The barrier option in deposit insurance with bankruptcy cost
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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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I introduction 1 12
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


I introduction 2 12
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)


I introduction 3 12
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)


I introduction 4 12
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??


I introduction 5 12
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


I introduction 6 12
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


I introduction 7 12
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


I introduction 8 12
I. INTRODUCTION (8/12)

  • bankruptcy cost

    • direct bankruptcy costs

    • indirect bankruptcy costs


I introduction 9 12
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%


I introduction 10 12
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)


I introduction 11 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


I introduction 12 12
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


Ii some general considerations 1 7
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


Ii some general considerations 2 7
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)


Ii some general considerations 3 7
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



Ii some general considerations 5 7
II. SOME GENERAL CONSIDERATIONS(5/7)

  • The value of the call provision

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


Ii some general considerations 6 7
II. SOME GENERAL CONSIDERATIONS(6/7)

  • Risk based closure rules

    • Reasonable?? Or Unreasonable ??

  • ā = 0.97+0.05σ (Reasonable!)


Ii some general considerations 7 7
II. SOME GENERAL CONSIDERATIONS(7/7)

  • ā = 0.97+0.5σ

  • Unreasonable!!


Iii the di pricing model with bc 1 14
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.


Iii the di pricing model with bc 2 14
III. The DI Pricing Model with BC (2/14)

  • pbc(a,∞;1) satisfies the following ordinary differential equation:


Iii the di pricing model with bc 3 14
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


Iii the di pricing model with bc 4 14
III. The DI Pricing Model with BC(4/14)

  • The value of the noncallable perpetual American put with BC

  • Relationship between pbc and kx



Iii the di pricing model with bc 6 14
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:


Iii the di pricing model with bc 7 14
III. The DI Pricing Model with BC(7/14)

  • Solution

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







Iii the di pricing model with bc 13 14
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!)


Iii the di pricing model with bc 14 14
III. The DI Pricing Model with BC(14/14)

  • kā = 1-0.25σ

  • Reasonable!


Iii 5 conclusion
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”.


Iv 1 the bo in di without bc 1 9
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


Iv 1 the bo in di without bc 2 9
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.


Iv 1 the bo in di without bc 3 9
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


Iv 1 the bo in di without bc 4 9
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)


Iv 1 the bo in di without bc 5 9
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


Iv 1 the bo in di without bc 6 9
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.”


Iv 1 the bo in di without bc 7 9
IV.1. The BO in DI without BC (7/9)

  • Rebate

    where τis the first passage time for at to hit the ā


Iv 1 the bo in di without bc 8 9
IV.1. The BO in DI without BC (8/9)

  • Close form solution


Iv 1 the bo in di without bc 9 9
IV.1. The BO in DI without BC (9/9)

where


Iv 2 application of the bo to pricing di with bc 1 11
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


Iv 2 application of the bo to pricing di with bc 2 11
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.


Iv 2 application of the bo to pricing di with bc 3 11
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




Iv 2 application of the bo to pricing di with bc 6 11
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







V conclusion
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.


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