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Jose Nguyen
Jose Nguyen

The Impact of Collateralization on Derivatives Pricing: Insights from Piterbarg's Model


Introduction




Cooking with collateral is a term coined by Vladimir Piterbarg, a leading expert in derivatives pricing and risk management, to describe a new paradigm for valuing and managing derivatives contracts in a world without a risk-free rate and with all assets traded on a collateralized basis. This paradigm emerged as a response to the global financial crisis of 2007-2009, which exposed the fragility and limitations of the traditional derivatives pricing theory that assumed the existence of a risk-free rate as a matter of course.




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Vladimir Piterbarg is a managing director and head of quantitative analytics at Barclays Capital. He has over 20 years of experience in derivatives research and development, having worked at Bankers Trust, Deutsche Bank, Goldman Sachs, Lehman Brothers, and Nomura. He is also an adjunct professor at NYU Courant Institute of Mathematical Sciences and a co-author of the three-volume book "Interest Rate Modeling". He has published numerous papers on various topics in derivatives pricing and risk management, including cooking with collateral.


Cooking with collateral is not only a theoretical concept, but also a practical reality that affects all participants in the derivatives markets. The post-crisis regulatory reforms, such as Basel III, Dodd-Frank, EMIR, and MiFID II, have increased the demand for high-quality collateral to mitigate counterparty credit risk and systemic risk. Moreover, the unprecedented monetary policies implemented by central banks around the world have created significant distortions and dislocations in the money markets, leading to negative interest rates, cross-currency basis spreads, and funding liquidity premia. These developments have profound implications for the valuation and hedging of derivatives contracts, as well as for the optimal choice of collateral currency.


In this article, we will review the main ideas and results of Piterbarg's cooking with collateral framework, as well as its practical implications and empirical evidence. We will also discuss some of the challenges and opportunities that cooking with collateral presents for derivatives practitioners and researchers.


Cooking with collateral: a theoretical framework




The starting point of Piterbarg's cooking with collateral framework is to recognize that there is no such thing as a risk-free rate in reality. Even government bonds cannot be considered credit risk-free, as evidenced by the sovereign debt crisis in Europe. Hence, using a risk-free money-market account or a zero-coupon bond as a foundation for asset pricing theory needs revisiting.


Instead, Piterbarg proposes to model an economy where all assets are traded on a collateralized basis, meaning that any trade involves posting and receiving collateral to secure the performance of the contract. The collateral can be cash or securities, and it can be posted continuously or discretely. The collateral account has a unit value that grows deterministically at a rate that depends on the collateral currency and the collateral agreement. The collateral agreement specifies the terms and conditions of the collateral exchange, such as the initial margin, the variation margin, the haircuts, the thresholds, the minimum transfer amounts, and the eligible collateral assets.


The key ingredients of Piterbarg's model are:


  • A set of traded assets, such as stocks, bonds, currencies, commodities, etc., whose dynamics are driven by a set of risk factors, such as interest rates, exchange rates, volatilities, etc.



  • A set of numeraire assets, such as bank accounts or zero-coupon bonds denominated in different currencies, whose dynamics are also driven by the same risk factors.



  • A set of collateral accounts, one for each currency, whose dynamics are driven by the collateral rates that depend on the collateral currency and the collateral agreement.



  • A set of pricing measures, one for each numeraire asset, under which the discounted prices of all traded assets are martingales.



  • A set of valuation adjustments (XVAs), such as funding valuation adjustment (FVA), credit valuation adjustment (CVA), debit valuation adjustment (DVA), margin valuation adjustment (MVA), etc., that capture the costs and benefits of trading on a collateralized basis.



Piterbarg's model differs from the traditional derivatives pricing theory in several ways:


  • It does not assume the existence of a risk-free rate or a risk-free asset. Instead, it allows for multiple numeraire assets and multiple pricing measures that reflect the different funding costs and credit risks of different market participants.



  • It does not assume that all trades are funded at the same rate. Instead, it allows for different funding rates for different trades depending on the collateral currency and the collateral agreement.



  • It does not assume that all trades are free of counterparty credit risk. Instead, it allows for different credit risks for different trades depending on the exposure profile and the recovery rate of the counterparties.



  • It does not assume that all trades are free of basis risk. Instead, it allows for different basis risks for different trades depending on the mismatch between the trade currency and the collateral currency.



Piterbarg's model can be extended to a cross-currency setting, where trades can involve multiple currencies and multiple collateral accounts. In this case, the choice of collateral currency becomes an important decision variable that affects the value and risk of the trade. Piterbarg develops a model of multi-currency collateral choice that takes into account the following factors:


  • The expected return and volatility of each collateral currency relative to the trade currency.



  • The correlation between each collateral currency and the trade currency.



  • The cross-currency basis spread between each collateral currency and the trade currency.



  • The transaction costs and operational risks associated with each collateral currency.



Piterbarg shows that the optimal choice of collateral currency depends on the trade characteristics, such as the maturity, the payoff profile, and the delta sensitivity. He also shows that there is no universal optimal choice of collateral currency that applies to all trades. Rather, each trade should be evaluated individually based on its own merits and constraints.


Cooking with collateral: practical implications




Cooking with collateral has significant implications for the valuation and hedging of derivatives contracts in the post-crisis environment. In particular, it affects: