the delta shows the ratio of price change in dollars to a unit change in the yield.

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• Fisher-Weil Duration: The Fisher-Weil duration is the generalized ver-sion of Macaulay duration as it measures the price sensitivity of fixed-income assets based on the zero-coupon yield curve rather than the bond’s own yield. The formula of Fisher-Weil duration is given below:

DF W = 1 P V " N · tn (1 + yn)n + n X i=1 C · ti (1 + yi)i # ,

where P V is the present value of bond, yi is the zero-coupon yield, N

is the principal amount, C is the coupon size and ti is the time period

where i is the time to maturity.

2.2 Introduction to Interest Rates

The interest rate is the percentage of the principal amount that the lender gets from the borrower for using the money. We can also say that it is the rate at which the borrower agrees to compensate the lender at the specific decided time period. There are different kinds of interest rates that are used in the market. This section covers some of the interest rates with a brief description.

• Coupon Rate: The coupon rate is the interest rate that is paid by fixed income security such as bonds or notes. This rate is fixed and it is calculated on the face value of the bond. The bond purchaser receives the coupon rate from the bond issuer (which can be government or company). These coupons are usually received one, two or four times per year [7].

• Benchmark Rate: The benchmark interest rate is the minimum rate that an investor will pay to invest in a risk-less security [7]. These rate are given as yield curve usually built from government securities with different time to maturities.

• Yield to Maturity: The yield to maturity (YTM) is the rate that in-vestor gets from holding a bond until it matures. It is the percentage rate of return paid on a bond, note or other fixed-income security where the calculation is based on the factors like coupon rate, length of time to maturity, and market price [5]. The calculation of yield to maturity

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helps to know if the bond is a good investment or not. The return (that is ytm) is paid if the investor buys and holds the bond to its maturity date. The comparison of yield to maturity with the yield helps to know if the bond is a good buy or not.

Figure 2.3: YTM for Synthetic 2Y, 5Y and 10Y bonds

Figure 2.3 shows the yield to maturity for three synthetic bonds with same coupon rate. The synthetic bonds have a maturity of 2-years, 5-years, and 10-years.

• Zero Coupon Rate: The zero coupon rate is the yield to maturity on a zero coupon bond i.e. a bond that pays no coupons [7]. These rates are often used for discounting cash flows or by risk managers to calculate the risk.

• Spot Rate: The spot rate is a theoretical profit given by a zero-coupon bond [7]. The spot rate rspot is also known as short rate and we use

spot rate to calculate the amount that we will get at time t1 if we invest

X in zero coupon bond at time t0. The short rate is the percentage of

the amount that we get at time t1 invested in zero coupon bond.

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and the present value of Xt is:

P V (Xt1) =

1 (1 + rspot)t1

Xt1.

The spot rate is calculated by fitting the yield curve i.e. bootstrapping [7].

• Forward Rate: The forward rate is the prediction of future rate that is calculated from the spot rate or the yield curve [5]. It is the rate that is applied between two future dates. We can represent forward rate as f (ti, tj)f orward and it is the percentage invested in zero-coupon bond at

time t0 that we will get between time ti and tj in the future dates. The

relationship between spot rate and forward rate is as follows: (1 + r_tspot₁ )t1_{(1 + f}f orward

(t2,t1) )

(t2−t1) _{= (1 + r}spot

t2 )

t2

We can rearrange the terms to represent the forward rate between years t1 and t2 as r_(tf orward 2,t1) = (1 + rspot_t₂ )t2 (1 + rspot_t₁ )t1 !_t2−t11 − 1,

This is a forward rate at which we can sign a contract today to borrow or lend between the time period t1 and t2. The easy way to show a

forward rate using a discount factor is as follow: p(0, t1) · p(t1, t2) = p(0, t2), p(t1, t2) = p(0, t2) p(0, t1) = p(t2) p(t1) .

where p(0, t) is zero-coupon bond at time 0 with maturity t. In terms of continuous compounding we have:

e−r(t1)·t1 _{· e}−f (t2,t1)·(t2−t1)_{= e}−r(t2)·t2_, −r(t1) · t1· −f (t2, t1) · (t2− t1) = −r(t2) · t2, f (t2, t1)(t2− t1) = r(t2) · t2− r(t1) · t1, f (t2, t1) = r(t2) · t2− r(t1) · t1 (t2− t1) .

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Figure 2.4: Rates for Synthetic 2Y, 5Y and 10Y bonds

The forward rate is the relationship between the two spot rates. The rate for a period of more than three years is extracted using the boot-strapping technique i.e. using the zero-coupon bonds with yield whereas for short time period forward rate agreements are preferred [7].

Figure 2.4 illustrates the government rate for three synthetics bonds. It shows how these rates have been varying over a period of one year (starting from 2019-04-09 to 2020-04-08). The graph reflect the market view of the risk-free zero-coupon rate with tenors 2, 5 and 10Y. • Discount Rate: The discount rate is also known as the capitalization

rate and it shows the time value of money. It refers to either the interest rate used to discount the future cash flows to the present value or it is the rate that is charged by the Federal Reserve to the banks when they face a shortage of money [5]. This rate is not uniquely defined in fact it depends on the deal and the counter-party. The discount rate also depends on the type of deal for example for a collateral agreement the collateral rate mentioned in the agreement should be used as a discount

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rate.

2.3 Aspects Of Volatility

The volatility of bonds and the stock market are important topics for the financial sector such as traders as they often make their decisions based on the volatility. It is the key concept in the financial market, and it shows how much are the fluctuation in the price and the performance of the security varies over time. The volatility is as a rapid change in the price of a security over a short period of time. The lower volatility means that the price of security doesn’t fluctuate dramatically whereas high volatility indicates more fluctuations in the price of a security. In other words, the more volatile the market is the bigger the price movements are.

2.3.1 Risk

The risk is something that we can control. It is a personal matter that only depends on the investor that how much loss or financial uncertainty he can control or bear. The risk is a potential loss of an investment and it is a degree of uncertainty of return on an asset [5]. When someone invests money in the stock market one must be aware of the risk that is associated with the security. The financial risk is divided into several classes namely legal risk, market risk, credit risk, liquidity risk and operational risk.

• Legal Risk: The first type of risk is the legal risk which is the risk of financial loss that can result from a lack of information on the change of laws and rules.

• Market Risk: The market risk refers to the risk that changes in interest rates, exchange rates, and equity prices which leads to a decline in the values of the bank’s net assets [6]. The market risk is associated with the movement in the price of financial instruments mainly caused due to the change in demand and supply.

• Credit Risk: The credit risk is the risk that occurs when the counter-party fails to fulfill its obligations. The credit risk arises from the possibility of a default by the counter-party when the value of the contract to the financial institution is positive [4].

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• Liquidity Risk: The liquidity means how easily the assets can be sold and converted to cash on the other hand liquidity risk occurs due to the lack of marketability or sale-ability of an investment.

• Operational Risk: The last type of risk is the operational risk that occurs due to human error, management failure, incorrect system, or technical faults. We can avoid the risk that is associated with the investment by understanding and identifying them and handling them in a proper way.

2.3.2 Volatility

It is important to know the difference between volatility and risk as it is easy to mix up and get confused. The volatility σ is a measure of risk and it is based on the standard deviation of the asset return. The volatility and risk are not same. The volatility of a stock price is a measure of how uncertain we are about future stock price movements. It is a variable that appears in option pricing formulas, where it denotes the volatility of the underlying asset return from now to the expiration of the option [5]. The volatility is measured on the scale of 1-9 and the higher the scale means higher the volatility. As volatility increases, the chance that the stock will do well or poor increases [4]. Volatility refers to the upward and downward movements in the security price and that is the reason why the investors have no control over volatility.

2.3.3 Standard Deviation

The standard deviation is a measure of the dispersion of a collection of data from its mean [5]. In finance, volatility is measured in two ways that is beta or standard deviation [2]. The first way to measure volatility is beta which is calculated using regression analysis where beta is the slope of the regression data points. The other way is using standard deviation which measures the variation in the price of a security, and we use the symbol sigma to represent volatility. Standard deviation measures the spread of a security’s return from its mean. The high standard deviation implies that the data is spread out over a wide range whereas the lower standard deviation implies that the values of spread are near to its means. The standard deviation is basically

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the square root of the variance and it is denoted as follows: σday=

√ σ2_.

2.3.4 Yield- and Price Volatility

It is important to know the difference between the volatility terms like price volatility and yield volatility. The volatility for the yield is used in the process for short rate whereas the volatility for the prices is used in the process of bond prices [7]. The price volatility describes the fluctuations in the price of a security or we can say that the price volatility measures the daily variations in the price of an asset. It is a percentage change in the security price to the percentage change in interest rates. In finance, the term yield means the return that an investor gets from investing in the financial securities. In other words, the return from the investment is also known as the interest rate. The yield volatility can be define as the fluctuations or variations in the interest rate or we can say that the yield volatility is a percentage change in the daily yield. The estimation of yield volatility plays an important role in the valuation of bonds with embedded and interest rate options and there are two ways to model yield volatility i.e. to estimate the yield volatility using some time-series models and by estimating yield volatility based on the observed prices of interest rate derivatives [2]. The process for short rate is given as:

dr(t) = µdt + σrdW (t),

and the zero-coupon price is given by F (t, T, r). By using Itˆo formula we get

dF = ∂F ∂tdt + ∂F ∂rdr + 1 2 ∂2_F ∂r2(dr) 2 = Ftdt + Fr[µdt + σrdW ] + 1 2σ 2 rFrrdt = Ft+ µFr+ 1 2σ 2_F rr dt + σrFrdW = αF dt + σpF dW

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where α = Ft+ µFr+ 1 2σ 2_F rr F σp = σrFr F dF = αF dt + σpF dW,

