Telekom Equity-Based Bonus Certificate Pricing and Portfolio Replication via Exotic Options

A Bonus Certificate is a type of structured derivative that offers investors conditional protection against moderate price declines, while still allowing participation in positive performance. We assume that any fraction of the security is tradable and that the risk-free interest rate r is constant at 2% per annum. Our analysis focuses on understanding the product’s payoff structure, analytically deriving and modeling its fair price under the Black-Scholes framework, and confirming the analytical price with a Monte Carlo simulation. In this article, we investigate Bonus Certificates whose underlying asset is Telekom’s stock price from 2014 to 2024.

We begin by loading the historical closing prices of Telekom stock from the dataset `Telekom.csv`. The data spans 10 years, from June 17, 2014 to June 16, 2024. After parsing the dates, we visualize the stock price evolution over time.

Next, we compute the log-returns of the Telekom stock based on the daily closing prices. The log-return at time step i is defined as

Log-returns are widely used in financial modeling because of their desirable properties — notably, time additivity, along with the fact that they tend to be approximately normally distributed over short intervals. Again, we visualize the evolution over the full 10-year period to gain some insight into the volatility and the dynamic of the asset.

To capture the key statistical features of the log returns, we report standard summary metrics such as the mean, standard deviation, minimum, maximum, and selected quantiles.

                                count        mean        std        min        25%        50%        75%        max
             Closing Price       2541        16.3       2.59       10.4       14.8       15.6       17.5       23.3
             Log return          2541      0.0002      0.013      -0.11     -0.006          0      0.008       0.06

Now that we have gotten a feel for the underlying statistics of our dataset, we proceed to fit the classical Black-Scholes model to the historical price data of the Telekom stock. This involves estimating the model's core parameters, namely the drift and volatility by using the empirical log-returns derived from the time series. We assume that prices follow a geometric Brownian motion, given by the stochastic differential equation:

where B_t and P_t are the risk-free bank account and the stock price at time t, μ is the drift (constant), σ is the volatility (constant) and W_t is a standard Brownian motion. Let Y_i denote the i-th log‐return and Y its sample mean. By Itô’s lemma, the log‐price under the Black–Scholes model satisfies the SDE:

Discretizing over a time step Δ = t_i+1 − t_i gives i.i.d. log‐returns.

Choosing a daily time‐step Δ = 1/252 (assuming 252 trading days per year), we obtain the following estimators.

Both of which, carry annual units thanks to our choice of Δ. In order to simulate the price path, we take our estimators and apply them to our aforementioned Gaussian distribution, using these parameters:

Thereby allowing us to simulate each incremental change via:

Which finally yields the well-known discrete-time price update:

This exact discrete‐time scheme is fully consistent with the continuous‐time Black–Scholes SDE. In a Monte Carlo setting, we generate M independent paths of length T⁄Δ = 252, each reflecting one possible stock evolution. As M → ∞, the diffusion noise averages out and the sample mean path converges to the deterministic drift trend driven by μ̂.

The following plots illustrate the effect of increasing M on the simulated average price paths, using P₀ = 12.59€, demonstrating convergence toward the expected price trajectory as the sample size grows.

By averaging over all simulated paths, the randomness contributed by the diffusion term smoothes out, allowing the underlying drift component of the Black-Scholes model to become clearly visible. This demonstrates how the expected trend drives the price evolution when stochastic fluctuations are averaged out. Furthermore, we can see the effect of the additive volatility corrective factor in our derivation of μ̂ by observing the offset between the historical (black) price and the monte carlo (yellow) line.

Now let us investigate the payoff profile these Certificates hold, so we can move onto modeling their profit and loss profile. Certificates are structured financial products that provide investors with the opportunity for enhanced returns compared to directly holding the underlying asset, usually a stock. These products blend characteristics of traditional investments and options, offering a minimum guaranteed payout (the bonus level) as long as the underlying asset’s price does not fall below a predefined barrier level during the certificate’s lifetime. These financial derivatives can be appealing in scenarios where the market's volatility is expected to be low, without holding the risk of high losses which can occur when shorting e.g. put and call option straddles.

