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## Über dieses Buch

Analytical Finance is a comprehensive introduction to the financial engineering of equity and interest rate instruments for financial markets. Developed from notes from the author’s many years in quantitative risk management and modeling roles, and then for the Financial Engineering course at Mälardaran University, it provides exhaustive coverage of vanilla and exotic mathematical finance applications for trading and risk management, combining rigorous theory with real market application.

Coverage includes:

• Date arithmetic’s, quote types of interest rate instruments • The interbank market and reference rates, including negative rates• Valuation and modeling of IR instruments; bonds, FRN, FRA, forwards, futures, swaps, CDS, caps/floors and others • Bootstrapping and how to create interest rate curves from prices of traded instruments• Risk measures of IR instruments• Option Adjusted Spread and embedded options• The term structure equation, martingale measures and stochastic processes of interest rates; Vasicek, Ho-Lee, Hull-While, CIR• Numerical models; Black-Derman-Toy and forward induction using Arrow-Debreu prices and Newton–Raphson in 2 dimension• The Heath-Jarrow-Morton framework• Forward measures and general option pricing models• Black log-normal and, normal model for derivatives, market models and managing exotics instruments• Pricing before and after the financial crisis, collateral discounting, multiple curve framework, cheapest-to-deliver curves, CVA, DVA and FVA

## Inhaltsverzeichnis

### 1. Financial instruments

Abstract
In the previous book, we studied derivatives in the equity markets and in this book, we will study the available instruments in the interest rate markets. First, we will shortly group the various instruments. In order to group the wide variety of instruments that exist adequately, it is necessary to break the interest rate asset classes into two subdivisions: long-term and short-term debts. In addition, it is necessary to divide the derivatives into two groups: standard derivatives and over-the-counter (OTC) derivatives.
Jan R. M. Röman

### 2. Interest Rate

Abstract
As we will see, there exists many different definitions of interest rates in the markets. A repo trader talks about the simple rate, an option trader of the continuous compounding rate and a bond trader of yield-to-maturity (YTM). We will briefly 0name some of the rates and give a short description. Some of these rates will be discussed in detail in later sections.
Jan R. M. Röman

### 3. Market Interest Rates and quotes

Abstract
In many, if not in all discussion about valuing financial instruments, especially interest rate derivatives, the risk-free interest rate is an important topic. The risk-free interest rate are used to discount projected or expected cash-flows to a present value. But, what rate should be used? A short answer should be that this depends on what instrument to value, the counterparty and the agreements made. A better answer might be that the rate should be chosen to reflect the funding cost of buying the instrument. In this section we will discuss how the market situations in the near future have changed the view about the risk-free interest rate.
Jan R. M. Röman

### 4. Interest Rate Instruments

Abstract
We will now describe some instruments in the interest rate markets, where there exist a huge number of different instrument types. To mention all variants is far out of the scope in this book, if possible at all. Some of these instruments are referred as Fixed Income instruments. The name refer to the fact that all income, that is, all cash flows, are known prior to the actual trade. Bonds are typical fixed income instruments since the coupon rate and the nominal amount are known.
Jan R. M. Röman

### 5. Yield Curves

Abstract
Ordering the current spot yields to maturities for any group of bonds. By maturity we get a so-called yield curve. This curve is often represented as a graph with time to maturity on the horizontal axis and yields on the vertical axis. The group is usually defined as bonds by the same issuer and/or the same credit rating. Thus, we speak of yield curves for government bonds, for mortgage bonds or for corporate bonds of the same credit rating. The word bond here is used in the academic sense which means bills, notes and bonds. Interest rates in international or domestic time deposit markets too can be ordered by maturity and credit class. Thus, we get London inter-bank offered rate (LIBOR) or XIBOR yield curves or yield curves for domestic deposits in any currency. There are many different yield curves.
Jan R. M. Röman

### 6. Bootstrapping Yield Curves

Abstract
We will now explain how to obtain zero-coupon yield curves from market data for coupon bonds or interest rate swaps. To do so, we begin with some simple examples and show how to use linear bootstrapping to find the spot rates and forward rates from a number of benchmark instruments. Also we will show how to use the derived zero-coupon yields to discount future cash flows. Finally, we will use some real market data, such as bonds, deposits, forward rate agreements (FRAs) and swaps in the bootstrap procedure.
Jan R. M. Röman

### 7. The Interbank Market

Abstract
We will now take a look at the Interbank market and different kind of spreads. We explain some of the details using the Swedish market (as Riksbanken, the Central bank in Sweden).
Jan R. M. Röman

### 8. Measuring the risk

Abstract
In this section we present some traditional risk measures based on the present value formula used in the markets for the quoting of prices and yields to maturity (ytms). These measures are calculated by trading software in order to at least partially manage the risk in instruments and portfolios.
Jan R. M. Röman

### 9. Risk management

Abstract
We will now give a short introduction of how to measure risk and how to define limits on risks for a portfolio with many different instruments. Such limits are used by financial institutions to control and minimize risks. There have been more and more focus on risk management, especially after the financial crises in 2007–2008.
Jan R. M. Röman

Abstract
A common method to value bonds, zero bonds and promissory loans with embedded options (that is, callable and putable instruments) is the use of option-adjusted spread (OAS). This model will use a spread on a benchmark curve to calculate bond prices for risky bonds, due to embedded options and since they are so called corporate bonds.
Jan R. M. Röman

