Collateralized Structured Products

Marcos EscobarDepartment of Mathematics,Ryerson University,254 Church Street, Toronto, Ontario , Canada,email: escobar@ryerson.ca,phone: +1 416 979 4867.

Mirco MahlstedtChair of Mathematical Finance,Technische Universität München,Parkring 11, 85748 Garching-Hochbrück, Germany,email: mirco.mahlstedt@tum.de,phone: +49 89 289 18789.

Sven PanzChair of Mathematical Finance,Technische Universität München,Parkring 11, 85748 Garching-Hochbrück, Germany,email: sven.panz@tum.de,phone: +49 89 289 17400.

Rudi ZagstChair of Mathematical Finance,Technische Universität München,Parkring 11, 85748 Garching-Hochbrück, Germany,email: zagst@tum.de,phone: +49 89 289 17401.

Abstract

In this paper multidimensional structured products with a collateral triggered by a default of theissuing company are studied. In the last decade, the volume of trades in structured products hasincreased tremendously. Particularly after the subprime and financial crisis with the default byLehman Brothers, the issue of default risk gained relevance worldwide. Since the early work ofBlack and Cox [1976], the default risk of a corporation is known to be a barrier-type product.Here, we present closed form solutions for arbitrary collateralized structured products (CSP) in theframework of n assets and two barriers, one representing default and the second a market-relatedoption.

Keywords

multiple barriers; default risk; structured products; collateral; vulnerable options.

2

After Black and Scholes [1973] derived an analytical solution for pricing and hedging contingentclaims, several extensions of that approach have been developed. The three main directions for theseextensions have been: options with more than one underlying (da Fonseca et al. [2008]); productsdepending on the path of the underlying either by means of the minimum/maximum or in generalby crossing thresholds/barriers (He et al. [1998]); and the inclusion of additional stochastic processesaccounting for stylized features such as stochastic volatility (Heston [1993]), as well as stochasticcorrelation (Escobar et al. [2013]). Our work in this paper involve the first two developments bypresenting and studying exotic multivariate path-dependent options inspired by a combination ofmarket and credit events. The volume of trades in exotic options has increased tremendously sincethe nineties (Walmsley [1997], Bouzoubaa and Osseiran [2010]). The payoff structures of many ofthese options depend on the maximum or minimum of its underlings. These exotic path-dependentoptions provide institutional investors with vehicles to meet their diverse financial needs, e.g. hedging,risk management, or speculation (He et al. [1998]). A particular case of these products are barrieroptions. These products care about the crossing of thresholds by their underlying and therefore thebehavior of the minimum or maximum value of the underlying process over a period of time. Barrierscan be embedded into market products as well as on credit products. In terms of market derivatives(Rubinstein and Reiner [1991], He et al. [1998] and literature therein) a barrier option, like a Down-and-Out, would be less expensive than a standard European option, therefore providing a convenientsubstitute or hedge in some risk management strategies. In term of credit derivatives, since the earlywork of Black and Cox [1976], the default of a corporation can be seen as a barrier-type product,where the debt is compared to the assets of the company continuously over time until maturity, anycrossing leading to a default. In other words, once the minimum of the assets falls below the debt,the company is declared bankrupt (see also Rich [1996]). A recent motivation for barrier productscomes with the recent subprime/financial crisis, here due to the default of Lehman Brothers, thetopic of default risk by option issuers gained more relevance. One example of products prone toissuer default are certificates (Zagst and Huber [2009]). Certificates, also referred to as structuredproducts, allow a customer to invest in a market index derivative, therefore from a legal point ofview those investments are bonds and the investors are creditors of the respective certificate issuer.The possibility of a defaultable issuer is commonly disregarded in the investment decision by manyinvestors. However, in the case of an insolvency of the issuer the investor may lose his total investmentregardless of the performance of the underlying of the certificate, in other words the investor has tobear the default risk of the certificate issuer. Many private investors were surprised that banks, whichhad sold Lehman certificates and products as an intermediary, had not covered the default risk ofthe issuer (Lehman) of these certificates (BGH [2011]). Hence, for a fair price, this default risk hasto be considered. Note that some of these certificates, which are market derivatives in nature, havepath-dependent payoff structures, an example is a Down-and-Out European options, therefore oncethe default of the issuer is included, the new product would involve two barriers, a barrier due to themarket component and a second one due to a credit event. The presence of two barriers is one of theingredients of a CSP product as proposed in this paper. The second main feature of a CSP productcomes from the fact that, as certificates are classical retail products, a separate collateral accountshould be explicitly added to the product. This “collateral” component should be triggered by thedefault of the issuer. In principle it could be provided by the issuer itself to the buyer as part ofthe contract using a marginal account or could be purchased separately by the buyer as an insurance

3

against a potential default. Issuer risk, and more generally, counterparty risk (default of any of theparties involved in the contract) has been studied before under the term Vulnerable or Defaultableoptions, see Bielecki et al. [2008], Klein and Inglis, Johnson and Stulz [1987] and Klein [1996] for areview. The existing literature fails to provide answers in a multidimensional setting in the contextof one or two barrier components. More specifically, the contributions of this paper are as follow:

• We provide a closed-form formula for the price of any general multidimensional derivative whosepayoff involves: two separate barriers on two of the underlying and n − 2 non-path dependentcomponents, for a total of n > 2 underlying. The barriers are considered time-dependent (forthe Credit component) and time-independent for the Market component.

• A further simplification of the pricing formula mentioned above is obtained leading to fewerintegrations and hence more efficient pricing which is compare to MonteCarlo approaches.

• We introduce and study a new financial product, the CSP. This product could involve up totwo barriers, one of them representing a structural credit default by the issuer which triggers acollateral component. The second barrier represents a general market barrier product.

• A comparison of prices with and without issuer risk/collateral as well as an analysis of sensitiv-ities is performed on 3 CSP products, dimensions n = 3 to 10.

Our work is built on those of Zhou [2001] and He et al. [1998] where the authors used a knownclosed-form solution for the joint distribution of maximum/minimum and maturity values for two as-sets in a constant volatility setting, applying it to a financial context. We, on the other hand, extendtheir work to allow for any number of underlying (any dimension n), keeping barriers for only twounderlying. To the best of our knowledge, this extension has not been seen in either the mathematicalor the financial literature.

In this paper we assume a generic multivariate framework of a geometric multivariate Brownianmotion with any possible constant correlation structure. For separable payoffs a simplification methodis introduced which significantly reduces the number of integrals. This becomes extremely efficient inhigher dimensions where a comparison to Monte Carlo methods is performed.

This paper is organized as follows. Section 2 describes the mathematical setting and notation. Section3 provides the main results for the pricing of financial derivatives. Section 4 compares differentproducts with the objective of highlighting the benefits of a collateralized product, while Section 5concludes.

