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Pricing and upper price bounds of relax certificates


Relax certificates are written on multiple underlying stocks. Their payoff depends on a barrier condition and is thus path-dependent. As long as none of the underlying assets crosses a lower barrier, the investor receives the payoff of a coupon bond. Otherwise, there is a cash settlement at maturity which depends on the lowest stock return. Thus, the products consist of a knock-out coupon bond and a knock-in claim on the minimum of the stock prices. In a Black-Scholes model setup, the price of the knock-out part can be given in closed (or semi-closed) form in the case of one or two underlyings only. With the exception of the trivial case of one underlying, the price of the knock-in minimum claim always has to be calculated numerically. Hence, we derive semi-closed form upper price bounds. These bounds are the lowest upper price bounds which can be calculated without the use of numerical methods. In addition, the bounds are especially tight for the vast majority of relax certificates which are traded at a discount relative to the corresponding coupon bond. This is also illustrated with market data.

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  1. Bonus certificates pay the maximum of the underlying value and a fixed payoff if the underlying never reaches a lower boundary until maturity. If the barrier is crossed, however, the investor instead receives the underlying.

  2. Cf. monthly reports of the Börse Stuttgart (EUWAX) and the monthly statistics of the Deutsche Derivate Verband (DDI).

  3. Similar products are also called Top-10-Anleihe, Easy Relax Express, Easy Relax Bonus, Multi-Capped Bonus or Aktienrelax. Furthermore, there are also relax certificates which bear some features of express certificates.

  4. Some examples for contracts which are currently traded will be given in Sect. 5.

  5. In the literature this minimum claim is also known as “cheapest-to-deliver”, i.e. an option on the worse of n assets, see Wilkens et al. (2001).

  6. There are also certificates where the investor can participate in the development of the underlying assets if the terminal value of the worst performing stock is larger than the face value of the coupon bond.

  7. The price bounds are calculated in a Black-Scholes model. For attractive relax certificates, however, the price bounds would be even lower if one takes the possibility of (downward) jumps or default risk of the issuer into account.

  8. This is in line with currently traded relax certificates where the minimum option is written on the return of the underlying stocks from time 0 to time t N .

  9. In the case of gap risk due to jump or liquidity risk the lowest stock price can be lower than m.

  10. Notice that the barrier feature causes some problems for the Monte Carlo simulation, see Boyle et al. (1997).

  11. To control the accuracy of the approximation, the simulation results for the survival probabilities and the prices of the knock-out component are compared to the exact closed-form solutions. These closed-form solutions, which are valid for one and two underlyings in the Black-Scholes model only, also allow for a quick calculation of the upper price bound.

  12. For all certificates, the implied volatilities of at least two underlying stocks as given in Table 1 are above 30%, so that a volatility of σ = 0.3 yields an upper bound for the survival probability. The same holds for the indices setting σ = 0.25.

  13. In 2009, CDS spreads of Commerzbank e.g. increased to more than 100 basis points. In a very rough approximation, this would reduce the prices of our certificates by around 1%. We thank an anonymous referee for pointing out this example.


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Correspondence to Judith C. Schneider.

Additional information

Earlier versions of this paper were presented at the University of Bonn, University of Duisburg-Essen and the 15th Annual Meeting of the German Finance Association in Münster, the 11th Symposium on Finance, Banking, and Insurance in Karlsruhe, the 10th Campus for Finance Research Conference in Vallendar, where it received the best paper award, and the 12th Conference of the Swiss Society for Financial Market Research in Geneva. The authors would like to thank the conference participants and discussants as well as two anonymous referee for useful comments and suggestions.


Appendix A: First hitting time: one-dimensional case

To derive the distribution of the first hitting time in the one-dimensional case, we use results given in He et al. (1998).They consider the probability density and distribution function of the maximum or minimum of a one-dimensional Brownian motion with drift. Along the lines of He et al. (1998), we define

$$ \underline{X}_t:=\mathop {\min}\limits_{0\leq s\leq t} X_s \qquad \overline{X}_t:=\max_{0\leq s\leq t} X_s $$

where X t  = αt + σW t ,  t ≥ 0 and α, σ are constants. W is a Brownian motion defined on some probability space.

