Dilip Madan - On Pricing Contingent Capital Notes.pdf

8/14/2019 Dilip Madan - On Pricing Contingent Capital Notes.pdf

1/24Electronic copy available at: http://ssrn.com/abstract=1971811

On Pricing Contingent Capital Notes

Dilip B. MadanRobert H. Smith School of Business

University of MarylandCollege Park, MD. 20742

Email: [email protected]

December 13, 2011

Abstract

A banks stock price is modeled as a call option on the spread of ran-dom assets over random liabilities. The logarithm of assets and liabilitiesare jointly modeled as driven by four variance gamma processes and thismodel is estimated by calibrating to quoted equity options seen as com-pound spread options. On dening riskweighted assets as asset value lessthe bid price plus the ask price of liabilities less the liability value weendogenize capital adequacy ratios following the methods of conic nancefor the bid and ask prices. All computations are illustrated on CSGN.VX,ADRed into USD on March 29 2011.

1 Introduction

Contingent capital notes are a nancial innovation occuring in response to thenancial crisis of 2008. The issuance of such securities was recommended by theSquam Lake Report (2010) and the authors of this report encouraged regulatorsto require nancial institutions to invest in regulatory hybrid securities. Theseare long-term debt obligations converting automatically to equity in times ofnancial stress for the issuing entity. Such securities are seen as providingavenues for automatic recapitalization in times of need (Due (2010)). A varietyof such notes are described in Madan and Schoutens (2011).

In November 2009, Lloyds Banking Group was the rst to issue such a secu-rity. It was a Lower Tier 2 hybrid capital instrument called Enhanced CapitalNotes. They include a contingent capital feature with the notes converting toordinary shares if Lloyds published consolidated core Tier 1 ratio falls below5%. In mid 2010 Rabobank issued a contingent core note and in October 2010, a

We thank Matthew Evans at Morgan Stanley for his encouragement on accomplishing theanalysis presented in this paper.

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2/24Electronic copy available at: http://ssrn.com/abstract=1971811

Swiss government-appointed panel, proposed the rst capital surcharge on too-big-to-fail banks. Switzerlands biggest banks are to hold total capital equal to

at least 19 percent of their assets. By 2019, the lenders need to have a commonequity ratio of at least 10 percent and the rest in contingent capital. In responseto these requirements Credit Suisse announced in February 2011 the issuanceof CHF 6 billion trigger tier 1 CoCos called buer capital notes. Regulatorsthroughout Europe are expected to provide further clarity on the use of CoCobonds later this year. It is anticipated that the market for such securities couldgrow to a trillion dollars in the coming years.

This activity has led to a demand for CoCo pricing models. There is apotential loss on conversion that is linked to the value of the underlying stockon the conversion date. However, the trigger for conversion is a balance sheetentity like a tier one capital ratio. The components of this ratio are the valueof equity, the level of risk weighted assets and the add ons to be applied torisky liabilities. Risk weighted assets are a measure of potential losses in assetvalues while liability add ons assess the risk of having to unwind risky liabilitiesunfavorably. We note in this regard the model with just risky assets that followa geometric Brownian motion process with a captial ratio trigger of equity toasset values studied in Glasserman and Nouri (2010), that also accomodatespartial conversion.

In this paper we generalize the Merton (1974, 1977) approach and treatequity as an option on the spread of risky assets over risky liabilities with astrike determined at the level of debt less cash on hand and a maturity set inthe distant future. Equity options are then compound spread options and weemploy the surface of traded equity options to infer the joint law of risky assetsand liabilities. We then employ the methods of conic nance (Cherny and Madan(2010)) that delivers models for bid and ask prices in two price economies. Risk

weighted assets are taken at the level of assets less a conservative bid price whileadd ons are modeled by the ask price for liabilities less the value of liabilities.The capital adequacy ratio is then determined endogeneously as the ratio ofequity values to the sum of risk weighted assets and liability add ons. We thushave access to the joint stochastic process for the stock price and the capital ratiothat we employ to price the CoCo note. The pricing procedures are illustratedon data for Credit Suisse.

