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1 Chapter 1 : Riccardo Rebonato Revolvy Interest-Rate Option Models: Understanding, Analysing and Using Models for Exotic Interest-Rate Options (Wiley Series in Financial Engineering) Second Edition by Riccardo Rebonato (Author). This first part also highlights the fundamental conceptual difference between interest-rate volatility on the one hand, and of FX and equity volatility on the other. The author then moves on to the problem of smiles, with considerable emphasis placed on option pricing when markets are incomplete. This second part focuses on the need for end users to take an approach, at the same time practical and theoretically sound, when it comes to implementing the various models which can account for smiles. To this effect, many existing models are reviewed and several new, original approaches are presented. The author points out that the temptation of being seduced by the elegance of mathematical models must be tempered by the need to look at the financial mechanisms driving the dynamics of the specific derivative product. The analysis of the third part deals with the role of volatility and correlation in the context of interest-rate models. In particular, it covers in detail practical and powerful calibration techniques to caplet volatilities and correlation surfaces of the state-of-the-art BGM approach, and suggests criteria to choose the functional form for the all-important instantaneous volatility functions. The new and updated material includes: The book is split into four sections. Part II deals with smiles in the equity and FX worlds. Beginning with a review of relevant empirical information about smiles, this part provides coverage of local-stochastic-volatility, general-stochastic-volatility, jump-diffusion and Variance-Gamma processes. Part II concludes with an important chapter that discusses if and to what extent one can dispense with an explicit specification of a process-based model, and can directly prescribe the dynamics of the smile surface. Part III focuses on interest rates, and part IV extends the setting used for the deterministic-volatility LIBOR market model in order to account for interest-rate smiles in a financially-motivated and computationally-tractable manner. In this final part the author deals, in increasing levels of complexity, with CEV processes, with diffusive stochastic volatility and with Markov-chain processes. Covering FX, equity and interest-rate products, "Volatility and Correlation" is a blend of theoretical and practical material and is designed for traders, risk managers, financial professionals and students. Rebonato has a knack for distilling the essence from a wide range of complex option pricing models. At times a colloquial stance is privileged over mathematical rigor and formalism, allowing a larger public to benefit from this book. It combines rigorous theory with practical knowledge of markets and models. Riccardo Rebonato uses his technical mastery to make the theory clear, and his wealth of experience to give insights into applications. Whatever your level of knowledge of these markets, you will learn from him. Both practitioners and academics will benefit from his teachings and advice. Fitting an Exogenous Smile Surface. IV Interest Rates -- Smiles. In his usual intuitive style he critically examine a variety of approaches to equity, currency and interest-rate options. This book is full of practical insights that reflect a wealth of experience in applying these models. Read it carefully and thoroughly. In this book, Riccardo Rebonato presents the subject in his characteristically elegant and simple fashion. He rightly emphasises the financial and economic assumptions which underpin the models, and gives salutary warnings against models which overfit the current structure of prices but which perform poorly in predicting future behaviour. A rare combination of intellectual insight and practical common sense. He has published several papers on finance in academic journals, and is on the editorial board of several journals. He is a regular speaker at conferences worldwide. Page 1

2 Chapter 2 : Riccardo Rebonato - Wikipedia Rebonato begins by presenting the conceptual foundations for the application of the LIBOR market model to the pricing of interest-rate derivatives. Next he treats in great detail the calibration of this model to market prices, asking how possible and advisable it is to enforce a simultaneous fitting to several market observables. Interest rate changes impact the overall economy, stock market, bond market, other financial markets and can influence macroeconomic factors. Which Interest Rate for Pricing Options? It is important to understand the right maturity interest rates to be used in pricing options. Most option valuation models like Black-Scholes use the annualized interest rates. Interest rate conversions over different time periods work differently than a simple up- or down- scaling multiplication or division of the time durations. How can you convert it to annual rate? Divide the monthly interest rate by to get 0. Add 1 to it to get 1. Raise it to the power of the time multiple i. Subtract 1 from it to get 0. Multiply it by, which is the annual rate of interest Other factors used in determining the option price like the underlying asset price, time to expiry, volatility, and dividend yield change more frequently and in larger magnitudes, which have a comparatively larger impact on option prices than changes in interest rates. How Interest Rates Affect Call and Put Option Prices To understand the theory behind the impact of interest rate changes, a comparative analysis between stock purchase and the equivalent options purchase will be useful. We assume a professional trader trades with interest-bearing loaned money for long positions and receives interest-earning money for short positions. Interest Advantage in Call Option: Yet the profit potential will remain the same as that with a long stock