We know the relationship between the price and yield volatility follows σp =

σrFr

F = σr· Dmod,

The relationship between price and yield volatility can be defined as the following:

σp = σr· Dmod. (2.5)

where Dmodis the modified duration. If the interest short rate is log-normally

distributed, that is,

dr(t) = µrdt + σrrdW (t),

then the price volatility is given as: σp = r

σrFr

F = rσrDmod,

The relationship between the price volatility and yield volatility is given as: σp =

σrFr

F ,

σp = rσrDmod. (2.6)

The derivation of formula to convert between price and yield volatility for general beta is given below. We know the process for rate is given as:

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and let zero-coupon price given by F (t, T, r) and using Itˆo formula we get dF = ∂F ∂tdt + ∂F ∂rdr + 1 2 ∂2_F ∂r2 (dr) 2 = Ftdt + Frµdt + σrβdW + 1 2σ 2_r2β_dtF rr = Ft+ µFr+ 1 2σ 2 r2βFrr dt + σrβFrdW = αF dt + σpF dW σp = σrrβFr F , σp = rβσrDmod,

where, Dmod is the modified duration given in Section 2.1.3. The price

volat-ility formula using beta and yield volatvolat-ility is:

σp = rβσrDmod. (2.7)

2.3.5 Historical Volatility

The first way to estimate the yield volatility is by using some time-series models such as ARCH and GARCH [2]. The yield volatility calculated using this method is called historical volatility. It is also known as backward-looking volatility as it shows how volatile security has been in the past. The historical volatility is defined as the standard deviation of the movements in the price of an asset [6]. The fluctuations in the price of an asset refer to the variation of returns of an asset. The historical volatility is calculated over a fixed time interval i.e. daily, weekly, monthly or yearly and it is used to make comparison with current market volatility. The first step to calculate the historical volatility empirically is to find the daily log return of an asset [4]. The log returns help to measure the variation in the price of an asset.

Rt= ln

pt

pt−1

. (2.8)

where, pt is the price at time t and pt−1 is the price at time t − 1. The next

step is to calculate σ that is the standard deviation of the log return Rt. The

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σ2 = E(X2) − E(X)2, σ2 = n P t=1 (xt− ¯x)2 n − 1 . where ¯x is ¯ x = 1 n n X t=1 (xt).

and standard deviation is the square root of variance

σday = v u u u t n P t=1 (xt− ¯x)2 n − 1 . (2.9)

where xt is the relative price change (logarithmic value), ¯x is the sample

mean, and n is the total number of observations which. The calculation of log return and the standard deviation is given as daily measure but the volatility is quoted on an annual basis which means we need to convert the daily measure to the annual by multiplying with the annual trading. We have assumed there were 250 days in a year however this number varies between 250 and 260 depending on public holidays and weekends. comparison is based on the yearly volatility. The volatility is the given as:

σannual= σday

√

d. (2.10)

where, d is the number of trading days in a year.

2.3.6 Volatility Curve

The volatility curve is a graph where we plot the strike price of an option against the implied volatility. The strike price is plotted on the x-axis whereas the implied volatility is on the y-axis of the graph. The implied volatility depends on the time to maturity and the strike price. The options with dif-ferent strike prices but the same underlying have difdif-ferent implied volatility. The volatility curve can be either in the shape of a smile or a skew [6]. The

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volatility smile shows the relationship between the option strike price and implied volatility. The volatility skew is a term that is used when the volat-ility smile is non-symmetric. The volatvolat-ility smile for equity options tends to be downward sloping which means in-the-money call options and out-of-the-money put options have high implied volatility while out-of-the-out-of-the-money call options and in-the-money put options have low implied volatility [4]. The volatility skew shows that all options with same underlying and expiration can have different implied volatility.

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Chapter 3 Methodology

This section covers some basic information about the models that will be used in the empirical analysis. The models that we have selected to work on include Black Model, Normal Black Model, Hull-White Model and SABR Model. This section will help the reader to revise some primary knowledge and theory related to the models.

3.1 The Black Model

The Black model is also known as the Black-76 model and it is the most common model used to price interest rate options, caps, floors and swaptions (an option on a swap). This model was developed in 1976 by Fischer Black who is one of the authors of the Black-Scholes equation. The formula was developed to price options on forwards and assumes the underlying asset is log-normally distributed [7]. When the model is used to price a cap or a swaption the underlying forward rate of a cap and the underlying swap rate is assumed to be log-normal. The Black-76 model is a generalization of the Black-Scholes model because if someone applies the Black model to an equity option where S(T ) = S(0)erT_{, the result is the Black-Scholes formula. The}

basic assumptions under the Black model are as follow:

• The underlying forward rate or swap rate is log-normally distributed. • The volatility of the underlying asset is constant.

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• There is continuous trading in all instruments.