In the following, we consider the payoff profile of a Classic Bonus Certificate (neglecting surcharges) as an illustrative example. We assume a barrier level H = 15 and a bonus level B = 25. The underlying asset price at maturity P_T is varied between 0 and 40 to visualize the corresponding payoff of the certificate. This visualization highlights how the payoff behaves when the barrier is not breached during the lifetime of the derivative and shows the position of the terminal price relative to the bonus level.

Investors may be drawn to Bonus Certificates because they offer the potential for higher returns compared to directly holding the underlying asset - especially when the stock price stays between the barrier and the bonus level. These instruments provide conditional downside protection: As long as the barrier is not breached, the investor is protected against moderate losses. In addition, if the underlying remains above the barrier at maturity, the investor receives a fixed bonus payout, which can exceed the return from simply holding the asset. This structure combines elements of capital preservation with the opportunity for enhanced yield in sideways or moderately bullish markets.

However, the main risk lies in the (possible) high volatility of the underlying asset’s price. Due to significant price fluctuations, there is an increased probability that the price will fall below the safety threshold at some point during the term. If the barrier is breached, the bonus mechanism expires, and the investor loses the protection, facing the full downside risk of the underlying.

This barrier risk is a key concern, as it can significantly affect the certificate’s payoff. Additionally, because Bonus Certificates are structured and more complex financial products, there are potential liquidity risks - it may be harder to sell them quickly or at a fair price compared to more straightforward investments.

In summary, while Bonus Certificates can be appealing for enhanced returns with partial protection, investors must carefully consider the volatility-driven barrier risk and the complexities that might affect liquidity.

Okay… now that we understand the payoff profile, how do we create a portfolio replicating this payoff?

As stated before, the payoff of a Bonus Certificate depends on the price of the corresponding underlying at maturity T and on whether the barrier H was breached at any time during its lifetime. More precisely, given a predetermined bonus level B, the payoff of the Bonus Certificate equals B if and only if the barrier H was not touched or breached during the product's lifetime and the terminal price P_T ≤ B. In all other cases, the payoff will simply be the terminal price P_T itself. We can formalize this by expressing the payoff D as:

This says:

• If the barrier is not breached and terminal price ≤ B, you get B
• Else, you get P_T

This is equivalent to:

Where (B − P_T)⁺ = max(B − P_T, 0). We can therefore replicate the payoff of the Bonus Certificate by taking a long position in both the down-and-out put option with strike B and the underlying asset P_t.

Now we have found and constructed a valid replication of our Classic Bonus Certificate by using known exotic options (which hold analytical solutions which we can use) along with the underlying spot price as our building blocks. Our objective is now to find a fair value for the Bonus Certificate. Because the market model is complete, the price of any contingent claim coincides with the price of its replicating portfolio. We will use this fact along with risk-neutral pricing to price the Bonus Certificate.

Now, since (B − P_T)⁺ = (P_T − B)⁺ − (P_T − B), we further obtain that:

Where C(P₀, 0) denotes the vanilla call option price at t = 0, KOD(P₀, 0) the knockout discount by which the knockout barrier H lowers the price, and Q^(1,2) as given on the task sheet. Note that we make use of the formulas for a down-and-out call, which has a well-known analytical closed-form.

We now explicitly compute the price of a Bonus Certificate on the Telekom stock with P₀ equal to the price quoted on 14.06.2024, barrier level 16, bonus level 24, and maturity of 1 year. We will use the market parameters from above, and since we are valuing with respect to the risk-neutral measure, we will use the risk-free rate r as drift.

