## Of r hf 2 1 r ff

The quantity (jFr'sher was used in the previous chapter in the numerical correlation uncertainty study. We notice that the (l-r2) factor makes < 7F,sher vanish at 1. Naturally as TV > , the statistical uncertainty is zero and the value r p is recovered. We re-emphasize that this entire framework assumes that there exists in fact one given value for the correlation. Perversely, in the real world, correlations have important non-stationary intrinsic instabilities that have nothing to do with...

## Calculation of Issuer Risk Generic Case

We can determine the issuer credit risk due to downgrade and default by straightforward simulation. We merely run through the portfolio9 one bond at a time10, get the distribution of losses for the portfolio, pick out the portfolio loss at some given confidence level, subtract the average loss, and write up a nice report for the management with some pretty color plots11. We will explicitly consider a pedagogical simple case of a portfolio of plain-vanilla bonds, a single transition default...

## Path Integral Warmup The Black Scholes Model

The reader may already be familiar with the Black-Scholes (BS) model. The goal here is partly to put old wine in new bottles and to exhibit the path integral formalism for those who are unfamiliar with it. We start with demonstrating the compatibility of path integrals with the standard no-arbitrage framework. At the end of the section, we re-derive the same results using a more compact and more straightforward approach in which no-arbitrage appears as a simple parameter specification. Appendix...

## Attacks on Economic Capital at High CL

Various critiques have been levied at Economic Capital, sometimes by smart traders pushing back, and sometimes by quants. The main issue is that Economic Capital is expensive, focuses on rare events that are hard to measure, and might be used in assessments of risk that influence compensation. We present three arguments, which we call attacks on EC, because that is what they are. These arguments have varying degrees of relevance. First Attack Misses Lascaux Cave Paintings and an Ergodic...

## J1il

1 Path Integrals and Term Structure Interest-Rate Modeling Dash, J. W., Path Integrals and Options II One-Factor Term-Structure Models, Centre de Physique Th orique, CNRS, Marseille preprint CPT-89 PE.2333, 1989. Hull, J. and White, A., Pricing Interest Rate Derivative Securities. Review of Financial Studies 3,4 1990. Pp. 573-92, and references therein. Jamshidian, F., An Exact Bond Option Pricing Formula, Journal of Finance, Vol 44, 1989, Pp. 205-209. Jamshidian, F., Pricing of Contingent...

## Historical Simulations and Hedging

An historical simulation can be run to test the efficacy of hedging strategies. We briefly outline the formalism. Take a portfolio V x,t of the security C x,t and hedge H xk,tk at a number of successive points in the past xk,tk with historical data used for xk. The hedge has a superscript because of a hedge rebalancing strategy for example delta hedging , so the hedge function itself changes. At i ,f 1 the new hedge is the old hedge Hplus the tf ' 1 W . V k W 1 Ww 15.1 Revalue the portfolio Vk...

## LE

1 More About Skew A lot more information about skew for equity options is in the next chapter. E jm 0.71 GBP I USD that pays off if r lt E '. The fear that the USD might depreciate this much can also place a premium on this volatility o E7 'Jnv . Since these two options are the same thing, ct ET Low a E hh . Thus, vol premia for both low and high strikes can exist this is the smile. Because the fear intensities are generally not equal, the currencies generally being of different strengths, the...

## Enhanced Stressed Var Esvar

The ES-VAR, or Enhanced Stressed VAR, is the most refined version of VAR that we shall consider. The Stressed attribute means that risks further out in the tails will be considered. The Enhanced attribute means that other attributes lending a more realistic aspect to the VAR will be included. The ES-VAR includes the refinements of the IPV-VAR given above, plus the items in the following table We discuss these enhancements one at a time. Higher VAR CLs for Stressed VAR and Economic Capital The...

## Principal Component Risk Measures Tilt Delta etc

We have been describing principal-component yield-curve options designed to hedge against various movements of the yield curve. In order to systematize the discussion of yield-curve movement risk, a class of risk measures can be constructed with principal components13. For example, we can define first-order quantities tilt delta or flex delta arising from constructing a tilt or flex movement of the yield curve and observing the change in the portfolio value under such a movement. Similarly,...