The Black formula is similar to Black-Scholes except the spot price of un-derlying asset is replaced by a forward price F . In Black model the forward rate F is given as:

dF = σBF dW (t),

where σB is the Black volatility, F is forward rate, and W (t) is the Wiener

process. The price function for a European call and put option is: CB = e−rT [F · N (d1) − K · N (d2)] .

PB = e−rT [K · N (−d2) − F · N (−d1)] .

where T is time to maturity, F is the forward or future price, K is the strike price of an option, σ is the volatility or standard deviation, r is the risk free interest rate which is assumed to be constant and N is the cumulative standard normal distribution function. The d1 and d2 are defined as:

d1 = ln(F/K) + (σ_B2/2)T σB √ T , d2 = ln(F/K) − (σ2 B/2)T σB √ T , d2 = d1− σB √ T ,

If we have Normal volatility we can find Black (log-normal) volatility using the conversion formula given below:

σB = 2 √ TN −1 σN 2F r T 2π + 1 2 ! . The Normal volatility must satisfy the following condition:

σN ≤ F

r 2π

T

The Black formula is considered as theoretically inconsistent because both the forward rate and spot rate cannot be log-normal simultaneously. However, the reason behind its popularity indicates that the inconsistency is negligible. The traders based on their experience balance the inconsistency by adjusting the volatility [7].

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3.2 The Normal Black Model

The Normal Black model is the extension of the Black-Scholes model and it is also known as Bachelier’s model [7]. The Normal Black model follows normal distribution which means it deals with the negative interest rates. The main difference between the normal and log-normal distribution is the shape of the curve. The normal distribution is symmetrical and bell shape as compared to a log-normal distribution[4]. The log-normal models cannot be used when the forward rate L or the strike price K is near zero or negative because the terms d1 and d2 in the Black-Scholes formula include logarithm

i.e. ln(L/K). In this case, the market will use the normal distribution to quote the prices [7]. On the other hand, the traders sometimes think that the normal distribution can describe the market more effectively. The Black model with normal volatility is known as the Black-Normal model. The normal Black model is derived from the risk-neutral stochastic process for the forward rate F given as:

dF = σNdW (t),

where, σN is the Normal volatility of forward rate and W (t) is the Wiener

process. The price for a put and call option (i.e a caplet) is given as:

CN = e−rT " (F − K)N (d) + σN √ T √ 2π e −d2 2 # . PN = e−rT " (K − F )N (−d) + σN √ T √ 2π e −d2 2 # . where d = F − K σN √ T, If K = F , d = 0 gives: CN = e−rT σN √ T 2π .

There is a possibility to convert from one volatility type to another for ex-ample from Black volatility to Normal volatility. The conversion between

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Normal volatility and Black volatility is given as: σN = F r 2π T " 2N σB √ T 2 ! − 1 # .

where σN is the Normal volatility and σB is the Black volatility.

3.3 Term Structure Models

The term structure models are used to price fixed income derivatives i.e. the securities or assets that depends on the interest rate level such as options, futures and forwards. These models describe the transformation of all zero-coupon interest rates [4]. The term structure of interest rates indicates the relationship between the interest rate and different time to maturities and it is in the form of a graph which is known as the yield curve [5]. The essential components that are involved in term structure models are drift, the structure of volatility and the distribution. The short rate process is given as:

dr(t) = µ(t, r(t))dt + σ(t, r(t))dW (t).

where, r is the yield or short-term interest rate that is a stochastic process, µ is the expected change or drift and W (t) is a standard Brownian motion and the change in W is normally distributed. The drift and volatility are functions of interest rate r and time t.

3.4 One-Factor Term Structure Models

The one-factor model indicates that in a short time period the rates move in the same direction but it doesn’t mean they move by the same amount [4]. The rate here means the short rate r at which a person can borrow money or it is a profit that is earned from a zero-coupon bond. As the name suggests the one-factor model focus on one variable (that is the interest rate in this case) to observe the result. This section doesn’t include all one-factor models, in fact, it incorporates two normal models i.e. Vasicek model and Hull-White [7].

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3.4.1 Vasicek Model

The Vasicek model was proposed by Oldrich Alfons Vasicek in 1977 and it is a one-factor term structure model that is used to measure the interest rate movements [7]. The model is used for the valuation of interest rate derivat-ives and it has two assumptions i.e. the short rate is normally distributed (symmetric about the mean) and the short rate is described as a mean re-version which means the interest rates will return towards the mean. This is the first model that deals with the mean reversion property. The property of normal distribution means that the model allows negative interest rates. The mean reversion is the important attribute of interest rate that makes it different from other financial prices. This means the interest rate doesn’t move drastically (like stocks) and it moves in a limited range. The model is arbitrage-free which means it doesn’t allow arbitrage opportunities. The dynamics of Vasicek model under the risk neutral measure is given as:

dr = a(b − r)dt + σdW (t). or

dr = (b − ar)dt + σdW (t).

