School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
Received 2 November 2016; Revised 24 December 2016; Accepted 13 March 2017; Published 27 March 2017
Copyright © 2017 Li Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper gives analytical formulas for lookback and barrier options on underlying assets that are exposed to a counterparty risk. The counterparty risk induces a drop in the asset price, but the asset can still be traded after this default time. A novel technique is developed to valuate the lookback and barrier options by first conditioning on the predefault and the postdefault time and then obtain the unconditional analytic formulas for their prices.
Lookback and barrier options are among the most popular path-dependent options traded in exchanges and over-the-counter markets. A standard floating lookback call (put) option gives the holder the right to buy (sell) an asset (e.g., stock, index, exchange rate, and interest rate) at its lowest (highest) price during the life of the contract. In other words, the payoffs of a floating lookback call/put are, respectively, and . A barrier option is a financial derivatives contract that is activated (“in”) or deactivated (“out”) when the price of the underlying asset crosses a certain level (i.e., a barrier) from above (“down”) or below (“up”). Lookback and barrier options have been useful in the “real options" literature. For example, Longstaff  approximates the marketability values of a security as a continuous-time lookback option. Barrier options are inherent in the reduced-form approach to credit risk, where the default time of a company is modeled as the first hitting time, or barrier breaching time, of the asset process of the company; see Merton , Black and Cox , Leland and Toft , and Chen and Kou . For a review of the literature of both continuous and discrete lookback options and barrier options, please refer to Kou  and the references therein.
In the financial market, a counterparty default usually has important influences in various contexts. In terms of credit spreads, one observes in general a positive jump of the default intensity which is called the contagious jump (see, e.g., Jarrow and Yu ). In terms of asset (or stock) values for a firm, the default of a counterparty will in general induce a drop of its value process (see, e.g., El Karoui et al. ). Jiao and Pham  analyzed the impact of this counterparty risk on the optimal investment problem. In this paper, we study the impact of the counterparty risk on option pricing problems. In particular, we focus on the pricing of lookback and barrier options when the underlying asset is subject to the default of a counterparty and the instantaneous loss of the asset at the default time.
The explicit valuation of vanilla European options with this counterparty default risk was partly given by Ma et al. . However, the derivation of the analytic formula for pricing lookback and barrier options with this default risk model has not been done in the previous literature. The main difficulty lies in that the distribution of the first passage time is difficult to derive due to the default and the continuous trading of the underlying asset after the default time. We consider the conditional density approach of default, which is particularly suitable to study what goes on after the default and was adopted by Jiao and Pham  for the optimal investment problem. We derive the explicit distribution of the first passage time and then obtain the analytic formulas for valuation of the lookback and barrier options.
The outline of the paper is organized as follows. In Section 2, we introduce the financial models. In Section 3, we derive the distribution of the first passage time and then the formula for pricing lookback options. In Section 4, we derive the formula for pricing barrier options. Conclusions are given in the final section.
2. Financial Models
In this section, we consider a financial market model with a risky asset subject to counterparty risk: the dynamics of the risky asset is affected by the possibility of the counterparty defaulting. However, this stock still exists and can be traded after the default.
Let be a Brownian motion with horizon on the probability space and denote by the natural filtration of . Let , an almost surely finite nonnegative random variable on , represent the default time. Before , the filtration represents the information accessible to the investors. After , the investors add this new information to the reference filtration . Introduce , and let be the filtration generated by this jump process, and the progressively enlarged filtration , representing the structure of information available for the investors over .
The stock price process is governed by the following dynamic:where , , and are -predictable processes. From , any -predictable process can be written in the form:with -adapted, and is measurable with respect to , for all . Therefore, the dynamic (1) can be rewritten aswhere , , , are -adapted processes, and , , are -measurable functions for all . The process represents the (proportional) loss on the stock price induced by the default of the counterparty; by misuse of notation, we shall thus identify in (1) with the -adapted process in (5). When the counterparty defaults, the drift, and diffusion coefficients of the stock price switch from to , the postdefault coefficients may depend on the default time . Here for simplicity we assume thatwhere are nonnegative constants, , are only deterministic functions of , in , for example , , which have the following economic interpretation. The ratio between the pre- and postdefault rate of return is smaller than one and increases linearly with the default time: the postdefault rate of return drops to zero, when the default time occurs near the initial date and converges to the predefault rate of return, when the default time occurs near the finite investment horizon. We have a similar interpretation for the volatility whose ratio is larger than one, which decreases linearly with the default time, converging to the double (resp., initial) value of the predefault volatility, when the default time goes to the initial (resp., terminal horizon) time. The distribution of is fixed. Moreover , , and are independent and is an exponential variable with parameter . For more general set-ups, see Jiao and Pham .
