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Acknowledgements
INTRODUCTION, OVERVIEW, AND EXERCISE
Introduction and Outline
This book is primarily for PhD scientists and engineers who want to learn about quantitative finance, and for graduate students in finance programs. This book will enable you to gain an understanding of practical and theoretical quantitative finance and risk management.
Summary Outline: Book Contents
Overview (Tech. Index 1/10)
In this review, we look at some general aspects of quantitative finance and risk management.
Objectives of Quantitative Finance and Risk Management
While human error should not be underestimated, the main problem in finance often lies in uncertainties and incompleteness in the models and/or the data. We will discuss one example in determining the uncertainty in risk when we discuss the uncertainties in the components of risk (CVARs) that lead to a given total risk (VAR) at a given statistical level.
Tools of Quantitative Finance and Risk Management
Some of the hedging variables may have little to do with the portfolio variables, and thus represent a good deal of additional risk on their own. Knowing changes in historical data in different time frames plays a big role in risk assessment, especially abnormally large movements, as well as the magnitudes of movements at different statistical levels.
The Traditional Areas of Risk Management
Traditional credit risk assessment by rating agencies includes financial statistics (balance sheet, cash flows), comparisons and in-depth analysis. Conversely, if credit risk is increasing due to economic weakness, markets will not be bullish.
When Will We Ever See Real-Time Color Movies of Risk?
Many People Participate in Risk Management
In a single-tier risk structure, the corporate risk management department is in direct contact with traders at a designated desk. Risk management structure: The paradigm adopted depends on the risk management philosophies of the trading offices and corporate risk management.
Quants in Quantitative Finance and Risk Management
A message to PhD scientists and engineers who want to become quants For model building and risk management, you must be able to program fluently in at least one language (C, C++ or Fortran). If you learn too much about quantitative analysis, finance and systems - and if you know how to manage people - you can become the head of the Quant Group.
An Exercise (Tech. Index 1/10)
The idea is not just to read the exercise and chuckle, but to actually try it. Please note that the footnotes provide a running commentary and expansion of the topics in the main text.
Part #1: Data, Statistics, and Reporting Using a Spreadsheet
You shouldn't skip it, because if you can't write what you did clearly, you might not get paid as much. Again, you shouldn't skip it, because if you can't clearly describe verbally what you did, you won't get paid as much.
Part #2: Repeat Part #1 Using Programming
You've just done the same exercise in "production mode" as opposed to a "prototype mode" spreadsheet.
Messages and Advice
PART I1
RISK LAB (NUTS AND BOLTS OF RISK MANAGEMENT)
Equity Options (Tech. Index 3/10)
Just for balance, we cover some topics in the FX options chapter that are directly applicable to stock options and vice versa.
Pricing and Hedging One Option
Definition of a model: Including the types of parameters as part of the model is not an empty formality. You may now need to attend meetings to check the status of the system.
American Options
Once the critical path is determined, the fraction of paths that cross the critical path at each time interval can be found, for example by Monte Carlo simulation. The figure above has multiple bins in time for ten paths with a stylized (constant) critical path.
Basket Options and Index Options
Determining the current critical path is done using a chain-back procedure from the time of the last exercise along with a profit maximization dictum - if at a given time, exercising the option is more profitable than the expected return of holding the option, the option is exercised. . In practice, basket volatility can use the above formula for small custom basket deals.
Other Types of Equity Options; Exotics
Portfolio Risk (Introduction)
Scenario Analysis (Introduction)
FX Options (Tech. Index 4/10)
We discuss quant options and correlations, FX options in the presence of stochastic interest rates, and comment on numerical codes and sanity checks.
FX Forwards and Options
The result should be the same as if we kept the USD in the USD bank at domestic rate rd. My definition for 5 is the same as in Andersen's book ’ and is like any price, e.g. Cross FX rates UVW I X Y Z are treated in exactly the same way.
Some Practical Details for FX Options
However, say broker-dealer BD (which sells USD settled on X) is in the US. Then BD must buy GxN pesos at expiration t* with a number of GID = GxN /q* dollars, where q* is the exchange rate at t*.
Hedging FX Options with Greeks: Details and Ambiguities
Variations in procedure may be apparent when volatility is low and/or the option is far from ATM. The hold (ie the term from the moving interest rates rd ,rf) occurs from settlement to the settlement date of the option.
FX Volatility Skew and/or Smile
Risk Return Convention: The price of a risk return is the difference between the call and put prices in USD. The market is also quoted in terms of the difference in implied USD call and put volatilities.
Pricing Barrier Options with Skew
It makes the barrier seem more distant. From the point of view of a drunken gremlin staggering along such a path defined by a given set of corresponding random n ~ m b e r s ’ ~, the barrier at Kt. Option to increase: This means that if the underlying value exceeds the limit at any time before the option expires, the call option ceases to exist.
Double Barrier Option: Practical Example
The 100 bp increases in the interest rates are very large for the 3-month period of the option. A "discount" may be part of the contract, which provides for a fixed amount to be paid to the option holder in case a barrier is crossed.
The “Two-Country Paradox”
If one of the barriers is removed, the double barrier option becomes a single barrier option. Again, this is due to the mismatch in the drift for l n q using the change of variable relative to interest rate parity.
Quanto Options and Correlations
Consider e.g. interest term structure constraints in the case that a physical interest rate r ( t ) is derived from a change of variable from another variable y ( t ). The term structure constraints from zero-coupon bonds are used to determine the operation of the physical rates r(r).
FX Options in the presence of Stochastic Interest Rates
Numerical Codes, Closed Form Sanity Checks, and Intuition
Equity Volatility Skew (Tech. Index 6/10)
What this graph indicates is that if a single volatility is plugged into the usual Black-Scholes option formula, then that volatility falls to reproduce market option prices as a function of the option's strike. Thus, a premium exists for OTM puts with low strikes to protect the downside [fear], relative to.
Put-Call Parity: Theory and Violations
Call vol - Put vol
Even these relatively simple exotics are highly illiquid and there is little misinformation readily available. Options on individual stocks usually do not have enough information even to construct a volatility surface (see below).
The Volatility Surface
It is therefore common practice to model the skew, using a model for barrier options that matches the skewed information of the standard options market4. However, the word "share" will be used in this section instead of "index" for clarity.
Dealing with Skew
Local volatility function: Make the volatility in the stock diffusion process at time t a function of the stock price oSkewFi, [ S(t), t] such. Intuitive models: these are called “sticky strike” or “sticky delta / sticky moneyness”; they are sometimes used to describe the stock price's dependence on the skew as time progresses.
Perturbative Skew and Barrier Options
Suppose the call UO has strike E , barrier K and time to expiration or maturity Ti,. Again, the idea is that the tilt of this portfolio is used to generate the tilt of the barrier option.
Static Replication
OTM call option payouts above the barrier are carried back using the standard Green function without the barrier. The standard Green's function (including discounting) for propagation between times t j and t, > tj in the absence of an obstacle is written as G ( x j ,tj;x,tr).
Stochastic Volatility
Now if we claim that the volatility takes value o with probability F [ o ] in do, we can define the volatility-probability-average call option CAvg -' as. Note that although the above two different integrals yield the same C A v g - ", this ends the discussion of stochastic volatility.
Local Volatility and Skew
The Skew-Implied Probability Distribution
Local vs. Implied Volatility Skew; Derman's Rules of Thumb
Derman's "l S t Rule of Thumb" states that local volatility fluctuates with the stock price about twice as fast as the implied volatility fluctuates with the strike. Because the skewed slope is negative, local volatility decreases as the stock price moves up.
Option Replication with Gadgets
This reduction is the negative change in Crall caused by the negative change in volatility Socall for the given increase 6s. Here the weight function G ( R ) ( y l , yg-, ;dtg-, ) is written in terms of the strikes, and will be determined to make the gadget value equal to zero (so that the.
Intuitive Models and Different Volatility Regimes
The Macro-Micro Model and Equity Volatility Regimes
Jump Diffusion Processes
Appendix A: Algorithm for “Perturbative Skew” Approach
Erasing the skewness correction with the probability of elimination: We use the academic formula for the probability of elimination. This is because we are just wiping out the skewness correction that comes from the average elimination time.
Forward Curves (Tech. Index 4/10)
In this chapter we discuss the construction of the forward interest rate curves that are mainly needed for the pricing of interest rate derivatives. We begin with a discussion of the input rates for the forward curve structural models, and then discuss the mechanisms.
Market Input Rates
Exchange rates in other currencies (e.g. Euro, GBP, JPY) provide similar restrictions on pricing agreement data in those currencies. For the Libor curve, Treasuries are actually only used to represent exchange rates as exchange spreads on Treasuries.