The CoCo notes are USD dollar denominated while the underlying equityoption surfaces are in CHF. We therefore rst quanto the underlying optionsurfaces into USD. The notes are however not quantoed and take the currencyrisk at conversion. We therefore ADR (American Depository Receipt) the quan-toed surface to build the surface for options on the dollar cost of foreign stockswith dollar strikes. We then calibrate a synthetic dollar denominated asset and

liability process from the surface of CHF equity options ADRed into USD.The specic model for the stochastic evolution of risky assets and liabilities

is a linear mixture of independent Lvy processes. We allow for the existenceof idiosyncratic shocks to assets and liabilities along with compensating andcompounding shocks that reduce assets and raise liabilities simultaneously. Wetherefore employ four independent Lvy processes, two idiosyncratic, one com-

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pensating and one compounding. The specic Lvy processes used are thevariance gamma model with three parameters for each of the four processes.

This yields a twelve parameter model for the joint law of assets and liabilitiesthat lies in the LG class as dened in Kaishev (2010).

Once the asset and liability model has been calibrated to the USD ADRedsurface of CHF equity options, we price the CoCo by simulating the law of assetsand liabilities, pricing equity on this path space using a spread option model,evaluating risk weighted assets and add ons by determining the bid price ofassets and the ask price of liabilities a year later. We then evaluate the capitalratio and if a conversion is triggered we evaluate the loss on conversion. Thestress level for bid and ask prices is determined to match the initial or startingreported capital ratio.

The steps in the procedure are

1. Calibrate the option surface in the foreign currency.

2. Calibrate the surface of FX options on CHF as a dollar denominated asset.

3. Quanto the surface into USD.

4. ADR the Quantoed surface to USD.

5. Calibrate the compound spread option model on equity option data forthe joint law of assets and liabilities.

6. Calibrate the conic stress level to the initial capital ratio.

7. Simulate time paths for assets, liabilities, stock prices and capital ratios.

8. Price the CoCo.

We present the details for each of these steps in separate sections with anapplication to data on Credit Suisse. Though the trigger on the capital ratio is7% with a conversion stock price oored at20 the market is trading closer tothe these triggers being at 6% and 19 respectively.

2 The Foreign Equity Option Surface

The rst step is to parsimoniously represent the risk neutral distributions forthe stock price at all maturities with a few parameters. There are many optionpricing models one may use for this purpose and they include Lvy processes(Schoutens (2003), Cont and Tankov (2004)), stochastic volatility models (He-

ston (1993), Carr, Geman, Madan and Yor (2003)) with and without jumpsand Sato processes (Carr, Geman, Madan and Yor (2007)). Lvy processes areparticularly suited to options at a single maturity but as theoretically excesskurtosis and skewness decrease like the reciprocal of maturity and its squareroot respectively, while in data they are relatively constant, these models donot provide a good synthesis when multiple maturities are involved (Konikov

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and Madan (2002)). Stochastic volatility models on the other hand introducethe complexity of a second stochastic dimension for volatility when we already

know that in the absence of static arbitrage, option prices must be consistentwith a one dimensional Markovian model (Carr and Madan (2005), Davis andHobson (2007)). The Sato process provides us with a particularly simple fourparameter model capable of synthesizing option prices at a point of time acrossboth strike and maturity. We employ here the Sato process based on the vari-ance gamma law at unit time (Madan and Seneta (1990), Madan, Carr andChang (1998)).

Let G be a gamma variate with unit mean, variance and density

f(g) =

1

1 g

11e

g

1

; g >0:The variance gamma variate Xis the law of

X= G + pGZwhere Z is a standard normal variate independent of G: The law for X isinnitely divisible with characteristic function

E

eiuX

=

1

1 iu+ 22 u2

! 1

:

The Lvy process associated with Xis a pure jump process with Lvy measure

k(x)dx = 1

expx2 Bjxjjxj dx;

B = 1s2

+ 2

2:

The variance gamma law is also a self decomposable law as is evidenced byobserving thatjxjk(x)is decreasing in x for x >0 and increasing in x for x