Proposition 7

LetG(xt;α) andg(yxt1) be defined as

$$ \begin{aligned} G(x,t;\alpha)&:=N\left(\frac{x-\alpha t}{\sigma \sqrt{t}}\right)-e^{\frac{2\alpha x}{\sigma^2}}N\left(\frac{-x-\alpha t}{\sigma \sqrt{t}}\right),\\ g(y,x,t;\alpha_1)&:=\frac{1}{\sigma\sqrt{t}} N^\prime \left(\frac{x-\alpha_1 t}{\sigma\sqrt{t}}\right) \left(1-e^{-\frac{4x^2-4x4y} {2\sigma^2t}}\right) \end{aligned} $$

where N denotes the cumulative distribution function of the standard normal distribution andN′(z) the density of the standard normal distribution. Forx ≥ 0, it holds

$$ P(\overline{X}_t\leq x)=G(x,t;\alpha),\qquad f_{\tau_m^{(1)}}^P=g(y,x,t;\alpha_1)dy. $$

Forx < 0, it holds

$$ P(\underline{X}_t \geq x) =G(-x,t;-\alpha),\qquad P(X_1(t)\in dy,\underline{X}_1(t)\geq x)=g(-y,-x,t;-\alpha_1)dy. $$


C.f. Theorem 1 of He et al. (1998). and the proof given here. \(\square\)

Corollary 2


$$ S_t = S_0 e^{\left(\mu -\frac{1}{2} \sigma^2\right)t + \sigma W_t} $$

where μ, σ (σ > 0) are constants. W is a Brownian motion defined on some probability space. For the first hitting time\( \tau_m:={\rm inf}\{t\geq0 \vert S_t\leq m\}, \, (m < S_0)\)it holds that

$$ \begin{aligned} P(\tau_m \leq t) &= N\left(\frac{ln\frac{m}{S_0}-\left(\mu-\frac{1}{2}\sigma^2\right)t} {\sigma\sqrt{t}}\right) +e^{2 \frac{\mu-\frac{1}{2}\sigma^2} {\sigma^2}ln \frac{m}{S_0}} N\left(\frac{ln\frac{m} {S_0}+\left(\mu-\frac{1}{2}\sigma^2\right)t} {\sigma\sqrt{t}}\right),\\ f_{\tau_m^{(1)}}^P &=\frac{-ln \frac{m}{S_0}} {\sqrt{2\pi\sigma^2t^3}}e^{-\frac{1}{2}\frac{\left(ln\frac{m} {S_0}-(\mu-\frac{1}{2}\sigma^2)t\right)^2}{\sigma^2t}}dt. \end{aligned} $$


Note that

$$ \tau_m:=\inf\left\{t\geq0 \left\vert S_t\leq m\right\} = \inf\left\{t\geq 0\left\vert \ln \frac{S_t}{S_0}\leq \ln\frac{m} {S_0}\right\}.\right.\right. $$

Let X t denote the logarithm of the normalized asset price, i.e. \( X_t:=\ln\frac{S_t}{S_0}=\left(\mu-\frac{1}{2}\sigma^2\right)t+\sigma W_t\) and set \(\alpha = \mu-\frac{1}{2}\sigma^2.\) The stopping time τ m is related to the first hitting time of a one-dimensional Brownian motion with drift α. With \(x:=\ln\frac{m}{S_0}<0\) it follows

$$ P(\tau_m\leq t)=P(\underline{X}_t\leq x)=1-P(\underline{X}_t\geq x). $$

According to Proposition 7, we have

$$ \begin{aligned} 1-P(\underline{X}_t\geq x) &= 1-G(-x,t;-\alpha) = 1-N\left(\frac{-x+\alpha t}{\sigma\sqrt{t}}\right) + e^{\frac{2\alpha x}{\sigma^2}} N\left(\frac{x+\alpha t} {\sigma\sqrt{t}}\right) \\ &=N\left(\frac{x-\alpha t}{\sigma\sqrt{t}}\right) +e^{\frac{2\alpha x}{\sigma^2}}N\left(\frac{x+\alpha t} {\sigma\sqrt{t}}\right). \end{aligned} $$

Inserting α and x gives the distribution function. To derive the density function, define

$$ f(t):=N\left(\frac{x-\alpha t}{\sigma\sqrt{t}}\right) +e^{\frac{2\alpha x} {\sigma^2}}N\left(\frac{x+\alpha t} {\sigma\sqrt{t}}\right). $$

This implies

$$ \begin{aligned} f'(t)&=N'\left(\frac{x-\alpha t} {\sigma\sqrt{t}}\right) \times\left(\frac{-\alpha\sigma\sqrt{t}-\frac{\sigma} {2\sqrt{t}}(x-\alpha t)} {\sigma^2t}\right) \\ &\quad + e^{\frac{2\alpha x} {\sigma^2}}\times N'\left(\frac{x+\alpha t}{\sigma\sqrt{t}}\right) \times\left(\frac{\alpha\sigma\sqrt{t}-\frac{\sigma} {2\sqrt{t}}(x+\alpha t)}{\sigma^2t}\right) \end{aligned} $$

Using \(N'(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2},\) we get

$$ \begin{aligned} e^{\frac{2\alpha x}{\sigma^2}}N^{\prime}\left(\frac{x+\alpha t}{\sigma\sqrt{t}}\right) &= \frac{1}{\sqrt{2 \pi}}e^{\frac{2\alpha x}{\sigma^2}-\frac{1}{2}\left(\frac{x+\alpha t}{\sigma\sqrt{t}}\right)^2} = \frac{1}{\sqrt{2 \pi}}e^{\frac{1}{\sigma^2}[2\alpha x-\frac{1}{2t}(x+\alpha t)^2]} \\ &= \frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}\frac{(x-\alpha t)^2}{\sigma^2t}} = N^{\prime}\left(\frac{x-\alpha t}{\sigma \sqrt{t}}\right). \end{aligned} $$

Inserting this in the above equation for f′(t) gives

$$ f'(t) = N'\left(\frac{x-\alpha t}{\sigma\sqrt{t}}\right) \times \frac{-\sigma}{2\sqrt{t}\sigma^2 t}\left(x-\alpha t+x+\alpha t\right) = \frac{-x}{\sqrt{2\pi \sigma^2 t^3}}e^{-\frac{1} {2}\frac{(x-\alpha t)^2}{\sigma^2 t}}. $$

Using \(\alpha=\mu-\frac{1}{2}\sigma^2\) and \(x=\ln\frac{m} {S_0}\) gives the result. \(\square\)

Appendix B: First hitting time: two dimensional case

The distribution of the first hitting time of a two-dimensional arithmetic Brownian motion is given in He et al. (1998) and Zhou (2001):

Proposition 8

Let\(X^{(j)}_t=\alpha_j t+\sigma_j W_t^{(j)}\) (j = 1, 2), where α j and σ j are constants. W(1),   W(2)are two correlated Brownian motions with < W(1)W(2) >  t  = ρ t. Then, the probability thatX(1) and X(2)will not hit the upper boundariesx(1) > 0 and x(2) > 0 up to time t is given by

$$ \begin{aligned} Q\left(\overline{X}_t^{(1)}\leq x^{(1)},\overline{X}_t^{(2)}\leq x^{(2)}\right) &=\frac{2}{\alpha t}e^{a_1 x_1 + a_2 x_2 + b t} \sum_{n=1}^{\infty}{\rm sin}{\left(\frac{n\pi\theta_0}{\alpha}\right)}\\ &\quad e^{-\frac{r_0^2}{2t}}\int\limits_{0}^{\alpha}{\rm sin}{\left(\frac{n\pi\theta} {\alpha}\right)}g_n(\theta)d\theta \end{aligned} $$

where\(\overline{X}_t:=\max_{0\leq s\leq t} X_s.\)The parameters are defined by

$$\begin{array}{*{20}l} a_1 = \frac{-\alpha_1\sigma_2 + \rho\alpha_2\sigma_1} {(1-\rho^2)\sigma_1^2\sigma_2} & \quad a_2 =\frac{-\alpha_2\sigma_1 + \rho\alpha_1\sigma_2}{(1-\rho^2)\sigma_2^2\sigma_1} \\d_1= a_1\sigma_1+a_2\sigma_2\rho &\quad d_2= a_2\sigma_2\sqrt{1-\rho^2}\\\end{array} $$

and by

$$ \begin{aligned} b &= \alpha_1 a_1 + \alpha_2 a_2 + \frac{1}{2} \sigma_1^2 a_1^2 + \frac{1}{2} \sigma_2^2 a_2^2 + \rho \sigma_1 \sigma_2 a_1 a_2\\ \alpha&=\left\{\begin{array}{cc}{\rm tan}^{-1}\left(-\frac{\sqrt{1-\rho^2}}{\rho}\right) &{ if } \rho<0\\ \pi+ {\rm tan}^{-1}\left(-\frac{\sqrt{1-\rho^2}}{\rho}\right)& otherwise \\ \end{array}\right.\\ \theta_0&=\left\{\begin{array}{cc} {\rm tan}^{-1}\left( \frac{\frac{x2}{\sigma_2}\sqrt{1-\rho^2}} {\frac{x1}{\sigma_1}-\rho\frac{x2}{\sigma_2}}\right) &{ if (.)} >0\\ \pi+ {\rm tan}^{-1}\left(\frac{\frac{x2}{\sigma_2}\sqrt{1-\rho^2}}{\frac{x1}{\sigma_1}-\rho\frac{x2}{\sigma_2}}\right)& otherwise \\ \end{array}\right.\\ r_0&=\frac{x_2}{\sigma_2}/ sin(\theta_0). \end{aligned} $$

The functiong n is defined as

$$ g_n(\theta)= \int\limits_0^\infty r e^{-\frac{r^2}{2 t}} e^{d_1 r \sin\left( \theta-\alpha\right) -d_2 r \cos\left( \theta-\alpha\right)} I_{\frac{n \pi}{\alpha}}\left( \frac{rr_0} {t} \right) dr. $$

I v (z) is the modified Bessel function of order v.