Construction of the Forward-Rate Curve
The correction can also be supported numerically by applying the general consistency of the forward rate curve in the futures region with other data”. Admittedly, the construction of the forward rate curve is not well presented in a mathematical sense.
We can find an equivalent 6M forward rate that earns the same interest as using the first 3M forward rate and reinvesting the interest in the mix with the second 3M forward rate. Thus, if the IMM forward rate starts at tl-l and f the IMM forward rate starts at t, then the 3M futures rate f a starts at the intermediate time tl-l < t, < tl and decreases to the corresponding limits at the end points of the supply.
Swaps: Pricing and Risk
From the broker-dealer BD's point of view, the swap in the photo is a fixed-fee swap. The change in forward rates (10 bps) is assumed to be the same for all forward rates – a “parallel shift”.
Interest Rate Swaps: Pricing and Risk Details
Below is a graph of the first 3 years of the delta ladder hedge required for replacement. At the first reset date to the FRN definition is (no fictitious and with rates in decimal).
Counterparty Credit Risk and Swaps
There is therefore a maximum point for the value of the forward swap on this interest rate path, which is around 1.5 years. Such counterparty risk calculations can be a huge undertaking, both in calculation and in collecting the data.
Calculations usually have to be made over a long period of time, corresponding to the possible times when the counterparty may default in the future, until the end of the transactions. Since securities depend on many types of variables, a firm-wide simulator must include multivariate statistics to generate paths.
Types of Bonds
Bonds are obligations of ABC to repay the borrowed money (called the "notional" or "pari" amount) to X on the maturity date of the bond and in some cases earlier. The quantitative analysis of such bonds is complicated; we will examine some of the features in this book.
Bond Issuance
Muni issuance in 2002 was record high following the recent turmoil in the stock markets coupled with recent low interest rates. Most mortgage loans are issued by agencies (FHLMC and FNMA, with GNMA slightly less); there is also a small non-agency or "private label".
Bond Trading
Selling pressure increases dramatically as more and more people abandon strategies and follow the flight to quality. Phase Transitions and Collective Panic: Theoretically, the flight to quality can be thought of as a phase transition, where we move from a highly disordered state (buyers and sellers of different securities in comparable numbers) to a highly ordered state (in essentially just sellers with pending buyers).
Bond Math
For a full description of the intricacies, check out The Bloomberg (press the GOVT button and then type DES and HELP). According to the benchmark curve, OAS is therefore a translation of the bond price, including the effect of options.
Introduction to Caps
This volatility is placed in the lognormal process of the forward rate corresponding to the caplet option expiration date tl*. This leads to inconsistencies in the way the desk views its risk (using the trading model) and the way the risk is reported to corporate risk management (using the reporting model).
The Black Caplet Formula
A procedure is used to obtain a term structure of the volatility for the caplets with different volatilities o(imu'), varying the volatility of the caplets until the market capitalization prices are obtained. A useful analytical approach to cap implied volatility involves averaging, where the weights are the caplet values5.
Non-USD Caps
Relations between Caps, Floors, and Swaps
So in practice, the implied vols on the floor are not exactly the same as the implied cap volumes. As interest rates rise significantly, a ceiling becomes deep ITM, while with the same kinematics the floor becomes deep OTM and worthless.
Hedging Delta and Gamma for Libor Caps
Therefore, the implied volatilities of caps and floors should be identical, provided the transaction kinematics are the same.
Hedging Volatility and Vega Ladders
An implied volatility is sometimes said to give the "market expectation in the future" of the realized volatility of the forward rate. For example, as we saw in the last chapter, the forward rates can change in such a way that the exchange rates do not change.
Matrices of Cap Prices
Vol can be marked at the intermediate level and a reserve is taken for the expected amount to settle the position. The reserve can be incorporated directly into the reporting by e.g. marking the long volume positions to bid volume (i.e. under valuation in relation to mid-volume).
Prime Caps and a Vega Trap
Using the Libor full oLibor and Libor rate RLibor together with an equivalent Libor strike obtained by subtracting the Prime-Libor spread from the Prime strike, ELib(,r - Ehime - sprime-Libor. If we, just by a change in the semantics, we redefine "input vol" to mean the volume used as input for the Black model, then we will change the Prime vol by 1%.
CMT Rates and Volatility Dependence of CMT Products
The CMT swap is much more complicated than the regular swaps discussed in the previous chapter due to the dependence of the CMT interest rate on volatility. However, the short course grid in the CMT rate calculation is lognormal and no negative rates appear in the grid.
