The Barrier Option in Deposit Insurance with Bankruptcy

The Barrier Option in Deposit Insurance with Bankruptcy Cost Dar Yeh Hwang 2006. 12. 14 1

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

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

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

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

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

I. INTRODUCTION (6/12) n Warner (1977) n n bankruptcy cost exists Williamson (1988) n n n 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 7

I. INTRODUCTION (7/12) n Cummins et. al. (1995) n n 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) n Bank’s bankruptcy cost exists 8

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

I. INTRODUCTION (9/12) n direct bankruptcy costs n Warner (1977) n n n 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) n 4. 3% 10

I. INTRODUCTION (10/12) n direct bankruptcy costs n Weiss (1990) n n n 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) 11

I. INTRODUCTION (11/12) n indirect bankruptcy costs n Altman (1984) n n n Andrade and Kaplan (1988) n n 4. 5% for retail business 10. 5% for industry indirect bankruptcy lie in between 10� ~17� including 31 high financial leverage firms Rajan (1996) n 4. 2% for bank 12

I. INTRODUCTION (12/12) n bankruptcy cost n direct bankruptcy costs n n indirect bankruptcy costs n n about 3% ~4. 5% of the firm market value about 4. 2% ~17% of the firm market value Bank’s bankruptcy costs n n about 10. 2% Bordo , Rocko, and Redish (1996): higher than 40% in Canada between 1880 and 1925 13

II. SOME GENERAL CONSIDERATIONS n n (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 14

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

II. SOME GENERAL CONSIDERATIONS n (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 16

II. SOME GENERAL CONSIDERATIONS (4/7) 17

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

II. SOME GENERAL CONSIDERATIONS n Risk based closure rules n n (6/7) Reasonable? ? Or Unreasonable ? ? ā = 0. 97+0. 05σ (Reasonable!) 19

II. SOME GENERAL CONSIDERATIONS n ā = 0. 97+0. 5σ n Unreasonable!! (7/7) 20

III. The DI Pricing Model with BC (1/14) n Assume n n n 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. 21

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

III. The DI Pricing Model with BC(3/14) n n 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 23

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

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

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

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

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

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

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

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

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

III. The DI Pricing Model with BC(13/14) n n Risk based closure rules => OUT Risk based bankruptcy cost => IN n n Reasonable? ? Or Unreasonable ? ? kā = 1 -0. 05σ (Reasonable!) 33

III. The DI Pricing Model with BC(14/14) n kā = 1 -0. 25σ n Reasonable! 34

III. 5. Conclusion n n 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”. 35

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

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

IV. 1. The BO in DI without BC (3/9) n self-closure policy? n Allen and Saunders (1993) n n n “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 38

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

IV. 1. The BO in DI without BC (5/9) n n 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 40

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

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

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

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

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

IV. 2. Application of the BO to Pricing DI with BC (2/11) n Kaufman (1992) n n 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) n Forbearance is granted whenever the FDIC fails to enforce its known regulatory closure point. 46

IV. 2. Application of the BO to Pricing DI with BC (3/11) n n n (1 - kā) is the asset regulatory point chosen by deposit insurer (1 - ka) is the bankruptcy cost on the selfclosure 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 47

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

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

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

IV. 2. Application of the BO to Pricing DI with BC (7/11) n Close form solution and 51

IV. 2. Application of the BO to Pricing DI with BC (8/11) 52

IV. 2. Application of the BO to Pricing DI with BC (9/11) 53

IV. 2. Application of the BO to Pricing DI with BC (10/11) 54

IV. 2. Application of the BO to Pricing DI with BC (11/11) 55

V. CONCLUSION n n 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. 56