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The characteristic function for the logartihm of the stock price is easily obtainedin closed form and option prices may then be computed using Fourier inversion

as described in Carr and Madan (1999).We take the surface for Credit Suisse as quoted in Zurich, CSGN:V X on

March 29 2011. The number of options is 169 across 11 maturities and wepresent a graph of the actual option prices in Figure 1 along with the actualand tted prices in Figure 2. The parameter estimates for the variance gammaSato process and t statistics of the root mean squared error (rmse)the averageabsolute error (aae)and the average percentage error (ape)are

= 0:2689

= 0:2922

= 0:2880 = 0:6527

rmse = 0:1455

aae = 0:1046

ape = 0:0611

3 The FX Option surface

In order to Quanto and ADR a surface from CHF into USD we also need therisk neutral law of the USD quoted in CHF. For this purpose we also t the Satoprocess based on the variance gamma law to these FX options. The parameterestimates for this t on March 29 2011 were

= :1203

= :1105

= :0254

= :5224

The t statistics were rmse= :000648, aae = :0004858, and ape = :0315:

4 Quantoing CSGN.VX from CHF to USD

We rst explain in a subsection the general procedure employed for quantoing anoption surface from one currency into another. This requires the specication of

a joint risk neutral law for the stock and the currency with risk neutral marginalsas already estimated by our Sato process associated with the variance gammalaw at unit time. A separate subsection details the joint law employed. A thirdsubsection applies these methods to construct the quantoed surface that weADR in the next section.

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20 25 30 35 40 45 50 55 600

2

4

6

8

10

12

strike

OptionPrice

csgn on 20110329

T=4.7260

T=3.7288

T=2.7315

T=2.2329

T=1.7342

T=1.2164

T=0.7178

T=0.9671

T=0.4685

T=0.2192T=0.1425

Figure 1: 169 Option Prices at 11 maturities for CSGN on March 29 2011.

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20 25 30 35 40 45 50 55 60-2

0

2

4

6

8

10

12actual and fitted prices for csgn on 20110329

strike

option

price

Figure 2: Actual prices in circles with tted prices in dots of the same color formatching maturities.

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4.1 General Principles for Quantoing Option Surfaces

We wish to quanto a foreign asset with foreign price S with foreign exchangerate B in foreign currency per dollar into dollars. We assume the existence ofa foreign risk neutral joint law, (S; B)that prices all joint claims on (S; B)inforeign currency by pricing c(S; B)at

w= erFTZ10

Z10

c(S; B)(S; B)dSdB:

We now dene the exchange rate the other way around by A = B1 thatis a dollar denominated asset and consider the issuance of quantoed securitiespayingec(S; A) in dollars. The initial values for the exchange rates for the twodirections as A0; B0: The quantoed joint risk neutral law (S; A) prices thisclaim in dollars at

ew= erDT Z10

Z10

ec(S; A)(S; A)dSdA:The price w is in foreign currency whileew is a dollar price.

We may hedgeec(S; A)by buying c(S; B)in the foreign market and deningc(S; B)such that

ec(S; A) = AcS; 1A

;

c(S; B) = BecS; 1B

:

The cost of this in foreign currency is

w = erFT Z10

Z10

c(S; B)(S; B)dSdB

= erFTZ10

Z10

BecS; 1B

(S; B)dSdB:

and the dollar cost is

ew= A0erFT Z10

Z10

BecS; 1B

(S; B)dSdB:

We now write this in the desired form to identify (S; A): We write

ew = erDTA0e

(rDrF)TZ10

Z10

B

ec

S;

1

B

(S; B)dSdB

= erDTA0e(rDrF)T Z1

0

Z10

1A3ec (S; A) S; 1

A dSdA:

It follows that

(S; A) = A0e(rDrF)T 1

A3

S;

1

A

:

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We verify that we have a joint density as

Z10

Z10

(S; A)dSdA = Z10

Z10

A0e(rDrF)T 1

A3S; 1

A dSdA

= A0e(rDrF)T

Z10

Z10

B(S; B)dSdB

= A0e(rDrF)TB0e

(rFrD)T

= 1:

The quantoed marginal is

H(S) =

Z10

(S; A)dA

= A

0e(rD

rF)T Z

1

0

1

A3 S; 1A dA= A0e

(rDrF)TZ10

B(S; B)dB

=

R10

B(S; B)dB

B0e(rFrD)T :

4.2 The joint law employed

At each maturity we have the marginal law for both the logarithm of the stockand the currency as a variance gamma law. We may write the logarithm for thestock in the form

s= s0

+ (rF

q)t + !s

+ s

gs

+ sp

gs

Zs

:

We may also write similarly for the logarithm of the exchange rate quoted asCHF per USD that

x= x0+ (rF rD)t + !x+ xgx+ xpgxZx:

If we wish to price the quanto option by just correlating the Brownian motionswe simulate s; x on the above joint law with correlation between Zs and Zx:For a call option with strike Kwe average

erDt (es K)+ e!x+xgx+xpgxZx ;

where the martingale in the exchange rate now serves as a measure change that

reweights the paths. Given vgssd parameters for the logarithm of the stockand the logarithm of the exchange rate along with a correlation value we mayconstruct this quanto surface.

We have a few options in building the joint law given the two marginals.We may correlate the Brownian motions. We may also correlate the gammaprocesses. Basically the marginal gammas are obtained in terms of standard

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gammas or the law of a unit scale standard gamma process (t) simulated at(s) for gs and an independent gamma process taken at (x)for gx:

Apart from correlating the gammas we can build a copula for the joint lawfor the two marginal martingales

Ms = exp (!s+ sgs+ sp

gsZs) ;

Mx = exp(!x+ xgx+ xp

gxZx);

asC(FMs(Ms); FMx(Mx)):

In particular one could sample

Ms = F1Ms

(N(Zs));

Mx = F1Mx

(N(Zx));

for correlated Zs; Zx:In the current implementation we merely correlate the Brownian motions

to build a quanto surface. As an input we then require a term structure ofcorrelations that species the correlation between the standard normal variatesto be employed at each maturity. In our example we have just used a atcorrelation.

4.3 Quantoing CSGN.VX into USD

We employed a at correlation of15% at each maturity to build the surface ofquantoed option prices. The169 options on the Zurich exchange were quantoedinto USD using the joint law based on correlating the normal variates. Figure3 presents the data on these quantoed option prices.

5 ADR the Quantoed Surface

We present rst in a subsection a general procedure for how we ADR a surface.The next subsection presents the results on CSGN ADRed into USD.

5.1 General Procedure to ADR a surface

Let Y =S A where Sis the quantoed asset and A is the currency as U SD perCHF: We may build the joint law ofY; A from the joint law of S; A and herewe must have and will show that we do have the monotonicity in convex orderfor the joint law on (Y; A): Let this joint density be (Y; A): By the change of

variables we get that

(Y; A) =

Y

A; A

1

A

= 1

A4

Y

A;1

A

:

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20 25 30 35 40 45 50 55 60 650

2

4

6

8

10

12

Dollar S trike

OptionPrice

csgn quantoed into usd at flat 15% c orrelation

Figure 3: CSGN 169 options at 11 maturities quantoed into usd at at 15%correlation

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Consider now a convex function c(Y; A)and the expectation

Z10

Z10

c(Y; A)(Y; A)dY dA = Z10

Z10

c(Y; A) 1A4

YA

; 1A dY dA

=

Z10

Z10

c(S

B;

1

B)B(S; B)dSdB;

on making the change of variables

S = Y

A;

B = 1

A:

We now show that for any convex function c(Y; A) the function v(S; B) =

c SB ; 1B B is convex. This short proof was communicated by Marc Yor. Con-sider and two martingales M; N and note that by convexity of c; c(M; N) isincreasing in expectation. Now change measure to Q using the martingale N:Under Q, the pair of processes

MN

; 1N

are martingales. Hence under Q the

expectation ofcMN

; 1N

is increasing in expectation. It follows that under the

original probability N cMN

; 1N

is increasing in expectation or the expectation

ofv (M; N)is increasing in expectation for all martingales. Now take M; N tobe continuous martingales driven by correlated Brownian motions with constantvolatilities and correlations, apply Itos lemma and deduce that v is a convexfunction. So the implied joint surface of the ADR and the exchange rate isincreasing in the convex order.