European Swaptions
The discount factors serve to discount the payouts of the overdue cash flows in the forward swap back to the value date, today through today. Note that the A ladder will only be significant during the period of the forward swap.
BermuddAmerican Swaption Pricing
Swings in vega for discrete codes can be observed as a function of the amount of change in volatility 60. This is partly because then only a few nodes of the algorithm can be used to determine the swap.
Delta and Vega Risk: Move Inputs or Forwards?
The noise is also increased if the curve generation itself leads to discontinuities in the forward rates, i.e. if the forward rates are not smoothed in the curve generation algorithm itself. Now, as an example, consider moving the 5-year swap rate RS-Yr up, say 10 bp, and holding all other input rates (futures, other swaps) fixed.
Swaptions and Corporate Liability Risk Management
So, to anticipate this, ABC has already built into the swap contract the option to cancel the swap at the same times as the bond call schedule. In this way, the link and the rest of the exchange disappear at the same time.
Practical Example: A Deal Involving a Swaption
Once bought by ABC, BD cannot sell the Bermuda exchange because there is no market for it. If rates decrease, BD will exercise the Bermuda swap, remove the hedges, and receive the fixed E above market coupon from ABC.
Miscellaneous Swaption Topics
Portfolios and Scenarios (Tech. Index 3/10)
Introduction to Portfolio Risk Using Scenario Analysis
Definitions of Portfolios
Each transaction has its own file listing the details, including the definition of the transaction, the curves used for pricing, the hedging results, etc. Economic capital: See the chapter on economic capital for a discussion of the standalone and diversification risk issues.
Definitions of Scenarios
Note that not doing a 20 year scenario in practice means you are assuming that the Libor rate will be unchanged 20 years from now, which is itself a scenario. Here's the question: What happens to the value of a cap when the forward rate curve steepens.
Many Portfolios and Scenarios
We can either "slide down the forward curve" or not (see box below). After specifying the forward curve of the scenario, we can slide down the curve to define the final scenario, or not.
A Scenario Simulator
We actually have two possibilities for keeping rates "fixed" as time changes from "today" to the scenario date. Either the forward rate on a given date remains fixed, or the forward rate on a given time interval from the value date remains fixed.
Risk Analyses and Presentations
PART I11
EXOTICS, DEALS, AND CASE STUDIES
A Complex CVR Option (Tech. Index 5/10)
CVR will be discussed in some depth to give you an idea of the complexities that sometimes arise. We use the present tense to dramatize the situation as it happened at the time.
The M&A Scenario
This chapter provides a case study of a complex stock option, called a CVR, that was a key part of an M&A deal.
CVR Starting Point: A Put Spread
CVR Extension Options and Other Complications
On the third exercise date, the maximum of the average is compared to the strike prices instead of the median of the average. The specifications included an exchangeable portion of the debt for preferred stock at Viacom's option if the acquisition of Blockbuster was not completed by a certain date.
The Arbs and the Mispricing of the CVR Option
A Simplified CVR: Two Put Spreads with Extension Logic
The same phenomenon occurs with the complete CVR expansion logic, although it is more difficult to visualize due to its complexity. The following graph shows the non-uniqueness of the indifference point in the example with two well distributions:
Non-Academic Corporate Decision for Option Extension
We do not attempt to estimate ,QcorpDecEx, , and simply price the CVR both ways - with and without the corporate decision to extend. This graph shows that we can use the upper indifference point as corresponding to the rational market procedure, and choose the lower indifference point as corresponding to the low stock price $Scor,DecExl at which the corporate decision to expand can be made.
The CVR Option Pricing
Class-8 tracks will stop at t; by the rational market logic and pay the maximum amount $E2upper - $E2Lower. Class-9 roads are a continuation of class-4 roads, which resulted from a previous corporate decision to expand by time tl*.
Analytic CVR Pricing Methodology
The feedback chaining logic for CVR is a direct extension of the logic presented in the example for two sales spreads. In a feedback loop process, the results provide standard academic benchmarks that tell ABC when to disburse and when to extend.
Some Practical Aspects of CVR Pricing and Hedging
The CVR component parts to be priced correspond to the payout logic diagram above. If, hypothetically, the company's decision to extend is made, the value of the CVR corresponding to the above parameters increases by $0.44 to $9.40.
