The Barrier Option in Deposit Insurance with Bankruptcy Cost

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
• FDICs regulatory closure policy
• banks self-closure policy
• Callable perpetual American put
• option
• bankruptcy cost??

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I. INTRODUCTION (5/12)

• Dreyfus, Saunders and Allen (1994)
• Claim that it may be costly to transfer of the
  banks assets or to make early liquidation if the
  banks is closed

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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, RD, 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
  firms owner die out
• Gendreau and Prince (1986),
• Berger, Kashyap and Scalise's (1995),
• Rajan (1996),
• Bordo , Rocko, and Redish (1996)
• Banks bankruptcy cost exists

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I. INTRODUCTION (8/12)

• bankruptcy cost
• direct bankruptcy costs
• indirect bankruptcy costs

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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

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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)

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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

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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
• Banks bankruptcy costs
• about 10.2
• Bordo , Rocko, and Redish (1996) higher than 40
  in Canada between 1880 and 1925

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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

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II. SOME GENERAL CONSIDERATIONS (2/7)

• Without bankruptcy cost
• the cost of insurer is max(0, 1-a)
• noncallable perpetual American putp(a,81)

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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
  ?/(1?) and

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II. SOME GENERAL CONSIDERATIONS (4/7)
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II. SOME GENERAL CONSIDERATIONS (5/7)

• The value of the call provision
• The value of the callable perpetual American put
  option (p-c)

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II. SOME GENERAL CONSIDERATIONS (6/7)

• Risk based closure rules
• Reasonable?? Or Unreasonable ??
• a 0.970.05s (Reasonable!)

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II. SOME GENERAL CONSIDERATIONS (7/7)

• a 0.970.5s
• Unreasonable!!

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III. The DI Pricing Model with BC (1/14)

• Assume
• The BC is 1-kx on the self-closure point
• The BC is 1-ka on the regulatory closure point
• To distinguish p(a,81), we assume pbc(a,81) is
  noncallable perpetual American put with BC.

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III. The DI Pricing Model with BC (2/14)

• pbc(a,81) satisfies the following ordinary
  differential equation

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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 ?/(1?) kx and

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III. The DI Pricing Model with BC(4/14)

• The value of the noncallable perpetual American
  put with BC
• Relationship between pbc and kx

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

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III. The DI Pricing Model with BC(7/14)

• Solution
• The value of the callable perpetual American put
  option (pbc-cbc)

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III. The DI Pricing Model with BC(8/14)
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III. The DI Pricing Model with BC(9/14)
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III. The DI Pricing Model with BC(10/14)
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III. The DI Pricing Model with BC(11/14)
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III. The DI Pricing Model with BC(12/14)
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III. The DI Pricing Model with BC(13/14)

• Risk based closure rules gt OUT
• Risk based bankruptcy cost gt IN
• Reasonable?? Or Unreasonable ??
• ka 1-0.05s (Reasonable!)

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

• ka 1-0.25s
• Reasonable!

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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.

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

• Allen and Saunders (1993)
• forbearance granted by the FDIC
• FDICs regulatory closure policy
• banks self-closure policy
• gt Callable perpetual American put option

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IV.1. The BO in DI without BC (2/9)

• Forbearance?
• Kane (1986)
• deposit insurer have enforced to forbear the
  assured banks closure point
• Allen and Saunders (1993)
• Forbearance is granted whenever the FDIC fails to
  enforce its known regulatory closure point.

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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 gt regulatory closure point
• self-closure point lt regulatory closure point

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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)

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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
• a is a regulatory closure point

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IV.1. The BO in DI without BC (6/9)

• Where is self-closure point?
• a lt aT lt1
• We call the interval as self-closure region.

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IV.1. The BO in DI without BC (7/9)

• Rebate
• where t is the first passage time for at to
  hit the a

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IV.1. The BO in DI without BC (8/9)

• Close form solution

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

• where

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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 banks closure
  point

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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.

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IV.2. Application of the BO to Pricing DI with BC
(3/11)

• (1- ka) 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

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IV.2. Application of the BO to Pricing DI with BC
(4/11)

• Close form solution

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

• where

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IV.2. Application of the BO to Pricing DI with BC
(6/11)

• self-closure region gt regulatory closure point
  ???
• We dont need it!
• Owing to the above reasons and considering the BC

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IV.2. Application of the BO to Pricing DI with BC
(7/11)

• Close form solution
• and

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IV.2. Application of the BO to Pricing DI with BC
(8/11)
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IV.2. Application of the BO to Pricing DI with BC
(9/11)
54
IV.2. Application of the BO to Pricing DI with BC
(10/11)
55
IV.2. Application of the BO to Pricing DI with BC
(11/11)
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V. CONCLUSION

• Our model can improve the unreasonable points in
  some aspects on barrier option applied on pricing
  deposit insurance directly.
• The bankruptcy cost is a key factor for pricing
  deposit insurance in the real world and should
  not be ignored.