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We shall now check that both sides of equality (2.51) coincide at time 7(1) on the event {7(1) < T}. To this end, we note that, on the event {7(1) < T}, where the auxiliary functions Y?: (0, 7] — R, i = 1, 2,3, are given by

Table 7 We shall now check that both sides of equality (2.51) coincide at time 7(1) on the event {7(1) < T}. To this end, we note that, on the event {7(1) < T}, where the auxiliary functions Y?: (0, 7] — R, i = 1, 2,3, are given by

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Before proceeding to the proof of Proposition 1.4.1, we will establish an elementary lemma.
Before proceeding to the proof of Proposition 1.4.1, we will establish an elementary lemma.
Lemma 1.4.1 For anya€ R andb>0 we have, for every y > 0, where n is the probability density function of the standard Gaussian law. Note also that
Lemma 1.4.1 For anya€ R andb>0 we have, for every y > 0, where n is the probability density function of the standard Gaussian law. Note also that
Given the optimal strategy of the stockholders, the price process of the firm’s debt (i.e., of a consol bond) takes the form, on the event {r* > t}, We end this section by mentioning that other important developments in the area of optimal capital structure were presented in the papers by Leland [116], Leland and Toft [117], Christensen et al. [50]. Chen and Kou [46] and Hilberink and Rogers [87] study analogous problems, but they model the firm’s value as a diffusion process with jumps. This extension was aimed to eliminate an undesirable feature of other models, in which the spread for corporate bonds converges to zero for short maturities.
Given the optimal strategy of the stockholders, the price process of the firm’s debt (i.e., of a consol bond) takes the form, on the event {r* > t}, We end this section by mentioning that other important developments in the area of optimal capital structure were presented in the papers by Leland [116], Leland and Toft [117], Christensen et al. [50]. Chen and Kou [46] and Hilberink and Rogers [87] study analogous problems, but they model the firm’s value as a diffusion process with jumps. This extension was aimed to eliminate an undesirable feature of other models, in which the spread for corporate bonds converges to zero for short maturities.
Recovery at Default is an H-martingale. It is also worth noting that the recovery can be formally interpreted as a dividend stream paid at the rate Zy up to time TAT. The discounted price is here an H-martingale under the risk-neutral probability Q and the price S$ does not vanish (unless Z equals zero).
Recovery at Default is an H-martingale. It is also worth noting that the recovery can be formally interpreted as a dividend stream paid at the rate Zy up to time TAT. The discounted price is here an H-martingale under the risk-neutral probability Q and the price S$ does not vanish (unless Z equals zero).
We will now prove an integral representation theorem for any G-martingale stopped at 71) with respect to some finite collection of G-martingales stopped at 7(1). To this end, we define, for every 4—1.2....n. 2.5.2 First-to-Default Representation Theorem
We will now prove an integral representation theorem for any G-martingale stopped at 71) with respect to some finite collection of G-martingales stopped at 7(1). To this end, we define, for every 4—1.2....n. 2.5.2 First-to-Default Representation Theorem
so that the conditional tail equals, for u € [v, co],
so that the conditional tail equals, for u € [v, co],
Lemma 2.5.6 The pre-default risk-neutral value of an FTDC equals The next result extends Proposition 2.4.2 to the multi-name setup. Its proof is similar to the proof of Lemma 2.5.5 and thus it is omitted. By the pre-default risk-neutral value associated with a G-adapted process 7, we mean the function m such that mel rer} = TO) gr<r,)} for every t € [0,7]. Direct calculations lead to the following result, which can also be deduced from Proposition 2.5.1. Proposition 2.5.2 The pre-default risk-neutral value of an FTDC satisfies where the last two terms represent the past dividends. Let us stress that the risk-neutral valuation of an FTDC will be later supported by replication arguments (see Theorem 2.5.1) and thus risk-neutral value 7 of an FTDC will be shown to be its replication price.
Lemma 2.5.6 The pre-default risk-neutral value of an FTDC equals The next result extends Proposition 2.4.2 to the multi-name setup. Its proof is similar to the proof of Lemma 2.5.5 and thus it is omitted. By the pre-default risk-neutral value associated with a G-adapted process 7, we mean the function m such that mel rer} = TO) gr<r,)} for every t € [0,7]. Direct calculations lead to the following result, which can also be deduced from Proposition 2.5.1. Proposition 2.5.2 The pre-default risk-neutral value of an FTDC satisfies where the last two terms represent the past dividends. Let us stress that the risk-neutral valuation of an FTDC will be later supported by replication arguments (see Theorem 2.5.1) and thus risk-neutral value 7 of an FTDC will be shown to be its replication price.
Proof. From the previous remarks, we obtain Under the assumption of independence of default times, we also have that Si j (si) = Si(«,) for i,j =1,2 andi 4 j. Furthermore, from Example 2.4.1, we have that Si(«;) = 0 for t € (0, 7] and thus the matrix N(t) in Theorem 2.5.1 reduces to The replicating strategy can be found easily by solving the linear equation N(t)d(t) = h(t) where h(t) = (hi(t), ha(t)) with the function h; given by the formula
Proof. From the previous remarks, we obtain Under the assumption of independence of default times, we also have that Si j (si) = Si(«,) for i,j =1,2 andi 4 j. Furthermore, from Example 2.4.1, we have that Si(«;) = 0 for t € (0, 7] and thus the matrix N(t) in Theorem 2.5.1 reduces to The replicating strategy can be found easily by solving the linear equation N(t)d(t) = h(t) where h(t) = (hi(t), ha(t)) with the function h; given by the formula
We are in a position to determine the replicating strategy. Under the standing assumption that kK; = AiO; for i = 1,2 we still have that $?(«;) = 0 for i = 1,2 and for t € [0,7]. Hence the matrix N(t) reduces to
We are in a position to determine the replicating strategy. Under the standing assumption that kK; = AiO; for i = 1,2 we still have that $?(«;) = 0 for i = 1,2 and for t € [0,7]. Hence the matrix N(t) reduces to
The expression for S*,(K2) can be found by analogous computations. By solving the equation
The expression for S*,(K2) can be found by analogous computations. By solving the equation
In order to find the replicating strategy, we proceed as in the proof of Proposition 2.6.2. Under the present assumptions, we obtain the following expression for S 1 (K1) By the symmetry of the model, a similar expression is valid for the value Sai (K2)- This completes the proof of the proposition.
In order to find the replicating strategy, we proceed as in the proof of Proposition 2.6.2. Under the present assumptions, we obtain the following expression for S 1 (K1) By the symmetry of the model, a similar expression is valid for the value Sai (K2)- This completes the proof of the proposition.
where the third equality follows from formula (3.5) and where we denote, for every r € R4, Assume that F' admits the Doob-Meyer decomposition F' = N+C, where the process C is absolutely continuous with respect to the Lebesgue measure, so that Cy = fo c, du for some F-progressively measurable process c.
where the third equality follows from formula (3.5) and where we denote, for every r € R4, Assume that F' admits the Doob-Meyer decomposition F' = N+C, where the process C is absolutely continuous with respect to the Lebesgue measure, so that Cy = fo c, du for some F-progressively measurable process c.
We have thus shown that the process M" is an F-martingale. Moreover, the F-adapted process @, given by (3.17) is manifestly continuous, and thus it is F-predictable. By virtue of uniqueness of the Doob-Meyer decomposition, we conclude that M* = m and formula (3.17) is valid. is an F-martingale. Clearly, the process M¥" is integrable and F-adapted. Moreover, for every t < s, Corollary 3.1.3 Let us denote ¢ = Eg(c |F,). The process
We have thus shown that the process M" is an F-martingale. Moreover, the F-adapted process @, given by (3.17) is manifestly continuous, and thus it is F-predictable. By virtue of uniqueness of the Doob-Meyer decomposition, we conclude that M* = m and formula (3.17) is valid. is an F-martingale. Clearly, the process M¥" is integrable and F-adapted. Moreover, for every t < s, Corollary 3.1.3 Let us denote ¢ = Eg(c |F,). The process
where 6 and « are G-predictable processes, with k > —1. This means the process 7 is the Doléans exponential, more explicitly where € and ¢ are G-predictable stochastic processes. Since 7 is a strictly positive martingale, by setting 0, = €n;_ and K, = G:n;_", we obtain
where 6 and « are G-predictable processes, with k > —1. This means the process 7 is the Doléans exponential, more explicitly where € and ¢ are G-predictable stochastic processes. Since 7 is a strictly positive martingale, by setting 0, = €n;_ and K, = G:n;_", we obtain
Let us stress that the market CDS spread «(s, 7’) is not defined neither at the moment of defaul nor after this date, so that we shall deal in fact with the pre-default value of the market CD spread. Observe that «(s, 7’) is represented by an F,-measurable random variable. In fact, it follow immediately from (3.51) that «(s,7’) admits the following representation, for every s € [0, T], In what follows, we shall write briefly «, instead of «(s,7T). The next result furnishes a convenient representation for the price at time t of a CDS issued at some date s < t, that is, the marked-to- market value of a CDS that exists already for some time (recall that the market value of the just issued CDS is null). It should be noted that CDSs are quoted in terms of spreads. At any date t, one can take at no cost a long or short position in the CDS issued at this date with the fixed spread equal to the actual value of the market CDS spread for a given maturity and a given reference credit name.
Let us stress that the market CDS spread «(s, 7’) is not defined neither at the moment of defaul nor after this date, so that we shall deal in fact with the pre-default value of the market CD spread. Observe that «(s, 7’) is represented by an F,-measurable random variable. In fact, it follow immediately from (3.51) that «(s,7’) admits the following representation, for every s € [0, T], In what follows, we shall write briefly «, instead of «(s,7T). The next result furnishes a convenient representation for the price at time t of a CDS issued at some date s < t, that is, the marked-to- market value of a CDS that exists already for some time (recall that the market value of the just issued CDS is null). It should be noted that CDSs are quoted in terms of spreads. At any date t, one can take at no cost a long or short position in the CDS issued at this date with the fixed spread equal to the actual value of the market CDS spread for a given maturity and a given reference credit name.
Recall that € = 1y¢<;}7. It is thus clear that the pricing functions v(-,0) and v(-;1) satisfy th PDEs given in the statement of the proposition. L From (4.41), it follows that the process Cisa martingale under Q!'. Therefore, the continuous finite variation part in the above decomposition necessarily vanishes, and thus we get It should be stressed that in what follows we only examine the form of a replicating strategy prior to default time. Proposition 4.4.2 The replicating strategy ¢ for the claim Y is given by formulae
Recall that € = 1y¢<;}7. It is thus clear that the pricing functions v(-,0) and v(-;1) satisfy th PDEs given in the statement of the proposition. L From (4.41), it follows that the process Cisa martingale under Q!'. Therefore, the continuous finite variation part in the above decomposition necessarily vanishes, and thus we get It should be stressed that in what follows we only examine the form of a replicating strategy prior to default time. Proposition 4.4.2 The replicating strategy ¢ for the claim Y is given by formulae
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