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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.

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