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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:


  • The discounting rate: Traditionally, derivatives contracts were discounted at a risk-free rate or a Libor rate. However, in a world without a risk-free rate and with Libor being phased out, derivatives contracts should be discounted at a rate that reflects their funding cost and credit risk. This rate depends on the collateral currency and the collateral agreement of each contract. For example, a swap contract that is collateralized with cash in US dollars should be discounted at an overnight indexed swap (OIS) rate in US dollars. A swap contract that is collateralized with cash in euros should be discounted at an OIS rate in euros. A swap contract that is uncollateralized or partially collateralized should be discounted at a blended rate that reflects its FVA and CVA.



the spot exchange rate and the interest rate differential between US dollars and euros, but also the OIS rate differential and the cross-currency basis spread between US dollars and euros. The cross-currency basis spread is a measure of the relative shortage or abundance of a currency in the market relative to its demand. It can be positive or negative, and it can vary across different maturities. For example, a negative cross-currency basis spread of -25 basis points for a 3-month EUR/USD swap means that the counterparty borrowing US dollars in a swap pays 25 basis points more than the 3-month US dollar Libor, while the counterparty borrowing euros pays the 3-month euro Libor.


  • The hedging strategy: Traditionally, derivatives contracts were hedged by replicating their cash flows with a portfolio of risk-free assets or Libor-based instruments. However, in a world with multiple funding rates and cross-currency basis spreads, derivatives contracts should be hedged by replicating their cash flows with a portfolio of collateralized assets or OIS-based instruments. This portfolio should also take into account the XVAs that arise from trading on a collateralized basis. For example, a swap contract that is collateralized with cash in US dollars should be hedged with OIS-based instruments in US dollars. A swap contract that is collateralized with cash in euros should be hedged with OIS-based instruments in euros. A swap contract that is uncollateralized or partially collateralized should be hedged with a combination of OIS-based instruments and credit default swaps.



Cooking with collateral also introduces new sources of risk and uncertainty in the valuation and hedging of derivatives contracts. These include:


  • Collateral funding risk: This is the risk that the cost of funding the collateral posted or received changes over time due to market movements or contractual terms. For example, if the collateral rate increases, the value of the collateral account decreases, and vice versa. This affects the net present value of the trade and creates a funding gap that needs to be covered by borrowing or lending additional funds. Collateral funding risk can be measured by FVA, which is the difference between the value of a trade assuming it is funded at the risk-free rate and the value of a trade assuming it is funded at the actual funding rate.



  • Counterparty credit risk: This is the risk that the counterparty defaults on its obligations under the trade and causes a loss to the non-defaulting party. For example, if the counterparty defaults when the trade has a positive mark-to-market value for the non-defaulting party, the non-defaulting party suffers a loss equal to the difference between the mark-to-market value and the recovery value of the trade. Counterparty credit risk can be measured by CVA, which is the expected loss due to counterparty default over the life of the trade.



the mismatch between the Libor rate and the OIS rate. The Libor-OIS spread is a measure of the difference between the interbank lending rate and the risk-free rate. It can reflect the credit risk and liquidity risk in the interbank market. For example, a positive Libor-OIS spread of 25 basis points for a 3-month swap means that the interbank lending rate is 25 basis points higher than the risk-free rate.


To measure and manage these risks, derivatives practitioners need to use appropriate models and tools that can capture the complexities and nuances of cooking with collateral. Some of these models and tools include:


  • Collateralized valuation frameworks: These are valuation frameworks that incorporate the effects of collateralization on the discounting rate, the forward rate, and the XVAs. For example, a collateralized valuation framework for a swap contract would take into account the collateral currency, the collateral agreement, the FVA, the CVA, and the basis risk.



  • Collateral optimization algorithms: These are algorithms that determine the optimal choice of collateral currency for a given trade or portfolio of trades. For example, a collateral optimization algorithm for a cross-currency swap contract would take into account the expected return, volatility, correlation, cross-currency basis spread, transaction costs, and operational risks of each collateral currency.



  • Collateral risk analytics: These are analytics that quantify and monitor the collateral funding risk, counterparty credit risk, and basis risk of a trade or portfolio of trades. For example, a collateral risk analytic for a swap contract would calculate and report the sensitivity of the trade value to changes in the collateral rate, the counterparty default probability, and the cross-currency basis spread.