position. Hence, an increase in interest rates will lead to either saving in outgoing interest on the loaned amount or an increase in the receipt of interest income on the saving account. Interest Disadvantage in Put Option: Theoretically, shorting a stock with an aim to benefit from a price decline will bring in cash to the short seller. Buying a put has similar benefit from price declines, but comes at a cost as the put option premium is to be paid. This case has two different scenarios: With an increase in interest rates, shorting stock becomes more profitable than buying puts, as the former generates income and the latter does the opposite. Hence, put option prices are impacted negatively by increasing interest rates. The Rho Greek Rho is a standard Greek a computed quantitative parameter that measures the impact of a change in interest rates on an option price. Similarly, the put option price will decrease by the amount of its rho value. How a change in interest rates affects call and put option prices? The call price and put price has changed by almost the same amount as the earlier computed call rho 0. The fractional difference is due to BS model calculation methodology, and is negligible. In reality, interest rates usually change only in increments of 0. The other numbers are the same as in Case 1. As can be observed, the changes in both call and put option prices are negligible after a 0. Over the course of the year, other factors can vary with much higher magnitudes and can significantly impact the option prices. Similar computations for out-of-the-money OTM and ITM options yield similar results with only fractional changes observed in option prices after interest rate changes. Arbitrage Opportunities Is it possible to benefit from arbitrage on expected rate changes? Usually, markets are considered to be efficient and the prices of options contracts are already assumed to be inclusive of any such expected changes. Also, a change in interest rates usually has an inverse impact on stock prices, which has a much larger impact on option prices. Overall, due to the small proportional change in option price due to interest rate changes, arbitrage benefits are difficult to capitalize upon. Call option and put option premiums are impacted inversely as interest rates change. Trading Center Want to learn how to invest? Get a free 10 week series that will teach you how to start investing. Delivered twice a week, straight to your inbox. Page 2

3 Chapter 3 : How and Why Interest Rates Affect Options Investopedia Interest-Rate Option Models: Understanding, Analysing and Using Models for Exotic Interest-Rate Options / Edition 2 The modelling of exotic interest-rate options is such an important and fast-moving area, that the updating of the extremely successful first edition has been eagerly awaited. Interest Rate Models and Negative Rates October 13, In our last blog we reported on the relationship between the level of interest rates and their volatility. However, all of the data underlying the analyses that we reported on in that blog came from or earlier. Since then we have seen rates decline and, in many jurisdictions, become negative. In this blog we will discuss the models that can be used for calculating the price of European style interest-rate options such as caps and swap options when rates are low or negative. There are four related models that can be used to calculate the price of European style interest-rate options such as caps or swap options. In this model the future forward rates are lognormally distributed. The option price is expressed as an interest rate. To turn it into a dollar value it must be multiplied by the principal amount underlying the option. For options with a particular maturity the volatilities generally form a downward sloping smile: The second model is the Bachelier normal model in which the forward rate is normally distributed. The third model is the constant elasticity of variance CEV model. The final model is the SABR model. This is a much more complicated model in which both the forward rate and the volatility of the forward rate are random. It also reflects smile dynamics reasonably well. However, the volatility parameters are different for different option maturities. These models are all closely related. The volatility parameters in these models have different meanings. Of the four models only the Bachelier model allows rates to become negative. Brokers now adopt one of two approaches. These are usually expressed in basis points. The following table shows broker quotes based on the Bachelier model for Euro-denominated caps of various maturities and strikes. The Euro term structure is negative out to about the eight-year point. If the broker wants to base the quotes on a displaced log-normal model it is necessary to choose an appropriate displacement. Since the lowest forward rate is about â 0. As the size of the displacement is reduced the implied volatilities rise and vice versa. If the displacement is reduced to 0. The shape of the volatility surface is also affected by the choice of the displacement value. For a very high displacement, the reverse is true, higher strikes have lower volatilities. The size of the displacement is also important when the SABR model is calibrated to all the options with a particular maturity. Larger displacement values result in a flatter volatility smile which is easier for the SABR model to capture. As a result the pricing errors resulting from the calibration to market data are smaller. This means that we are implicitly assuming a numeraire equal to a zero-coupon bond with the same life as the option. The Bachelier model for caplets is subtly different from the Hull-White model in that in the Bachelier model it is the rate that has a compounding frequency equal to the accrual period that is normally distributed while in the Hull-White model it is an instantaneous rate that is normally distributed. The result of this is that caplet pricing formulas are different in the two models and