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The Barrier Option in Deposit Insurance with Bankruptcy Cost

The Barrier Option in Deposit Insurance with Bankruptcy Cost. Dar Yeh Hwang 2006.12.14. I. INTRODUCTION (1/12). Deposit insurance (or DI) is an important instrument defense financial crisis stabilize financial system Purposes of DI prevent run on a bank ensure depositor.

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The Barrier Option in Deposit Insurance with Bankruptcy Cost

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  1. The Barrier Option in Deposit Insurance with Bankruptcy Cost Dar Yeh Hwang 2006.12.14

  2. I. INTRODUCTION (1/12) • Deposit insurance (or DI) is an important instrument • defense financial crisis • stabilize financial system • Purposes of DI • prevent run on a bank • ensure depositor

  3. I. INTRODUCTION (2/12) • Default events of U.S. banks in 1980 • Almost exhaust insurance fund in 1988 • FDIC has to take the bank closure risk and bankruptcy cost (hereafter, BC)

  4. I. INTRODUCTION (3/12) • Pricing of DI • OPM model • Merton (1977) : put option • Applied and spanned OPM • Merton (1978), Marcus & Shaked (1984), • Ronn & Verma (1986), Pyle (1986), • Pennacchi (1987a, b), Thomson(1987) • Allen& Saunders (1993), Duan & Yu (1999)

  5. I. INTRODUCTION (4/12) • Allen and Saunders (1993) • forbearance granted by the FDIC • FDIC’s regulatory closure policy • bank’s self-closure policy • Callable perpetual American put option • bankruptcy cost??

  6. I. INTRODUCTION (5/12) • Dreyfus, Saunders and Allen (1994) • Claim that it may be costly to transfer of the bank’s assets or to make early liquidation if the banks is closed

  7. I. INTRODUCTION (6/12) • Warner (1977) • bankruptcy cost exists • Williamson (1988) • also pointed out the bankruptcy cost problems of the corporate special assets. • The special assets like intangible assets, such as brand, R&D, charter value, advertisement …etc. • the intangible assets are short of liquidity when the corporate fall into the bankruptcy

  8. I. INTRODUCTION (7/12) • Cummins et. al. (1995) • the franchise value or charter value hold by firm’s owner die out • Gendreau and Prince (1986), Berger, Kashyap and Scalise's (1995), Rajan (1996), Bordo , Rocko, and Redish (1996) • Bank’s bankruptcy cost exists

  9. I. INTRODUCTION (8/12) • bankruptcy cost • direct bankruptcy costs • indirect bankruptcy costs

  10. I. INTRODUCTION (9/12) • direct bankruptcy costs • Warner (1977) • observed 11 railroad firms • found the direct bankruptcy cost is the 4% of the market value on the one year before the bankruptcy time • Altman (1984) • 4.3%

  11. I. INTRODUCTION (10/12) • direct bankruptcy costs • Weiss (1990) • observed 37 bankruptcy firms of New York from November, 1979 to December, 1986 • 3.1% • Franks and Torous (1994) ：4.5% • Betker (1997) ：3.51% • Branch (2002) ：3.1﹪~4.3﹪ • Gendreau and Prince (1986) : 6% (bank)

  12. I. INTRODUCTION (11/12) • indirect bankruptcy costs • Altman (1984) • 4.5% for retail business • 10.5% for industry • Andrade and Kaplan (1988) • indirect bankruptcy lie in between 10﹪~17﹪ including 31 high financial leverage firms • Rajan (1996) • 4.2% for bank

  13. I. INTRODUCTION (12/12) • bankruptcy cost • direct bankruptcy costs • about 3% ~4.5% of the firm market value • indirect bankruptcy costs • about 4.2% ~17% of the firm market value • Bank’sbankruptcy costs • about 10.2% • Bordo , Rocko, and Redish (1996): higher than 40% in Canada between 1880 and 1925

  14. II. SOME GENERAL CONSIDERATIONS(1/7) • We assume the value of bank assets follow a logarithmic diffusion process and assets are normalized by deposit (denote, a) • Asset/deposit ratio line

  15. II. SOME GENERAL CONSIDERATIONS(2/7) • Without bankruptcy cost • the cost of insurer is max(0, 1-a) • noncallable perpetual American put:p(a,∞;1)

  16. II. SOME GENERAL CONSIDERATIONS(3/7) • Solution where x is denoted the maximum asset value for which it is optimal for the bank to prematurely exercise its deposit insurance put. Then x =γ/(1+γ) and

  17. II. SOME GENERAL CONSIDERATIONS(4/7)

  18. II. SOME GENERAL CONSIDERATIONS(5/7) • The value of the call provision • The value of the callable perpetual American put option (p-c)

  19. II. SOME GENERAL CONSIDERATIONS(6/7) • Risk based closure rules • Reasonable?? Or Unreasonable ?? • ā = 0.97+0.05σ (Reasonable!)

