## Term Structure Modeling

The framework outlined in this chapter demonstrates the links between swap, bond, and FRA markets. We will now discuss the term structure implications of the derived formulas. The set of formulas we studied implies that, given the necessary information from two of these markets, we can, in principle, obtain arbitrage-free prices for the remaining market.10 We discuss this briefly, after noting the following small, but significant, modification. Term structure models concern forward rates as...

## Case Study Danish Mortgage Bonds

Danish mortgage bonds are liquid, volatile, and complex instruments. They are traded in fundamentally sound legal and institutional environments and attract some of the best hedge funds. Here, we look at them only as an example that teaches us something about swaps, swaptions, options, and the risks associated with modeling household behavior. To answer the questions that follow, you need to have a good understanding of these notions and instruments 1. Plain vanilla interest-rate swaps 3....

## Forward Contracts

Forwards, futures contracts, and the underlying interbank money markets involve some of the simplest cash flow exchanges. They are ideal for creating synthetic instruments for many reasons. Forwards and futures are, in general, linear permitting static replication. They are often very liquid and, in case of currency forwards, have homogenous underlying. Many technical complications are automatically eliminated by the homogeneity of a currency. Forwards and futures on interest rates present more...

## Which Volatility

This chapter dealt with four notions of volatility. These must be summarized and distinguished clearly before we move on the discussion of the volatility smile in the next chapter. When market professionals use the term volatility, chances are they refer to Black-Schole's implied volatility. Otherwise, they will use terms such as realized or historical volatility. Local volatility and variance swap volatility are also part of the jargon. Finally, cap-floor volatility and swaption volatility are...

## Ln SK r 2ff2T t

The C(t) is the value of the vanilla call given by the standard Black-Scholes formula, and the J(t) is the discount that needs to be applied because the option may disappear if St falls below H during t, T . But we now know from the contractual equation that a long knock-in and a long knock-out call with the same strike K and threshold H is equivalent to a vanilla call 12 It is important to remember that these assumptions preclude a volatility smile. The smile will change the pricing. Using...

## InArrears Swaps and Convexity

Although an overwhelming proportion of swap transactions involve the vanilla swap, in some cases parties transact the so-called Libor in-arrears swap. In this section we study this instrument because it is a good example of how Forward Libor volatilities enter pricing directly through convexity adjustments. But first we need to clarify the terminology. In a vanilla swap, the Libor rates Lti are assumed to set at time ti whereas the floating payments are made in arrears at times ti+1.30 In the...

## Strategies

Hedge funds are classified according to the strategies they employ. The market a hedge fund uses is normally the basis for its strategy classification. Global macro funds bet on trends in financial markets based on macroeconomic factors. Positions are in general levered using derivatives. Such derivatives cost a fraction of the outright purchase. Managed futures. These strategies speculate on market trends using futures markets. Often computer programs that use technical analysis tools like...

## Smile

Options were introduced as volatility instruments in Chapter 8. This is very much in line with the way traders think about options. We showed that when we deal with options as volatility instruments mathematically we arrived at the same formula, in this case the same partial differential equation PDE as the Black-Scholes PDE. Mathematically the approach was identical to the standard textbook treatment that considers options as directional instruments.4 Yet, although the interpretation in...

## Dv01

In general for small changes in c the annuity factor will be close to the DV01. These defaultable DV01 and annuity factors are important tools to credit traders and play equivalent roles in default-free swap and bond trading. This completes the discussion of the fixed leg of the CDS contract. We can similarly calculate the expected payments, denoted by Pay by the protection seller using the same concepts. We obtain E Pay B to,t1 p B to,t2 1 p p B io,is 1 - p 2p B to,t4 1 - p 3p B to, t5 1 - p...

## Info

Then we can express the cash flows of an equity swap as the exchange of at each ti. The di being an unknown percentage dividend yield, the market will trade this as a spread. The market maker will quote the expected value of di and any incremental supply-demand imbalances as the equity swap spread. 5. The swap will involve no upfront payment. This construction proves that the market expects the portfolio Sti to change by Lti-1 - dti each period, in other words, we have, This result is proved...

## What Is a Synthetic

The notion of a synthetic instrument, or replicating portfolio, is central to financial engineering. We would like to understand how to price and hedge an instrument, and learn the risks associated with it. To do this we consider the cash flows generated by an instrument during the lifetime of its contract. Then, using other simpler, liquid instruments, we form a portfolio that replicates these cash flows exactly. This is called a replicating portfolio and will be a synthetic of the original...

