In this paper we consider several parametric assumptions for the instantaneous covariance structu... more In this paper we consider several parametric assumptions for the instantaneous covariance structure of the Libor market model. We examine the impact of each different parameterization on the evolution of the term structure of volatilities in time, on terminal correlations and on the joint calibration to the caps and swaptions markets. We present a number of cases of calibration in the Euro market. In particular, we consider calibration via a parameterization establishing a controllable one to one correspondence between instantaneous covariance parameters and swaptions volatilities, and assess the benefits of smoothing the input swaption matrix before calibrating. D. Brigo, F. Mercurio, M. Morini: Joint Calibration of the Libor model 3
Financial market infrastructures (FMIs) have evolved as core elements of highly intermediated fin... more Financial market infrastructures (FMIs) have evolved as core elements of highly intermediated financial markets partly due to the technological limitations of the time when they were first designed. Organizations and firms were unable to share records without having to entrust a single party to manage them; hence this phenomenon of intermediation has led to significant information silos. Simultaneously, it has driven the structure of business models, as well as regulatory supervision and oversight, in ways that furthered intermediation and also created a misalignment of incentives and risk taking between entities now categorized as systemically important financial institutions (SIFIs) and systemically important financial market infrastructures. Over time, this consolidation has led to highly concentrated FMIs and with it, concentrated risks. Some of these risks go beyond the credit risks of just one or two institutions, becoming instead systemic risks that are continuously monitored...
The LIBOR Market Model (LMM) is rapidly becoming the industry standard approach for pricing inter... more The LIBOR Market Model (LMM) is rapidly becoming the industry standard approach for pricing interest rate derivatives like caps and swaptions. But while fast and exact calibration of model parameters to market cap quotes is possible, the problem is much more ...
Viii Workshop on Quantitative Finance, Jan 25, 2007
SABR model emerged as a reference stochastic volatility framework for modelling swaption implied ... more SABR model emerged as a reference stochastic volatility framework for modelling swaption implied volatility. However, many implications of model behaviour in a market context are overlooked. We test empirically the model behaviour when practical considerations on implementation feasibility and parameter stability lead to specific constraints. We compare SABR behaviour with the traditional stochastic volatility models, and assess consequences on pricing non-vanilla structures. Secondly SABR leaves aside issues about modelling jointly different assets. In this paper we compute and analyze these dynamics when SABR distributional assumptions are applied to a Libor market model (BGM) framework. Thirdly, we perform analysis of the hedging behaviour of SABR for different parametric assumptions, in particular considering explicitly correlation between volatility and rates. Results are not in line with the general market wisdom.
We show that when a derivative portfolio has different correlated underlyings, hedging using clas... more We show that when a derivative portfolio has different correlated underlyings, hedging using classical greeks (first-order derivatives) is not the best possible choice. We first show how to adjust greeks to take correlation into account and reduce P&L volatility. Then we embed correlation-adjusted greeks in a global hedging strategy that reduces cost of hedging without increasing P&L volatility, by optimization of hedge re-adjustments. The strategy is justified in terms of a balance between transaction costs and risk aversion, but, unlike more complex proposals from previous literature, it is completely defined by observable parameters, geometrically intuitive, and easy to implement for an arbitrary number of risk factors. We test our findings on a CVA hedging example. We first consider daily re-hedging: in this test correlation-adjusted greeks allow to reduce P&L volatility by more than 30% compared to standard deltas. Then we apply our general strategy to a context where a CVA portfolio is exposed to both credit and interest rate risk. The strategy keeps P&L volatility in line with daily standard delta hedging, but with massive cost-saving: only six rebalances of the illiquid credit hedge are performed, over a period of six months.
Abstract: This work focuses on the swaptions automatic cascade calibration algorithm (CCA) for th... more Abstract: This work focuses on the swaptions automatic cascade calibration algorithm (CCA) for the LIBOR Market Model (LMM) first appeared in Brigo and Mercurio (2001). This method induces a direct analytical correspondence between market swaption volatilities and LMM ...
We analyze the market gaussian copula framework for default correlation. First, we show that only... more We analyze the market gaussian copula framework for default correlation. First, we show that only a total rethinking of the approach to correlations can make the model compatible with a realistic representation of risk. Secondly, we show that however the choice of a copula framework implies unrealistic probability of losses clustered in a short period of time. Thirdly we show that the market approach to bespokes was bounded to underestimate the risk, and hint at a solution. The analysis can be seen as a thorough stress-test of the market approach.
In order to evolve beyond bitcoins, which are still speculative, volatile and small in terms of m... more In order to evolve beyond bitcoins, which are still speculative, volatile and small in terms of market cap, cryptocurrencies need decentralized financial intermediaries. In this work we first show that one fundamental role they can take regards price stability. The Economist and the Center for Financial Stability focus on the importance of regulating money supply for stable prices in a digital currency, and http://ssrn.com/abstract=2425270 designs a cryptocurrency where money supply is regulated to match money demand. This first proposal, however, has a limitation: the mechanism for stable prices implies that the amount of money in every account is continuously modified. This way the instability of prices typical of bitcoins is transferred to accounts, so that the purchasing power of savings is still less protected from inflation/deflation than in standard fiat currencies.The solution is to give wallet owners the freedom to choose how much they want to be affected by changes of money supply, by introducing two types of wallets: Inv wallets and Sav wallets. Money in Sav wallets is protected from instability thanks to the presence of Inv wallets, that take the risk, and the reward, of being the target of changes in money supply. This role of channel and cushion for monetary policy is taken by banks in fiat currencies, and is needed also in digital currencies, where it can be taken by decentralized financial intermediaries. This is a starting point to analyze the other roles that financial intermediaries can have in a digital currency, linking the ownership of Inv wallets to proof-of-stake validation of transactions and to lending.The appendix explores the possibility of currencies indexed to property prices registered in a block chain.
In this paper we consider several parametric assumptions for the instantaneous covariance structu... more In this paper we consider several parametric assumptions for the instantaneous covariance structure of the Libor market model. We examine the impact of each different parameterization on the evolution of the term structure of volatilities in time, on terminal correlations and on the joint calibration to the caps and swaptions markets. We present a number of cases of calibration in the Euro market. In particular, we consider calibration via a parameterization establishing a controllable one to one correspondence between instantaneous covariance parameters and swaptions volatilities, and assess the benefits of smoothing the input swaption matrix before calibrating. D. Brigo, F. Mercurio, M. Morini: Joint Calibration of the Libor model 3
Financial market infrastructures (FMIs) have evolved as core elements of highly intermediated fin... more Financial market infrastructures (FMIs) have evolved as core elements of highly intermediated financial markets partly due to the technological limitations of the time when they were first designed. Organizations and firms were unable to share records without having to entrust a single party to manage them; hence this phenomenon of intermediation has led to significant information silos. Simultaneously, it has driven the structure of business models, as well as regulatory supervision and oversight, in ways that furthered intermediation and also created a misalignment of incentives and risk taking between entities now categorized as systemically important financial institutions (SIFIs) and systemically important financial market infrastructures. Over time, this consolidation has led to highly concentrated FMIs and with it, concentrated risks. Some of these risks go beyond the credit risks of just one or two institutions, becoming instead systemic risks that are continuously monitored...
The LIBOR Market Model (LMM) is rapidly becoming the industry standard approach for pricing inter... more The LIBOR Market Model (LMM) is rapidly becoming the industry standard approach for pricing interest rate derivatives like caps and swaptions. But while fast and exact calibration of model parameters to market cap quotes is possible, the problem is much more ...