### 11. Stochastic Processes

Abstract
Modern pricing models generally use one of two powerful approaches; equilibrium pricing or relative pricing. In an equilibrium framework, certain market characteristics, such as a price risk, are estimated and the model can be used to predict prices for securities in the market. There is no guarantee that the model will price any security at its observed market price. In the relative pricing framework, some observed market prices are used as a starting point, and then other securities are priced relative these.
Jan R. M. Röman

### 12. Term Structures

Abstract
We will now consider the problem where we will model price processes on an arbitrage-free market of zero coupon bonds. On this market we will model the short rate, r(t) under the real probability measure P.
Jan R. M. Röman

### 13. Martingale Measures

Abstract
From now on, we will consider the filtrated probability space $$({\rm{\Omega }},{\cal F},P,\underline {\cal F} )$$ as given where W is a $${\cal F}$$-Wiener process on $$[0,T]$$.
Jan R. M. Röman

### 14. Pricing of Bonds

Abstract
As we have seen the price of a zero coupon bond at $$t = 0$$ and time to maturity T is given by
$$p(0,T) = E_{t,r}^Q \left[ {\exp \left\{ { - \int\limits_0^T {r(s)ds} } \right\}} \right]$$
Jan R. M. Röman

### 15. Term-Structure Models

Abstract
Let us again study an interest rate model where the P-dynamics of the short rate of interest are given by
$$dr(t) = \mu (t,r(t))dt + \sigma (t,r(t))dW(t)$$
Jan R. M. Röman

### 16. Heath-Jarrow-Morton

Abstract
Up to this point we have studied interest models where the short-rate r is the only explanatory variable. The main advantages with such models are as follows.
Jan R. M. Röman

### 17. A new Measure – The Forward Measure

Abstract
In previous sections, we have used two probability measures: the objective (real) probability measure P, and the “risk-neutral” martingale measure Q. In this section we will introduce a whole new class of probability measures, so-called forward measures, including Q as a member of that class. These probability measures are connected to a technique called change of numeraire. They are of great importance both in the understanding and for practical calculations since the amount of computational work needed in order to obtain a pricing formula can be drastically reduced by a suitable choice of numeraire. Especially the forward measures simplify the calculations of prices on bond options.
Jan R. M. Röman

### 18. Exotic Instruments

Abstract
For some exotic instruments, we can use the forward measure pricing described in the previous chapter. We will now describe methods of how we can calculate prices for such kinds of derivatives.
Jan R. M. Röman

### 19. The Black Model

Abstract
The Black-76 modified Black-Scholes model has become the standard model for valuing over-the-counter (OTC) interest rate options, caps, floors and European swaptions. The formula was originally developed to price options on forwards and assumes that the underlying asset is lognormal distributed.
Jan R. M. Röman

### 20. Converibles

Abstract
A convertible bond is a security issued by a company that may be converted from debt to equity (and vice versa) at various prices and stages in the life cycle of the contract (e.g. the time to maturity). There are many types of convertible bonds with various conversion properties and complex structures.
Jan R. M. Röman

### 21. A New Framework

Abstract
Ten years ago if you had suggested that a sophisticated investment bank did not know how to value a plain vanilla interest rate swap, people would have laughed at you. But that isn’t too far from the case today. We will now give an introduction to yield curve constructions and how this has been changed since after the financial crisis.
Jan R. M. Röman

### 22. CVA and DVA

Abstract
For years, a practice in the financial industry has been to mark derivatives portfolios to market without considering counterparty risk. All cash flows were discounted using the LIBOR or another “risk-free” curve. However, the true portfolio value must incorporate the possibility of losses due to counterparty default. This observation has gained wider recognition following the high-profile defaults of 2008. The credit value adjustment (CVA) is by definition the difference between the risk-free portfolio and the true portfolio value which should take into account the possibility of counterparty defaults. In other words, CVA represents the monetized value of the CCR.
Jan R. M. Röman

### 23. Market Models

Abstract
One of the general disadvantages of short rate and HJM models is that they focus on unobservable instantaneous interest rates. The so-called market models, that was developed in the late 90’s tries to overcome this problem by instead modelling observable market rates such as LIBOR and swap rates. These models can be calibrated to the market and have gained a widespread acceptance from practitioners.
Jan R. M. Röman

### 24. A Model for Exotic Instruments

Abstract
In the following section, I refer to articles by Patrick Hagan. The adjustors described here are called “Hagan adjustors”. This is a method of turning bad prices into good prices, We will study the need of pricing and trading an exotic derivative, but because of limitations in our pricing systems, we cannot calibrate on the “natural set” of hedging instruments. Instead, we have to calibrate on some other set of vanilla instruments, which provide only a poor cash-flow replication of the exotic.
Jan R. M. Röman

### 25. Modern Term Structure Theory

Abstract
From the bootstrapped Swap curve, see Section 6.1.4e have a given yield curve. This curve is used for discounting all cash flows. We define the discount function D(t) as
$$D(t) = \exp \left\{{ - \int\limits_0^t {f(0,{T}')d{T}'}} \right\}.$$
Jan R. M. Röman

### 26. Pricing Exotic Instruments

Abstract
In this chapter we will give an introduction of valuation of exotic interest rate derivatives in a Gaussian framework and how to calibrate such models to market data of plain vanilla instruments.
Jan R. M. Röman

### Backmatter

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