Assume that is a risk-free interest rate. Let us define the -adapted process
Preface To The First Edition.
Section One. Derivatives And Their Markets.
Essay 1. The Structure Of Derivative Markets.
Essay 2. A Brief History Of Derivatives.
Essay 3. Why Derivatives?
Essay 4. Forward Contracts And Futures Contracts.
Essay 5. Options.
Essay 6. Swaps.
Essay 7. Types Of Risks.
Section Two. The Basic Instruments.
Essay 8. Interest Rate Derivatives: FRAs And Options.
Essay 9. Interest Rate Derivatives: Swaps.
Essay 10. Currency Swaps.
Essay 11. Structured Notes.
Essay 12. Securitized Instruments.
Essay 13. Equity Swaps.
Essay 14. Equity-Linked Debt.
Essay 15. Commodity Swaps.
Essay 16. American Versus European Options.
Essay 17. Swaptions.
Essay 18. Credit Derivatives.
Essay 19. Volatility Derivatives.
Essay 20. Weather And Environmental Derivatives.
Section Three. Derivative Pricing.
Essay 21. Forward And Futures Pricing.
Essay 22. Put-Call Parity For European Options On Assets.
Essay 23. Put-Call Parity For American Options On Assets.
Essay 24. Call Options As Insurance And Margin.
Essay 25. A Nontechnical Introduction To Brownian Motion.
Essay 26. Building A Model Of Brownian Motion In The Stock Market.
Essay 27. Option Pricing: The Black-Scholes-Merton Model.
Essay 28. Option Pricing: The Binomial Model.
Essay 29. Option Pricing: Numerical Methods.
Essay 30. Dynamic Option Replication.
Essay 31. Risk-Neutral Pricing Of Derivatives: I.
Essay 32. Risk-Neutral Pricing Of Derivatives: II.
Essay 33. It's All Greek To Me.
Essay 34. Implied Volatility.
Essay 35. American Call Option Pricing.
Essay 36. American Put Option Pricing.
Essay 37. Swap Pricing.
Section Four. Derivative Strategies.
Essay 38. Asset Allocation With Derivatives.
Essay 39. Protective Puts And Portfolio Insurance.
Essay 40. Misconceptions About Covered Call Writing.
Essay 41. Hedge Funds And Other Privately Managed Accounts.
Essay 42. Spreads, Collars, And Prepaid Forwards.
Essay 43. Box Spreads.
Section Five. Exotic Instruments.
Essay 44. Barrier Options.
Essay 45. Straddles And Chooser Options.
Essay 46. Compound And Installment Options.
Essay 47. Digital Options.
Essay 48. Geographic Options.
Essay 49. Multi-Asset Options.
Essay 50. Range Forwards And Break Forwards.
Essay 51. Lookback Options.
Essay 52. Deferred Start And Contingent Premium Options.
Section Six. Fixed Income Securities And Derivatives.
Essay 53. Duration.
Essay 54. Limitations Of Duration And The Concept Of Convexity.
Essay 55. The Term Structure Of Interest Rates.
Essay 56. Theories Of The Term Structure: I.
Essay 57. Theories Of The Term Structure: II.
Essay 58. Simple Models Of The Term Structure: Vasicek And Cox-Ingersoll-Ross.
Essay 59. No-Arbitrage Models Of The Term Structure: Ho-Lee And Heath-Jarrow-Morton.
Essay 60. Tree Pricing Of Bond And Interest Rate Derivatives: I.
Essay 61. Tree Pricing Of Bonds And Interest Rate Derivatives: II.
Essay 62. Tree Pricing Of Bonds And Interest Rate Derivatives: III.
Essay 63. Tree Pricing Of Bonds And Interest Rate Derivatives: IV.
Essay 64. Tree Pricing Of Bonds And Interest Rate Derivatives: V.
Section Seven. Other Topics And Issues.
Essay 65. Stock Options.
Essay 66.Value At Risk.
Essay 67. Stock As An Option .
Essay 68. The Credit Risk Of Derivatives.
Essay 69. Operational Risk.
Essay 70. Risk Management In An Organization.
Essay 71. Accounting And Disclosure Of Derivatives.
Essay 72. Worst Practices In Derivatives.
Essay 73. Best Practices In Derivatives.
Answers To End-Of-Essay Questions.