Viii Workshop on Quantitative Finance, Jan 25, 2007
SABR model emerged as a reference stochastic volatility framework for modelling swaption implied ... more SABR model emerged as a reference stochastic volatility framework for modelling swaption implied volatility. However, many implications of model behaviour in a market context are overlooked. We test empirically the model behaviour when practical considerations on implementation feasibility and parameter stability lead to specific constraints. We compare SABR behaviour with the traditional stochastic volatility models, and assess consequences on pricing non-vanilla structures. Secondly SABR leaves aside issues about modelling jointly different assets. In this paper we compute and analyze these dynamics when SABR distributional assumptions are applied to a Libor market model (BGM) framework. Thirdly, we perform analysis of the hedging behaviour of SABR for different parametric assumptions, in particular considering explicitly correlation between volatility and rates. Results are not in line with the general market wisdom.
We show that when a derivative portfolio has different correlated underlyings, hedging using clas... more We show that when a derivative portfolio has different correlated underlyings, hedging using classical greeks (first-order derivatives) is not the best possible choice. We first show how to adjust greeks to take correlation into account and reduce P&L volatility. Then we embed correlation-adjusted greeks in a global hedging strategy that reduces cost of hedging without increasing P&L volatility, by optimization of hedge re-adjustments. The strategy is justified in terms of a balance between transaction costs and risk aversion, but, unlike more complex proposals from previous literature, it is completely defined by observable parameters, geometrically intuitive, and easy to implement for an arbitrary number of risk factors. We test our findings on a CVA hedging example. We first consider daily re-hedging: in this test correlation-adjusted greeks allow to reduce P&L volatility by more than 30% compared to standard deltas. Then we apply our general strategy to a context where a CVA portfolio is exposed to both credit and interest rate risk. The strategy keeps P&L volatility in line with daily standard delta hedging, but with massive cost-saving: only six rebalances of the illiquid credit hedge are performed, over a period of six months.
Abstract: This work focuses on the swaptions automatic cascade calibration algorithm (CCA) for th... more Abstract: This work focuses on the swaptions automatic cascade calibration algorithm (CCA) for the LIBOR Market Model (LMM) first appeared in Brigo and Mercurio (2001). This method induces a direct analytical correspondence between market swaption volatilities and LMM ...
We analyze the market gaussian copula framework for default correlation. First, we show that only... more We analyze the market gaussian copula framework for default correlation. First, we show that only a total rethinking of the approach to correlations can make the model compatible with a realistic representation of risk. Secondly, we show that however the choice of a copula framework implies unrealistic probability of losses clustered in a short period of time. Thirdly we show that the market approach to bespokes was bounded to underestimate the risk, and hint at a solution. The analysis can be seen as a thorough stress-test of the market approach.
In order to evolve beyond bitcoins, which are still speculative, volatile and small in terms of m... more In order to evolve beyond bitcoins, which are still speculative, volatile and small in terms of market cap, cryptocurrencies need decentralized financial intermediaries. In this work we first show that one fundamental role they can take regards price stability. The Economist and the Center for Financial Stability focus on the importance of regulating money supply for stable prices in a digital currency, and http://ssrn.com/abstract=2425270 designs a cryptocurrency where money supply is regulated to match money demand. This first proposal, however, has a limitation: the mechanism for stable prices implies that the amount of money in every account is continuously modified. This way the instability of prices typical of bitcoins is transferred to accounts, so that the purchasing power of savings is still less protected from inflation/deflation than in standard fiat currencies.The solution is to give wallet owners the freedom to choose how much they want to be affected by changes of money supply, by introducing two types of wallets: Inv wallets and Sav wallets. Money in Sav wallets is protected from instability thanks to the presence of Inv wallets, that take the risk, and the reward, of being the target of changes in money supply. This role of channel and cushion for monetary policy is taken by banks in fiat currencies, and is needed also in digital currencies, where it can be taken by decentralized financial intermediaries. This is a starting point to analyze the other roles that financial intermediaries can have in a digital currency, linking the ownership of Inv wallets to proof-of-stake validation of transactions and to lending.The appendix explores the possibility of currencies indexed to property prices registered in a block chain.
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