In this paper, we introduce an alternative framework using the Brownian Bridge technique for pricing the autocallable structured product with knock-in (KI) feature. After the revision of the Securities and Exchange Law in 2003, Equity-linked security (ELS) has been one of the most popular financial derivatives and the market of ELS has been grown rapidly. As shown in
With the remarkable growth of ELS market, a lot of studies have discussed on the pricing of ELS. Among them, most studies for pricing ELS rely on the numerical analysis method such as Finite Difference Method (FDM) or Monte Carlo simulation method.
Our findings are summarized as follows: First, we rewrite the probability formula including minimum value in
The remainder of the paper is organized as follow. Section 2 summarizes the framework of pricing autocallable structured product suggested by
The autocallable structured product with knock-in (KI) feature is one of the popular investment products which pays coupon in addition to principal if the underlying asset touches predetermined barriers. In this section, we introduce a closed-form pricing of the product derived by
With
Now, let us introduce the closed-form pricing formula for the autocallable product.
As the activating condition met, the investor would receive principal plus interest (1 +
Consider the process {
(i)
(ii)
(iii)
See
For 6 subperiods, we can easily compute
We need to compute the probability and expectation which the process
Another advantage of using BB is that it reduces bias caused by discretization. MC is a simulating method by discretizing the continuous flow of underlying index, so a systematic bias occurs accordingly. This is called monitoring bias or discretization bias. Systematically, in terms of pricing barrier option, price of Knock-Out option is overestimated, but price of Knock-In option is underestimated. When we divide computational interval into
For more details, we refer to
In this section, we introduce the explicit formula for the exit probability with BB and use it to calculate
For convenience, we denote (
Now, we need to transform
See
We would compute expectation of conditional probabilities including minimum value in (
After generating index paths for all scenarios, extract only scenarios that meet the condition
We can also compute
By following the process listed so far, we can obtain all the elements needed for pricing the autocallable ELS.
In this section, we provide numerical examples of the time-0 price and breakeven coupon rates of autocallable ELS using three different methods, which is the explicit formula, BB and MC. To verify the superiority of BB, we also provide the results of relative error and some summary statistics of the distributions of
In
As shown in
Also, by setting the time-0 price equals to initial level of underlying index, we can obtain the breakeven point of the coupon rates
where
In
Finally, given
The robustness of BB can also be conformed through the summary statistics of the simulation values in
In this paper, we illustrate how to obtain the price of autocallable structured product with KI feature using conditional probabilities and the Brownian Bridge technique. The conditional probabilities make the pricing formula simpler than the old one and the technique makes it possible to compute the probabilities quickly and accurately. Also, the technique is far superior to the Crude Monte Carlo method in terms of accuracy, robustness, and computational speed.
We focus on pricing the product with one underlying index. However, as for the market shares of issuance by the number of underlying assets, are one-, two-, three-, and more than four-asset account for 3.90%, 3.08%, 88.83%, and 4.19%, respectively. Future research could examine the pricing frameworks for generalized investment product such as increasing the number of underlying index or diversifying knock-in barrier types to meet the diverse needs of derivatives investors.
To meet the diverse needs of derivatives investors, pricing frameworks for generalized investment product such as increasing the number of underlying index or diversifying knock-in barrier types are needed.