where, a, b and σ are constants. The constant a indicates the mean reversion speed and to maintain stability we need this variable positive, b is the long-term mean level, t is the current time, σ is the volatility of the interest rate, d is the derivative or change and W is the market risk factor presented in the form of Wiener process. If the difference between b and r is greater it means that the expected change in the short term rate towards b is greater [9]. The term (b − ar) is the drift factor which shows the expected change in the interest rate. The drift factor can be positive or negative based on the movement of interest rate for example if the rate is below equilibrium the drift is positive. In the Vasicek model the price at time t of a bond that pays $1 at the maturity T is given as [4]:

p(t, T ) = A(t, T )e−B(t,T )r(t). r(t) is the short rate at time t, where

B(t, T ) = 1 − e

−a(T −t)

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and

A(t, T ) = exp (B(t, T ) − T + t)(a

2_{b −} σ2 2 ) a2 − σ2B(t, T )2 4a ,

If the volatility is given in terms of yield we need to convert the yield volatility to price volatility which is the topic of our thesis. The bond price volatility in the Vasicek model is given as [7]:

σp =

σr

a (1 − e

−a(T −t)

).

where, σp indicates the price volatility, T is the maturity. The change in the

sign is made to deal with the negative interest rates.

3.4.2 Hull-White Stochastic Volatility Model

The Hull-White model is the extended Vasicek model and it was introduced by John Hull and Allan White in 1990 [7]. The model lies in the category of no-arbitrage models which means the traders cannot make a risk-free profit. The H-W model has time-dependent parameters (which means the paramet-ers are a function of time) which provide an accurate fit to the initial term structure [4]. The H-W model has the same assumptions as of the Vasicek model i.e. normal distributed short rates and mean reversion. This model is getting popularity because it deals with the negative interest rates due to the assumption of normal distribution. The stochastic differential equation for Hull-White model is written as follows:

dr = (b(t) − a(t)r)dt + σ(t)dW (t). or dr = a b(t) a − r dt + σ(t)dW (t).

where, the parameter b(t) is the deterministic function of time, a is the mean reverting parameter, σ is the volatility parameter and W is the Brownian motion. The prices at time t of a bond maturity at T is given as:

p(t, T ) = A(t, T )e−B(t,T )r(t). where

B(t, T ) = 1 − e

−a(T −t)

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and A(t, T ) = p(0, T ) p(0, t)exp B(t, T )f (0, t) −σ 2 4a(B(t, T ) 2_{(1 − e}−2at ) ,

The H-W model can be described as the Vasicek model with a time-dependent reversion level or it can be described as Ho-Lee model with mean reversion at rate a [4]. The short rate reverts to b(t)_a at the rate a and time t.

If we are given yield volatility we need to find the price volatility to know the fluctuations in the price. We need a conversion formula to convert between the yield and price. So, the price volatility of the Hull-White model is same as the Vasicek model which is given below:

σp = σr a (1 − e −a(T −t) ). (3.1)

3.5 SABR Model

The Stochastic Alpha Beta Rho model was proposed by Hagan et al.[3] in 2002. According to Hagan et al, the local volatility models predict the wrong dynamics of the volatility curve and the SABR model accurately predicts the volatility smile of different volatility models [3]. The model is used in the valuation of financial securities like caps, floors and swaptions. The model attempts to capture the volatility smile in derivatives market [7]. In this model, the price of an asset and the volatility is correlated. The model specifies a single forward rate F and the volatility of forward rate by the parameter σ. The SABR model is simple compared to other volatility models and it can be attained from the Black’s formula. This model is widely used in the interest rate market because it provides a good fit to the implied volatility and it also captures the dynamics of the volatility smile accurately [3].

3.5.1 The Original Formula

In this model, the variables F and σ are stochastic variables and the model is given by the following SDEs:

dFt = σtFtβdW (t),

F (0) = f

dσt = νσtdZ(t),

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where, W (t) and Z(t) are two correlated Wiener processes with correlation coefficient −1 < ρ < 1 and ρ, ν and β are constants.

dW (t)dZ(t) = ρdt.

The variable Ft is the underlying forward rate, f is the current underlying

forward rate, σt is the volatility, α is the current or initial volatility that is

constant, β is the skewness parameter usually 0 ≤ β ≤ 1, ν is the vol-of-vol (volatility of volatility) and ρ is the correlation between forward rate and volatility and it controls the volatility skew. The model is specified after the stochastic parameters α, β and ρ. The choice of parameter β is not crucial since it has little effect on the volatility curve produced by the SABR model. The β parameter allows the model to switch between normal and log-normal model. If β = 1 the model is suppose to be log-normal and for β = 0 the model is normal. The log-normal and normal means the forward rate is normally distributed or log-normal distributed. There is a special case in which we assume β = 0.5 which means the CIR stochastic mode.