  20. II. SOME GENERAL CONSIDERATIONS(7/7) • ā = 0.97+0.5σ • Unreasonable!!

  21. III. The DI Pricing Model with BC(1/14) • Assume • The BC is 1-kx on the self-closure point • The BC is 1-kā on the regulatory closure point • To distinguish p(a,∞;1), we assume pbc(a,∞;1) is noncallable perpetual American put with BC.

  22. III. The DI Pricing Model with BC (2/14) • pbc(a,∞;1) satisfies the following ordinary differential equation:

  23. III. The DI Pricing Model with BC(3/14) • Solution • Similarly, where x is denoted the maximum asset value for which it is optimal for the bank to prematurely exercise its deposit insurance put. Then x =γ/[(1+γ) kx] and

  24. III. The DI Pricing Model with BC(4/14) • The value of the noncallable perpetual American put with BC • Relationship between pbc and kx

  25. III. The DI Pricing Model with BC(5/14)

  26. III. The DI Pricing Model with BC(6/14) • The call provision with BC will satisfies the following ODE: subject to the following boundary conditions:

  27. III. The DI Pricing Model with BC(7/14) • Solution • The value of the callable perpetual American put option (pbc-cbc)

  28. III. The DI Pricing Model with BC(8/14)

  29. III. The DI Pricing Model with BC(9/14)

  30. III. The DI Pricing Model with BC(10/14)

  31. III. The DI Pricing Model with BC(11/14)

  32. III. The DI Pricing Model with BC(12/14)

  33. III. The DI Pricing Model with BC(13/14) • Risk based closure rules => OUT • Risk based bankruptcy cost => IN • Reasonable?? Or Unreasonable ?? • kā = 1-0.05σ (Reasonable!)

  34. III. The DI Pricing Model with BC(14/14) • kā = 1-0.25σ • Reasonable!

  35. III.5. Conclusion • Using the insights of bankruptcy cost, we model the deposit insurance with forbearance as a callable put option. • This paper solved the unreasonable results, risk based closure rules, in Allen and Saunders (1993) by using “risk based bankruptcy cost”.

  36. IV.1. The BO in DI without BC (1/9) • Allen and Saunders (1993) • forbearance granted by the FDIC • FDIC’s regulatory closure policy • bank’s self-closure policy => Callable perpetual American put option

  37. IV.1. The BO in DI without BC (2/9) • Forbearance? • Kane (1986) • deposit insurer have enforced to forbear the assured bank’s closure point • Allen and Saunders (1993) • Forbearance is granted whenever the FDIC fails to enforce its known regulatory closure point.

  38. IV.1. The BO in DI without BC (3/9) • self-closure policy? • Allen and Saunders (1993) • “even if the option expires in the money, bank shareholders may choose not to exercise the put since exercise implies voluntary bank closure” • self-closure point > regulatory closure point • self-closure point < regulatory closure point

  39. IV.1. The BO in DI without BC (4/9) • Brockman and Turtle (2003) • They take the equity and debt claims as a barrier option. • regulatory closure point = lower barrier • So pricing the deposit insurance could be viewed as a barrier option (hereafter, BO)

  40. IV.1. The BO in DI without BC (5/9) • We assume the value of bank assets follow a logarithmic diffusion process and assets are normalized by deposit (denote, at) • down-and-out put (hereafter, DOP) option DOP = where ā is a regulatory closure point

  41. IV.1. The BO in DI without BC (6/9) • Where is self-closure point? • ā < aT <1 • We call the interval as “self-closure region.”

  42. IV.1. The BO in DI without BC (7/9) • Rebate where τis the first passage time for at to hit the ā

  43. IV.1. The BO in DI without BC (8/9) • Close form solution

  44. IV.1. The BO in DI without BC (9/9) where

  45. IV.2. Application of the BO to Pricing DI with BC(1/11) • Buser, Chen, and Kane (1981) • The FDIC has to take the bank closure risk and bankruptcy cost • Kane (1986) • the FDIC which consider supervise cost has enforced to forbear the assured bank’s closure point

  46. IV.2. Application of the BO to Pricing DI with BC(2/11) • Kaufman (1992) • recommended the government and superintendent would not hope to see the circumstance that the bank is easy to close for some society stable. • Allen and Saunders (1993) • Forbearance is granted whenever the FDIC fails to enforce its known regulatory closure point.

  47. IV.2. Application of the BO to Pricing DI with BC(3/11) • (1-kā) is the asset regulatory point chosen by deposit insurer • (1-ka) is the bankruptcy cost on the self-closure point chosen by bank • Then the value of deposit insurance premium where MDOP means the modify down-and-out put option in the rule of the rebate

  48. IV.2. Application of the BO to Pricing DI with BC(4/11) • Close form solution

  49. IV.2. Application of the BO to Pricing DI with BC(5/11) where

  50. IV.2. Application of the BO to Pricing DI with BC(6/11) • self-closure region > regulatory closure point ??? • We don’t need it! • Owing to the above reasons and considering the BC

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