Pricing Principal protected JPY Mixed Note
July 22, 2006
Recently, a financial institution issued a 5 year coupon Japanese Yen chooser note to a select group of private investors and hedge funds which had rather attractive terms. We will see how such a note can be priced. The note guaranteed the principal at maturity plus it allowed the investors to participate in the movement of the Nikkei225 index on both sides. This means, whether the Nikkei225 goes up or down the investors will be able to benefit from the absolute movement of the index.
The note, which had a principal value of Yen 1,000 paid no coupons and upon maturity (at the end of five years) it paid a redemption amount of:
What should be the fair value of this note today? We will use two methods (a) closed form solution and (b) Monte Carlo simulation method.
Closed Form Pricing of the Note:
The payoff of the note can be decomposed as follows:
Therefore, the note can be decomposed into a zero coupon bond plus a certain factor times the sum of a 5 year call option on Nikkei and a 5 year put option on Nikkei. If the strike on the Nikkei is 15,000 (say, the value today), the risk free rate is assumed to be 2.5% and a volatility of 15% is assumed (say, the forecast 5 year realized volatility using GARCH) then the following values are obtained using Black-Scholes valuation model:
Call = 2,889 points, Put = 1,127 points and Zero = Yen 882.5
Thus the value of the 5 year Chooser note comes out to be 98.96%*.
Monte Carlo Simulation of the Note
We can use the differential form of the stochastic asset price equation for Nikkei225 and simulate the one year path using the random normal distribution:
Else, we could also use the integral form of the stochastic asset price equation for simulating the 5 year path of Nikkei225:
Then using the participation factor of 40% and the Nikkei strike of 15,000 we calculate the payoff (redemption value) at the end of 5 years and then discount the terminal value back to the present and using the risk free rate of 2.5%. This will be a single simulation run. We do this over at least 20,000 simulation runs to get the value of the note. Our results are summarized here:
Runs = 20,000 Riskfree = 2.5%, Volatility = 15%, Note Value = 99.4% .
Thus using Monte Carlo simulation the value of the 5 year JPY Chooser note comes out to be 99.0%. Actually, one needs to run the simulation for at least 50,000 runs to get an accurate value.
*We are extremely thankful to Paul Hsueh of NCCU, Taiwan who pointed out an embarrassing typo in the previous version of this article. The values are restated again .
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