## Resettable Options Cliquets

Resettable options are another type of option whose strike is not fixed. They are generally baskets of component options. Note that a basket of options is different from an option on a basket, so we are talking about sets of options, not options on a composite variable. The component options have the feature that their strikes are unknown at the start, and are dependent on the behavior of the underlying variable at fixed times in the future. That is, the strikes are reset, so these options...

## Variance at t is a2aj I dlxcAti dtxa

9 Third Method for Stressed Correlations Using Principal-Components There is, theoretically, a third method for getting random correlations using principal components. The idea is to generate changes in the eigenvalues, along with rotations of the eigenfunctions. However, this method is far removed from the changes in the physical correlations, because many variables contribute to each principal component. For this reason, we prefer the methods in the text dealing directly with correlations....

## Diagram for one swaplet

Now we consider the change in the characteristics of a swap under a hypothetical simple scenario. The scenario is that all forward rates f J regardless of maturity are raised by 10 bp keeping time fixed, i.e. df mrio 10bp. Sometimes rates can change suddenly by this magnitude, e.g. in less than 1 day. We could also envision a time-dependent scenario, although the simplest and most common procedure is to separate out the time dependence by moving time forward while keeping rates fixed. The...

## The CVAR Volatility with Two Variables

Here, we restrict our attention to two variables. We begin with the CVAR volatility. Here is a picture of the geometry The CVAR volatility' turns out to be the same for both variables. Both triangles with CVAR uad, CVARf'ad have a common leg, the CVAR volatility lt 7cvar Writing the correlation pn cos 6, the CVAR volatility is 1 Synopsis For those of you who just tuned in, CVAR volatility measures the uncertainty in the contribution of risk of the corresponding variable to the total VAR. The

## The CVAR Volatility Triangle

The CVAR Volatilities for the Nonlinear Risk Case The existence of the CVAR volatility is not at all limited to the linear case. The uncertainty in the components of risk for a given total risk is a general concept. Moreover, it turns out that even for somewhat non-linear portfolios, the CVAR volatility formula 28.23 is a reasonably good approximation. This has been explicitly checked by Monte-Carlo generated CVAR volatility with convexity approximated by using a grid. Only for highly convex...

## Rr I Irurrr I814

So, for a given swap, the off diagonal gamma matrix elements are approximately given by the square root of the geometric average of the two diagonal matrix elements. In order to motivate this, note that yir involves the variation of the discount factors. The relatively simple form of the discount factors then leads after some algebra to the above approximate relation. We also need to insert the sign. We assume that the gamma matrix has elements all of the same sign. For a swap, this sign is...

## Interest Rate Swaptions Tech Index 510

We described interest-rate caps in the last chapter. A cap is a collection or basket of options caplets , each written on an individual forward rate. A swaption, on the other hand, is one option written on a collection or a basket of forward rates, namely all the forward rates in a given forward swap1. The fact that the swap option is written on a composite object means that correlations between the individual forward rates are critical for swaptions. Swaptions are European if there is only one...

## J

For example, if the r2 convention is Q360 with f2 - 4, Ndays 2 - 360, we get r2 0.0490lyr. This is ten basis points less than for the r, convention. Such an amount is definitely significant, being larger than the bid-ask spread for many swaps in the market. See the footnote for another convention25 called 30 360. Compounding rates is related to rate conventions, but the emphasis is different. Instead of coming up with two representations of the same rate, we wish to generate a composite rate...

## J a b v

However, interchange of the multiple jda jdx integrals is problematic if the volatility becomes infinite even if large volatilities are suppressed , because the Gaussian spatial damping for large x is not uniform. It makes a difference whether we let jc for fixed a or we let T gt at fixed x , or we take a scaled limit as both variables become infinite. The non-uniformity in the two variables x and lt r just described would seem to pose a problem for stochastic volatility models that do not have...