Cooking with collateral: empirical evidence




Cooking with collateral has been empirically observed and tested in various derivatives markets. Some of the empirical findings are:


  • Cooking with collateral explains some of the market anomalies and dynamics that emerged after the financial crisis. For example, cooking with collateral can account for the negative swap spreads observed in some markets, where the fixed rate of a swap contract is lower than the yield of a government bond with a similar maturity. This can happen when the swap contract is collateralized with cash at a lower OIS rate than the government bond yield. Cooking with collateral can also account for the divergence of implied volatilities across different options markets, where options on different underlying assets or with different maturities have different implied volatilities. This can happen when options are priced under different pricing measures that reflect different funding costs and credit risks.



  • Cooking with collateral affects the relative pricing and hedging performance of derivatives contracts. For example, cooking with collateral implies that derivatives contracts that are denominated in different currencies or have different maturities should not be priced or hedged using a single discount curve or a single forward curve. Rather, they should be priced or hedged using multiple discount curves or multiple forward curves that reflect their specific funding costs and credit risks. Cooking with collateral also implies that derivatives contracts that are subject to different XVAs should not be priced or hedged using a single valuation framework or a single hedging strategy. Rather, they should be priced or hedged using customized valuation frameworks or hedging strategies that incorporate their specific XVAs.



the complexity and volatility of derivatives valuation and hedging. For example, cooking with collateral creates opportunities for arbitrage and relative value trading by exploiting mispricing or inefficiencies in different derivatives markets. Arbitrage and relative value trading are trading strategies that seek to take advantage of price differentials between related financial instruments, such as stocks and bonds, by simultaneously buying and selling the different securities. Relative value trading is also referred to as pairs trading, because it involves investing in a pair of related securities that have high correlations, meaning they tend to move in the same direction at the same time. Cooking with collateral also creates challenges for risk management and regulation by increasing the exposure to collateral funding risk, counterparty credit risk, and basis risk, which require appropriate models and tools to measure and manage.


Conclusion




In this article, we have reviewed the main ideas and results of Piterbarg's cooking with collateral framework, as well as its practical implications and empirical evidence. We have seen that cooking with collateral is 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.


We have seen that cooking with collateral has significant implications for the valuation and hedging of derivatives contracts, as well as for the optimal choice of collateral currency. We have also seen that cooking with collateral introduces new sources of risk and uncertainty in the valuation and hedging of derivatives contracts, which require appropriate models and tools to measure and manage. We have also seen that cooking with collateral explains some of the market anomalies and dynamics that emerged after the financial crisis, and affects the relative pricing and hedging performance of derivatives contracts.


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.


Cooking with collateral is an important and relevant topic for derivatives practitioners and researchers. It offers new opportunities and challenges for arbitrage and relative value trading, risk management and regulation, and innovation and development. It also poses new questions and problems for further research and exploration.


FAQs




Here are some frequently asked questions about cooking with collateral and their brief answers:


  • Q: What is cooking with collateral?A: Cooking with collateral is a term coined by Vladimir Piterbarg 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.



  • Q: What are the main ingredients of Piterbarg's model?A: The main ingredients of Piterbarg's model are: a set of traded assets, a set of numeraire assets, a set of collateral accounts, a set of pricing measures, and a set of valuation adjustments.



and the cross-currency basis spread; it affects the hedging strategy by making it depend on the collateralized assets or OIS-based instruments.


  • Q: What are the main sources of risk and uncertainty in cooking with collateral?A: The main sources of risk and uncertainty in cooking with collateral are: collateral funding risk, counterparty credit risk, and basis risk.



  • Q: What are some of the models and tools that can measure and manage these risks?A: Some of the models and tools that can measure and manage these risks are: collateralized valuation frameworks, collateral optimization algorithms, and collateral risk analytics.



  • Q: What are some of the empirical findings on the impact of cooking with collateral on derivatives markets?A: Some of the empirical findings on the impact of cooking with collateral on derivatives markets are: cooking with collateral explains some of the market anomalies and dynamics that emerged after the financial crisis, such as negative swap spreads and divergent implied volatilities; cooking with collateral affects the relative pricing and hedging performance of derivatives contracts, such as swaps and options; cooking with collateral creates new opportunities and challenges for arbitrage and relative value trading, risk management and regulation, and innovation and development.



Q: What are some of the future research directions on cooking with collateral?A: Some of the future research directions on cooking with collateral are: developing more realistic and robust models of collateral choice, funding cost, credit risk, and basis risk; exploring the interactions and feedback effects between cooking with collateral and other market factors, such as liquidity, volatility, and regulation; investigating the welfare and efficiency implications of cooking with collateral for market part


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