3.5.2 Conversion Between Log-Normal and Normal

Volat-ility

The β = 1 gives the Black-Scholes model which is a log-normal model. The Black log-normal model is:

dF = σBF dW (t),

F (0) = f

where the parameter f is the current forward rate and σB is the implied

Black log-normal volatility. The Normal Black model is: dF = σNdW (t),

F (0) = f

where the parameter σN is absolute or normal volatility. The beta equals to

zero represents the stochastic normal model or the Normal Black model. The model is used in the trading markets where the forward rates are negative or zero [3]. The normal volatility σN as log-normal volatility σB is:

σN = σB f − K ln(f /K) · 1 1 + ₂₄1 (1 −₁₂₀1 [lnf /K)]2_σ2 Bτ + 1 5760σ 4 Bτ2 .

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where, f is the current forward rate, K is the strike price and τ is the time to exercise date in years. When f → K the formula goes to ”0 over 0”. To avoid this we will use the alternative formula given below:

σN = σB p f · K 1 + 1 24[ln(f /K)] 2 1 + ₂₄1σ2 Bτ + 1 5760σ 4 Bτ2 . (3.2) when f − K K < 0.001.

3.5.3 Conversion Between Normal and Log-Normal

Volat-ility

When we are given absolute normal volatility and we want to evaluate the log-normal (Black) volatility we need to solve the above equation for σB. To

find σB when σN is given:

σB = σN ln (f /K) f − K 1 + 1 24 1 − 1 120ln[f /K] 2 σ2N · τ f · K . when f − K K ≥ 0.001. and σB = σN √ f K 1 + ₂₄1σ2 Bτ 1 + 1 24log 2 f K . when f − K K < 0.001.

3.5.4 Conversion Between Normal and CEV Volatility

The constant elasticity of variance (CEV) model is a stochastic volatility model with skewness parameter β and parameter α which refers to volvol (the log-normal volatility of the volatility parameter σ). The model is given as follows:

dF = αFβdW (t), F (0) = f

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The following formula can be used to convert CEV volatility α into a normal (absolute) volatility. σN = α (1 − β)(f − K) f1−β _{− K}1−β 1 1 + 1− 2−2β+β2 120 [lnf /K]2β(2−β)α2τ 1−(1−β)2₁₂ [lnf /K]2_{24(f K)}1−β .

In the case when f is near to K or when β is near to 1, we need to replace the formula with one that doesn’t have the singularity at β = 1 or f = K. To avoid this possibility we will use the alternative formula given below:

σN = α(f K) β 2 1 + ₂₄1[lnf /K]2 1 + (1−β)₂₄ 2[lnf /K]2 1 1 + 1− 2−2β+β2 120 [lnf /K]2β(2−β)α2τ 1−(1−β)2₁₂ [lnf /K]2_{24(f K)}1−β . (3.3) when (1 − β) f − K K < 0.001.

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Chapter 4 Empirical Analysis and

Discussion

This section covers the empirical findings and discussion about the results that we have achieved using the models.

4.1 Data Selection

On the courtesy of Jan R¨oman, I have received the market data to perform the empirical analysis. The bonds SE GEV 2Y, SE GEV 5Y, and SE GEV 10Y are used in the study and they are not real tradable bonds but synthetic bonds. In the calculation of normal bonds, the number of days to maturity decreases day by day for example a bond with two-year a maturity today will have maturity two year minus one day tomorrow. The time-series data consists of two sets:

• Bootstrapped government curves in SEK.

• Prices of the bonds 2Y, 5Y, and 10Y expressed as YTM.

The time-series is based on bootstrapped curves. It means every day at Swed-bank they feed all government instruments such as bills, notes, and bonds to a risk system. The data is then used to bootstrap a risk-free interest rate zero-coupon curve with predefined nodes (e.g. 1Y, 2Y, ...., 10Y). Since the data that I have received is already a time-series of bootstrap data therefore we do not have to bootstrap the curve our-selves. Table B.1 shows the data

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of bootstrapped government curve for a period of one month.

The rates and prices are given in the currency SEK. The data is selected for one year i.e. from 2019-04-09 to 2020-04-09. The bond coupon rate is considered as 6% whereas the face value of a bond is 100 SEK. The prices of 2Y, 5Y, and 10Y expressed as YTM are given in the appendix Table B.2.

4.2 Bond Prices

The dirty price is the price that we pay to purchase a bond. The first step is to calculate the dirty prices of these synthetic bonds because the prices are given in yield to maturity. We know that the dirty price is equal to the clean price plus accrued interest [7]. The first step is to calculate the bond prices, and for this, we will refer to the formula given in Section 2.1.1. The formula is used to calculate the prices for 2Y, 5Y, and 10Y synthetic bond and we have considered the coupon rate as 6%. The 2Y bond will give two coupons, the 5Y bond will give five coupons, and similarly, the 10Y bond will have ten coupons. The small example for the calculation of dirty prices is given below:

• Bond Price for 2Y maturity

P = 6 (1 + ytm₁₀₀)1 + 106 (1 + ytm₁₀₀)2 = 6 (1 + −0.486₁₀₀ )1 + 106 (1 + −0.486₁₀₀ )2 = 113.067182 • Bond Price for 5Y maturity