## Summary Outline Book Contents

Qualitative Overview of Risk A qualitative overview of risk is presented, plus an instructive and amusing exercise emphasizing communication. II. Risk Lab for Derivatives Nuts and Bolts of Risk Management The Risk Lab first examines equity and FX options, including skew. Then interest rate curves, swaps, bonds, caps, and swaptions are discussed. Practical risk management including portfolio aggregation is discussed, along with static and time-dependent scenario analyses. This is standard...

## Hybrid 2Dimensional Barrier Options Examples

The two-dimensional barrier options are single barrier options that are dependent on two different variables. They are part of a class of options also called hybrid options14. Hybrid options can be formulated in a variety of markets. Typically, what happens is that an option depending on one variable is knocked in or knocked out at a barrier depending on a second variable. The two variables are correlated together15. In this section, we give an example. The mathematical apparatus 16 is given in...

## Tips

Which depends on inflation and yield changes 8g and Sy, along with the correlation2 p p 8g g ,8y y . Increased inflation 8g gt 0 increases the TIPS value, while increased yields Sy gt 0 decreases the TIPS value. Now if a yield change Sy occurs, we expect on the average a correlated inflation change as S g 58 y where J3 - lognormal inflation, yield volatilities3. This particular idea is rather similar to that used in the CAPM model in another context1. The above expression for 8g is the parallel...

## Caplet Kinematics one forward rate

Diffusion of forward rate fl starts with value fi i0 at time tQ, and ends at maturity t . The strike of the corresponding caplet is E. Paths in at out of the money are notated as ITM ATM OTM, respectively. We first discuss standard Libor caps, and then go on to discuss Prime caps and CMT caps. Other rates e.g. CP, Muni are also used'. We present the standard market model that uses lognormal forward rates. The new ingredient for options is, of course, volatility. The picture above illustrates a...

## Forward Rate Kinematics

Today's forward rate curve is the set of f T gt tn AT for different T, already been placed in portfolios where they are held to term. Therefore, these get less weight in the fit. By the way, government securities are known as Govies. 13 Sliding Down the Yield Curve Because securities that have already been issued do not have their original time to maturity, the jargon is that the securities have slid down the yield curve from the point where they were issued. The picture is that the yield...

## Swaplet reset times located between IMM dates

Date until the payment date for the entire cash flow. However, if the swap is sold, the cash to be paid from the reset to the transaction date is sometimes set on an accrual basis not including discounting. 12 The End-Year Effect There is an end-year effect that happens at the end of the year which leads to anomalies in the money market and which needs to be hedged separately. For the year 2000, this effect was very large. 13 Hedging with ED Futures The liquidity of the ED futures is greatest...

## Municipal Derivatives Muni Issuance Derivative Hedging

Muni derivatives are treated with formalism similar to that of Libor-based derivatives. The main assumption is that the Muni forward rates at different maturities T are fractions lt 1 of Libor forward rates f or gt v'z fLl, T fair The fractions can be written as lt r 1 - r.jTJ , with a forward tax rate rj - The idea of course is that municipal cash bond interest is Trader's Intuition Before the TIPS were issued, one influential trader forcefully promoted the idea that the TIPS would remain at...

## The Arbs and the Mispricing of the CVR Option

Significant arbitrageur activity for a potential M amp A deal is often present, where arbs bet on the deal going or not going through with appropriate positions in the ABC stock. This can have an undesirable effect from the point of view of ABC. If the arbs own the CVR, which is some form of put, they want the ABC stock to go down near the exercise or payoff time. Therefore, near payoff, they may start to sell ABC stock or go short. This puts downward pressure on the stock, increasing the CVR...

## Receivers Swaption Deal

ABC gets cash Cswt now, sells swaption Broker-Dealer buys swaption for price CSW, In strategic terms, ABC thinks rates are low now and thus E is relatively low even given the extra spread , making it desirable to act now. BD gets the extra spread. Thus, the deal can appear attractive from both sides. 14 ABC In fact ABC was the Metropolitan Transportation Authority in New York. The announcement was in 1999. See Bloomberg News ref. 15 Beauty Contest The activity described in the text is called a...