Cf. Proposition 1 of Zhou (2001) and the proof given there. \(\square\)

Corollary 3

For two stocksS(k)andS(l)with volatilities σ k and σ l and correlation ρk,l, the distribution function of the first hitting time min{τm,k, τm,l} under the risk-neutral measure is given by

$$ Q\left({\rm min}\{\tau_{m,k},\tau_{m,l}\} \leq t \right)\\ =1-\frac{2}{\alpha t}e^{a_k \ln\left(\frac{S_0^{(k)}}{m}\right) + a_l \ln\left(\frac{S_0^{(l)}}{m}\right)+ b t} \sum_{n=1}^{\infty}{\rm sin}{\left(\frac{n\pi\theta_0}{\alpha}\right)} \cdot e^{-\frac{r_0^2}{2t}}\int\limits_{0}^{\alpha}{\rm sin}{\left(\frac{n\pi\theta}{\alpha}\right)}g_n(\theta)d\theta $$


$$ \begin{aligned} a_k &= \frac{\left(r-0.5\sigma_k^2\right)\sigma_l - \rho_{k,l} \left(r-0.5\sigma_l^2\right)\sigma_k}{\left(1-\rho_{k,l}^2\right)\sigma_k^2\sigma_l}\qquad \qquad a_l = \frac{\left(r-0.5\sigma_l^2\right)\sigma_k - \rho_{k,l} \left(r-0.5\sigma_k^2\right)\sigma_l} {\left(1-\rho_{k,l}^2\right)\sigma_l^2\sigma_k}\\ d_k&= a_k\sigma_k+a_l\sigma_l\rho_{k,l} \qquad\qquad d_l= a_l\sigma_l\sqrt{1-\rho_{k,l}^2} \end{aligned} $$


$$ \begin{aligned} b &= -\left(r-0.5\sigma_k^2\right) a_k - \left(r-0.5\sigma_l^2\right) a_l + \frac{1}{2} \sigma_k^2 a_k^2 + \frac{1}{2} \sigma_l^2 a_l^2 + \rho_{k,l} \sigma_k \sigma_l a_k a_l \\ g_n(\theta) &= \int\limits_0^\infty r e^{-\frac{r^2}{2 t}} e^{d_k r \sin\left( \theta-\alpha\right) -d_l r \cos\left( \theta-\alpha\right)} I_{\frac{n \pi} {\alpha}}\left( \frac{rr_0} {t} \right) dr. \end{aligned} $$

I v (z) is the modified Bessel function of order v. α,   θ0, andr0 are given in Proposition 8 in Appendix B for the case wherek = 1 andl = 2.


The stock prices are given by

$$ S_t^{(j)} = S_0 e^{\left(r-\frac{1}{2}\sigma_j^2\right)t+\sigma_j W^{(j)}_t} \quad j=k,l. $$

The first hitting time of the lower boundary \(m_j< S_0^{(j)}\) by the geometric Brownian motion \(S_t^{(j)}\) is

$$ \tau_m^{(j)} := \inf\left\{t\geq 0 \left\vert S_t^{(j)}\leq m_j\right\} \right. = \inf\left\{t\geq 0 \left\vert -\ln\frac{S_t^{(j)}}{S_0^{(j)}}\geq \ln\frac{S_0^{(j)}} {m_j}\right\}\right.. $$

With the definition of the arithmetic Brownian motion

$$ X_t^{(j)}:= - \ln \frac{S_t^{(j)}}{S_0^{(j)}}=-\left(r-\frac{1} {2}\sigma_j^2\right)t-\sigma_j W^{(j)}_t, $$

the first hitting time can be rewritten as \(\tau_m^{(j)}=\inf\left\{t\geq 0 \left\vert X_t^{(j)} \geq \ln\frac{S_0^{(j)}} {m_j}\right\}.\right.\) Using the relation \(\{\tau_{m,j} > t\}=\left\{\overline{X}_t^{(j)} < \ln\frac{S_0^{(j)}}{m_j}\right\} \) we can conclude

$$ Q\left(\min\{\tau_{m,k},\tau_{m,l}\}>t \right) = Q\left( \overline{X}_t^{(k)} < \ln\frac{S_0^{(k)}}{m_k}, \overline{X}_t^{(l)} < \ln\frac{S_0^{(l)}}{m_l} \right). $$

Since both, \(\ln\frac{S_0^{(k)}}{m_k}>0 \) and \(\ln\frac{S_0^{(l)}}{m_l}>0,\) the result follows from Proposition 8. \(\square\)

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Branger, N., Mahayni, A. & Schneider, J.C. Pricing and upper price bounds of relax certificates. Rev Manag Sci 5, 309–336 (2011).

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  • Certificates
  • Barrier option
  • Price bounds

JEL Classification

  • G13