P = 6 (1 + −0.223₁₀₀ )1 + 6 (1 + −0.223₁₀₀ )2 + . . . + 106 (1 + −0.223₁₀₀ )5 = 131.324247 • Bond Price for 10Y maturity

P = 6 (1 + 0.39₁₀₀)1 + 6 (1 + 0.39₁₀₀)2 + . . . + 106 (1 + 0.39₁₀₀)10 = 154.915192

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Figure 4.1: Bond Prices for 2Y bond

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Figure 4.3: Bond Prices for 10Y bond

Figure 4.1 shows the bond prices for the 2Y bond where 2Y means a bond with a two-year maturity. The x-axis of the graph indicates the dates whereas the y-axis indicates the bond prices. Figure 4.2 shows the bond prices for the 5Y bond whereas Figure 4.3 shows the bond price for the 10Y bond. We can clearly see from the graph that the bond prices for all three bonds went down after the pandemic outbreak in March 2020.

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4.3 Zero-Coupon Prices

The next step is to strip the coupons for 2Y, 5Y, and 10Y using the rates. This step is important because the formula that is derived in [7] uses zero-coupon prices, therefore we need to calculate the zero-zero-coupon prices. To calculate the zero-coupon price we first need the present value of coupons. An example below shows the calculation of present value PV of coupon:

• PV of Coupon 2Y P V = 6 (1 + −0.00456)1 + 6 (1 + −0.00456)2 = 12.083 • PV of Coupon 5Y P V = 6 (1 + −0.00175)1 + 6 (1 + −0.00175)2 + . . . + 6 (1 + −0.00175)5 = 30.158 • PV of Coupon 10Y P V = 6 (1 + 0.0034)1 + 6 (1 + 0.0034)2 + . . . + 6 (1 + 0.0034)10 = 58.889

Now, we have both values i.e. the dirty price of the bond and the present value of the coupon that we need to calculate the zero-coupon price. The formula for the zero-coupon price is as follow:

• ZCP 2Y

ZCP = P − P V (coupons) = 113.067 − 12.082 = 100.985

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• ZCP 5Y ZCP = 131.324 − 30.158 = 101.1661924 • ZCP 10Y ZCP = 154.915 − 58.889 = 96.026

Figure 4.4 shows the zero-coupon prices of bonds with maturity 2- years, 5-years, and 10-years. The zero-coupon price of the 10Y bond fluctuates from 96 SEK and going up to 105 SEK.

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4.4 Annual Price Volatility

The calculation of rates and prices will be different because there is a dif-ference between the rates and the price. The prices depend on the previous days, which means the bond prices have a memory, and we have to com-pare today’s price to the last day price. The first step in calculating the price volatility is to find the logarithmic return Rt using Equation (2.8). If

we have pt equal to 113.067 and pt−1 equal to 113.085 then the logarithmic

return will be given as follow:

Rt= ln

113.067 113.085 = −0.00016

We know the volatility can be measure in terms of variance or standard

Figure 4.5: Plot of 2Y Log-Returns

deviation, and we can calculate variance using Equation (2.9). The other way to calculate the standard deviation is to use the function STDEV.S (divided by T-1) in Excel. The function STDEV.S is used when our data represents the sample of the population. For the 2Y bond, we have the

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standard deviation of 0.00036 using STDEV.S. The next step is to calculate the annual price volatility and, we will use Equation (2.10) to find the annual volatility of the bonds.

σannual = 0.000363 ×

√ 250 = 0.573%

In Figure 4.5, we plot the log returns of 2Y bond over 1-year period. We can see that the mean of log-return is almost zero. We can see large moves in the end of 2019 and in March 2020. Table 4.1 shows the calculation of annual price volatility using log-returns method.

Table 4.1: Annual Price Volatility Using Log-Returns Prices STDEV.S Annual Volatility

2Y 0.00036 0.573%

5Y 0.00123 1.936%

10Y 0.00321 5.072%

The other way to find price volatility is to find simple returns using the given formula.

Rt = Pt/Pt−1− 1

where, Ptis the price at time t and Pt−1 is the price at time t − 1. If we have

pt equal to 113.067 and pt−1 equal to 113.085 then the price change will be:

= (113.067/113.085) − 1 = −0.000156

The next step is similar to the above, where we will find the standard devi-ation using Excel and multiply with the number of days in a trading year. The annual price volatility using Equation (2.10) is given as follows:

σannual = 0.00036 ×

√ 250 = 0.573%

Table 4.2 shows the annual price volatility of bonds using simple returns method.

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Table 4.2: Annual Price Volatility Using Simple Returns Prices STDEV.S Annual Volatility

2Y 0.00036 0.573%

5Y 0.00122 1.936%

10Y 0.00321 5.068%

Figure 4.6: Plot of 2Y Simple Returns

The same calculations have been performed on zero-coupon prices to find the price volatility. Table 4.3 shows the zero-coupon bond price volatility using the log-returns method.