## Eur Usd Quanto Swap

In a quanto swap, one party would like to receive 6-month USD Libor and pay 6-month JPY Libor for 1 year. However, all payments are made in USD. For example, if the first settlement is according to the quotes given in the table, in 6 months this party will receive 30,000,000 .0164 i - 30,000,000 0.00185 1 - 30,000,000 1 c 76 where the c is a constant spread that needs to be determined in the pricing of this quanto swap. Note that the JPY interest rate is applied to a USD denominated principal....

## The Swap Logic

Swaps are the first basic tool that we introduced in Chapter 1. It should be clear by now that swaps are essentially the generalization of what was discussed in Chapters 1, 3 and 4. We start this chapter by providing a general logic for swaps. It is important to realize that essentially all swaps can be combined under one single logic. Consider any asset. Suppose we add to this asset another contract and form a basket. But, suppose we choose this asset so that the market risk, or the volatility...

## Figure 1014

Up to this point, the chapter has dealt with option strategies that used only plain vanilla calls and puts. The more complicated volatility building blocks, namely straddles and strangles, were generated by putting together plain vanilla options with different strike prices or expiration. But the use of plain vanilla options to take a view on the direction of markets or to trade volatility may be considered by some as outdated. There are now more practical ways of accomplishing similar...

## Bond PDE

A partial differential equation consisting of the gains from convexity of long bonds and costs of maintaining the volatility position can be put together. Under some conditions, this PDE has an analytical solution, and an analytical formula can be obtained the way the Black-Scholes formula was obtained. First we discuss the PDE informally. We start with the trading gains due to convexity. These gains are given by the continuous adjustment of the hedge ratio ht, which essentially depends on the...

## Equity Repos

If we can repo bonds out, can we do the same with equities This would indeed be very useful. Equity repos are becoming more popular, but, from a financial engineering perspective, there are potential technical difficulties 1. Equities pay dividends and make rights issues. There are mergers and acquisitions. How would we take these events into consideration in a repo deal It is easy to account for coupons because these are homogeneous payouts. But mergers, acquisitions, and rights issues imply...

## Currency Swaps versus FX Swaps

We will now compare currency swaps with FX-swaps introduced in Chapter 3. A currency swap has the following characteristics 1. Two principals in different currencies and of equal value are exchanged at the start date t0. 2. At settlement dates, interest will be paid and received in different currencies, and according to the agreed interest rates. 3. At the end date, the principals are re-exchanged at the same exchange rate. A simple example is the following. 100,000,000 euros are received and...

## Real Life Complications

Real-life complications make dynamic replication a much more fragile exercise than static replication. The problems that are encountered in static replication are well known. There are operational problems, counterparty risk, and the theoretically exact synthetics may not be identical to the original asset. There are also liquidity problems and other transactions costs. But, all these are relatively minor and in the end, static replicating portfolios used in practice generally provide good...

## Figure 107

For small movements in the exchange rate, the position is neutral, but for large movements it represents a hedge similar to a futures contract. Of course, such a position could also be taken in the futures market. But one important advantage of the risk reversal is that it is composed of options, and hence involves, in general, no daily mark-to-market adjustments. The class of option strategies that we have studied thus far is intended for creating synthetic short and long...

## Ad Hoc Synthetics

Then how can we replicate the two-period bond There are several answers to this question, depending on the level of accuracy a financial engineer expects from the synthetic. An accurate synthetic requires dynamic replication which will be discussed later in this chapter. But, there are also less accurate, ad hoc, solutions. As an example, we consider a simple, yet quite popular way of creating synthetic instruments in the fixed-income sector, referred to as the immunization strategy. In this...

## What Is Repo

We begin with the standard definition. A repo is a repurchase agreement where a repo dealer sells a security to a counterparty and simultaneously agrees to buy it back at a predetermined price and at a predetermined date. Thus, it is a sale and a repurchase written on the same ticket. In a repo, the dealer first delivers the security and receives cash from the client. If the operation is reversed that is to say, the dealer first buys the security and simultaneously sells it back at a...

## Figure 813

The Greeks seen thus far are not the only sensitivities of interest. One can imagine many other sensitivities that are important to market professionals and investors. In fact, we can calculate the sensitivity of the previously mentioned Greeks themselves with respect to St, a,t, and r. These are higher-order cross partial derivatives and under some circumstances will be quite relevant to the trader. Two examples are as follows. Consider the gamma of an option. This Greek determines how much...

## Libor and Other Benchmarks

We first need to define the concept of Libor rates. The existence of such reliable benchmarks is essential for engineering interest rate instruments. Libor is an arithmetic average interest rate that measures the cost of borrowing from the point of view of a panel of preselected contributor banks in London. It stands for London Interbank Offered Rate. It is the ask or offer price of money available only to banks. It is an unsecured rate in the sense that the borrowing bank does not post any...

