**Decomposition of Structured Product through Pay-Off Diagrams**

Neeraj Kumar

Kotak Mahindra Bank, Mumbai, India

July 24, 2007

The following example explains how a derivative structure with complex payoff can be broken down into plain vanilla put, call, digital option and bond. The procedure can be applied to any structure to arrive at its building blocks.

Let's look at a fairly complex Dollar-Yen FX payoff that is attached to a zero coupon bond. We'll ignore the bond payoff and only analyze the embedded derivative payoff. The structure was originally done in the late eighties and was a five year structure.

However, let's analyze this as a general asset structure and call the final maturity value of the asset as **S **.

Derivative Payoff

- If S < 84.5 then payoff = 0
- If S > = 84.5 and S < = 169 then payoff = (169/S) -1
- If S > 169 then payoff = 0

Decomposition

- Plot the payoff :

The plot of the above payoff will look like - Break the payoff into parts where the slope of the payoff is 0 and where it has some value. Thus we can break the above payoff into three parts part 1 and 3 where the slope of the payoff is 0 and part 2 where the slope of the payoff is negative.
- First analyze the payoff in part 2 and replicate it through combination of call and put options.

- The payoff in part 2 is negative sloping and the minimum is 0. We know that such payoff can be replicated by long put.
- But a long put will give payoff even when S < 84.5, thus we have to negate the payoff on the option for all S < 84.5. This can be done by shorting a put at 84.5.
- The combined pay off this is shown below. Thus to create a payoff in part 2 of the structure we are going long one put at strike of 169 and short one put at strike of 84.5. This is also called a 'put bear spread'.
- The payoff of the 'put bear spread' shows that its payoff is 0 for S > 169. Thus this spread also replicates the payoff of part 3.
- However the 'put bear spread' generates a constant pay of 84.5 (169-84.5)for all S < 84.5.
- We know that the structure pays 0 for S < 84.5 but the ¡¥put bear spread' pays 84.5 for S < 0. Thus now we need to negate this payoff. This payoff can be negated by going short 84.5 digital puts at strike = 84.5.
- Thus now we have three options -

- One Long put at strike = 169
- One Short put at strike = 84.5
- 84.5 short digital put at strike = 84.5.

The combined payoff of these three options is shown below. The payoff is shown by the solid blue line. This payoff exactly replicates the payoff of the structure. Thus the price of the structure should be equal to the price of these options.

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