To build the ADR surface we price options as

erDt Z10

Z10

(Y K)+(Y; A)dY dA

= erDtZ10

Z10

(Y K)+ 1A4

Y

A; 1

A

dY dA

= erDtZ10

Z10

S

B K

+B(S; B)dSdB

= erDtEh

SB K+ Bi

B0e(rFrD)t :

so from a joint distribution ofS; B we simulate the above expectation to deter-mine the ADR surface.

5.2 CSGN.VX ADR into USD

We employed the procedure described in the section 5.1 to ADR the surface ofCSGN.VX as quoted in Zurich into USD. Figure 4 presents the prices of all 169options after they have been ADRed into USD.

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20 25 30 35 40 45 50 55 60 650

2

4

6

8

10

12

strike

option

price

csgn adr into usd

Figure 4: CSGN 169 options at 11 maturities ADRed into usd at at 15%correlation

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6 The Compound Spread Option Model for the

law of the balance sheetWe present in a subsection the theoretical model for equity options as a com-pound option on the spread of assets over liabilities with a strike given by adebt face value of F less cash reserves of M and a distant maturity. The re-sults of calibrating this model on the surface of CSGN.VX ADRed into USD ispresented in a following subsection.

6.1 Equity Options as Compound Spread Options

The risk neutral law ofA(T)L(T) may be modeled as the dierence of twoexponential Lvy processes that we may simulate forward in time. On this pathspace we may evaluate the path space of equity prices computed as a spread

option with an expected payo under a risk neutral t conditional expectationoperator Et as

J(t) = Et

h(A(T) L(T)(F M))+

i: (1)

The spread option computation is a two dimensional Fourier inversion thatintegrates out the random elements in the asstes and liabilities. We employfor the purpose the algorithm proposed by Hurd and Zhou (2009). The pathspace of assets and liabilities is transformed into a path space of equity pricesupon applying the spread option computation at the various levels of assetsand liabilities reached at time t in the asset liability simulation. This equityprice path space is then used to construct equity option prices for strike Kandmaturity t by averaging over the path space to estimate the equity option value

reported in the ADR surface as

w(K; t) = ertEh

(J(t) K)+i

:

We determine the parameters of the joint and correlated risky asset and liabilityvalue process to best t the surface of the equity option surface as seen on theADR surface.

We take the risk neutral risky asset and the risky liability as exponentialLvy processes with

A(t) = A(0)exp(X(t) + (r+ !X)t) ;

L(t) = L(0)exp(Y(t) + (r+ !Y)t) ;

where we now allow for a rich dependence in these processes. If we take a linearmixture of just two independent Lvy processes we get jumps occurring on tworays from the origin. If the independent processes are variance gamma V Gprocesses for example then we have a variance gamma process running in logspace on a particular ray from the origin with the asymmetry parameter on thisray being the skewness parameter of this particular variance gamma process.

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Given that we operate in a two sided way for each independent Lvy process,we need to cover180degrees of possible directions of motion. We take 4variance

gamma processes with 12 parameters placed at the degrees 0; 45; 90; and 135.This gives us two idiosyncratic rays at the angles of0 and 90 where assets andliabilities are independently aected. The angle of45 allows for compensatingeects with assets and liabilities moving together while the angle 135 allows forcompounding eects on the two sides of the balance sheet.