## Options Definition and Notation

Option contracts are generally divided into the categories of plain vanilla and exotic options, although many of the options that used to be known as exotic are vanilla instruments today. In discussing options, it is good practice to start with a simple benchmark model, understand the basics of options, and then extend the approach to more complicated instruments. This simple benchmark will be a plain vanilla option treated within the framework of the Black-Scholes model. The buyer of an option...

## Case Study Japanese Loans and Forwards

You are given the Reuters news item below. Read it carefully. Then answer the following questions. 1. Show how Japanese banks were able to create the dollar-denominated loans synthetically using cash flow diagrams. 2. How does this behavior of Japanese banks affect the balance sheet of the Western counterparties 3. What are nostro accounts Why are they needed Why are the Western banks not willing to hold the yens in their nostro accounts 4. What do the Western banks gain if they do that 5....

## Volatility as an Asset Class and the Smile 439

Introduction to Volatility as an Asset Class 439 5. Application to Option Payoffs 444 6. Breeden-Litzenberger Simplified 446 7. A Characterization of Option Prices as Gamma Gains 450 8. Introduction to the Smile 451 10. A First Look at the Smile 453 11. What Is the Volatility Smile 454 13. How to Explain the Smile 462 14. The Relevance of the Smile 469 17. Exotic Options and the Smile 471 18. Conclusions 475 Suggested Reading 475 Exercises 476

## Figure

One difference between securities lending and repo is in their quotation. In securities lending, a fee is quoted instead of a repo rate. Nominal GBP 10 million 8.5 12 07 05 is lent for 2 weeks Collateral GBP 10.62 million 8 10 07 06 Total fee 50 bp x 14 days 360 x GBP10 million Obviously, the market value of the collateral will be at least equal to the value of borrowed securities. All other terms of the deal will be negotiated depending on the credit of the borrowing counterparty and the term....

## Market Conventions

Market conventions often cause confusion in the study of financial engineering. Yet, it is very important to be aware of the conventions underlying the trades and the instruments. In this section, we briefly review some of these conventions. Conventions vary according to the location and the type of instrument one is concerned with. Two instruments that are quite similar may be quoted in very different ways. What is quoted and the way it is quoted are important. As mentioned, in Chapter 1 in...

## Engineering Interest Rate Swaps

We now study the financial engineering of swaps. We focus on plain vanilla interest rate swaps. Engineering of other swaps is similar in many ways, and is left to the reader. For simplicity, we deal with a case of only three settlement dates. Figure 5-13 shows a fixed-payer, three-period interest rate swap, with start date t0. The swap is initiated at date t0. The party will make three fixed payments and receive three floating payments at dates t1 ,t2, and t3 in the same currency. The dates t1...

## Exercises

A dealer needs to borrow EUR 30 million. He uses a Bund as collateral. The Bund has the following characteristics Collateral 4.3 Bund, June 12, 2004 a How much collateral does the dealer need b Two days after the start of the repo, the value of the Bund increases to 101. How much of the securities will be transferred to whom c What repo interest will be paid 2. A dealer repos 10 million T-bills. The haircut is 5 . The parameters of the deal are as follows Maturity of T-bills 90 days Repo...

## Principles of Dynamic Replication

We now go back to the issue of creating a satisfactory synthetic for a short bond B t0, T2 using the savings account Bt and a long bond B t0,T3 . The best strategy for constructing a synthetic for B t0,T2 consists of a clever position taken in Bt and B t0, T3 such that, at time t1, the extra cash generated by the Bt adjustment is sufficient for adjusting the B t0,T3 . In other words, we give up static replication, and we decide to adjust the time-t0 positions at time t1, in order to match the...

## The Greeks and Their Uses

The Black-Scholes formula gives the value of a vanilla call put option under some specific assumptions. Obviously, this is useful for calculating the arbitrage-free value of an option. But a financial engineer needs methods for determining how the option premium, C t , changes as the variables or the parameters in the formula change within the market environment. This is important since the assumptions used in deriving the Black-Scholes formula are unrealistic. Traders, market makers, or risk...

## Forward Rate Agreements

A forward loan contract implies not one but two obligations. First, 100 units of currency will have to be received at time ti, and second, interest Ft0 has to be paid. One can see several drawbacks to such a contract 1. The forward borrower may not necessarily want to receive cash at time 11. In most hedging and arbitraging activities, the players are trying to lock in an unknown interest rate and are not necessarily in need of cash. A case in point is the convergence play described in Section...

