Thomas S. Y. Ho PhD President THC October 26, 2009 Tom.ho@thomasho

1. Applications of Arbitrage-free Models: New Frontiers in Interest Rate, Credit and Energy RisksThird Annual Bloomberg Lecture in Finance Thomas S. Y. Ho PhD President THC October 26, 2009 Tom.ho@thomasho.com

2. Arbitrage-free Term Structure Models Valuation models Derivative pricing (relative valuation) under interest rate, credit and other risk drivers Applications Trading Portfolio management Enterprise risk management Impacts on the markets Price discovery process Regulatory policies in the financial markets Introduction

3. Questions Addressed What are the model’s economic principles that make the model popular and fundamental? What are the frontiers of applications of the model in going forward? What are my cautionary notes on the use of the model? Detail discussions are available in the references Introduction

4. References Amin, Kaushik I., and Andrew J. Morton, 1994, “ Implied Volatility Functions in Arbitrage-free Term Structure Models, “ Journal of Financial Economics, 35 (2), 141-180 Benth, Fred Espen, Lars Ekeland, RagnerHauger and Bjorn Fredrik Nielsen 2003 “A Note on Arbitrage-free Pricing of Forward Contracts in Energy Market” Applied Mathematical Finance 10, 325-336 Eydeland, Alexander and Krzysztof Wolyniec 2003 Energy and Power Risk Management, Wiley Finance Harrison, J Michael, and David M. Kreps, 1979 “Martingales and Arbitrage in Multiperiod Securities Markets<” Journey of Economic Theory, 20(3), 381-408 Ho, Thomas S. Y. 1992 “Key rate durations: measures of interest rate risks” Journal of Fixed-Income, 2(2), 19-44 Ho, Thomas S. Y. and Sang-Bin Lee 2003, The Oxford Guide to Financial Modeling, Oxford University Press Ho, Thomas S. Y. and Sang-Bin Lee 1986, “Term Structure Movements and the Pricing of Interest Rate Contingent Claims,” Journal of Finance, 41 (5), 1011-1029 Ho, Thomas S. Y. and Sang Bin Lee,2009 “ Valuation of Credit Contingent Claims: An Arbitrage-free Credit Model” Journal of Investment Management vol 7 No 5 Ho, Thomas S. Y. Ho and Sang Bin Lee, 2009 ”A Unified Credit and Interest Rate Arbitrage-Free Contingent Claim Model” Journal of Fixed-Income Ho, Thomas S. Y. and Blessing Mudavanhu,2007 “Stochastic Movement of the Implied Volatility Function” Journal of Investment Management 4th quarter Ho, Thomas S. Y. and Sang Bin Lee, 2007 ““Generalized Ho-Lee Model: A Multi-factor State-Time Dependent Implied Volatility Function Approach” Journal of Fixed Income 4th quarter Ho, Thomas S. Y. 2007 “Managing Interest Rate Volatility Risk: Key Rate Vega” Journal of Fixed Income 4th quarter Ho, Thomas S. Y. and Sang-Bin Lee 2009 “ A Unified Model: Arbitrage-free Term Structure Movements of Flow Risks” Ho, Thomas S. Y. and Sang Bin Lee “ Pricing of Contingent Claims on Natural Gas” working paper Nawalkha, Sanjay K., Natalia A. Beliaeva and Gloria M Soto 2007 Dynamic Term Structure Modeling Wiley Finance References

5. Outline Salient features of the model A perspective of arbitrage-free term structure models A framework to explore new frontiers in applications Apply the framework to … Interest rate risk Credit risk Energy risk Going forward: Managing model risks Edwards Deming approach to risk management Introduction

6. The Black-Scholes ModelComponents of an Arbitrage-free Model Valuation component C( S, t) Specify the contingent claim dS = r(t) Sdt + σ (t) Sdz Specify the underlying risk process Application component Delta: dynamic replication Calibration to determine the implied volatility Arbitrage-free Models

7. Arbitrage-free Term Structure ModelValuation Component of the Model C = C( r , p(t, T), t) Contingent claims on the discount function dr = F( p(t, T), t)dt + σ (r, t) dw Short rate model Forward rate model Market model X(n-1,i) = 0.5 p(n,i)(B(n,i) + B(n, i+1)) Rolling back adjusted by the discount rate B(n-1, i) = max ( X(n-1, i), K) Arbitrage-free models

8. Key Rate Duration – Dynamic Replication Application Component of the Model Callable bond Maturity 2020-10-15 SA fixed coupon rate 5.65% Bermudan callable at par Used for hedging, risk management, and investment Arbitrage-free Models

9. Implied Volatility FunctionCalibration: Application Component of the Model Arbitrage-free Models

10. Term Structure Models and the Black-ScholesModel: a Comparison The term structure model: Time dimension, rate, a “flow concept” From the economic modeling perspective, the Black Scholes model is not a special case of a term structure model – hence “term structure” Correlations of the securities of the term structure: Principal component methods Ho, Thomas S. Y. and Sang-Bin Lee 2009 “ A Unified Model: Arbitrage-free Term Structure Movements of Flow Risks” Arbitrage-free models