We shall let the calibration determine the relative variance placed on eachof the four rays. For the four anglesj ; j = 1; 4;we have the jumps in assetsand liabilities as

xj = ujcos(j);

yi = ujsin(j);

whereui is the jump in the j th V Gprocess with parameters j ; j ; j: We then

have that

X(t)Y(t)

=

cos(1) cos(2) cos(3) cos(4)sin(1) sin(2) sin(3) sin(4)

2664U1(t)U2(t)U3(t)U4(t)

3775 ;and our joint law is the linear mixture of4independentV GLvy processes witha prespecied mixing matrix. The joint characteristic function is

E[exp(iuX(t) + ivY(t))] =4Y

j=1

0

@ 1

1 i(u cos(j) + v sin(j))jj+ 2

jj

2 (u cos(j) + v sin(j))2

1

A

= (u; v):

The value of

!X =4X

j=1

1

jln

1cos(j)jj

2jjcos2(j)

2

!;

!Y =4X

j=1

1

jln

1sin(j)jj

2jjsin2(j)

2

!;

and the characteristic function of the logarithm of assets and liabilities is

Eheiu ln(A(t))+iv ln(L(t))

i= (u; v) exp(iu ln(A(0))+iv ln(L(0))+iu(r+!X)t+iv(r+!Y)t):Our equity value at any date t given a simulation ofA(t); L(t) is the price

of a spread option with some strike and maturity using this joint characteristicfunction with initial valuesA(t); L(t)and time to maturity Tt:For the initialvalue of risky assets and risky liabilities excluding debt, we take these magni-tudes from the balance sheet but permit some option market adjustment factor

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to match the stock price. The adjustment factor is calibrated by equating thevalue of equity computed as a spread option at the strike of debt less initial cash

equivalent reserves with the initial stock price at market close on the calibrationdate.

6.2 Results of Calibrating Compound Spread Option Model

on the ADR surface

We estimated the asset and liability process from ADRed equity options seen ascompound spread options. The assets and liabilities are linear mixtures of fourindependent variance gamma processes running on the angles 0; 45; 90; 135:Theestimated parameter values for the ADR surface with 169 options are as givenin Table 1.

TABLE 1Results of Calibrating

Compound SpreadOption ModelParameter Value(0) 0:0175(0) 0:0326(0) 0:0709(45) 0:0083(45) 0:0804(45) 0:1790(90) 0:2237(90) 0:1383(90) 0:2450(135) 0:0890(135) 0:0495(135) 0:1412

We present a graph in Figure 5 a graph of the t of this compound spreadoption model to the ADR surface.

We have not converged in this calibration but we stopped at 2500 functionevaluations. The t is quite good for some of the intermediate maturities but thelonger maturities are not that well calibrated. We are aware of the limitations ofa Lvy model with regard to tting a full surface (Konikov and Madan (2002)).As an additional check on the convergence we computed the left and rightderivatives at the calibration point and these are reported in Table 2. Most ofthese, excepting (0); (45); are negative and positive as they should be for alocal minimum. We accepted this calibration for our subsequent analysis of the

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20 25 30 35 40 45 50 55 60 650

2

4

6

8

10

12

Strike

OptionPrice

Compound 4 VG Mixture Spread Option Model Calibration to CSGN.VX ADRed in to USD

Figure 5: Calibration of Compound Spread Option Model with Assets andLiabilities as a linear mixture of four independent VG processes. Data in circles.

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CoCo as reported in the remaining sections.

TABLE 2LEFT RIGHT GRADIENTS

LEFT RIGHT(0) 0:4791 0:0383(0) 1:5068 0:1751(0) 0:0900 0:0259(45) 0:0351 0:0330(45) 0:0364 0:6542(45) 0:2110 0:0364(90) 2:0140 4:7936(90) 0:0488 0:2379(90) 0:1822 0:2515(135)

2:9507 0:3911

(135) 0:3645 0:0486(135) 0:0857 0:0283

7 Calibrate the Conic Stress Level

For the capital adequacy ratio(CAR)we need to construct risk weighted assets.For this purpose we simulate assets and liabilities out one year and we thendene risk weighted assets as the loss on an unfavorable unwind of assets andliabilities. This is computed as asset value less the bid price of assets one yearlater plus the ask price of liabilities one year later less the liability. We maywrite this as