Example of structure of ELS
Payoff | Autocallable date | Activating condition | |
---|---|---|---|
1 | 1+ | Pr ( | |
2 | 1+ | Pr ( | |
⋮ | ⋮ | ⋮ | ⋮ |
6 | 1+ | ||
7 | 1+ | ||
8 |
Numerical example of time-0 price of autocallable ELS
Explicit Forumla | Brownian Bridge | Crude Monte Carlo | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
3% | 5.0% | 100.42 | 98.79 | 96.76 | 100.50 | 98.82 | 96.76 | 100.48 | 99.02 | 96.70 |
6.5% | 101.50 | 99.91 | 97.87 | 101.59 | 99.94 | 97.88 | 101.57 | 100.15 | 97.82 | |
8.0% | 102.59 | 101.03 | 98.99 | 102.68 | 101.06 | 98.99 | 102.66 | 101.27 | 98.93 | |
4% | 5.0% | 99.81 | 98.28 | 96.33 | 99.81 | 98.35 | 96.26 | 99.84 | 98.55 | 96.41 |
6.5% | 100.86 | 99.38 | 97.43 | 100.86 | 99.44 | 97.36 | 100.89 | 99.66 | 97.52 | |
8.0% | 101.92 | 100.47 | 98.53 | 101.92 | 100.54 | 98.45 | 101.95 | 100.77 | 98.62 | |
5% | 5.0% | 99.21 | 97.77 | 95.89 | 99.15 | 97.85 | 95.79 | 99.28 | 97.97 | 96.06 |
6.5% | 100.24 | 98.85 | 96.98 | 100.18 | 98.92 | 96.87 | 100.31 | 99.05 | 97.14 | |
8.0% | 101.26 | 99.92 | 98.06 | 101.21 | 100.00 | 97.95 | 101.34 | 100.13 | 98.23 |
Numerical example of breakeven coupon rate
Explicit Forumla | Brownian Bridge | Crude Monte Carlo | |||||||
---|---|---|---|---|---|---|---|---|---|
3% | 4.42% | 6.62% | 9.36% | 4.30% | 6.58% | 9.36% | 4.33% | 6.31% | 9.44% |
4% | 5.27% | 7.35% | 10.01% | 5.28% | 7.26% | 10.11% | 5.23% | 6.96% | 9.88% |
5% | 6.15% | 8.11% | 10.69% | 6.24% | 8.00% | 10.85% | 6.05% | 7.82% | 10.45% |
Numerical examples of relative error of the time-0 price for autocallable ELS
Brownian Bridge | Crude Monte Carlo | ||||||
---|---|---|---|---|---|---|---|
3% | 5.0% | 0.08% | 0.03% | 0.00% | 0.06% | 0.23% | 0.06% |
6.5% | 0.08% | 0.03% | 0.00% | 0.06% | 0.24% | 0.06% | |
8.0% | 0.08% | 0.03% | 0.00% | 0.06% | 0.24% | 0.06% | |
4% | 5.0% | 0.00% | 0.07% | 0.07% | 0.03% | 0.28% | 0.09% |
6.5% | 0.00% | 0.07% | 0.07% | 0.03% | 0.29% | 0.09% | |
8.0% | 0.00% | 0.07% | 0.07% | 0.03% | 0.29% | 0.09% | |
5% | 5.0% | 0.06% | 0.08% | 0.11% | 0.07% | 0.20% | 0.17% |
6.5% | 0.06% | 0.08% | 0.11% | 0.07% | 0.21% | 0.17% | |
8.0% | 0.06% | 0.08% | 0.11% | 0.07% | 0.21% | 0.17% |
Numerical example of relative error of breakeven coupon rate
Brownian Bridge | Crude Monte Carlo | |||||
---|---|---|---|---|---|---|
3% | 2.64% | 0.61% | 0.05% | 1.98% | 4.76% | 0.80% |
4% | 0.02% | 1.24% | 0.98% | 0.80% | 5.33% | 1.32% |
5% | 1.33% | 1.34% | 1.51% | 1.76% | 3.62% | 2.24% |
Summary description of the distribution of
Description | Computational time | |||||
---|---|---|---|---|---|---|
BB | MC | BB | MC | BB | MC | |
Min. | 0.65% | 0.57% | 0.67% | 0.60% | 44.36 sec. | 811.78 sec. |
Mean | 0.72% | 0.79% | 0.78% | 0.73% | 45.20 sec. | 833.70 sec. |
Max. | 0.81% | 1.05% | 0.88% | 0.86% | 56.95 sec. | 1181.31 sec. |
St.Dv | 0.04% | 0.09% | 0.04% | 0.05% | 1.64 sec. | 49.71 sec. |