## Case Study DECS and Synthetic Convertibles

Some simple forms of convertibles are synthetic convertibles called DECs. A convertible bond is a security has both bond and embedded equity options with possibility of conversion to stock1. A synthetic convertible is generally simpler than a true convertible bond. A synthetic convertible example is the DECS Debt Exchangeable for Common Stock or Dividend Enhanced Convertible Stock . It is common to use DEC for short pronounced deck the plural is then somewhat inconsistently DECs. DECs are...

## The Black Caplet Formula

The classic Black formula for the caplet value evaluated at time t0 is4 Here f t , where we use carets A indicating that the forward rates are determined today. The time interval to the caplet maturity is r,. is the th notional, in cursive to avoid confusion with the normal functions N dl . Also is the zero coupon bond price for discounting in arrears payment made after reset , dt is the time interval between resets 1 4 year for 3 month Libor , E is the rate strike which can also depend on the...

## Interest Rate Swaps Pricing and Risk Details

In this section, we follow up the introduction to swaps with some more detail. Consider a broker-dealer swap desk BD that transacts a new swap with a customer ABC. Typically, the new swap risk to BD will be hedged immediately. This new deal also may be considered in the context of a portfolio of deals already held by BD. The actual hedge put on by BD for the new deal may depend on what happens to be in its book, as well as the risk appetite and view if any of BD regarding the market, perhaps...

## Info

Include for example buying bonds or buying ED contracts or both ' . As we saw, the total delta or DV0127 from a swap model for this swap is short 1714 equivalent ED contracts28. This means that if ED futures were to constitute 24Bond Futures and the CTD A bond future as opposed to ED futures requires delivery of a bond to the holder of the bond future from a party that is short the bond future. However, this does not mean delivery of a definite bond, but rather of any bond chosen at liberty...

## Counterparty Credit Risk and Swaps

A counterparty of a deal is the just other party to the deal, for example the other side of a swap. We will look at the risk from the point of view of a brokerdealer BD. The counterparty will be called ABC. The counterparty risk for BD is the risk that the counterparty ABC defaults on some condition of the deal. Counterparty risk can be calculated using multi-step Monte-Carlo MC simulations of underlying variables, along with models for the securities at future times V1. The MC simulations move...

## Bermuda American Swaption Pricing

A Bermuda swaption has a discrete exercise schedule usually every 6 months after a lockout period during which no calls are allowed . Swaptions and callable bonds are closely related. A 10NC3 bond means a 10-year bond that is callable after 3 years. The call option embedded in the bond corresponds to a Bermuda swaption, which can be exercised after 3 years. The exercise prices usually vary with time in a schedule. For a non-amortizing swap, we have seen that we can rewrite a receiver's swap as...

## Miscellaneous Swaption Topics

Liquidity and Basis Risk for Swaptions As mentioned already, Bermudas and Americans are highly illiquid in the secondary market. Thus, considerable basis risk exists with European volatility. Typically, limits will be set depending on the risk tolerance for this basis risk. Fixed Maturity vs. Fixed Length Forward Swap Generally, the forward swap arising from swaption exercise has a fixed maturity date. Sometimes the forward swap lasts for a fixed time period after the date of exercise. For...

## Yjyjtt aj2dt

Because y t - y xe xf_x from the change of variable, we find Noting that dyf dE f jEs_x and dx f dSf S t we can cast this equation into a notation closer to Derman's. This ends the discussion of local volatility. Intuitive Models and Different Volatility Regimes The heavy artillery of the local volatility described above contrasts with two simple intuitive models for the movement of the implied volatilities with changes in the stock price. These are called the sticky-strike model and the...

## Acknowledgements

First, I owe a big debt of gratitude to Andy Davidson, Santa Federico, and Les Seigel, all super quants, for their support. The work of two colleagues contributed to this book Alan Beilis who collaborated with me on the Macro-Micro model , and Juan Castresana who implemented numerical path-integral discretization . I thank them and the other members of the quant groups I managed over the years for their dedication and hard work. Many other colleagues helped and taught me, including quants,...

## Hedging Volatility and Vega Ladders

We naturally have vega ladders to describe the details of the volatility dependence of interest-rate products. Each caplet has its own vega, defined as the sensitivity of the caplet value to its own volatility. These vegas correspond to the maturities of the caplets. An illustrative vega ladder for a 5-year forward cap Vega futures equivalents Vega futures equivalents The vega normalization here is in futures equivalents, defined as one future equivalent 2500, for a change in vol of 1 . If a...

## Prime Caps and a Vega Trap

The Prime rate is the interest rate that US banks charge to their most creditworthy customers. The Prime rate generally changes across the banking industry, depending on macroeconomic conditions, when a major bank decides to change it. Prime caps provide insurance against increases in the Prime rate. Now because the Prime rate is only changed sporadically, it has a behavior that does not look like diffusion at all. Rather the Prime rate has a step-like behavior where it is fixed for a...

## Basket Options and Index Options

We have used the word stock as if it were a single stock. However the most common stock options correspond to stock index options, e.g. options on the S amp P 500 index. The index B is a linear combination, or basket, of stocks Sa with certain weights wa , viz B t waSa t . Hence, an index stock option is actually an option on a basket. The first and still most common theoretical assumption made for stock price movements is lognormal namely relative changes or returns 2Za t dS a t S a t of a...

## U

Here a looks like a discrete label and for the trinomial model a 1,2,3 but will actually be continuous for our derivation, i.e. we will integrate over the variable . We use the subscript i because Derman's idea is to use gadgets at Note that this is not static replication. As we saw above, static replication is concerned with the determination of a set of options that can be used to replicate a boundary condition, e.g. a barrier for an up and out call. Derman used a trinomial model to...

## Vol Skew Example

What this graph indicates is that if a single volatility is put into the usual stock-option Black-Scholes formula, then in order to reproduce the market option prices this volatility decreases as a function of the strike of the option. Physically, this condition seems imply that fear is greater than greed. Thus, a premium for OTM puts with low strikes to protect the downside exists fear , relative to 1 Data Data are the averaged call and put midpoint DEC 1997 S amp P index option vols on 5 14...

## Quants in Quantitative Finance and Risk Management

First, what is a Quant This is a common though not pejorative term mostly applied to PhDs in science, engineering, or math doing various quantitative jobs on Wall Street4, also called The Street. We start with jobs involving models. Risk is measured using models. Here the standard paradigm is that models are developed by PhD quants writing their own code, while systems programmers develop the systems into which models are inserted. A quant writing a model to handle the risks for a new product...

## N

This procedure clearly only works if the off-diagonal gamma matrix elements are small compared to the diagonal terms. This can produce misleading results with non-parallel yield-curve shifts. How can we see this Imagine a forward rate 34 So Which Delta Ladder does Your Report Show In the text, we have exhibited several possibilities. Maybe the guy who programmed the risk system has just left the firm to become a junior trader. Did he document the formula that he programmed Hmm 35 Sick Gamma...

## Hedging FX Options with Greeks Details and Ambiguities

The Greeks are used for option risk management. It is important to understand that the Greeks do not have unique definitions. Although this information is in this section on FX options, the same remarks in this section apply to equity options. For FX, there are ambiguities depending on the normalization, as above. Because the spot changes day-by-day indeed minute by minute , if the option is divided by the spot there is a change in the formula for the delta hedging of the option. Besides the...

## Hedging a Swap with Treasuries and ED Futures

Total 5-10 yr risk Hedge with 10 yr treasury Total 0-2 yr risk Hedge with first 8 ED futures The hedge components that hedge the deal in the various buckets can be found using a computer minimization routine. The result amounts to a back-chaining algorithm. In practice, a simpler hedge could be executed e.g. a rough hedge only with 10-year treasuries and later adjusted to a complicated hedge. The algorithm indicated in the drawing above is as follows The 10-15 year tail risk of the 15-year swap...