11. Proposed Perspective of Term Structure Models “Stock” versus “Flow” Arbitrage-free models have two components Valuation component (some examples) Multi-factor models Time and state dependent implied volatility function Unspanned stochastic volatility function Application component Effectiveness of dynamic hedging and implications of the implied volatilities Arbitrage-free Models

12. Credit Term Structure Valuation of fixed-income instruments with credit risk Reduced form and structural models Credit default swap (CDS) Survival function vs discount function State and time dependent survival rate s(n,i) • Ho, Thomas S. Y. and Sang Bin Lee,2009 “ Valuation of Credit Contingent Claims: An Arbitrage-free Credit Model” Journal of Investment Management vol 7 No 5 Applications: Credit Risk Modeling

13. Valuing Credit Contingent ClaimsValuation Component of the Model Make-whole Option X(n-1,i) = 0.5 p(n) s(n,i) (B(n,i) + B(n, i+1)) Rolling back adjusted by the survival rate B(n-1, i) = max ( X(n-1, i), K) Boundary and terminal conditions p(n) one period time value discount factor K strike price (an example), present value of the yield adjusted Treasury bonds Applications: Credit Risk Modeling

14. Valuation of Embedded Credit Options Applications: Credit Rsk Modeling

15. Implications of the Credit Term Structure Application Component of the Model Determine the credit key rate durations for credit hedging Specify the precise dollar credit exposure in the term structure Identify the implied credit volatilities Use of the structural models Interest rate and credit risk relationship Applications to a callable instruments Applications: Credit Risk Modeling

16. Applications of the Term Structure Credit Model Dynamic movements of the term structure of credit Specify the embedded make whole option in commercial mortgages Impact of the correlation to the interest rate level Relating a reduced form model to the structural model Multi-factor credit model Applications: Credit Risk Modeling

17. Natural Gas Futures Term Structure Basic economics of natural gas: Henry Hub data Well head cost, gathering and processing costs Storage and cost to carry Demand: Weather affects heating; power generation Injection season: April - October Withdrawal season: November – March Ho, Thomas S. Y. and Sang Bin Lee “ Pricing of Contingent Claims on Natural Gas” working paper Applications: Energy Risk Modeling

18. Importance of Modeling NG Term Structure Deregulation of power industry Supply: Horizontal rigs Power prices Depending on the bid stack and power demand Bid stack depends on the fuel price and the outage Use of derivatives to manage energy risk and capital investments Applications: Energy Risk Modeling

19. Modeling of the Natural Gas Price Process Abadic, Luis M and Jose Chamorra (2006) 2 factor model with the stochastic fuel price mean reverting to a stochastic long term price The stochastic long term price mean reverting to a constant price Eydeland, Alexander and Krzysztof Wolyniec (2003) and Benth et al (2003) Use arbitrage-free models Ho and Lee (2009) Identify the term structure “flow risk” and the “stock risk” Applications: Energy Risk Modeling

20. Data and Methodology Futures and spot daily prices from 1/3/2006 -12/27/2006 Futures delivery dates: monthly from 1/1/2007 and 1/1/2010 Implied cost of carry c(t, T) = (1/(T-t))ln F(t,T)/S(t) Use the principal component approach to specify the movements Data Source: Logical Information Machines (LIM) Applications: Energy Risk Modeling

21. Henry Hub $ MMBtu (12/12/05-8/7/09) Applications: Energy Risk Modeling

22. Term Structure of Henry Hub Futures Prices10/16/2009 Applications: Energy Risk Model

23. Natural Gas Futures Term Structure Movements Applications: Energy Risk Modeling

24. Arbitrage-free Natural Gas ModelValuation Component of the Model Contingent claims on the NG spot and futures prices Implied cost to carry: The term structure Futures contracts determining the implied cost of carry The one period cost to carry is equivalent to the survival rate Dynamics of the term structure: The term structure of cost to carry and the spot rates dS = c(t) S dt + σ(t) S dz dc(t) = F( c(t, T), t) dt + σ* dw Applications: Energy Risk Modeling

25. Preliminary Results on Dynamic Hedging Application Component of the Model The futures term structure movements have two factors: “Price” (85%), “Cost to Carry” (14%), 3rd vector (0.3%) The cost of carry movements has one factor: level movement (99.5%) and 2nd factor (0.4%) Correlation of the spot price and cost to carry: low The use of key rate durations on the cost to carry and the spot price for hedging Applications: Energy Risk Modeling

26. 1st Principal Movement of the Cost to Carry Applications: Energy Risk Modeling

27. 2nd Principal Movement of the Cost to Carry Applications: Energy Risk Modeling

28. Calibrate the Model to the Implied VolatilitiesApplication component of the Model Applications: Energy Risk Modeling

29. Implications to Energy Trading Quantitative analysis of NG contracts and the changing shape of the implied cost to carry curve Relate the NG futures option prices and to other derivatives Applications to the calendar basis trades Use of the multi-factor model to determine the correlations of the cycles Correlation of interest rates with the cost to carry curve Power stack function: relation to coal and crude oil Applications: Natural Gas Contracts

30. Going Forward A lesson learnt from the financial crisis Mispricing and hence misallocation of resources Justification for the dynamic replication and volatility calibration Revisiting the application component of an arbitrage-free model Internal consistency of the model Manage Model Risks

31. Managing Model Risk Example of mis-valuation: The CDO copula model The use of implied correlations How to manage model risk? Managing Model Risks

32. A Solution to Managing Model Risk (Deming) Deming: Statistical approach to quality control Defects are often traced directly to the cause. But… Statistical approach is more objective Catching the defects when they are small Return attributions of Treasury futures Cheapest to deliver Delivery options, timing options, end of month options Managing Model Risks

33. Return Attribution and Risk Management Data source: BGCantor Market Data

34. Analysis of the Model Risk Explanatory power of the model? Mean reversion behavior of the residuals? Effectiveness of the dynamic replication? Detect “defects” in the time series in relation to events Manage Model Risks

35. Analysis of the Residuals over the Roll Month September and December 5 year futures over the month of September Market price = Mid quotes 10 minute intervals from 7:00am till 5:30 pm Both the explanatory power and mean reversion rate decline by mid month This behavior varies across the contracts Manage Model Risks

36. Explanatory Power Metric : R squaredp(market)= a + b p(model) + e Data source: BGCantor Market Data

37. Explanatory Power Metric : R squaredp(market)= a + b p(model) + e Data source: BGCantor Market Data

38. Mean Reversion Metric: ( 1 -b )ch/rh(n) = a + b ch/rh(n-1) + e Data source: BGCantor Market Data

39. Mean Reversion Metric: ( 1 -b )ch/rh(n) = a + b ch/rh(n-1) + e Data source: BGCantor Market Data

40. Replication Metric : R squared p(model) = a + b TSY returns + e Data source: BGCantor Market Data

41. Summary: Proposed Perspective Term structure models deal with “rates”, flows ( interest rate, default rate, cost to carry, inflation rate ) Contrast to the Black-Scholes Valuation and application components to the model Similar to the Black-Scholes Manage model risk: Statistical approach (Deming) Examples of term structure Interest rate, credit, energy Natural gas shows the use of both price and rate models Conclusions

42. Conclusions and Implications Trading Return attributions on performance measure Model risk Statistical Approach: explanatory, mean reversion, replication Securities valuation Interest rate, default rate, inflation rate, liquidity, cost to carry Hybrid models Identifies the price formation process From the basic valuation building blocks to exotic structures Regulatory policy on market transparency and the role of exchanges Conclusions

43. References Amin, Kaushik I., and Andrew J. Morton, 1994, “ Implied Volatility Functions in Arbitrage-free Term Structure Models, “ Journal of Financial Economics, 35 (2), 141-180 Benth, Fred Espen, Lars Ekeland, RagnerHauger and Bjorn Fredrik Nielsen 2003 “A Note on Arbitrage-free Pricing of Forward Contracts in Energy Market” Applied Mathematical Finance 10, 325-336 Eydeland, Alexander and Krzysztof Wolyniec 2003 Energy and Power Risk Management, Wiley Finance Harrison, J Michael, and David M. Kreps, 1979 “Martingales and Arbitrage in Multiperiod Securities Markets<” Journey of Economic Theory, 20(3), 381-408 Ho, Thomas S. Y. 1992 “Key rate durations: measures of interest rate risks” Journal of Fixed-Income, 2(2), 19-44 Ho, Thomas S. Y. and Sang-Bin Lee 2003, The Oxford Guide to Financial Modeling, Oxford University Press Ho, Thomas S. Y. and Sang-Bin Lee 1986, “Term Structure Movements and the Pricing of Interest Rate Contingent Claims,” Journal of Finance, 41 (5), 1011-1029 Ho, Thomas S. Y. and Sang Bin Lee,2009 “ Valuation of Credit Contingent Claims: An Arbitrage-free Credit Model” Journal of Investment Management vol 7 No 5 Ho, Thomas S. Y. Ho and Sang Bin Lee, 2009 ”A Unified Credit and Interest Rate Arbitrage-Free Contingent Claim Model” Journal of Fixed-Income Ho, Thomas S. Y. and Blessing Mudavanhu,2007 “Stochastic Movement of the Implied Volatility Function” Journal of Investment Management 4th quarter Ho, Thomas S. Y. and Sang Bin Lee, 2007 ““Generalized Ho-Lee Model: A Multi-factor State-Time Dependent Implied Volatility Function Approach” Journal of Fixed Income 4th quarter Ho, Thomas S. Y. 2007 “Managing Interest Rate Volatility Risk: Key Rate Vega” Journal of Fixed Income 4th quarter Ho, Thomas S. Y. and Sang-Bin Lee 2009 “ A Unified Model: Arbitrage-free Term Structure Movements of Flow Risks Ho, Thomas S. Y. and Sang Bin Lee “ Pricing of Contingent Claims on Natural Gas” working paper Nawalkha, Sanjay K., Natalia A. Beliaeva and Gloria M Soto 2007 Dynamic Term Structure Modeling Wiley Finance References