Black implied volatility parameters in the two models are slightly different. References Hull, John, and Alan White. Lesniewski, and Diana E. Pricing, calibration and hedging for complex interest-rate derivatives. His research has an applied focus and is concerned with risk management, bank regulation, and valuation of derivatives. He is best known for his books Risk Management and Financial Institutions now in its 3rd edition, Options, Futures, and Other Derivatives now in its 9th edition, and Fundamentals of Futures and Options Markets now in its 8th edition. His books have been translated into many languages and are widely used in trading rooms throughout the world, as well as in the classroom. He earned his Ph. He is well known for his work with Rotman Professor John Hull on stochastic volatility models, the Hull-White interest rate model, credit risk and the valuation of structured products. In addition to the theoretical developments he has contributed to the development of numerical procedures used to evaluate the models in practice. These models are widely used by financial engineers in trading rooms around the world to value a wide variety of derivative products. He is the Associate Editor of the Journal of Derivatives. Page 3

4 Chapter 4 : Short-rate model - Wikipedia An accessible, first-rate overview of interest rate dependent options for traders and institutional investors Until now market professionals seeking to exploit the profit potential of interest rate dependent options were forced to hunt through esoteric journals for a crumb or two of practical knowledge about their use. For intermediate values of the rate, the behavior is approximately normal. Thus Deguillaume et al contended that rates tended to be lognormal when they are less than 1. Their results were based on historical data. They were therefore estimating the process followed by rates in the real world. In Hull and White we decided to test this out by finding which piecewise linear representation of the short rate standard deviation, similar to that shown in Figure 1, best fit cap data. We first developed a procedure for rounding the corners of the volatility function so that the function was both continuous and differentiable. We calibrated the model as closely as possible to the prices of caps observed on December 2, An optimizer was used to find the corner points that minimized the objective function. Because of the large number of degrees of freedom in the calibration, the fit to the market prices is very good. The result of this is also shown in Figure 2. Two different functional forms are used. As another test we calibrated the model to cap prices observed between and Market data for caps observed on the last trading day of March in each year were used. To minimize the effect on the calibration of very high and very low strike options the cap quotes used for a particular maturity were those where the cap rate was within 1. As a result we divided the cap data into two groups: In each calibration, the model was fitted to about 50 caps with different strikes and maturities. The results are illustrated in Figures 3 and 4. The volatility structure implied by cap prices is similar to that observed in the real world. The structures calibrated from cap prices provide a measure of the volatility structure at a point in time. As the results show, this changes from year to year and probably from day to day. Overall, our results are supportive of the conjecture that the volatility function used implicitly or explicitly by market participants when pricing interest rate caps is similar to the volatility function derived by Deguillaume et al. What is more, this was true even before the Deguillaume et al research was first available as a working paper. Since rates in Europe and elsewhere have dipped below zero. This means that the lognormal structure for rate volatilities for low rates found by us and Deguillaume et al will have to be modified. Dealing with negative rates will be the subject of a future blog. White, "Numerical procedures for implementing term structure models I," Journal of Derivatives, 2 Fall, His research has an applied focus and is concerned with risk management, bank regulation, and valuation of derivatives. He is best known for his books Risk Management and Financial Institutions now in its 3rd edition, Options, Futures, and Other Derivatives now in its 9th edition, and Fundamentals of Futures and Options Markets now in its 8th edition. His books have been translated into many languages and are widely used in trading rooms throughout the world, as well as in the classroom. He earned his Ph. He is well known for his work with Rotman Professor John Hull on stochastic volatility models, the Hull-White interest rate model, credit risk and the valuation of structured products. In addition to the theoretical developments he has contributed to the development of numerical procedures used to evaluate the models in practice. These models are widely used by financial engineers in trading rooms around the world to value a wide variety of derivative products. He is the Associate Editor of the Journal of Derivatives. Chapter 5 : Interest Rate Volatility Derivatives Risk Management Software & Pricing Analytics FINCAD Extra info for Interest-rate option models. Example text. 1 December Japanese yen futures. 2. Interest-rate option models by Riccardo Rebonato. by Anthony Chapter 6 : Interest Rate Options By Riccardo Rebonato. ISBN ISBN The modelling of unique interest-rate strategies is such a tremendous and fast-moving region, that the updating of the tremendous winning first variation has been eagerly awaited. Page 4

5 Chapter 7 : Volatility and Correlation : Riccardo Rebonato : Find helpful customer reviews and review ratings for Interest-Rate Option Models: Understanding, Analysing and Using Models for Exotic Interest-Rate Options (Wiley Financial Engineering) by Riccardo Rebonato () at blog.quintoapp.com Read honest and unbiased product reviews from our users. Page 5