4

Notation and Setting

In general collateralized structured products (CSP) could have n underlying. For example, assume3 underlying: S1, S2, S3 that have values on a comparable range: Hence, a CSP should have thefollowing general payoff structure:

Payoff = g1(S1(T )) · 1τ1>T1τ2>T + g2(S3(T )) · 1τ1>T1τ2<T

where τi is defined as a first passage time of Si:

τi = inf t ∈ [0, T ] | Si(t) < Mi(t)

In this work, we allow for either a constant (Mi(t) = M∗i ) or a time dependent barrier (Mi(t) =M∗i exp rt). The former applies to market products, i.e. S is a stock in the market, while the latterapplies to credit products hence S represents the assets of a company, Mi the debt and the eventτ < T represents a default in a Black-Cox setting. g1(S1) (the certificate component) is a payofffunction depending on a stock or index S1, which the certificate pays if S1 satisfies a barrier condition.The barrier condition on S2 monitors the certificate issuer’s (S2) default state and in case of a default,the payoff is determined by a payoff function g2(S3) (the collateral component) depending on somecollateral S3. The collateral component has to be held by the certificate issuer in a separate account.

The structure for the certificate component g1(S1) can be very general, in principle any certificatewhich depends on a continuous barrier condition would be acceptable. Next, two examples for out-performing certificates are provided:

g1(S1(T )) = [S1(T ) + c · (S1(t)−K)+]g1(S1(T )) = [K + (S1(T )−K)+]

The structure for the collateral component g2(S3) requires two components, the collateral S3 and thefunction g2(.). The collateral S3 could be any additional asset, like for example, a bond index. Itshould be pointed out that the payoff of the certificate also triggers the payoff from the collateral.This means that in the case of a default by the certificate issuer (S2), the certificate buyer gets apayment from the collateral account if and only if the buyer would have received a payment from thecertificate component in case of survival of the certificate issuer (S2). Therefore, the payoff functiong2(.) should appear together with the barrier condition for S1. g2(S3(T )) itself could be a simplefunction, for instance:

g2(S3(T )) = α · S3(T )

where α is a constant which is related to the product, for example: α = [g1(S1(0))S3(0) ]. Another possibility

for g2(S3(T )) is to take the theoretical payoff of the certificate component at t = 0 and invest this in

5

the collateral account hence:

g2(S3(T )) = α · V · S3(T )S3(0)

where V is the price of the certificate and α (0 < α < 1) is a scaling factor. The idea behind thisis that at least a portion of the price the certificate holder pays is initially invested in the collateralaccount.

Many collateralized structured products are possible, satisfying a variety of purposes. We couldeven think of generalizations where we allow the certificate payoff to depend on some other marketvariables Si, i ≥ 4. As an example take a compensation payment in case that the market variablesSi underperform their individual barriers Ki, i ≥ 4. This could be an individual asset portfolio, abond index, or a commodity price to give some examples. In this paper, the following collateralizedstructured products are studied in detail:

Payoff C1(T ) = (S1(T )−K)+ · 1τ1>T · 1τ2>T + α · S3(T ) · 1τ1>T · (1− 1τ2>T)Payoff C2(T ) = K · 1S4(T )<K4 · 1τ1≤T · 1τ2>T + S3(T ) · 1τ1≤T · (1− 1τ2>T)

More generally, in dimension n, we consider the following product:

Payoff C3(T ) = K · 1mini=4...nKi−Si(T )>0 · 1τ1≤T · 1τ2>T + S3(T ) · 1τ1≤T · (1− 1τ2>T)

These products will be studied within an empirical analysis in Section 4. The variety of products showshow important the tracking of the minimum and maximum value of underlying is in a high dimensionalsetting. In general, any basket or mountain range option (e.g. Everest, Atlas or Himalayan) fits inour framework.

Mathematical Framework

The model framework and the general notation are stated in this section. The system of processesor market variables is defined on a filtered probability space (Ω, (Ft)t≥0, Q,F) where F0 containsall subsets of the (Q−) null sets of F and (Ft)t≥0 is right-continuous. The processes under therisk-neutral measure Q are defined as follow.

dSiSi

= rdt+ σidWi,t i = 1, 2, . . . , n (1)

< dWi,t, dWj,t >= ρi,jdt (2)

This is a common multivariate lognormal process. Here, Si accounts for either stock prices or

6

asset value or indexes, depending on the application. We allow the barriers M1 and M2 to fol-low deterministic processes with either constant or with risk-free rate r growth, Mi(t) = M∗i orMi(t) = M∗i · ert, i = 1, 2.

Definition 1(Running minimum and running maximum)Let Si(t) be a stochastic process. The running minimum Si(t) and the running maximum Si(t) aredefined by

Si(t) = min0≤s≤t

Si(s)

Si(t) = max0≤s≤t

Si(s)

As for the derivatives of interest, it is sufficient to focus on the case of down-and-out double bar-rier derivatives, whose payoffs depend on the minimum of the first two assets and the end pointsof all n underlying. The rationale for this is two-fold: first indicator functions with ”τ ≤ T”as for example 1τ1≤T,τ2>T, 1τ1>T,τ2≤Tand 1τ1≤T,τ2≤T can be expressed in terms of ”τ > T”:(1τ2>T − 1τ1>T,τ2>T), (1τ1>T − 1τ1>T,τ2>T) and (1 − 1τ1>T − 1τ2>T + 1τ1>T,τ2>T) re-spectively. Secondly, payoffs with fewer barriers (for instance g(S)1τ2>T or g(S)1τ1>T) can becomputed from the general solution very efficiently by simply assigning a very small value of theproper barrier (i.e. M1 ∼ 0 or M2 ∼ 0). This way we gain in simplicity and long closed-formsolutions are avoided.

The payoff of interest in this paper would be of the form

g (S(T )) 1S∗1(T )>M∗1 ,S∗2(T )>M∗2 (3)

or equivalentlyg (S(T )) 1τ1>T,τ2>T

where S(T ) = (S1(T ), . . . , Sn(T )), S∗i (t) = Si(t)e−rt if Mi(t) = M∗i · ert, otherwise S∗i (t) = Si(t). Dueto the particular expression of the payoff in (3), the main objective of this paper is the correspondingconditional joint density/distribution of the n assets. This is a density with respect to the endpointvalues of the n assets and a distribution function of the running minimum values of the first twoassets being above the thresholds M1 and M2:

P (S(T ) ∈ dS, S∗1(T ) > M∗1 , S∗2(T ) > M∗2 ), n ≥ 3 (4)

We follow the notation in He et al. [1998], and for simplicity, the probability in Equation (4) iswritten as

p(T, S,M1,M2)dS

7

This conditional joint density/distribution will be key in deriving the prices of derivatives with payoffas in (3). This function can be used to derive payoffs depending on maximums and/or combinationsof minimum and maximum, i.e. P (S(T ) ∈ dS, S∗1(T ) < M∗1 , S

∗2(T ) > M∗2 ), P (S(T ) ∈ dS, S∗1(T ) <

M∗1 , S∗2(T ) > M∗2 ), which can be easily seen from properties of Brownian motions (see He et al.

[1998]).

The next section will provide the main results of this paper, the solution for the derivatives of interestwill be studied by means of the Backward Kolmogorov Equation (BKE). This is why the notation ofp(t, S,M1,M2) is extended to:

p(t, S,M1,M2; t, S, µ,Σ). (5)

Here µ and Σ represent the drift vector and the covariance matrix of the n assets, (S, t) are thebackward variables, and thus (S, t) are the forward variables.

Pricing of Collateralized Structured Products

In this section closed-form expressions for the price of the financial derivatives are derived. First wegive the results for a general, arbitrary covariance. In order to simplify the expression of the price, amethod useful for the case of separable payoffs is presented which significantly reduces the complexityof the solution.

The following result is about the pricing of a product with the general payoff given in Equation (3).Due to the fact that, in a credit setting, barriers are time dependent, whereas for market indicators,time independent barriers are more common, here we consider both types of barriers on a singleproduct. We will assume that the threshold M is time dependent for the underlying in group A (Siwith i ∈ A) and time independent for all others (Si with i /∈ A).

Theorem 2Given the market model in (1)–(2) with constant volatilities, the price at time t of a payoff as in (3)is given as:

C(t, S,M1,M2) =∫ ∞m1

∫ ∞m2

∫Rn−2

g(S) · p(T, x,m1,m2;T − t, x)dx (6)

where

p(T, x,m1,m2; τ, x) = ePni=1 αi(xi−bxi)+(b−r)(T )|det(J−1)|·h(T, ω, θ, z3, . . . , zn,m1,m2; τ, ω, θ, z3, . . . , zn)

h(T, ω, θ, z3, . . . , zn,m1,m2; τ, ω, θ, z3, . . . , zn)

= (2βτ

)∞∑N=1

e−ω2+ω2

2τ · sin(Nπθ

β)sin(

Nπθ

β) · INπ

β(ω · ωτ

) · 1

(2πτ)n−2

2

· e−Pni=3(bzi−zi)2

2·τ

8

Here INπβ

is the fundamental solution of the second kind for the Bessel equation. The variables andparameters are defined as follow:

τ = T − t, xi(t) =

ln( bSi(bt)bS(0)

)for i /∈ A

ln( bSi(bt)e−rbtbS(0)

)for i ∈ A

, mi = ln

(M∗i

S(0)

)

(similar with the forward variables, xi(t))

b =n∑i=1

Biαi +12

n∑i=1

n∑j=1

σiσjρijαiαj,

α1

...αn

= −Σ−1 ·B

where

B = (B1, .., Bn)′ , Bi =

σ2i2 if i ∈ A

σ2i2 − r if i /∈ A

z1.........zn

= J−1 ·

x1 −m1

x2 −m2

x3

...xn

,

z1.........zn

= J−1 ·

x1 −m1

x2 −m2

x3

...xn

J is the lower triangular matrix of the Cholesky decomposition of the covariance matrix Σ and JNπ

β

is the fundamental solution of the first kind for the Bessel equation.

ω =√z21 + z2

2 , tan θ =z1z2, θ ∈ [0, β]

ω =√z21 + z2

2 , tan θ =z1z2, θ ∈ [0, β]

tan β = −√

1− ρ212

ρ12, β ∈ [0, π]

The proof is given in the Appendix.

Note that once the price of a product with payoff (3) is known then other payoffs with differentindicator functions can be similarly derived. For example a result similar to Theorem 2 could bederived for a payoff involving only one barrier, hence of the type g · 1τ1>T. Note that the relationg · 1τ1>T = limM2→0 g · 1τ1>T · 1τ2>T allows for a good approximation of the price using a lowvalue of M2 hence avoiding implementing new formulas. Similarly payoffs of the form g ·1τ1<T,τ2<Tor g ·1τi>T,τj<T could be analyzed by simply using the findings in Theorem 2 and the approximationsfor g · 1τ1>T and g · 1τ2>T.

Simplifications of the formula could be potentially derived in two directions. The first direction isbased on the method of images as utilized by He et al. [1998]. This is good only for very specific

9

correlations between S1 and S2, those of the form ρ12 = −cos(π/N). All other pairs of underlyingcould have any correlation. This simplification could eliminate the infinite sum over Bessel functions.A second direction can be obtained by taking advantage of factor models and separable, in terms ofthe underlying, payoffs g(S). A general result for a one factor model is provided next. The benefits ofthis particular setting can be seen in dimensions higher than 3, as the number of integrals is reducedfrom n to 3.Corollary 3(Simplification to one factor) Given the following market model:

dSiSi

= rdt+ σi

(ρidWt +

√1− ρ2

i dBi,t

)i = 1, 2, . . . , n,

where Wt and Bi,t are independent Brownian motions. Assume the payoff is separable: g(S) =n∏i=1

gi(Si) then the price of a derivative with payoff as in 3 is given by.

C(t, S,M1,M2)

=∫

R

(∫R2

2∏i=1

hi(xi)K2(x1, x2)φ(bi; 0, 1)db1db2

)n∏i=3

(∫Rhi(xi)φ(bi; 0, 1)dbi

)φ(w; 0, 1)dw

where:xi = σi

(ρiw +

√1− ρ2

i bi

)+ xi −Biτ

K2(x1, x2) = K2(x1, x2, x1, x2) = e

P2i=1(bzi−zi)2

2·τ (2βτ

)∞∑N=1

e−ω2+ω2

2τ · sin(Nπθ

β)sin(

Nπθ

β) · INπ

β(ω · ωτ

)

h1(x1) := h1(x1, x1) = 1x1>m1g1(S1)eα1(x1−bx1)+ bnτ

h2(x2) := h2(x2, x2) = 1x2>m2g2(S2)eα2(x2−bx2)+ bnτ

hi(xi) := hi(xi, xi) = gi(Si)(2πτ)eαi(xi−bxi)+ bnτ , i = 3, ..n

and φ(x; 0, 1) is the standard Gaussian density. All variables and parameters are defined as in theprevious Theorem.

When using a special case of one factor models, where the common factor coincide with one of thebarrier underlying then the solutions can be reduced even further from 3 to 2 integrals.Remark 4In the setting of 3, if S1 is chosen as the common factor, then the following simplification is obtained:

C(t, S,M1,M2)

=∫

Rh1(x1)

(∫Rh2(x2)K2(x1, x2)φ(b2; 0, 1)db2

) n∏i=3

(∫Rhi(xi)φ(bi; 0, 1)dbi

)φ(w; 0, 1)dw

10

where:

x1 = σ1w + x1 −B1τ

xi = σi

(ρiw +

√1− ρ2

i bi

)+ xi −Biτ , i = 2, .., n

Empirical analysis

Three collateralized structured products are studied in this section, a Double Barrier CSP Option,a Double Barrier Digital CSP Option and an n-dim Double Barrier Digital CSP Option. A detailedanalysis of the influence of the collateral and the various barriers is performed. Also, a sensitivityanalysis with respect to volatilities and correlations is presented.

Double Barrier CSP Option.

First C1(T ) is investigated:

C1(T ) = (S1(T )−K)+ · 1τ1>T1τ2>T + S3(T ) · 1τ1>T1τ2<T

Here S2 represents the assets of the option issuer, hence the barrier is considered time dependent.S3 represents the collateral, and S1 is a market index with constant barrier. Two simpler productswithin C1(T ) should be distinguished:

C11(T ) = (S1(T )−K)+ · 1τ1>T (7)

C12(T ) = (S1(T )−K)+ · 1τ1>T1τ2>T (8)

Each of these products, C11(T ) and C12(T ) keeps a feature of interest but are not a full CSP. C11(T )only considers the market indicator disregarding the default risk of S2, and therefore any collateral.C12(T ) additionally includes the default risk of the issuing company without a collateral. C1(T ),on the other hand, is a full CSP, including a market indicator, the default risk, and the collateral.The price differences of these products will be investigated. In our analysis we use the following toyscenario:

S1(0) = S2(0) = 100, S3(0) = 60,K = 40, r = 0.04, T = 1,σ1 = σ2 = 0.2, σ3 = 0.1,ρ12 = 0.3, ρ13 = ρ23 = 0.1M1 = 80,M2 = 70

Being the collateral only S3 could have a lower initial value than S1 − K. With increasing initialvalues of S3, the issuer has to tie up a lot of capital. Additionally, we assume the collateral S3 to be

11

very low correlated to S1 and S2, and that it has a lower volatility, σ3 = 0.1.

A comparison of the analytical price and the price calculated via Monte Carlo simulation for the toyscenario as well as several particular cases of it is provided next in Exhibit 1:

Price analytical formula Monte Carlo simulationstepsize = 0.01 P = 10,000, dt = 1

250 P = 2,000,000, dt = 11000

C1(t) 37.34 37.86 37.55M2 = 0 37.50 37.93 37.71T = 2 34.75 35.42 35.05

Table 1 Analytical formula versus Monte Carlo pricing

This comparison is performed for three scenarios, the toy scenario described before, the same scenariobut withM2 = 0 (i.e. payoff (7)), and the toy scenario with T = 2. The analytical formula is computedwith a stepsize in the approximation of the integral of 0.01 and limits of integration between lnS(0)−5and lnS(0) + 5. The simulations are performed in two cases, a simple one and a robust one. Thesimple one is done for 10000 paths (P ) with a delta t in the Euler discretization of the processes of1

250 (dt), while the robust one is performed for 2 million paths with a thinner dt ( 11000).

Price analytical formula Monte Carlo simulationstepsize 0.02 0.01 P 10K 2M 6M 10M 14M 14MN 20 30 dt 1

2501

10001

20001

40001

80001

12000

C1(t) 37.336 37.342 37.852 37.571 37.592 37.447 37.413 37.408Relative Error 0.02% real price 1.40% 0.65% 0.44% 0.31% 0.22% 0.21%time [min] 1.7 21.5 0.0 6.7 39.9 127.7 314.2 478.5

Table 2 Study of Monte Carlo simulations

Next we study in more detail the computational time between Monte Carlo simulations and analyticalformulas, we select C1 as an example. It is well known that by increasing the number of simulationpaths (P) and decreasing the dt, the relative error at Monte Carlo simulation decreases. Whereas forthe analytical computation the accuracy of the grid (stepsize) and the number of summands of theinfinity sum (N) are the key parameters for the precision of prices. As illustrated in Figure 3, ournumerical analysis has shown, that with 35 summands and a stepsize of 0.01 the results are absolutelystable and reliable.

Furthermore, the bias from discretization leads to an underestimation of the probabilities of hittinga barrier. This can be seen more clearly in Exhibit 2, here up to 14 million paths were simulatedwith a delta t down to 1

12000 , this allows to reduce the relative error from to 1.4% to 0.21%. Thisobservation explains the performance of a simpler simulation as well as the usefulness of closed-formformulas. The more robust the simulation the closer the simulated value to the closed-form solution

12

Figure 3 Numerical analysis of accuracy of the grid (stepsize) and the number of summands of theinfinity sum

in all cases. By comparing the computational time of the analytical formula and the computation byMonte Carlo simulations, as shown in Exhibit 2, the advantage of the analytical formula is highlighted.The analytical formula is extremely efficient in terms of computational time. The analytical formulasassumes approximately 1

275 of the computational time and achieve a ten times smaller relative errorin that time.

Next, a comparison between C1(t), C11(t) and C12(t) is performed in Exhibit 4 with respect to thethresholds: M1 and M2. Exhibit 4(b) shows a similar behavior between the main product C1 andthe simplest one, C11, the difference comes from large values of M2, making C1 more expensive. Thedifference among the products is quite notable when comparing C1 and C11 versus C12, Exhibits 4(a)and 4(c) respectively, confirming the strong influence of default risk (τ2) in the price.

If an investor is interested in the payoff (S1(T ) − K)+ · 1τ1>T, he should be aware that the priceof this product is usually determined without any regard for default risk of the issuing company andhence misleading. The true product should explicitly account for a possible default of the issuingcompany, this is represented in C12(T ). Obviously, the price for C11(t) exceeds the price of C12(t)due to the presence of a second down-and-out barrier condition in the payoff of C12(t). The pricedifference between these two products is significant, which clearly indicates that the default risk of theissuing company is a factor which should not be neglected. On the other hand, the new product C1(t)explicitly includes a correction protecting against the default of the issuer company. In conclusion,for our choices of S3 and default probabilities from τ2, C1 has a similar value as that of C11 whileprotecting against defaults.

In Exhibit 5, a comparison of C12(T ) and C1(T ) with respect to the influence of the Collateral inthe price is shown. It can be seen that the collateralized structured product C1(T ) is more expensivethan C12(T ), which is reasonable due to the addition of the collateral. In the case of a default of theissuing company the collateral S3(T ) is an adequate substitute for (S1(T ) − K)+. C1(T ) is alwaysmore expensive than C12(T ), and these differences are getting bigger with an increase in M2 andS3(0) because it is more likely that the collateral gets paid. An increasing initial value S3(0) alsoleads to increasing prices of C1(T ) because in a case of default of the option issuer, the investor would

13

((a)) C11 (top) versus C12 (bottom) ((b)) C11 (top) versus C1 (bottom) ((c)) C1 (top) versus C12 (bottom)

Figure 4 Comparison of C1(t), C11(t) and C12(t): Influence of default risk

Figure 5 Comparison between C12(T ) (bottom) and C1(T ) (top) with respect to Thresholds andCollateral

receive a higher collateral.

Sensitivity Analysis.

The increase of the volatilities affects the price of C1(T ) in two different ways, see Exhibit 6(a). Onthe one hand, an increase in larger values for σ1 leads to a significant increase in S1, which would resultin higher values for the call component (S1(T )−K)+, on the other hand there is a higher probabilitythat S1 fall below its barrier resulting in a worthless product. This second effect is stronger drivingthe price down. The Exhibit also shows that the volatility of S3 has a limited influence in the priceof C1(t). A similar analysis could be draw using σ2 instead of σ1.

As it can be seen in Exhibit 6(b), high correlation values between S1 and S2 lead to higher productprices while a decrease in correlation between S1 or S2 and S3 leads to an increase in product prices.The price is mainly driven by the correlation between S1 and S2, because these stocks are related tothe barriers.

14

((a)) Influence of volatilities on C1 ((b)) Influence of correlations on C1

Figure 6 Sensitivity of C1 to Volatilities and Correlations

The stronger the negative correlation, the lower the price of C1(T ). This is due to the combinationof two down-and-out barrier options in this product. The volatility of S3 has little influence in theoverall price, while an increase in volatilities for S1 or S2 lead to lower prices due to less likelihood ofreaching the barriers.

Double Barrier Digital CSP Option.

The financial crisis has shown, that in crashes there may occur a so called "correlation breakdown", i.e.in the downward movement of the market "diversification effects" disappear and all underlying behavesimilarly. With C2(T ) we present a product which could provide a hedge against this situation:

C2(T ) = K · 1K4>S4(T ) · 1τ1<T · 1τ2>T + α · S3(T ) · 1τ1<T · (1τ2<T )

Here S1 is a market index. A crises is identified whenever this index falls below a given threshold. Inthis situation the investor is interested in receiving a payoff. S2 represents again the issuing companyand monitors it’s default state. In case of a default by the issuing company (τ2 < T ), a digitalcollateral depending on S3 has to be provided. S4 represents the assets of the investor, which triggersa payoff of K only if they are below a threshold at maturity (K4 > S4(T )). If the assets of theinvestor are in good state then there is no need for the option. Here, the market index S1 is associ-ated with a constant barrier, whereas the issuing company S2 is compared to a time dependent barrier.

As in the analysis before, the following products within C2(T ) are interesting to study but are notfull CSP.

15

((a)) C21 (top) versus C22 (bottom) ((b)) C2 (top) versus C21 (bottom) ((c)) C2 (top) versus C22 (bottom)

Figure 7 Comparison of C2(t), C21(t) and C22(t): Influence on the price of the consideration ofdefault risk

C21(T ) = K · 1K4>S4(T ) · 1τ1<T (9)

C22(T ) = K · 1K4>S4(T ) · 1τ1<T · 1τ2>T (10)

In our analysis we use the following toy scenario:

S1(0) = S2(0) = S3(0) = S4(0) = 100,K4 = 120, r = 0.04, T = 1,K = 100, α = 1σ1 = 0.3, σ2 = σ4 = 0.2, σ3 = 0.1,ρ12 = ρ14 = 0.4, ρ13 = ρ23 = ρ24 = ρ34 = 0.2M1 = 80,M2 = 60

The market index S1 has a higher volatility than the assets of the option issuer S2 and the assets ofthe investor S4. The collateral component S3 is again a low volatile asset, for example a bond index.Furthermore the market index is strongly correlated to the assets of the issuer and the investor.

Next, a comparison between C2(t), C21(t) and C22(t) is performed in Exhibit 7. If an investor wantsto receive a payoff in a downward movement of turbulent markets, he should be fully aware thatthere is an increasing default probability of the option issuer. Exhibit 7 shows significant differencesin the prices of C2(T ), C21(T ) and C22(T ). In particular, Exhibit 7(a) shows that C21(T ) is moreexpensive than C22(T ) due to the additional consideration of the default risk of the issuer on thelatter. This in an indication of an overpriced product (C21) due to wrong assumptions in the market(no default on the issuer). The CSP product C2 is slightly more expensive than C21 particularlyfor large values of M1 and M2. As expected the differences between these products increase withan increasing down-and-in barrier M1 and an increasing down-and-out-barrier barrier, respectivelydefault probability, M2.

16

Figure 8 Comparison between C22(T ) and C2(T ), Thresholds and Collateral

Exhibit 8 compares C22(T ) and C2(T ) and shows the influence of the Collateral in the price of theCSP. The product C22(T ) takes the default risk of the issuer into account, but does not provide anykind of collateral in case the issuer defaults. This collateral is considered in C2(T ) and is representedby a scaled digital put depending on S3. In case of a default of the issuer S2, the investor receivesthe collateral payment. Therefore the product C2(T ) is always more expensive than C22(T ).

Sensitivity Analysis

As in the Double Barrier CSP Option before and as you can see in Exhibit 9(a), the correlation ρ12 isthe main driver of the price in terms of correlations. Only this correlation affects the price significantly,because it relates the involved assets in the barriers. The more negative the correlation the lower theprice of the products. Furthermore there are different effects of increasing volatilities. In our settingonly the volatilities σ1,σ2 of the barrier related stocks have a significant influence on the price (seeExhibit 9(b). With increasing volatility σ1 it is more likely that S1 falls below the knock-in barrierand thereby the option gets more valid. S2 is related to a knock-out barrier, so with increasing σ2 theprobability decreases that the investor receives the payoff K · 1K4>S4(T ) · 1τ1<T · 1τ2>T , otherwiseis getting more likely that he receives the collateral. The other effect of increasing volatility σ4 isthat it becomes more likely that the stock at maturity gets lower than the strike price K4, which is acondition of a valid option. The volatility σ4 has however a small influence in the price of C2(T ) asthe collateral compensate for large movements of S4.

n-dim Double Barrier Digital CSP Option.

In this section a n-dimensional product, where the simplifications of Corollary 4 applies, is considered.This product is a generalization of the Double Barrier Digital CSP Option we introduced before.

C3(T ) = K · 1mini=4...nKi−Si(T )>0 · 1τ1<T · 1τ2>T + α · S3(T ) · 1τ1<T · (1− 1τ2>T )

17

((a)) Influence of volatilities on C2 ((b)) Influence of correlations on C2

Figure 9 Sensitivity of C2 to Volatilities and Correlations

As before S1 can be seen as an overall market index hence τ1 denotes the time of a crisis. If τ1 < Tthen an investor is interested in receiving a payoff (either K or S3(T )). τ2 monitors the default timeof the issuing company. S4, S5, ... and Sn represent indexes for n − 3 sectors in the economy. Apayoff to the investor with value K is triggered if three conditions are satisfied, first these sectors areperforming below a threshold at maturity (K4 > S4(T ), ..., Kn > Sn(T )), second the issuing companyhas not defaulted and third the market index S1 goes below its threshold. If the assets of the issuingcompany are in good state then there is no need to use a collateral. On the other hand if the issuingcompany default (τ2 < T ) then a collateral, S3(T ), has to be provided to the investor. Here, themarket index S1 is associated with a constant barrier, whereas the assets of issuing company S2 arecompared to a time dependent barrier.