## Delta Ladder Swap

Here, the total delta is A A, and the total gamma is y yir. We now consider the delta ladder A, . We will see that there are several possible definitions. First, the variation of the discount factors, while complicated, contributes as a rule of thumb only around 10 to delta. So to get an approximate but reasonable estimate of delta, you can ignore the discount factors. In this approximation, delta A, in ED contracts for the l'h forward rate is obtained by two steps 1 Differentiate the swaplet...

## Irp

Since IRP is fundamental, it is necessary to define the drift for each variable separately, consistent with IRP, and inconsistently with 77 l lt f. It further adds to the confusion that the change of variables 77 in the standard FX option formula expressed in terms of does in fact lead to the standard option formula expressed in terms of 77. This is due to a fortuitous cancellation. The cancellation does not occur for digital options or rebates for example. It is not clear to me whether...

## Data Statistics and Reporting Using a Spreadsheet

Find the 3-month cash Libor rate and the interest rates corresponding to the prices of the first twelve Euro-dollar 3-month futures1. Keep track of each of these thirteen rates every day2 for two weeks3 using the spreadsheet program Excel4'1 and note the rate changes each day. 1 Libor and ED Futures There are a number of different interest rates used for different purposes. You need to spend some time learning about the conventions and the language. Libor is probably the most important to...

## Scenario Analysis Introduction

It should be clear that the use of the Greeks will not work well if large moves in the stock price occur. For example, a short11 put option with a large notional that is far OTM7 might contribute negligibly to the risk. However, if the market suddenly tanks12, this option can suddenly become ATM with large risk. Conversely, an apparent large risk may not mean much. For example, a short put option ATM near expiration with large negative gamma may have little real downside risk13. For this...

## Bare replicating portfolio for Up Out Call intrinsic value shown for simplicity

Again, the idea is that the skew of this portfolio is used in order to generate the skew of the barrier option. The procedure is in App. A. 8 Skew for other barrier options Other single barrier options can be obtained using sum rules. For example, the down and out DO call can be obtained from the DO put and the DO forward. Up and in UI options can be obtained from the standard options and the DO options. Double-barrier option skew can be obtained approximately by using a perturbative approach...

## Some Practical Details for FX Options

There are several ways to quote the results for options and a number of different conventions for reporting the Greeks. There are also modifications of the option formula to take account of the specific features of the FX options market. The overall normalization factor for the option formula is important. We need to express the option price in units of a definite currency and get rid of the FX currency ratio. The same option can be reported in four ways Method 1. Call on XYZ using the variable...

## Repeat Part 1 Using Programming

Instead of the spreadsheet, write a program in your favorite computer language along with file inputs to perform the same steps as in Exercise Part 1. Document your source code16 by clearly writing at the top what it is you are doing, in good English with complete sentences. If you skipped the memo and verbal communication, it's time to bite the bullet17. Print out your report and graph18. 14 Second Warning The Most Dangerous Word in the English Language is It Again, the probability is 100 that...

## And Options499

Path Integrals and Options Overview Tech. Index 4 10 501 42. Path Integrals and Options I Introduction Tech. Index 7 10 505 Introduction to Path Path-Integral Warm-up The Black Scholes Model 509 Connection of Path Integral with the Stochastic Equations 521 Dividends and Jumps with Path Discrete Bermuda American Appendix A Girsanov's Theorem and Path Integrals 538 Appendix B No-Arbitrage, Hedging and Path Integrals 541 Appendix C Perturbation Theory, Local Volatility, Skew 546 Figure...

## Table of Contents

PART I INTRODUCTION, OVERVIEW, AND EXERCISE 1 1. Introduction and Who How What, Tech. Index, Messages, Personal Note 3 Summary Outline Book 2. Overview Tech. Index Objectives of Quantitative Finance and Risk Management 7 Tools of Quantitative Finance and Risk The Traditional Areas of Risk When Will We Ever See Real-Time Color Movies of Risk 13 Many People Participate in Risk Quants in Quantitative Finance and Risk Management 15 3. An Exercise Tech. Index Part 1 Data, Statistics, and Reporting...