RW At = At b(At+1) + a(Lt+1) Lt: (2)The bid and ask prices are computed by concave distortions using the dis-

tortionminmaxvar whereby

(u) = 1(1 u 11+ )1+;

and the bid price of a random outcome X with distribution function F(x) forpositiveX is given by

b(X) =

Z10

xd(F(x));

while the ask price isa(X) =b(X):

In greater detail given the joint law of assets and liabilities one may simulate

paths of A(ti); Li(ti); forward in time quarterly for 13 years for quarters i =1; ; 52;and ti = ih; withh = 25;from initial dollar values per share reportedin the balance sheet. The per share initial dollar value of assets was1066:34while the per share dollar value of liabilities was 983:25: The strike per shareof debt less cash was 22:79 with a stock price of 42:93: We may then apply

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our spread option computation to transform the level of assets and liabilitiesattained to equity values J(ti):

For any level of assets A(ti) and liabilities L(ti) attained at any time pointtiwe may further simulate assets and liabilities forward from these levels by oneyear to get readings Ati+1;j ; Lti+1;j forj = 1; ; N: We then evaluate the bidprice of assets as a distorted expectation of Ati+1;j on arranging the readingsin increasing order Ati+1;(j) with the bid price being

bti

=Xj

Ati+1;(j)

j

N

j1

N

:

Similarly we determine the ask price for liabilities ati as the negative of the bidprice forLti+1;j:

The distortion (u) employed here is minmaxvar and we model the levelof risk weighted assets as per equation (2).

The equity price J(ti) is determined by applying the spread option model(1) and the capital adequacy ratio is computed as

CARti = J(ti)

RW Ati

:

The value ofwas estimated at0:425to calibrate the reported an initial capitalratio of13:82%:

8 Simulating assets, liabilities, stock prices and

capital ratios

We now hold the stress level xed at the initial calibration point and simulatethe paths for assets, liabilities, stock prices and capital ratios. We present inFigure 6 a graph of the stock price against the capital ratio at six year ends.

We observe that the relationship is nonlinear. The relationship is closer tolinear on a log log plot as shown in Figure 7.

We regressed the logarithm of the stock price against the logarithm of thecapital adequacy ratio at each quarter end. The results of the regressions areavailable a spreadsheet. The average relationship between the stock price andthe capital adequacy ratio is

S= 115:7115(CAR)0:6102 exp

:1865 z :03482

;

for a standard normal variate z: The conditional volatility of the stock givenCAR in this model is

115:7115(CAR)0:6102 exp

:0348

2

;

and this rises with the level of the C AR.

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Figure 6: Graph of the stock price against the capital ratio at six year ends.

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Figure 7: Log Stock vs Log capital ratio at six year ends

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We may run the regression the other way to get

CAR = :00053572 S1:5615 exp:2943z :294322

:One could use such a dependence in modeling the CoCo conversion and

periodically we may revise the dependence.

9 Pricing the CoCo

We simulate100000paths of assets and liabilities each quarter for 13 years andconvert these to stock prices via the spread option calculator. We also use astress level of0:425 and a starting value of CAR of13:82% and simulate riskweighted assets quarterly for 13 years.

The loss is modeled on a conversion occuring the rst time both

CAR < Cand Stock P rice < S:

We then evaluate the present value of the loss for a number of values ofS;C:The present value of the loss l(t) at time t is given by

l(t) = 1CAR


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cash equivalents and a distant maturity. The logarithm of assets and liabilitiesare jointly modeled as driven by four variance gamma processes, two of which

are idiosyncratic, one is compensating while the other is compounding in theirshock eects. The joint law is estimated by calibrating this law to quoted equityoptions that are seen as compound spread options. Once the law is calibratedit may be simulated for the time paths of assets and liabilities and the resultingstock prices seen as a spread option. Additionally we simulate at all timesthe random assets and liabilities out by one year to evaluate their riskiness byevaluating bid and ask prices for the assets and liabilities respectively. Deningriskweighted assets as asset value less the bid price plus the ask price of liabilitiesless the liability value we endogenize capital adequacy ratios. The resulting pathspace allows for an assessment of conversion losses on securities like CoCosthat trigger a stock priced based loss contingent on a capital ratio event. Anumber of such securities are issued as dollar denominated on European andSwiss underliers. Procedures are also developed for quantoing and ADRingforeign equity option surfaces into USD before estimating the compound spreadoption model driven by four independent variance gamma processes.

All computations are illustrated on CSGN.VX, ADRed into USD on March29 2011. We nd that the market trades the Credit Suisse CoCo at a capitalratio trigger strike at 6% with the stock oor at 19with listed strike and oorbeing 7% and 20:

References

[1] Carr, P. and D. B. Madan (1999), Option Valuation Using the Fast FourierTransform, Journal of Computational Finance, 2,61-73.

[2] Carr, P. and D. B. Madan (2005), A Note on Sucient Conditions for NoArbitrage,Finance Research Letters, 2, 125-130.

[3] Carr, P., H. Geman, D. Madan and M. Yor (2003), Stochastic volatilityfor Lvy Processes, Mathematical Finance, 13, 345-382.

[4] Carr, P., H. Geman, D. Madan and M. Yor (2007), Self Decomposabilityand Option Pricing, Mathematical Finance, 17, 31-57.

[5] Cont, R. and P. Tankov (2004),Financial Modelling with Jump Processes,CRC Press, Chapman and Hall, Boca Raton.

[6] Cherny, A. and D. B. Madan (2010), Markets as a counterparty: AnIntroduction to conic nance, International Journal of Theoretical and

Applied Finance, 13, 1149-1177.

[7] Davis, M. and D. Hobson (2007), The Range of Traded Options,Math-ematical Finance, 17, 1-14.

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[8] Due, Darrell (2010), A Contractual Approach to Restructuring FinancialInstitutions, inEnding Government Bailouts as We Know Them, Eds. G.

Schultz, K. Scott and J. Taylor, Hoover Institute Press, Stanford University.

[9] French, K., M. Baily, J. Campbell, J. Cochrane, D. Diamond, D. Due, A.Kashyap, F. Mishkin, R. Rajan, D. Scharfstein, R. Shiller, H. S. Shin, M.Slaughter, J. Stein and R. Stulz (2010), The Squam Lake Report: Fixingthe Financial System, Princeton University Press, Princeton, NJ.

[10] Glasserman, P. and Behzad Nouri (2010), Contingent capital with a cap-ital ratio trigger, working Paper, Columbia University.

[11] Heston, S. L. (1993), A Closed-form Solution for Options with Stochasticvolatility with applications to bond and currency options, The Review ofFinancial Studies, 6, 327-343.

[12] Hurd, T. and Zhou, Z. (2009), A Fourier Transform Method for SpreadOption Pricing, SIAM Journal of Financial Mathematics, 1, 142-157.

[13] Kaishev, V. (2010), Lvy processes induced by Dirichlet (B-) splines:modelling multivariate asset price dynamics, forthcoming MathematicalFinance.

[14] Konikov, M. and D. Madan (2002), Stochastic Volatility via MarkovChains,Review of Derivatives Research, 5, 81-115.

[15] Madan, D., P. Carr and E. Chang (1998), The variance gamma processand option pricing, European Finance Review, 2, 79-105.

[16] Madan, D.B. and W. Schoutens (2011), Conic Coconuts: The Pricing of

Contingent Capital Notes using Conic Finance, Mathematics and Finan-cial Economics, 5, 87-106.

[17] Madan D. B. and E. Seneta, (1990), The variance gamma (VG) model forshare market returns, Journal of Business, 63, 511-524.

[18] Merton, R. C. (1974). On the pricing of corporate debt: The risk structureof interest rates, Journal of Finance29, 449470.

[19] Merton, R.C. (1977), An analytic derivation of the cost of loan guaranteesand deposit insurance: An application of modern option pricing theory,Journal of Banking and Finance, 1, 3-11.

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Probability Theory and Related Fields, 89, 285-300.

[21] Schoutens, W. (2003), Lvy Processes in Finance, John Wiley & Sons,Chichester, England.

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Dilip Madan - On Pricing Contingent Capital Notes.pdf