Pricing a Bull and Bear Note
(UBS Note linked to DJ Eurostoxx 50)
Dec 19, 2005
UBS has recently offered a 5 year capital guaranteed product in the German market which is a bull/bear note on DJ Eurostoxx 50. The payoff of the note is linked to the return on the index DJ Eurostoxx 50 and it offers a return to the investor both in the event of an upside and a downside in the index.
The payoff of the notes (certificate) is:
If ...the final index level is at or above the initial index level Then the product offers at maturity a minimum capital guaranteed return of 100% plus 70% of the rise of the index over the investment period;
Else If... the final index level is less than the initial index level but not less than 65% of the if the initial index level Then the product offers at maturity a minimum capital return of 100% plus 70% participation in the absolute value of the fall of the index over the investment period;
Otherwise, If ...the final index level is less than 65% of the initial index level then at maturity the product offers a capital return of 100%.
Method I : Simple Method of Pricing
Monte Carlo Simulation of the payoff:
Generate a set of 60 random normal numbers using a random number generator such as a Sobol sequence or Box-Muller algorithm (5 years times 12 months given 60 data points). Then generate the asset price path over these points using the integral stochastic equation such as this one:
Set the mean or the drift term as difference between the risk free rate and the constant dividend yield for DJ Eurostoxx 50, i.e. before simulating the asset price path. Use an estimate of local volatility (from the volatility surface) as sigma.
Then simply convert the IF...Then...Else analysis above into a payoff structure (and discount the payoff backwards) and run the simulation over 10,000 or better, 20,000 runs to get the price of the note. The concept is simple: if a notional payoff of the note is $100 upon maturity plus an upside define by the payoff function then what is the fair discounted value of the note today, given that the asset (DJ Eurostoxx 50) follows a Brownian motion with a mean mu and a volatility of sigma .
Method II : More Complex, but Elegant Method of Pricing
Decomposing the Derivative Instrument into Vanilla & Exotic Options
This method is given in the November 2005 issue of Structured Products . This magazine has given a pricing of this product.
The buyer of the bull/bear note can get an equivalent portfolio by (decomposing the derivative):
- Going long on a zero coupon bond that provides capital guarantee;
- Going long on a call option with strike of 100% and a participation of 70%;
- Going long on a put option with strike of 100% and participation of 70%;
- Going short on a put option with a strike of 65% and a participation of 70%;
- Going short on a binary option with a payoff of 24.5% if the final index level is less than 65% of the initial level;
This method is the method of the traders whereby a trader (bank) who sells such a structure needs to hedge the note. The note cannot be hedged as a single product, but rather as a combination of calls and puts (and binaries).
The above vanilla calls and puts can be prices in a simple Balck-Scholes-Merton framework whereby the drift term has to be risk free rate less the dividend yield and strike prices are as shown in the equation above. The binary option can be priced in a standard Black-Scholes framework as well and in a trading situation can be replicated as a put spread.
We have used the first method (Monte Carlo simulation of the payoff) to price this note by assuming a constant volatility of 18.7% and have used the same values used by the magazine Structured Products (November 2005 issue) After a 10,000 simulation runs we get a value of 99.15% for the note which is very close to one of the values mentioned in the magazine.
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