A large European bank has recently issued an outperformance equity note on Japanese stocks to European investors ( www.structuredproducts.ing.com ). The payoff of the note is linked to the outperformance 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 BlackScholes 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 
Jan06 
Maturity 
Dec07 
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.
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