# Pricing an "equity protection" derivative: a practical example

This is the derivative security (its underlying index is the S&P 500):

• time to expiry $=4.8$Y;
• payoff calculation (0): on the expiry date, give a look at S&P 500 and let its price to be $S_{T}$. Furthermore, let S&P 500 price on issue date was $S_{0}=2,078.36$;
• payoff calculation (scenario 1): if $S_{T}<2,078.36$, you will get a nominal amount equal to \$$2,000; • payoff calculation (scenario 2): if 2,078.36<S_{T}<3,034.4056, you will get a nominal amount equal to \$$2,000(\frac{S_{T}}{S_{0}})$, that is, \$$2,000 times S&P 500 performance from t=0 to t=T; • payoff calculation (scenario 3): if S_{T}>3,034.4056, you will get a nominal amount equal to \$$2,920$.

Practical example of payoff: let you paid it \$$2,000 on t=0 and today S_{t}=2,372.6, you would have following payoffs in each scenario... • (scenario 1) \$$[2,000 - 2,000=0]$; • (scenario 2) \$$[2,000(\frac{2,372.6}{2,078.36})-2,000=283.15] if S_{t}=S_{T}; • (scenario 3) \$$[2,920-2,000=920]$.

What kind of security is this? Very straightforward to describe: a zero coupon bond plus a vertical spread, the latter being made up by a long Call struck @ $2,078.36$ and a short Call struck @ $3,034.4056$ that acts like a "cap".

Forgive Italian language and give a look at following image to clarify:

So I thought this was easy to price: