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Thư viện số Văn Lang: Stochastics of Environmental and Financial Economics: Centre of Advanced Study, Oslo, Norway, 2014-2015

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Nguyễn Gia Hào

Academic year: 2023

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The purpose of this is a security to cover losses in the sense that in the event of liquidity problems it causes the recapitalization of the entity. In their work, Koziol and Lawrenz use the low level of the company's asset price as a conversion trigger. This Credit Suisse issue was made under the new regulatory regime in Switzerland.

In the structural approach to modeling the trigger, one models the stochastic variable that describes the conversion time and links it to the dynamics of the assets, debt or shares. In the structural approach, one can use a market trigger based on a low level of the equity value. Another option, in the structural approach, is to use a low value of the asset value as a trigger.

It will be assumed that the issuer of CoCopays yields according to a deterministic functionκ. In the case of CoCos with impairment, the investor does not receive a certain number of shares when the trigger event occurs. Instead, she receives only a fraction R ∈ (0,1) of the original face value K, assuming the issuer has not defaulted.

Indeed, from the definition of random timesστ andτj(see (14)) and barriers andjit it follows that.

Fig. 1 The graph illustrates the share price ( S t ) t≥0 , along with the barriers  j and its parameters L j and M j
Fig. 1 The graph illustrates the share price ( S t ) t≥0 , along with the barriers j and its parameters L j and M j

The Black-Scholes Model and the Greeks

On the one hand, it has been documented that by actively hedging equity risk, investors can inadvertently force the conversion by causing the share price to deteriorate and eventually trigger the conversion. Now it can be checked from the above expression for the DeltaΔ that it is strictly positive, and in fact it can be seen that the DeltaΔ increases sharply when the time to maturity T decreases. Thus, one of the conclusions from [13] is that the coupon cancellation function leads to a flatter behavior of the DeltaΔ, reducing the risk of a death spiral.

On the other hand, it can also be verified that Vegaν is strictly negative, this tells us that an increase in volatility translates into a decrease in prices. Such an observation is clearly in line with the intuition that a higher volatility will increase the probability of crossing the barriers and define the conversion and coupon cancellation events.

4 Advanced Models

Incorporating the Heston Stochastic Volatility Model

If we further accept the independence between the noises that drive the prices and their volatility, then we see that the PTj-Brown movement in (24) is independent of the (now stochastic) time change. 27) Thus, by a preconditioning argument, we get the following extension of Proposition4. From the price formula above, we can see that the CoCoprice is related to the price of binary options. Let us now give an explicit calculation of the probability above; it illustrates how the CoCoprice can be calculated.

For this case, we use the relation between the characteristic and distribution functions (see e.g. [38]), which allows us to write Therefore, the problem of calculating the probability above translates to the problem of finding an expression for ϕTj(t,u,v;ξ).

An Exponential Lévy Model

  • First-Passage Times and Wiener-Hopf Factorization
  • The One-Sided CGMY Lévy Process

Consequently, the knowledge of one of the factors in (34) allows us to establish the other. A special case of interest arises when (Xt)t≥0 is a spectrally negative process, since in this case it is known that the right factor in (34) is given by. This condition is imposed to facilitate the calculation of βλ (i.e. the solution of (35)) using appropriate integration contour change.

Here the parameters (A1,A2,l1,l2) are positive real numbers chosen large enough to control the aliasing error. The final Euler summation is used to improve the accuracy of the rough approximations N. This process has no continuous part, and only one-sided jumps where the Lévy measure is given by .

Before going any further, we should mention that the CGMY processes are also known as Tempered stableprocesses. On the other hand, by setting the parameter Y =0 (resp. Y =1/2), the CMY process becomes a Gamma process (resp. Inverse Gaussian process). To price CoCos, we need to understand the behavior of (Xt)t≥0 under both the risk-neutral measure P∗ and the equity measure P(S).

We note here that this technical assumption will, on the one hand, enable us to accommodate the Variance Gamma(VG) process discussed in [30]. Theorem 9 In the current setting, the price of aCoCoat time0≤t≤T can be efficiently approximated numerically by means of the expression. Thus, writing P0=P∗ and P1=P(S), we see by the reasoning in Section 4.2.1 that the Laplace transform of Fα(t, ξ):=Pα(Xt >−ξ) is given byFα as defined above .

As the noise driving stock prices, the authors consider the so-called Beta-Variance Gamma(β-VG) process – also referred to as Lamperti-Stableprocess by [3] – which exhibits the same exponential decay as the Variance Gamma process, leading to a smile-compliant model. For this β-VG process, the distribution of the variables Xe(λ) and Xe(λ) can be specified, obtaining the Wiener-Hopf factors ψλ+andψλ−. Taking (8) into account, combining knowledge of the (Xt,Xt) density with a Monte-Carlo technique due to [29], the authors provide an efficient numerical pricing of CoCos.

5 Triggering Conversion Under Short-Term Uncertainty

Pricing CoCos on a Black-Scholes Model Under Short-term Uncertainty

In this setting, Assumption(A2) is translated as the correlation structure between the noise driving(St)t≥0 and that of the new process. Instead of the process(Ut)t≥0 above, we must now consider the parametric family(Ut(ρ))t≥0, whose driving noise(Wtρ)t≥0 is also a Brownian motion, but correlated to(Wt∗) t≥0, in such a way that dWtρdWt∗ =ρdt. Furthermore, the time at which the jth coupon can be canceled is given by τj(ρ):=inf.

Since Gt ⊇Ft, and the equality holds only on the predetermined dates{tj, j ∈N}, we can think of (Zt)t≥0 as an extra source of noise that clears at update times (tj)j∈N. The fact that the extra noise is cleared at (tj)j∈N motivates the idea of ​​short-term uncertainty, and this has two important implications. On the other hand, in contrast to other structural models, the short-term uncertainty considered here prevents the switching timeτj(ρ) from being a stopping time with respect to the reference filtrationF, which is generated by the relevant state variables and the risk-free market.

Coupon Cancellation Probabilities Under Short-Term Uncertainty

On the one hand, our model differs from other partial or incomplete information models such as or [25] as the information structure is different. Proof Since under Gt the computation is known, for each t = tj we have, in {τx(ρ) >t},. In a way, the short-term uncertainty model considered here can be seen as a bivariate extension of [27].

For example, the survival probabilities obtained by [27] are constant between observation dates, i.e., within each interval [tj,tj+1). This difference relies on the fact that, even though within each interval [tj,tj+1) our knowledge of the short-term noise ζt is constant, we still fully observe the evolution of (St)t≥0 and all others fitted to F. state variables. To conclude this section, let us recall that in light of our discussion in Section 3.1, CoCoprices can be obtained in our current environment after computing expressions of the form.

It is worth noting that the joint Ft-conditional distribution of (τj(ρ),STj) = (infs≤TjUs(ρ),STj) cannot be calculated directly from Proposition 11 since the inputs of this vector are driven by two different (even why correlated) Brownian motions. An additional complication comes from the fact that the actual information about one of them may be incomplete.

6 Extension Risk

For the sake of clarity, let us note that in the present setting the barriers in (18) and their parameters become From the issuer's point of view, the question of whether or not the face value payment should be deferred depends on which alternative is cheaper. Thus, similar to the situation of Bermuda options (see for example [37]), the discount price of a CoCoBelonging to Additional Tier 1 capital category is equal.

It is important to note that although the general optimal stopping theory allows us to characterize the solution to the optimization problem in (48a), it is not sufficient to handle the entire pricing problem. Having a finite horizon allows us to obtain (Yk(n))k∈{n,..,N} by the following backward procedure. As it turns out, the lower envelope Snell(Yk(n))k∈{n,..,N} can be obtained in a fairly clear form.

For the first iteration of (49), ifτ >Tn, then what we have is the raw expression. Due to stock price Markovianity, the price πTN−1 can be viewed as a function of STN−1; we simply denote this function by . Now, as discussed in Sect.3.1, the CoCo has a positive Delta, thus the functionπTN−1(x) is increasing and we can find a valueSN∗−1 such.

In [14] it was shown that the calculation of the Snell envelope (Yk(n))k∈{n,..,N} can be performed using the above backward procedure. In fact, it can be seen that the payoff of aCoCo, which has a calling function, can be written in terms of Proof With an explicit description of the payoff corresponding to the risk of CoCowith expansion, we obtain the result as in Proposition 2, subject to the following identity.

In light of this theorem, obtaining a closed-form formula for the price CoCo with expansion risk requires knowledge of the conditional distribution of (τ,ST0,ST1, ..,STi) for i = 1, ..,N . In the Black-Scholes model this can be achieved by means of the following general lemma obtained in [14]. Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any non-commercial use, distribution, and reproduction in any medium, provided the original author(s) and source are acknowledged.

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Fig. 1 The graph illustrates the share price ( S t ) t≥0 , along with the barriers  j and its parameters L j and M j

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