## Consider A 2-year Currency Swap Between Usd And Eur Involving Floating Rates Only. The Eur Benchmark Is Selected As

d What is your net position in terms of present value e How would you hedge this with a 4-year swap Which position would you take, and what should the notional amount be f Where would you go to get this hedge g Can you suggest another hedge 4. Suppose at time t 0, we are given prices for four zero-coupon bonds B1 , B2,B3,B4 that mature at times t 1,2,3, and 4. This forms the term structure of interest rates. We also have the one-period forward rates f0, f1,f2,f3 , where each fi is the rate...

## Repo Market Strategies

The previous sections dealt with repo mechanics and terminology. In this section, we start using repo instruments to devise financial engineering strategies. The most classic use of repo is in funding fixed-income portfolios. A dealer thinks that it is the right time to buy a bond. But, as is the case for market professionals, the dealer does not have cash in hand, but he can use the repo market. A bond is bought and repoed out at the same time to secure the funds needed to pay for it. The...

## Additional Terminology

We would like to introduce some additional terms and instruments before moving on. A par swap is the formal name of the interest rate swaps that we have been using in this chapter. It is basically a swap structure calculated over an initial and final nominal exchange of a principal equal to 100. This way, there will be no additional cash payments at the time of initiation. An accrual swap is an interest rate swap in which one party pays a standard floating reference rate, such as Libor, and...

## Types of Repo

The term repo is used for selling and then simultaneously repurchasing the same instrument. But in practice, this operation can be done in different ways, and these lead to slightly different repo categories. A classic repo is also called a U.S.-style repo. This is the operation that we just discussed. A repo dealer owns a security that he or she sells at a price, 100. This security he or she immediately promises to repurchase at 100, say in 1 month. At that time, the repo dealer returns the...

## Short and Long Positions

A long position is easier to understand because it conforms to the instincts of a newcomer to financial engineering. In our daily lives, we often buy things, we rarely short them. Hence, when we buy an item for cash and hold it in inventory, or when we sign a contract that obliges us to buy something at a future date, we will have a long position. We are long the underlying instrument, and this means that we benefit when the value of the underlying asset increases. A short position, on the...

## Futures

Up to this point we considered forward contracts written on currencies only. These are OTC contracts, designed according to the needs of the clients and negotiated between two counterparties. They are easy to price and almost costless to purchase. Futures are different from forward contracts in this respect. Some of the differences are minor others are more important, leading potentially to significantly different forward and futures prices on the same underlying asset with identical...

## Show How To Construct A Synthetic 9-month Loan With Fixed Rate Beginning With

You have purchased 1 Eurodollar contract at a price of Q0 94.13, with an initial margin of 5 . You keep the contract for 5 days and then sell it by taking the opposite position. In the meantime, you observe the following settlement prices Q 1 94.23, Q2 94.03, Q3 93.93, Q4 93.43, Q5 93.53 60 a Calculate the string of mark-to-market losses or gains. b Suppose the spot interest rate during this 5-day period was unchanged at 6.9 . What is the total interest gained or paid on the clearing firm...

## Taxation Example

Now consider a totally different problem. We create synthetic instruments to restructure taxable gains. The legal environment surrounding taxation is a complex and ever-changing phenomenon, therefore this example should be read only from a financial engineering perspective and not as a tax strategy. Yet the example illustrates the close connection between what a financial engineer does and the legal and regulatory issues that surround this activity. In taxation of financial gains and losses,...

## Contractual Equation

Once an instrument is replicated with a portfolio of other liquid assets, we can write a contractual equation and create new synthetics. In this section, we will obtain a contractual equation. 5 In fact, bringing in a forecasting model to determine the Ft0 will lead to the wrong market price and may create arbitrage opportunities. 6 Remember the important point that, in practice, both the liquidity and the credit risks associated with the synthetics could be significantly different. Then the...

## Day Count Conventions

The previous discussion suggests that ignoring quotation conventions can lead to costly numerical errors in pricing and risk management. A similar comment can be made about day-count conventions. A financial engineer will always check the relevant day count rules in the products that he or she is working on. The reason is simple. The definition of a year or of a month may change from one market to another and the quotes that one observes will depend on this convention. The major day-count...

## Credit Indices and Their Tranches 547

Introduction to ABS and CDO 548 4. A Setup for Credit Indices 550 6. Tranches Standard and Bespoke 555 7. Tranche Modeling and Pricing 556 8. The Roll and the Implications 560 9. Credit versus Default Loss Distributions 562 10. An Important Generalization 563 12. Conclusions 568 Suggested Reading 568 Appendix 18-1 569 Exercises 570