      Historical P_0 (underlying):                    22.63
      Estimated Fair Bonus Certificate Price:         24.25

To confirm the pricing formula derived analytically in Task 5, we use a Monte Carlo simulation approach. The idea is to simulate multiple price paths of the underlying asset under the risk-neutral measure and evaluate the expected discounted payoff of the Bonus Certificate. Therefore, this time, we simulate price paths using the interest rate r = 0.02 as a drift to approximate the option price numerically.

We simulate 100,000 price paths of the underlying asset over a one-year horizon using the geometric Brownian motion model. At each timestep, we monitor whether the barrier H has been breached.

To achieve computational efficiency, we employ vectorized NumPy operations to:

1. Simulate all Brownian increments simultaneously.
2. Track barrier breaches for each path.
3. Calculate final discounted payoffs accordingly.

Additionally, we use antithetic variables to reduce the variance of our Monte Carlo estimator, improving the accuracy and convergence speed of the price estimate without increasing the number of simulations.

The price of the Bonus Certificate is then estimated as the average discounted payoff across all simulated paths. This approach allows us to approximate the fair value of the product under the risk-neutral measure efficiently and with high numerical stability.

To validate the analytical pricing formula, we compared it to the result obtained via Monte Carlo simulation. The absolute error and relative error between the two methods are calculated:

      Monte Carlo Estimate:                           24.27
      Analytical value:                               24.25
      Absolute error:                                 0.014
      Relative error:                                 0.057%

The small error observed confirms that the Monte Carlo method provides a close numerical approximation to the theoretical price. Such small deviations are normal due to the randomness of simulations. The result supports the correctness of the analytical pricing formula derived earlier. If higher accuracy is needed, the simulation can be refined by increasing the number of simulated paths or applying variance reduction techniques.

One might notice when conducting simulations, that the Monte Carlo price is always slightly higher than the analytical price. One might suggest that the absolute error between the analytical and numerical price might not only be caused by the randomness of the simulation. In fact the Monte Carlo price is always slightly overestimating the price of the bonus certificate, as it only checks if the barrier was breached in discrete time steps. In theory a price could breach the barrier between two time steps but not be detected by the Monte Carlo Payoff simulation algorithm, leading to more bonus-paying simulations and in this way driving up the price of the option. Therefore increasing the number of steps in the path simulation, would decrease the bias, but on the other hand grow the computational effort significantly.

We now simulate the evolution of the underlying price and the payoff of the Bonus Certificate under the physical probability measure. This allows us to investigate the risk characteristics of the instrument and compare them directly with those of the underlying asset. Specifically, we compare the distributional properties of the terminal asset price P_T and the corresponding certificate payoff. For both variables, we compute key summary statistics: the mean, variance, skewness, kurtosis, and the 5% quantile (i.e., Value-at-Risk at 95% confidence level).

                                 mean        var        Skew     Kurtosis        min        max       5% VaR
     Terminal Price Stats        24.5       27.5      0.635        0.648        11.4       50.9         16.9
             Payoff Stats        25.8       18.7      0.656         2.84        11.4       50.9         17.8

The results of the simulation reveal distinct differences between the underlying asset and the payoff of the structured product. The distribution of the terminal asset price P_T is more or less continuous and exhibits positive skewness, as expected from a geometric Brownian motion process. Its kurtosis is moderate, consistent with the thin tails of the log-normal distribution.

In contrast, the distribution of the certificate's payoff is more irregular. Due to the non-linear nature of the payoff function (influenced by the barrier and bonus level), the distribution is discontinuous and clustered around key values. We observe a concentration of mass at the bonus level B = 24, reflecting scenarios in which the barrier was not breached and the investor received the bonus. The kurtosis of the payoff distribution is significantly higher, showing that the payoff is more asymmetric and more prone to extreme outcomes than the terminal price itself.

Notably, the Value-at-Risk of the payoff distribution is slightly higher than that of the underlying asset. This reflects the downside protection provided by the barrier feature of the Bonus Certificate, which cushions losses in adverse scenarios.

These differences highlight the value added by structured products in tailoring risk-return profiles, but they also underline the complexity in analyzing and pricing such instruments.