Table 4.4 shows the price volatility of zero-coupon bond using simple returns method.

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Table 4.3: Zero-Coupon Price Volatility Using Log-Returns Prices STDEV.S Annual Volatility

ZCP 2Y 0.00038 0.604%

ZCP 5Y 0.00133 2.101%

ZCP 10Y 0.00392 6.198%

Table 4.4: Zero-Coupon Price Volatility Using Simple Returns Prices STDEV.S Annual Volatility

ZCP 2Y 0.00038 0.604%

ZCP 5Y 0.00133 2.100%

ZCP 10Y 0.00392 6.193%

4.5 Annual Yield Volatility

The calculation of annual yield volatility is different from the price volatility because rates have no memory, but the prices have a memory. The interest rates are independent, which means the interest rate today is independent of the foregoing rate. The first step is to calculate daily volatility or the standard deviation of the rates using Equation (2.9). For a 2Y bond, we have a standard deviation 0.00134. The next step is to calculate the annual yield volatility which is calculated using Equation (2.10).

σannual = 0.00134 ×

√ 250 = 2.118%

Table 4.5 shows the annual yield volatility of the bond: Table 4.5: Annual Yield Volatility Rates STDEV.S Annual Volatility

2Y 0.00134 2.118 %

5Y 0.00153 2.419 %

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4.6 Data Analysis

To get the feel of the data and to understand the distributions, we will plot the histogram. We will also plot two distributions i.e. normal distribution and log-normal distribution, with the histogram. The histogram is a graph that is in the form of a bar chart, and it helps to analyze the data distribu-tion. The histogram is not only the graphical representation of the frequency distribution but also identifies the tail behavior. In other words, we can say the histogram gives a scratch approximation of the actual distribution. The histogram is useful when we are working with a large data set, and it determ-ines how often a value occurs in a data set. There are different tools that we can use to plot the histogram like Excel and Python. The Jupyter Notebook is used in this study to make the graphs. The Python code is given in the appendix. We have the option to select the number of bins but, if we don’t add the number of bins the software will set the bins automatically. The bins are the intervals that divide all the data. In our case, the bins are equal to 50.

As mentioned above we have combined graphs for the comparison i.e. histo-gram with a normal distribution and histohisto-gram with a log-normal distribu-tion. These combined graphs help to study the shape of the distribudistribu-tion. The normal distribution is a bell-shaped curve and it is symmetrical which means if the graph is cut into half the one side of the graph is the mirror image of the other side. There are two parameters that define the normal distribution i.e. mean and standard deviation. The mean determines the location of the extreme value (known as peak) whereas the standard deviation defines how far the values tend to fall away from the mean in the normal distribution curve. The normal distribution shows how the values are distributed. The parameter mean defines the center of the distribution whereas the parameter standard deviation defines the spread. The log-normal distribution is used when the prices are not negative. The log-normal distribution takes only the positive values and that is the reason behind the long right-hand tail. There are two parameters that define the log-normal distribution i.e. location and scale. The location is the mean and the scale is the standard deviation of the logarithm.

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4.6.1 Histogram with Normal and Log-Normal

Distri-bution for 2Y, 5Y and 10Y

In figures 4.7, 4.8 and 4.9, we plot the frequency distribution of bonds simple returns over a 1-year period. The x-axis of the histogram indicates the simple returns of the prices, and the y-axis indicates the frequency. The simple returns are selected instead of prices because the prices give same plots for both distributions. It happens due to the fact that the prices have a memory. We didn’t use log-returns because the graphs with log-normal distribution and negatives prices are not acceptable as prices can never be negative. The frequency in the graph tells how many times the value has occurred.

Table 4.6 shows the descriptive statistics of a 2-year, 5-year, and 10-year bond prices. The descriptive statistics are a set of values that summarize the data-set and provide information about the data. We have information about total values in the data i.e. mean, standard deviation, minimum and maximum values, and quartiles.

Table 4.6: Descriptive Statistics of 2Y, 5Y, and 10Y Bond

2Y 5Y 10Y count 249 249 249 ¯ x 1.00004 1.00833 0.99989 σ 0.00047 0.00675 0.003203 min 0.99785 .99705 0.98841 25% 0.99990 1.00241 0.99825 50% 1.00000 1.00727 0.99958 75% 1.00015 1.01354 1.00151 max 1.00488 1.02225 1.01430

We can see in Figure 4.7 and 4.9 the distribution have a higher central peak. We can also observe the probability density functions do not differ a lot in the graphs. The reason for the similar probability density function is because the standard deviation is small.

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Figure 4.7: Normal and Log-Normal distribution of bond prices with matur-ity 2-years

Figure 4.8: Normal and Log-Normal distribution of bond prices with matur-ity 5-years

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Figure 4.9: Normal and Log-Normal distribution of bond prices with matur-ity of 10-years

Volatility- An investigation of the relationship between price- and yield volatility