A Structural Model with Jump-Diffusion Processes

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A structural model with jump-diﬀusion processes

ThanhBinhDAO PhD student at CEREG, University Paris IX Dauphine. Place de Lattre de Tassigny. 75016 Paris. Email: [email protected] Preliminary and Incomplete Version.

Abstract In this paper, we extend the framework of Leland’s 94 by examining corporate debt, equity and firm values with jump- difffusion processes. We choose two kinds of jumps such as the uniform and double exponential jumps to modelise the distribution of the log jump sizes. By this choice, we are able to derive closed-form results in both models for equity, debt and firm values. Our results have the same forms as those of Leland’s 94. However, in both our models, the spreads are modified significantly in comparison with those of Leland due to jumps’ assumption.

Acknowledgement 1 We want to express our thanks to Mme Jeanblanc for her precious help. Further thanks go to M. Quittard Pinon for introducing me to the subject.

1 A structural model with jump-diﬀusion processes

Abstract

In this paper, we extend the framework of Leland’s 94 by examining corporate debt, equity and firm values with jump-difffusion processes. We choose two kinds of jumps such as the uniform and double exponential jumps to modelise the distribution of the log jump sizes. By this choice, we are able to derive closed-form results in both models for equity, debt and firm values. Our results have the same forms as those of Leland’s 94. However, in both our models, the spreads are modified significantly in comparison with those of Leland due to jumps’ assumption.

1Introduction

Credit risk has raised a increasing interests for both academic and pro- fessional researchers for the past ten years. Although there have been many papers written on the subjects of credit risk and related credit derivatives, there are still many questions left unresolved and unsat- isfied. There are mainly three principal credit risk pricing approaches: the structural approach, the intensity approach and the rating based approach. The pricing of credit derivatives is mostly explicitly priced inside these three different approaches. The structural approach is marked by a series of papers by Leland [19] and Leland and Toft [20] which consider the question of the optimal capital structure of the firm and its endogenous default barrier. Although the ideas are very appealing, interesting and intuitive, the results of their models are unsatisfatory (for example the spread does not fit empirical data) due to the assumption that the underlying process follows a geometric Brownian motion with constant parameters as in the Black Scholes framework. Since then, many other solutions proposed to modify this model such as changing the underlying process, using the stochastic volatility or interest rate, or even introducing transaction costs. Among these propositions, the presence of jumps in the asset prices are further supported by empirical papers of Bates [1] and others.

In the tendency of using jump-diﬀusion processes for replacing the Black Scholes framework, we can refer to Merton [22], Zhou [29] for using

2 alognormaljump-diffusion. Hillberink and Rogers [15] first introduce a spectrally negative Lévy process into the framework of Leland’s model. However, the restriction of only negative jumps yields the results of the model less applaudable. Boyarchenski and Levendorskii [4] propose a structural model whose assets follow a regular Lévy process of exponential type. They propose the calculations mostly by the Wiener Hoft factorization. Although their model proposes two approximate formulations in extreme cases (very close to or very far from the bankruptcy level), these formulations are still too complicated to apply in general because the explicit calculation of the Wiener Hoft factorization is dif- ficult. The papers of Eberlein and Ozkan [28],[27] use the Lévy process for credit risk mostly in the intensity approach. (We do not cite a series of papers using the Lévy process or jump-diffusion process for pricing options).

Our paper is on the line of the Leland approach [19] using jump diffusions processes with two different kinds of jumps. One is the simple uniform density and the other is a special case of the Levy processes called the double exponential jump diffusion which is first proposed by Kou [16]. We choose these two kind of jumps which are totally different for two main reasons. The first one is to verify the importance of the different amplitude of jumps in the valuations of the firm’s value, its debtandequity.Thesecondreasonisthesetwokindsofjumpscan give us analytical solutions. With this reason, we can also choose the log normal jump-diffusion process as in Merton [22] or the bivariate jumps used by Gukhal [13]. Other reasons can be the fact that the uniform jump is the most simple jump process, however the derivative formula is not evident. While the double exponential jump diffusion for the assets process is a special case of the Lévy processes which has the two interesting properties of the exponential distribution1.Thefirst property is that the double exponential distribution has the leptokurtic feature2 of the jump size that provides the peak and tails of the return distribution found in reality. The second property is that the double exponential distribution has the memoryless property which makes it easier in the calculation of expected average and variance terms and in

1See Kou (2002) for more detail explications of the advantages of the double exponential jump diffusion model and the comparison with other models as the constant elasticity model (CEV), the normal jump diffusion model, model based on t-distribution, stochastic volatility model, affine jump difusion model, model based on Levy process. 2The leptokurtic feature means that the return distribution of assets may have a higher peak and two asymmetrically heavier tails than those of the normal distribu- tions.

3 the computation of the distribution of the first passage times. Thanks to this memoryless property, the problem of overshoots, a problem occurs when a jump diffusion process crosses a barrier level b, as sometimes it hits the barrier level exactly Sτ b and sometimes it incurs an ”overshoot” Sτ b over the barrier, can be resolved. b − We find that in general, the debt, equity and firm values in our models have the same forms as those in Leland’s 94 which confirm that his model is robust. However, the spreads in both our models are higher than those in Lelands due to the fact of jumps. Our contribution can be seen in two sides: mathematical and finance. In the mathematical side, we derive analytic formulas for the value of American perpectual option, ainsi que the formula for the debt, equity and firm values in two jump diffusion frameworks, the uniform and double exponential jumps. In the financial side, we test influences of jumps in the firm’s related values of a structural model with endogenous default barrier.

The paper is organized as follows. In section 2, we present the debt equity structure of the firm and also its general formulas of valuations. We present the two models of jump-diffusion for the assets’ process in section 3. Formulas for each process and also the results which are com- pared with the original Leland’s 94 model are presented in section 4. We end with conclusion in section 5. Technical results are reported in appendices. Appendix A derives the formula of a perpetual American put option in the double exponential jump diffusion framework. Ap- pendix B derives the equivalent formula for the uniform jump diffusion framework.

4 2 The debt-equity structure of the firm We suppose that the value of firm’s assets follows a jump diffusion under the risk neutral probability Q3 at time t as equation:

dV Nt t =(r δ λξ)dt + σdW + d (S 1) (1) V t i t − − Ã i=1 − ! − X where r is the riskless interest rate and δ is the payout ratio. This ratio δ is a proportional rate at which proﬁt is distributed to investors (both shareholders and bondholders). The other parameters will be pre- cise later in section 3.

Debt structure

Suppose that the firm is partly financed by debt. This debt promises a perpetual coupon payment C (C 0 non negative) until bankruptcy. If bankruptcy occurs (it means that≥ the value of the firm’s assets V is less than VB, the bankruptcy trigger level), a fraction of value of the firm’s assets α will be lost due to bankruptcy costs or to the reorganization and the absolute priority rule is respected.

Time of default

Default happens at the ﬁrst time τ that the value of the ﬁrm’s assets falls down to some level VB (default barrier that will be calculated optimally later).

τ =inf t>0,Vt VB (2) { ≤ } The value at default Vτ may be diﬀerent from the value of default barrier because of jumps.

Thevalueofthedebtisthe expected value of the actual value of coupon payment C if there is no default and the value at default (1 α) Vτ if there is default. We have, under the risk neutral probability Q −the formula of the debt value as follows:

τ ru rτ D = D (V,VB)=E Ce− du +(1 α) Vτ e− 1τ − ≺∞ ·µZ0 ¶ ¸

C rτ rτ 1 E e− 1τ + E (1 α) Vτ e− 1τ ⇐⇒ r − ≺∞ − ≺∞ 3Note that this risk¡ neutral¡ measure Q¢¢ is not unique£ and it depends on¤ the utility function of representative agent.

5 C rτ C D = E e− 1τ (1 α) Vτ (3) r − ≺∞ r − − · µ ¶¸ The value of the firm is the expected value of the firm’s assets (V ) plus the expected value of the tax benefit(TB) (the tax rate is κ) minus the value of expected bankruptcy cost value (BC). Thus we can writethevalueofthefirm as follows:

κC rτ rτ v = v (V,VB)=E V + 1 e− 1τ αVτ e− 1τ r − ≺∞ − ≺∞ · ¸ ¡ ¢ κC rτ rτ V + 1 E e− 1τ E αVτ e− 1τ ⇐⇒ r − ≺∞ − ≺∞ ¡ ¡ ¢¢ £ ¤ κC rτ κC v = V + E e− 1τ + αVτ (4) r − ≺∞ r · µ ¶¸ The value of ﬁrm’s equity which is equal to the ﬁrm’s value minus thedebtvalue,canbewrittendowneasily:

E = E (V,VB)=v (V,VB) D (V,VB) − Torearrangetheterm,wehavetheﬁnal formula of the ﬁrm’s equity value:

(1 κ) C rτ (1 κ) C E = V − + E e− 1τ − Vτ (5) − r ≺∞ r − · µ ¶¸ AndtheshareholdersobjectiveistomaximizethevalueofequityE by choice of Vτ or the moment of default τ.

Introducing K =(1 κ) C/r, the problem of maximization of the ﬁrm’s equity value E becomes:−

rτ rτ max E e− 1τ (K Vτ ) =supE e− 1τ (K Vτ ) (6) Vτ ≺∞ − τ ≺∞ − This formula£ looks quite familiar¤ to us£ and we realize that¤ is the formulation of a perpetual American put with the strike K. Recall the value of a perpetual American put is

rτ + sup E e− (K Vτ ) τ − £ ¤ + The only two diﬀerences between the two formulas are: (K Vτ ) − and 1τ ≺∞

6 It is well known that the optimal stopping time τ ∗ veriﬁes that • + (K Vτ ) 0i.e.,(K Vτ )=(K Vτ ) − ≥ − −

Indeed, suppose that τ ∗ is optimal and (K Vτ ) < 0 with positive − probability on τ ∗ < , we can always ﬁnd ∞

τ ∗∗ = τ ∗1(K Vτ )>01τ + 1(K Vτ )<01τ + 1τ= − ≺∞ ∞ − ≺∞ ∞ ∞ then τ ∗∗ is also a stopping time. We have:

rτ E e− ∗∗ (K Vτ )1τ − ∗∗ ∗∗≺∞

£ rτ ∗ ¤ = E e− (K Vτ )1τ 1(K Vτ )>0 − ∗ ∗≺∞ −

£ rτ ∗ ¤ > supτ E e− (K Vτ )1τ − ∗ ∗≺∞ £ + ¤ so if τ ∗ is optimal, it veriﬁes that (K Vτ ) −

We now demonstrate that •

rτ rτ E e− 1τ = E e− ≺∞ rτ This equality is trivial,£ in fact at¤ τ = £ ,e−¤ =0 ∞ Our question is now simple to look for the value of the perpetual American put!

Under the jump diﬀusion model, the rational expectation equilibrium price of an American put option under the risk neutral probability Q is given:

rτ + rτ Xτ + PA(V )=supE e− (K Vτ ) =supE e− K V0e τ − τ − ¡ ¢ ³ £ ¤ ´ The deﬁnition of Xτ will be presented in the section 3.

7 3TheTwoModelsofJumpDiﬀusion Processes 3.1 The General Jump Diﬀusion Model

Under the jump diffusion model, the dynamics of the firm’s assets has two parts, a continuous part driven by a geometric Brownian motion, and a jump part. The value of the firm’s assets following a jump diffusion model is written as follows, under the objective probability measure P:

dV Nt t =(µ δ)dt + σdW + d (S 1) (7) V t i t − Ã i=1 − ! − X where (Wt,t 0) is a one dimensional standard Brownian motion, ≥ (µ δ)isthedrift,δ is the payout ratio, σ is the volatility. (Nt,t 0) is − ≥ a Poisson process with rate λ and Si,i 0 a sequence of independent identically distributed (i.i.d) non negative{ ≥ random} variables.

In the model, the three sources of randomness, (Nt,t 0), (Wt,t 0) ≥ ≥ and (Si,i 1) are assumed to be independent. All parameters as the risk free interest≥ rate r, the drift µ, the payout ratio δ and the volatility σ are assumed to be constants. These assumptions, however can also be relaxed in general case, for the expense of heavier calculations.

To solve the stochastic diﬀerential equation (7), we consider the interval between two jumps. In each interval [Tn,Tn+1], the processus V is just as a geometric Brownian motion

σ2 Vt = VT µ δ (t Tn)+σ(Wt WT ) n − − 2 − − n µ ¶

and at Tn the jumps of V is VTn = STn VTn −

So we have the formula of Vt as follows:

σ2 Nt V = V exp µ δ t + σW S t 0 2 t i − − i=1 ½µ ¶ ¾ Y

σ2 Nt V exp µ δ t + σW + Y 0 2 t i ⇐⇒ ( − − i=1 ) µ ¶ X where V0 is the value at time 0,Yi =ln(Si).

8 We assume that the jump risk is diversiﬁed. That means the risk premium of jumps equals to zero. And there exists a risk neutral probability measure Q, in which the drift of the value of the ﬁrm’s assets is as follows:

RP : µ δ + λξ = r δ = RP = µ + λξ r − − ⇒ − with λξ is the expected mean of jump part and

Yi ξ = E [Si] 1=E e 1. − − £ ¤ To change the Brownian from under the objective probability measure P

dV RP Nt P : t =(µ δ RP)dt + σd W + + d (S 1) V t σ i t − − Ã i=1 − ! − µ ¶ X into under the risk neutral probability measure Q,wehaveanews dynamic of V :

N dV ˆ t Q : t =(r δ λξ)dt + σd W +d (S 1) (8) V t i t − − Ã i=1 − ! − X and using the same argument as under the probability P to solve the equation (8), we have:

σ2 Nt V = V exp r δ λξ t + σW + Y = V eXt (9) t 0 2 t i 0 ( − − − i=1 ) µ ¶ X where V0 is the value at time 0 and

σ2 Nt X = r δ λξ t + σW + Y t 2 t i − − − i=1 µ ¶ X

One important element in doing pricing with the jump-diﬀusion process is to ﬁnd out the moment generating function.

As the processus V has the Levy’s character, means the inﬁnitely divisible character of L´evy processes. From the Lemma on the Levy- Khinchine presentation in Bertoin [2], there exists a function G( )such βXt tG(β) · that e − is a martingale, so we have:

9 E eβXt =exp G(β)t (10) { } where £ ¤

σ2 σ2 G (β)=β r δ λξ + β2 + λ (υ 1) , υ = E eβY1 (11) − − 2 − 2 − µ ¶ ¡ ¢ The demonstration for this formula (11) is as follows:

From the equation (10), take the expectation with the ﬁltration (Fs,s t), we have: ≤

βXt tG(β) βXs sG(β) E e − Fs = e − | Arrange the term X¡ and G,wehave¢

β(Xt Xs) (t s)G(β) E e − Fs = e − (12) | Wecalculatenowtheleftsideoftheequation(12)¡ ¢

β(Xt Xs) acc.ind β(Xt Xs) βXt s E e − Fs = E e − = E e − | βXt s Replace e¡ − by its value¢ as equation¡ (9)¢ and¡ the jump’s¢ part E (Nt s)=n earn: −

Nt s 2 − βXt s σ E e − = E exp β r δ λξ (t s)+σWt s + Yi − − 2 − − − " Ãµ ¶ i=1 !# ¡ ¢ X σ2 E exp β r δ λξ (t s)+σWt s ⇐⇒ − − 2 − − − · µµ ¶ ¶¸

Nt s n n − ∞ λ(t s) (λ (t s)) E exp Yi E exp β Yi e− − − n! " i=1 # ⇐⇒ 1 " Ã i=1 !# X X X Arrange the terms, we have:

2 2 n n σ σ 2 ∞ n λ (t s) βXt s (t s) β r δ 2 λξ + 2 β λ(t s) βY1 E e − = e − − − − e− − E e − h ³ ´ i n! 1 ¡ ¢ X £ ¡ ¢¤ Note υ = E eβY1 , we can reduce the left side of the equation (12) into ¡ ¢ 10 σ2 σ2 2 βXt s (t s) β r δ 2 λξ + 2 β λ(t s) λυ(t s) E e − = e − − − − e− − e− − h ³ ´ i ¡ ¢ (t s)G(β) Recall the right side of the equation (12) is e −

We deduce the value of G (β)asfollows:

σ2 σ2 G (β)=β r δ λξ + β2 + λ (υ 1) (13) − − 2 − 2 − µ ¶ Until now we have not yet speciﬁed the distribution of jump sizes Si or its log jump size Yi. We focus next on the two special kinds of jumps. The log jump sizes Yi in ﬁrst case follows a double exponential jump and the second one follows an uniform jump.

3.2 The Double Exponential Jump Diﬀusion Model

Suppose that the log jump sizes Yi follows an asymmetric double exponential distribution with the density:

η1y η2y fE (y)=pη1e− 1 y 0 + qη2e 1 y<0 , η1 > 1, η2 > 0 { ≥ } { } where p, q 0,p+ q = 1, represent the probabilities of upward and downward jumps,≥ and

ξ+, with probability p log(S)=Y =d ξ−, with probability q ½− ¾ + where ξ and ξ− are exponential random variables with means 1/η1 d and 1/η2, respectively, and the notation = means equal in distribution.

The condition η1 > 1 is used to ensure that the assets’ value has ﬁnite expectation, means E(S) < and E(Vt) < .Thiscondition means that the average upward jump∞ cannot exceed∞ 100%.

We have

2 p q 1 1 p q E(Y )= η η and Var(Y )=pq η + η + η2 + η2 1 − 2 1 2 1 2 ³ ´ ³ ´ We now calculate the values of υ and ξ

βY ∞ βu υ = E e = e fY (u) du Z−∞ ¡ ¢ 11 0 ∞ βu η2u βu η1u e qη2e du + e pη1e− du ⇐⇒ 0 Z−∞ Z

pη qη υ = 1 + 2 η β η + β 1 − 2

pη qη ξ = E(eY ) 1= 1 + 2 1 − η 1 η +1− 1 − 2 To insert the values of υ and ξ into the equation (13), we have the equation of moment generating function in the case of double exponential jump-diﬀusion as follows:

2 2 σ 2 σ pη1 qη2 GE (β)= β + r δ λξ β + λ + 1 2 − − 2 − η1 β η2 + β − µ ¶ µ − (14)¶ We can show by graphs and by veriﬁcation of signal that the equation

GE(β)=α, α > 0 has exactly four roots: β1,α, β2,α, β3,α, β4,α where 0 < β < η < β < , 0 < β < η < β < − − 1,α 1 2,α ∞ 3,α 2 4,α ∞ By veriﬁcation of sign

β η2 η2+ 0 η1 η1+ + −∞ − − − − ∞

GE(β)+ + 0 + + ∞ & −∞ ∞ & % ∞−∞% ∞

So GE(β)=α, α > 0 has at least 4 roots. But as the equation is a polynomial equation with degree four, it has at most four real roots. It deduces that each interval , η2 , η2+, 0 , 0, η1 and η , + has only one root. −∞ − − − − 1+ ∞ ¡ ¢ ¡ ¢ ¡ ¢ By¡ graphic¢

Assume α = r =0.07, δ =0.01, σ =0.2, η1 =1/0.02, η2 =1/0.03, λ = 3, we plot in the Figure 1, the curves of GE (β)andα.

12 3.3 The Uniform Jump Diﬀusion Model

The log jump sizes Yi follows an uniform distribution with the density: 1 f (y)=c1 with c = U [a,b] b a −

We have

b+a (b a)2 E(Y )= 2 and Var(Y )= −12

We now calculate the values of υ and ξ

b 1 eβb eβa 1 υ = E eβY = eβu du = − b a b a β Za − ¡ − ¢ ¡ ¢ eb ea ξ = E(eY ) 1= − 1 − b a − − To insert the values of υ and ξ in the equation (11) ,the moment generating function G (β) in the uniform jump diﬀusion becomes:

2 2 βb βa σ 2 σ e e GU (β)= β + r δ λξ β + λ − 1 (15) 2 − − 2 − (b a) β − µ ¶ Ã¡ − ¢ !

For reasonable variables of a, b we can show that GU (β)=α, α > 0 has two roots: β , β 1,α − 2,α By graphic

Assume α = r =0.07, δ =0.01, σ =0.2,a= 0.5,b=0.7, λ =3,we − plot in the Figure 2, the curves of GU (β) , α.

13 4 Formulations of the option, equity, debt and ﬁrm’s values 4.1 The Double Exponential Jump Diﬀusion Model

Under the double exponential jump diﬀusion model, the rational expectation equilibrium price of a perpetual American put option is given:

Q rτ + rτ X(τ) + PA(V )=supE e− (K V (τ)) =supE e− K V0e Vτ − τ − ¡ ¢ ³ £ ¤ ´ Then the optimal stopping time is given by

τ =inf t 0,Vt VB { ≥ ≤ }

where VB is the optimal default barrier and Vt is as in equation (8).

And then the value of the perpetual American put option is given by PA(V )=u (V ),wherethevaluefunctionV is given by

K V ;ifVexercise boundary VB β β η2 +1 3,r 4,r VB = K η2 1+β3,r 1+β4,r the coeﬃcients A, B:

β3,r 1+β4,r β4,r A = V K VB > 0 B β β 1+β − 4,r − 3,r · 4,r ¸

β4,r 1+β3,r β3,r B = V VB K > 0 B β β − 1+β 4,r − 3,r · 3,r ¸ and the 1/η isthemeanofdownwardjumpsand β , β are 2 − 3,r − 4,r roots of the equation (14) : GE (β)=r. The demonstration of this formula is given in the appendix A

To insert the formulation of perpetual American put option into the formulas of equity, debt and ﬁrm’s values, equations (5, 3, 4) in the case V VB we have: ≥ The value of the ﬁrm’s equity is directly equal to:

14 (1 κ) C β β E (V )=V − + AV − 3,r + BV − 4,α − r Thevalueofthedebtcanbewritteneasilyas:£ ¤

C β β D (V )= (1 α) A0V − 3,r + B0V − 4,α r − − £ ¤ where the values A0,B0 and K0 are as follows:

β3,r 1+β4,r β4,r A0 = V K0 VB > 0 B β β 1+β − 4,r − 3,r · 4,r ¸

β4,r 1+β3,r β3,r B0 = V VB K0 > 0 B β β − 1+β 4,r − 3,r · 3,r ¸ K K0 = (1 α)(1 κ) − − The value of the ﬁrm is then:

κC β β v (V )=E (V )+D (V )=V + + A V 3,r + B V 4,α r 00 − 0 − £ ¤ where the values A00,B00 are as follows: β β β3,r 1+ 4,r κ 4,r A00 = VB K αVB − β4,r β3,r 1 κ 1+β4,r − − · − ¸

β4,r 1+β3,r κ β3,r B00 = VB αVB + K β4,r β3,r 1 κ 1+β3,r − · − ¸

4.2 The Uniform Jump Diﬀusion Model Under the uniform jump diﬀusion model, the rational expectation equilibrium price of a perpetual American put option is given by the formula:

Q rτ + rτ X(τ) + PA(V )=supE e− (K V (τ)) =supE e− K V0e τ − τ − ¡ ¢ ³ £ ¤ ´ Then the optimal stopping time is given by

τ =inf t 0,Vt VB { ≥ ≤ } 15 where VB is the optimal default barrier and Vt is as in equation (9). And then the value of the perpetual American put option is given by PA(V )=u (V )=u (ex),wherethevaluefunctionV is given by

K ex;ifex 0 1+β2,r

and β is root of the equation(15) : GU (β)=r. − 2,r The demonstration of this formula is given in the appendix B. To insert the formulation of perpetual American put option into the formulas of equity, debt and ﬁrm’s values, equations (5, 3, 4) in the case ex exB we have: ≥ The value of the ﬁrm’s equity is directly equal to:

(1 κ) C xβ E (V )=V − + Ae− 2,r − r Thevalueofthedebtcanbewritteneasilyas:

C xβ D (V )= (1 α) A0e− 2,r r − −

where the values A0 and K0 are as follows:

xBβ e 2,r K0 A0 = > 0 1+β2,r

K K0 = (1 α)(1 κ) − − The value of the ﬁrm is then:

κC xβ v (V )=E (V )+D (V )=V + A00e− 2,r r −

where the values A00 is written as: κ A00 = A 1 κ − 16 4.3 The results

In order to compare with the model of Leland’s 94, we take his inputs into our model. They are σ =20%, 15% or 25%,r =6%, κ =0.35, α =0.5 Other parameters of the double exponential jump:

p =0.3,q =0.7, η1 =1/0.02, η2 =1/0.03, λ =3

Other parameters of the uniform jump: a = 0.5,b=0.7, λ =3 −

GRAPHICS

Graphics of the moment generating function

Figure 1 : The cuvre of GE(β) and α Figure 2 : The cuvre of GU(β) and α

Leland’s 94 principal graphics

Leland's : Debt value as functio n o f the co upo n Leland's: Firm value as function of the coupon

140 140 120 120

100 100

80 80

60 60

40 40

20 20

0 0 02 468101214 0 2 4 6 8 10 12 14

Volatility 15% Volatility 20% Volatility 25% Volatility 15% Volatility 20% Volatility 25%

Figure 3 Figure 4

Leland's: Yield spreads as function of the coupon Leland's: Yield spreads as function of leverage

800 800

600 600

400 400

200 200

0 0 0% 30% 50% 70% 85% 95% 024681012

Volatility 15% Volatility 20% Volatility 25% Volatility 15% Volatility 20% Volatility 25% Figure 5 Figure 6

Graphics in the double exponential jump diffusion model

Debt value as function of coupon and volatility Firm value as function of coupon and volatility

140 120 120 100 100

80 80

60 60 40 40 20

20 0 0 2 4 6 7 8 9 1011121314 0 024 678 9101112131415 Sigma20 Sigma15 Sigma25 Si gma20 Si gma15 Si gma25

Figure 7 Figure 8

Debt value in fonction of leverage and volatility Firm value as fonction of leverage and volatility

120 140

120 100 100 80 80 60 60 40 40 20 20

0 0 0% 30% 50% 70% 80% 85% 90% 95% 100% 0% 30% 50% 70% 80% 85% 90% 95% 100%

Sigma=20% Sigma=15% Sigma=25% Sigma=20% Sigma=15% Sigma=25%

Figure 9 Figure 10

Credit spread as fonction of leverage and volatility Credit spread as fonction of leverage and volatility

1000 1600 900 1400 800 1200 700 600 1000 500 800 400 600 300 200 400 100 200 0 0 02467891011 0% 30% 50% 70% 80% 85% 90% 95% 100%

Sigma=20% Sigma=15% Sigma=25% Sigma=20% Sigma=15% Sigma=25%

Figure 11 Figure 12 The spreads in our models are much higher than thoses of Leland’s 94 Graphics in the uniform jump diffusion model

Debt value as functions of coupon and volatility Firm value as function of coupon and volatility

250 250

200 200

150 150

100 100

50 50

0 0 0246789101112131415 02467891011121314

Sigma20 Sigma15 Sigma25 Sigma20 Sigma15 Sigma25

Figure 13 Figure 14 I think have a problem of scale with these graphs Or these results due to the choice of variables [a,b] for the uniform distribution !

Debt value in f onction of leverage and volatility Firm value in fonction of leverage and volatility

300 300

250 250

200 200

150 150

100 100

50 50 0 0% 30% 50% 70% 80% 85% 90% 95% 100% 0 0% 30% 50% 70% 80% 85% 90% 95% 100%

Sigma=20% Sigma=15% Sigma=25% Sigma=20% Sigma=15% Sigma=25%

Figur 15 Figure 16

Credit spread in fonction of leverage and volatility Credit spread in fonction of leverage and volatility

148 145 146 143 144 141 142

140 139

138 137 136 135 134

132 133 1 3 5 6,5 7,5 8,5 9,5 10,5 0% 30% 50% 70% 80% 85% 90% 95%

Sigma=20% Sigma=15% Sigma=25% Sigma=20% Sigma=15% Sigma=25%

Figure 17 Figure 18

Appendix A: Demonstration of the formula of perpetual American put option in the double exponential jump diﬀusion model

The demonstration below strongly resembles of the Appendix 1 in Kou and Wang [18]. However, to clarify, we present it here in further detail.

Assumption: xB Suppose there exist some xB : e

2 1. u (x)isC on xB ,u(xB )and u (xB+)existingandu (x)isconvex <\ { } −

2. (Lu)(x) ru (x)=0for x xB − ∀ ≥

3. (Lu)(x) ru (x) < 0for x (K e ) for x xB − ∀ ≥ x + 5. u (x)=(K e ) for x

r(t τ τ ) e− ∧ ∧ ∗ u(X (t τ τ ∗)+x) Z, t>0,x, τ ∧ ∧ ≤ ∀ ∀

Results:

Then the optimal stopping time is given by

xB τ ∗ =inf t 0; Vt e { ≥ ≤ } And then the value of the perpetual American put option is given by PA(V )=u (V ) where the value function V is given by

x K e if x

+ 1 2 ∞ Lu (x)= σ u00 (x)+µu0 (x)+λ [u (x + y) u (x)] fY (y) dy 2 − Z−∞ 18 where u (x) is a twice continuously diﬀerential function and µ = r δ σ2 λξ . − − 2 − ³ ´ To compute the quantity (Lu)(x), we need to ﬁrst calculate the term + ∞ u (x + y) fY (y) dy −∞

R Since the value of u depends on the position with respect to xB,we shall consider two cases.

We ﬁrstconsiderthecasex x ≥ B In that case,

+ x x B x+y yη2 ∞ u(x + y)fY (y)dy = − (K e )qη2e dy −∞ −∞ −

R R0 (x+y)β (x+y)β yη + (Ae− 3 + Be− 4 )qη e 2 dy xB x 2 − R + ∞ (x+y)β3 (x+y)β4 yη1 + 0 (Ae− + Be− )pη1e− dy R = I1 + I2 + I3

The quantities Ii are obtained as follows: 1 yη2 xB x 1 x y(1+η2) xB x I1 = η qη2Ke − 1+η e qη2e − 2 |−∞ − 2 |−∞

(xB x) η2 xB η = qe − K e 2 − 1+η2 h i I = 1 e β3xqη Ae (η2 β3)y 0 + 1 e β4xqη Be (η2 β4)y 0 2 (η β ) − 2 − − xB x (η β ) − 2 − − xB x 2− 3 | − 2− 4 | − η η 2 β3x β3xB η2(xB x) 2 β4x β4xB η2(xB x) = (η β ) qA e− e− e − + (η β ) qB e− e− e − 2− 3 − 2− 4 − η η 1 £β3x (η1+β3)y ¤1 β4x £ (η1+β4)y ¤ I3 = (η +β ) e− pAe− 0∞ + (η +β ) e− pBe− 0∞ − 1 3 | − 1 4 |

η1 β x η1 β x = pAe− 3 + pBe− 4 (η1+β3) (η1+β4) We can now to compute (Lu)(x) ru (x): −

19 1 + 2 ∞ (Lu)(x) ru (x)=2 σ u00 (x)+µu0 (x)+λ [u (x + y) u (x)] fY (y) dy ru (x) − −∞ − − 1 2 2 xβ 2 Rxβ xβ xβ = σ Aβ e− 3 + Bβ e− 4 µ Aβ e− 3 + Bβ e− 4 2 3 4 − 3 4 ¡ xβ xβ ¢ xβ ¡ xβ ¢ r Ae− 3 + Be− 4 λ Ae− 3 + Be− 4 − − ¡ ¢ ¡ ¢ (xB x) η2 xBη qe − K e 2 1+η2 η − 2 β3x β3xB η2(xB x) + (η β ) qA e−h e− e i −  +λ 2η− 3 − 2 β4x β4xB η2(xB x) + (η β ) qB e− e− e −  2−η 4 £ − η ¤  1 β3x 1 β4x   + (η +β ) pAe− + (η +β ) pBe−   1 3 £ 1 4 ¤ recall the function G (x)is:  1 pη qη G (x)= σ2x2 + µx + λ 1 + 2 1 2 η x η + x − µ 1 − 2 ¶ we can pose

1 pη qη f (x)=G ( x) r = σ2x2 µx + λ 1 + 2 1 − − 2 − η + x η x − µ 1 2 − ¶ then we have (Lu)(x) ru (x) − xβ xβ (Lu)(x) ru (x)=Ae− 3 f (β )+Be− 4 f (β ) − 3 4

(xB x)η2 η2 xB η2 β3xB η2 β4xB +λqe − K 1+η e η β Ae− η β Be− − 2 − 2− 3 − 2− 4 h i As β , β are roots of G (β)=r so f (β )=0andf (β )=0 − 3 − 4 3 4 We can deduce that (Lu)(x) ru (x)=0implies − η η η 2 xB 2 β xB 2 β xB K e Ae− 3 Be− 4 =0 − 1+η − η β − η β · 2 2 − 3 2 − 4 ¸ We have three unknowns exB ,A,B so we need three equations in order to solve these unknowns.

When we recall the function u (x)atxB,wehavetheﬁrst equation:

xB xBβ xBβ K e = Ae− 3 + Be− 4 − 4 The smooth ﬁtcondition at xB gives us the second equation:

4The validation of smooth ﬁt condition can be found in Boyarchenko and Leven- dorskii (2002)

20 xB xBβ3 xBβ4 e = Aβ3e− + Bβ4e− And the third equation, we have from (Lu)(x) ru (x)=0: − η η η 2 xB 2 β xB 2 β xB K e Ae− 3 Be− 4 =0 − 1+η − η β − η β · 2 2 − 3 2 − 4 ¸ From the ﬁrst and second equation, we have the formulas of A and B: β K exB β exB A = 4 − 4 − − (β β ) e β3xB 3 − 4 − exB + β K β exB B = − 3 − 3 (β β ) e β4xB 3 − 4 − Inserting the values of A and B into the third equation generates these formulas:

1+β β A = exBβ3 4 4 K exB (16) β β 1+β − 4 − 3 µ 4 ¶ 1+β β B = eβ4xB 3 exB 3 K β β − 1+β 4 − 3 µ 3 ¶ η +1 β β exB = K 2 3 4 η2 1+β3 1+β4

So we have proven the condition n◦2.

The conditions n◦1, 4, 5 follow easily as we have u (xB+)=u (xB ) and 0 u (x) K − ≤ ≤ The only remaining condition that we have to prove is the condition n◦3. That is in the case x

+ 0 x+y yη2 ∞ u (x + y) fY (y) dy = (K e ) qη2e dy −∞ −∞ −

R RxB x x+y yη + − (K e ) pη e− 1 dy 0 − 1 R + (x+y)β (x+y)β yη + ∞ Ae− 3 + Be− 4 pη e− 1 dy xB x 1 − R ¡ ¢ = II1 + II2 + II3

21 The quantities IIi are obtained as follows:

η2 yη2 0 η2 x y(1+η2) 0 II1 = η qKe 1+η e qe 2 |−∞ − 2 |−∞ = Kq η2 exq − 1+η2 η η 1 yη1 xB x 1 x y(1+η1) xB x II2 = η pKe− 0 − (η 1) pe e− 0 − − 1 | − 1− | η η η1(xB x) 1 x 1 xB η1(xB x) = pK pKe− − (η 1) pe + (η 1) pe e− − − − 1− 1− η η II = 1 e β3xpAe (η1+β3)y + 1 e β4ypBe (η1+β4)y 3 (η +β ) − − x∞B x (η +β ) − − x∞B x − 1 3 | − − 1 4 | −

η1 β xB η (xB x) η1 β xB η (xB x) = e− 3 pAe− 1 − + e− 4 pBe− 1 − (η1+β3) (η1+β4)

To arrange the terms of the quantities IIi,wehave:

x pη1 qη2 II1 + II2 + II3 = K e η 1 + 1+η − 1− 2 h i η η η1(xB x) 1 xB 1 β3xB 1 β4xB pe− − K η 1 e η +β Ae− η +β Be− − − 1− − 1 3 − 1 4 h i

We can now compute (Lu)(x) ru (x)inthiscaseofx

x pη1 qη2 K e η 1 + 1+η +λ − 1− 2 1 η η  η1(xB x) xB h 1 βi3xB 1 β4xB  pe− − K η 1 e η +β Ae− η +β Be− − − 1− − 1 3 − 1 4 pη qη  h i 1 2 1 2 We have ξ = η 1 + η +1 and µ = r δ 2 σ λξ .Then(Lu)(x) 1 − 2 − − − − ru (x)isequal ¡ ¢ 1 η η x η (xB x) xB 1 β xB 1 β xB rK+δe λpe− 1 − K e Ae− 3 Be− 4 ⇐⇒ − − − η 1 − η + β − η + β · 1 − 1 3 1 4 ¸

To insert the values of A, B as in (16) into the third part

22 η η 1 xB 1 β3xB 1 β4xB K η 1 e η +β Ae− η +β Be− − 1− − 1 3 − 1 4

1 xB η1 1+β4 β4 xB = K η 1 e η +β β β 1+β K e − 1− − 1 3 4− 3 4 − ³ ´ η1 1+β3 xB β3 η +β β β e 1+β K − 1 4 4− 3 − 3 ³ ´ We can also rearrange the equation above with terms exB and K

xB 1 η1 1+β4 η1 1+β3 = e η 1 + η +β β β + η +β β β − 1− 1 3 4− 3 1 4 4− 3 ³ η1 β4 η1 β3 ´ +K 1 η +β β β + η +β β β − 1 3 4− 3 1 4 4− 3 ³ ´ Replacing exB with its value as in (16) generates:

η2+1 β3 β4 β3β4η1 = K η 1+β 1+β (η 1)(−η +β )(η +β ) 2 3 4 1− 1 3 1 4 +K β3 β4 ³ ´ η1+β3 η1+β4 β3 β4 η2+η1 = K η +β η +β η (η 1) − 1 3 1 4 2 1− Then, we can generate the reduced form of (Lu)(x) ru (x)which is: − β β η + η x η (xB x) 3 4 2 1 (Lu)(x) ru (x)= rK+δe +λpe− 1 − K 0 − − η + β η + β η (η 1) ≤ 1 3 1 4 2 1 − So the condition No3isproven. –––––––— Appendix B: Demonstration of the formula of perpetual American put option in the uniform jump diﬀusion case.

Work in advance...to complete

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