Out-performance Equity Note on Japanese Stocks
Team Latte
August 3, 2006
A large European bank has recently issued an out-performance equity note on Japanese stocks to European investors ( www.structuredproducts.ing.com ). The payoff of the note is linked to the out-performance of Nikkei index with respect to Hang Seng index ( Hong Kong stocks). There is a capital structure bond attached to the note and the payoff is given by:
The pricing of this note is pretty straightforward if one used Monte Carlo simulation. However, if we want to derive a closed form solution then we need to decompose this note into the embedded spread option and then price the spread option first.
In a closed form Black-Scholes framework the above payoff can be written as:
Where the parameters are:
A slice of the calculation from an Excel spreadsheet is shown below (the values of all the parameters are assumptions and shown here only for illustration. The input values may not be real or accurate):
| Model Input |
| Start Date |
Jan-06 |
| Maturity |
Dec-07 |
| NK Strike |
15000 |
| Div NK |
0.50% |
| Div HK |
3.15% |
| Risk free Target (EUR) |
2.00% |
| Risk free NK |
0.50% |
| Risk free HS |
3% |
| NK Vol |
19.40% |
| HS Vol |
20% |
| Correl |
0.5 |
| Calculations |
|
| Valuation Date |
8/3/2006 |
| Spot NK |
16000 |
| Spot HS |
15500 |
| Time to Maturity |
1.411 |
| q(HS) |
2.15% |
| q(NK) |
2.000% |
| vol |
0.197069 |
| lambda (NK) |
1.025641 |
| lambda (HS) |
1.033333 |
| d1 |
0.094173 |
| d2 |
-0.13993 |
| N(d1) |
0.537514 |
| N(d2) |
0.444359 |
| N'(d1) |
0.397177 |
| Model Output |
|
| Price of the bond |
82.63% |
| Out performance Call |
15.386% |
| Value of the Note |
98.02% |
| Greeks |
|
| Partial Vega (Nikkei) of the Note |
0.3815 |
| Partial Vega (Hang Seng) of the Note |
0.1856 |
| Correlation Sensitivity |
-0.1575 |
Thus using the closed form method we get the price of the note as 98.02%. Of course our favourite method remains Monte Carlo simulation using a Principal Components Analysis (PCA) method.
In all our training sessions and project work we stress to our trainees and clients that Monte Carlo simulation and other numerical techniques, such as finite difference method and numerical integration is far more appropriate and robust than a closed form solution, especially for notes with more than two degrees of complexity. Using a PCA Monte Carlo method and running only 10,000 simulation runs we get a value of the note as 97.25%. This note can also be easily priced using Numerical integration technique using a Gaussian quadrature method which will gives the most accurate value. We have also priced this note using numerical integration.
Any comments and queries can be sent through our web-based form.
Note: The above closed form pricing of the note is neither proprietary to Risk Latte nor is it used by the Company in any form of commercial training. This appears here for illustration only and the structure and its workings are taken from third party sources and/or public domain. Risk Latte shall not be liable for any factual errors or omissions and shall not be liable for any losses, direct or indirect, arising out of the use of the above.
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