Two simpler products within C3(T ) should be distinguished:

C31(T ) = K · 1mini=4...nKi−Si(T )>0 · 1τ1<TC32(T ) = K · 1mini=4...nKi−Si(T )>0 · 1τ1<T · 1τ2>T

In our analysis we use the following toy scenario in dimension 6 related to a one-factor model:

dSiSi

= rdt+ σi

(ρidWt +

√1− ρ2

i dBi,t

)i = 1, 2, . . . , 6

18

Si(0) = 100, i = 1, ..., 6Ki = 120, i = 4, 5, 6r = 0.04, T = 1,K = 100, α = 1σ1 = 0.3, σ3 = 0.1, σi = 0.2, i = 2, 4, 5, 6ρ1 = 1, ρ2 = 0.4, ρ4 = 0.5, ρi = 0.2, i = 3, 5, 6M1 = 80,M2 = 60

The difference in computational time between the analytical formula and Monte Carlo simulations(see Exhibit 2), would be even more pronounce in presence of Corollary 4. With this simplificationmethod the number of integrals is significantly reduced leading to an extremely efficient methodologyin terms of computational time. The analysis on the difference in prices as well as the sensitivity ofthis product is very similar to the case of a digital Double Barrier Digital CSP Option, because thisproduct is a generalization to a higher dimension in a similar setting.

((a)) C31 (top) versus C32 (bottom) ((b)) C3 (top) versus C31 (bottom) ((c)) C3 (top) versus C32 (bottom)

Figure 10 Comparison of C3(t), C31(t) and C32(t): Influence of default risk

In Exhibit 10 a comparison between C3, C31 and C32 is performed. The product C32 is only valid ifthe stock S1 falls during the maturity below the down-and-in barrier M1 and the stock S2 must notfall below the time dependent knock-out barrier M2. In contrast to C32, the product C31 does notdepend on the second barrier condition. By comparing the products C32 and C31 it is evident, thatthe price of C31 is always higher than the price of C32, similarly with C3 and C32. These differencesare influenced by the additional consideration of a second barrier condition in C32 while C3 offers acollateral to help with the the default and C31 does not consider a default at all.

Next we analyze the influence of the Collateral in the price of the CSP, therefore we compare C3(T )and C32(T ) in Exhibit 11. As we can see, observing the influence of the α, which is the scale factorof the digital collateral, the investor and the option issuer should find an α that is satisfactory forboth parties. With increasing α, the option issuer has to tie up more money and the investor wouldreceive a higher payoff, if the option issuer defaults. Therefore the price of the CSP C3 increases.

19

Figure 11 Comparison between C3(T ) and C32(T ), Thresholds and Collateral

Sensitivity Analysis

In the following we want to analyze the sensitivity of the CSP to the volatilities, the strike prices andto the number of dimensions, see Exhibit 12.

As it can be observed in Exhibit 12(a), the price of C3 is mainly driven by the volatilities σ1, σ2. Thisindicates, that the effect of the barrier related stocks is more dominant than the effect of the payoffdriving stocks. In general higher volatilities lead to higher probabilities that the barriers are crossed.In case of S1 a higher volatility σ1 leads to a higher likelihood that the option is activated. Onlywhen S1 falls below the threshold there is any payoff for the investor.

Accordingly, an increase of σ2 leads to higher default probability of the issuing company S2 andtherefore, the payoff is more driven by the collateral component of the CSP. As for the remainingvolatilities σ3, σ4, σ5, σ6, they have a less significant influence in the price of the product due in partto the non-path dependent nature of their respective underlying S3, . . . , S6. Exhibit 12(b) focus moreexplicitly on the difference between the effects of σ3 and σi, i = 4, 5, 6. It shows that the impact ofσi, i = 4, 5, 6 is greater, this is due to the presence of an indicator associated to stocks 4 to 6; whileσ3 has very little influence in the price.

The effect of increasing interest rates and increasing strikes is the same as in the Double BarrierDigital CSP Option we introduced before. Increasing interest rates leads to lower product prices,because the probability decreases that the down-and-in barrier will be broken and that the stocks atmaturity stay below their strikes.

Of peculiar interest is the influence of the dimension as illustrated in Exhibit 12(c). With increasingdimension, following the pattern of the toy scenario: Ki = 120, i = 4, ..., n, Si(0) = 100, i = 1, .., 6,σi = 0.2, i = 2, 4, 5..., n and ρi = 0.2, i = 3, ..., n, the probability that the stocks at maturity are allbelow the strike prices decreases. From this it follows that with increasing number of assets the priceof C3 decreases. Decreasing strike prices lead to the same effect.

20

((a)) Influence of σ1, σ2 on C3 ((b)) Influence of σ3 on C3 ((c)) Influence of number of dimen-sions and strike on C3

Figure 12 Sensitivity of C3 to changes in volatility, number of dimensions and strike.

Conclusion

In this paper we introduced collateralized structured products and worked with the multivariateframework of a geometric Brownian motion with n assets and any possible correlation structure. In aCSP, up to two of these assets are required to satisfy barrier constraints. We derived closed form so-lutions for the prices of these products hence extending the existing results in the literature (He et al.[1998]) from two assets and two barriers to n assets and two barriers. A simplification of the integralrepresentation for the solution was also obtained, reducing the number of integrals from n to onlytwo. In the numerical part, we analyzed the impact of the model parameters and compared simplebarrier products with those involving a possible default by the issuing company as well as productswhich also assumed a collateralized defaulting structure. The price differences clearly emphasize theimportance of considering default risk as well as the benefit of accounting for a collateral, pointingout the financial relevance of the constructed products.

Appendices

Theorem 2

Proofg(S(T )) denotes the non-path dependent payoff on the n assets while C(0, S,M1,M2) denotes theprice at time t = 0 with S = (S1, . . . , Sn) and S(T ) = (S1(T ), . . . , Sn(T )). We use for convenience S,t for backward and S, t for forward variables. In general the price of the option at a variable initialtime t, denoted as C(t, S,M1,M2), is determined via risk-neutral valuation:

C(t, S,M1,M2) = EQ[e−r(T−bt) · g(S(T )) · 1τ1>T,τ2>T | Fbt],with

τi = inft ∈ (t, T ] | Si(t) < Mi(t) i = 1, 2.

21

By using the Feynman-Kac Theorem the Kolmogorov Backward Equation is obtained:n∑i=1

n∑j=1

12ρijσiσjSiSj

∂2C

∂Si∂Sj+

n∑i=1

rSi∂C

∂Si+∂C

∂t− rC = 0,

with conditions:

C(t,M1, S2, . . . , Sn,M1,M2) = 0 = C(t, S1,M2, . . . , Sn,M1,M2),

C(T, S,M1,M2) = g(S(T ))

Next we perform the following transformations in both the forward (x, t) and backward (x, t) variables:

xi(t) =

ln( bSi(bt)bS(0)

)for i /∈ A

ln( bSi(bt)e−rbtbS(0)

)for i ∈ A

, mi = ln

(M∗i

S(0)

), (12)

G(t, x,m1,m2) = er(T−bt)C(t, S,M1,M2), (13)

with x = (x1, . . . , xn). This leads to:n∑i,j

12ρijσiσj

∂2G

∂xi∂xj−

n∑i=1

Bi∂G

∂xi+∂G

∂t= 0,

with

Bi =

σ2i2 if i ∈ A

σ2i2 − r if i /∈ A

Next the first order derivatives are removed by setting:

Z(t, x,m1,m2) = e−b(T−bt)ePn

i=1 αibxi ·G(t, x,m1,m2),

with

b =

n∑i=1

Biαi +12

n∑i=1

n∑j=1

σiσjρijαiαj

,

α1

...αn

= −Σ−1 ·

B1

...Bn

,

Note this leads to,

n∑i,j

12ρijσiσj

∂2Z

∂xi∂xj+

n∑i=1

Biαi +12

n∑i=1

n∑j=1

σiσjρijαiαj − b

·Z+n∑i=1

(−Bi−n∑j=1

σiσjρijαj)·∂Z

∂xi+∂Z

∂t= 0

22

and the choice for α eliminates the first order derivatives with respect to the underlying while thechoice of b eliminates the term Z:

n∑i,j

12ρijσiσj

∂2Z

∂xi∂xj+∂Z

∂t= 0

with boundary and terminal conditions:

Z(t, m1, . . . , xn,m1,m2) = 0 = Z(t, x1,m2, . . . , xn,m1,m2),

Z(T, x,m1,m2) = ePni=1 αibxig∗(x)

whereg∗(x(T )) = g(S(T ))

Using Feyman-Kac representation, the function Z can be written as:

Z(t, x,M1,M2) = EQ[e

Pni=1 αixi(T ) · g∗(x(T )) · 1τ1>T,τ2>T | Fbt

],

with x(T ) = (x1(T ), . . . , xn(T )) and

dxi = σidWi, E [dWidWj ] = ρijdt

τi = inft ∈ (t, T ] | xi(t) < mi i = 1, 2.

Next we look for the transition probability density function q(T, x,m1,m2, τ, x), which in the backwardvariables satisfies the following PDE:

∂q

∂τ=

n∑i=1

n∑j=1

12ρijσiσj

∂2q

∂xi∂xj

with τ = T − t, T > t and initial and boundary conditions:

q(T, x,m1,m2, 0, x) =n∏i=1

δ(xi − xi),

q(T, x,m1,m2; τ,m1, x2, . . . , xn) = 0, q(T, x,m1,m2; τ, x1,m2, . . . , xn) = 0.

The following relation between Z and q holds:

Z(t, x,M1,M2) =∫ ∞m1

∫ ∞m2

∫Rn−2

ePni=1 αixi · g∗(x) · q(T, x,m1,m2;T − t, x)dx3...dxndx2dx1

Next we explicitly find q. First the second order mixed term derivatives are eliminated using a Choleskydecomposition of the covariance matrix Σ = JJ ′. The variable transformation from x to z was the

23

inverse of the lower triangle matrix of the Cholesky decomposition of the covariance matrix (z′ =J−1x′). For the first two variables, after subtracting mi, this leads to the following transformation:

z1 =x1 −m1

σ1, z2 =

1√1− ρ2

12

(x2 −m2

σ2− ρ12

x1 −m1

σ1),

then (dz = J−1dx)q(T, x,m1,m2, τ, x) = h(T, z,m1,m2, τ, z)|det

(J−1

)|

and h satisfies a heat equation:∂h

∂τ=

12

n∑i=1

∂2h

∂z2i

, (16)

with boundary conditions:

h(T, z,m1,m2, 0, z) =n∏i=1

δ(zi − zi),

h(T, z,m1,m2, τ, L1) = h(T, z,m1,m2, τ, L2) = 0,

where

L1 = (z) : z1 = 0, L2 = (z) : z1 = −√

1− ρ212

ρ12· z2.

Next a transformation to polar coordinates on the first two backward variables (z1, z2) is performedallowing for the use of separation of variables:

h(T, ω, θ, z3, . . . , zn,m1,m2; τ, ω, θ, z3, . . . , zn) = R(ω,m1,m2) ·Θ(θ,m1,m2) · V (τ, z3, . . . , zn).

Inserting this into the PDE and using the Laplacian in polar coordinates(∂2h∂bz21 + ∂2h

∂bz22)

=(

1ω∂h∂ω + ∂2h

∂ω2 + 1ω2

∂2h∂θ2

)leads to:

2∂h

∂τ=

n∑i=3

∂2h

∂z2i

+(∂2h

∂z21

+∂2h

∂z22

)

2∂V

∂τ=

n∑i=3

∂2V

∂z2i

+ V (R′′

R+

1ω

R′

R+

1ω2

Θ′′

Θ)

2(∂V∂τ −

12 ·∑n

i=3∂2V∂bz2i)

V= (

R′′

R+

1ω

R′

R+

1ω2

Θ′′

Θ). (17)

it has to be constant. Furthermore, analogously to He et al. [1998], this constant has to be negative,because the solution must decay as τ →∞. This leads to the following equation for V :

2∂V∂τ −

12 ·∑n

i=3∂2V∂bz2i

V= −λ2

24

with initial condition:

V (0, z3, . . . , zn) =n∏i=1

δ(zi − zi)

This equation represents the Backward Kolmogorov Equation of a n−3 dimensional standard BrownianMotion with a killing rate of −λ

2

2 . A solution to this PDE has the form:

V (τ, z3, . . . , zn) = e−λ2

2·τ · 1

(2π)n−2

2 τn−2

2

· e−Pni=3(bzi−zi)2

2·τ

On the other hand R(ω,m1,m2) and Θ(θ,m1,m2) are found exactly as in He et al. [1998], pages223-224, leading to solutions:

R(ω,m1,m2) ∼ JNπβ

(ω · λ), N = 1, ...

Θ(θ,m1,m2) ∼ sin(Nπθ

β), N = 1, ...

where tanβ = −√

1−ρ212ρ12

, β ∈ [0, π] and JNπβ

is the fundamental solution of the first kind for theBessel’s equation. Substituting the expressions for R, Θ and V into equation (17) leads to a generalsolution of the form:

h(T, ω, θ, z3, . . . , zn,m1,m2; τ, ω, θ, z3, . . . , zn)

=1

(2πτ)n−2

2

· e−Pni=3(bzi−zi)2

2·τ ·∫ ∞

0

∞∑N=1

cN (λ) · sin(Nπθ

β) · JNπ

β(ω · λ) · e−

λ2τ2 dλ.

where cN (λ) are constants. This solution, by construction, fulfills the boundary conditions:

h(T, ω, θ, z3, . . . , zn,m1,m2; τ, ω, θ = 0, z3, . . . , zn) = 0,

h(T, ω, θ, z3, . . . , zn,m1,m2; τ, ω, θ = β, z3, . . . , zn) = 0,

This general solution has to satisfy the initial value condition for τ = 0:

h(T, ω, θ, z3, . . . , zn,m1,m2; τ, ω, θ, z3, . . . , zn) =1ωδ(ω − ω)δ(θ − θ)

n∏i=3

δ(zi − zi).

Moreover, it is already known that for τ = 0 the factor in front of the integral can be replaced by∏ni=3 δ(zi − zi). This leads to:

1ωδ(ω − ω)δ(θ − θ) =

∫ ∞0

∞∑N=1

cN (λ) · sin(Nπθ

β) · JNπ

β(ω · λ)dλ.

25

The terms cN (λ) are obtained as in He et al. [1998], page 225

cN (λ) =2λβsin(

Nπθ

β) · JNπ

β(ω · λ)

which produces a term:∫ ∞0

∞∑N=1

2λβsin(

Nπθ

β) · JNπ

β(ω · λ) · sin(

Nπθ

β) · JNπ

β(ω · λ) · e−

λ2τ2 dλ

Integrating with respect to λ and substituting in the expression for h leads to:

h(T, ω, θ, z3, . . . , zn,m1,m2; τ, ω, θ, z3, . . . , zn)

= (2βτ

)∞∑N=1

e−ω2+ω2

2τ · sin(Nπθ

β)sin(

Nπθ

β) · INπ

β(ω · ωτ

) · 1

(2πτ)n−2

2

· e−Pni=3(bzi−zi)2

2·τ .

We conclude that the price is:

C(t, S,M1,M2)

= e−r(T−bt) ∫ ∞

m1

∫ ∞m2

∫Rn−2

g∗(x) · ePni=1 αi(xi−bxi)+b(T−bt)|det(J)|

· h(T, ω, θ, z3, . . . , zn,m1,m2;T − t, ω, θ, z3, . . . , zn)dx

Corollary 3

ProofFrom Theorem 2 the price of the product would be:

C(t, S,M1,M2)

= e−r(T−bt) ∫ ∞

m1

∫ ∞m2

∫Rn−2

g∗(x) · ePni=1 αi(xi−bxi)+b(T−bt)|det(J)|

· h(T, ω, θ, z3, . . . , zn,m1,m2;T − t, ω, θ, z3, . . . , zn)dx

Let us define the following functions using g∗(x) =n∏i=1

g∗i (xi), g∗i (xi) = gi(Si):

K1(x) := K1(x, x) = 1x1>m11x2>m2

n∏i=1

g∗i (xi)ePni=1 αi(xi−bxi)+bτ

K2(x1, x2) := K2(x1, x2, x1, x2) = e

P2i=1(bzi−zi)2

2·τ (2βτ

)∞∑N=1

e−ω2+ω2

2τ · sin(Nπθ

β)sin(

Nπθ

β) · INπ

β(ω · ωτ

)

K3(x) := K3(x, x) = |det(J−1)| 1

(2πτ)n−2

2

· e−Pni=1(bzi−zi)2

2·τ = (2πτ)φ(x; x−Bτ,Σ)

26

References

where the relation between z, ω, θ and x is as in the previous Theorem, while φ(x; x − Bτ,Σ) is amultidimensional Gaussian density with mean x−Bτ and covariance Σ. The solution in terms of thenew functions is:

C(t, S1, . . . , Sn,M1,M2) =∫

RnK1(x)K2(x1, x2)K3(x)dx

We conveniently rewrite this as:

=∫

RnK2(x1, x2)

(n∏i=1

hi(xi)

)φ(x; x−Bτ,Σ)dx

with

h1(x1) := h1(x1, x1) = 1x1>m1g1(S1)eα1(x1−bx1)+ bnτ

h2(x2) := h2(x2, x2) = 1x2>m2g2(S2)eα2(x2−bx2)+ bnτ

hi(xi) := hi(xi, xi) = gi(Si)(2πτ)eαi(xi−bxi)+ bnτ , i = 3, ..n

Next we use the one factor model for S, which in terms of x can be written as:

xi = σi

(ρiW +

√1− ρ2

iBi

)+ xi −Biτ

This leads to:

C(t, S,M1,M2) =∫

R

∫

R2

2∏i=1

hi(xi)K2(x1, x2)φ(bi; 0, 1)db1db2

×n∏i=3

(∫R hi(xi)φ(bi; 0, 1)dbi

)φ(w; 0, 1)dw

References

[BGH [2011]] Federal Court of Justice of Germany (BGH), Bundesgerichtshofentscheidet über zwei Schadensersatzklagen von Lehman-Anlegern,Pressemitteilung 145/11 (2011).

[Black and Cox [1976]] F. Black and J. Cox, Valuing Corporate Securities: Some Effects ofBond Indenture Provisions, The Journal of Finance 31(2) (1976)351–367.

[Black and Scholes [1973]] F. Black and M. Scholes, The Pricing of Options and CorporateLiabilities, Journal of Political Economy 81(3) (1973) 637–654.

27

References

[Bielecki et al. [2008]] Bielecki, Tomasz R., et al. "Defaultable options in a Markovianintensity model of credit risk." Mathematical Finance 18.4 (2008):493-518.

[Bouzoubaa and Osseiran [2010]] M. Bouzoubaa and A. Osseiran, Exotic Options and Hybrids: AGuide to Structuring, Pricing and Trading, Wiley, 2010.

[da Fonseca et al. [2008]] J. da Fonseca, M. Grasselli and C. Tebaldi, A Multifactor VolatilityHeston Model, Quantitative Finance 8 (2008) 591–604.

[Escobar et al. [2013]] B. Götz, M. Escobar and R. Zagst, Closed Form Pricing of Two-Asset Barrier Options With Stochastic Covariance, submitted Ap-plied Mathematical Finance (2013).

[He et al. [1998]] H. He, W. Keirstead and J. Rebholz, Double Lookbacks, Mathe-matical Finance 8(3) (1998) 201–228.

[Heston [1993]] S. Heston, A Closed-Form Solution for Options with StochasticVolatility with Applications to Bond and Currency Options, TheReview of Financial Studies 6(2) (1993) 327–343.

[Hull and White [1987]] J. Hull and A. White, The Pricing of Options on Assets withStochastic Volatilities, The Journal of Finance 42(2) (1987) 281–300.

[Johnson and Stulz [1987]] Johnson, H., and Stulz, R. (1987). The pricing of options with de-fault risk. Journal of Finance, 42, 267-280.

[Klein [1996]] Klein, P. C. (1996). Pricing Black-Scholes options with correlatedcredit risk. Journal of Banking and Finance, 20, 1211-1229.

[Klein and Inglis] P. Klein and M. Inglis, Pricing vulnerable european options whenthe option payoff can increase the risk of financial distress, Journalof Banking and Finance. 25, 993–1012, (2001).

[Lipton [2001]] A. Lipton, Mathematical Methods for Foreign Exchange: A Finan-cial Engineer’s Approach, World Scientific: Singapore, 2001.

[Rich [1996]] D. Rich, The Valuation and Behavior of Black–Scholes Options Sub-ject to Default Risk, Review of Derivatives Research 1 (1996) 25–59.

[Rubinstein and Reiner [1991]] M. Rubinstein and E. Reiner (1991), Breaking down the barriers.Risk 4.8 (1991) 28-35.

[Walmsley [1997]] J. Walmsley. New financial instruments. John Wiley & Sons, Inc.,2nd edition, 1997.

[Zagst and Huber [2009]] R. Zagst and M. Huber, Zertifikate spielend beherrschen, Fi-nanzBuch Verlag: München, 2009.

28

References

[Zhou [2001]] C. Zhou, Default Correlation: An Analytical Result, The Review ofFinancial Studies 14(2) (2001) 555–576.

29