Replicating an Equity Protection Note with Bonds and Call Options

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


# 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:

- discount \$$2,000$ by a swap curve and issuer's Z-spread;

- transform underlying price $S_{t}$ to this security's "price" $S^{*}_{t}$: if $t=$ today you would have \$$2,000(\frac{2,372.6}{2,078.36})=2,283.15$;

- price a $4.8$Y Call struck @ $2,000$ using $S^{*}_{t}$;

- price a $4.8$Y Call struck @ $2,920$ using $S^{*}_{t}$;

- sum 1 to 3 and subtract 4.

Some kind of bicubic interpolation yields an S&P 500 implied volatility roughly equal to $22\%$ for 3 and $18\%$ for 4, hence Black & Scholes Call options NPVs should be \$$563$ and \$$158$; moreover, zero coupon clean price should be \$$1,850$ @ Z$+142$ bps (issuer's $5$Y CDS spread).

Instrument NPV$=$\$$[1,850+563-158=2,255]$.

Questions

- Do you think my reasoning is correct? If not, why?

- If it is correct, how would you explain that its market maker sells it @ \$$2,016$? I know who the market maker is, I would find extremely difficult that they're mispricing it.

Note: Call options have been priced according to a generalized Black & Scholes process assuming $0\%$ risk free rate and dividend yield.

## Answer by Quantuple (score 4, accepted)

https://quant.stackexchange.com/a/33058

The payoff you mention writes: $$ V_T = N \left( 1 + \frac{1}{S_0}(S_T - K_1)^+ - \frac{1}{S_0}(S_T - K_2)^+ \right) $$ with $K_1=S_0=2078.36 < K_2 = 3034.4056$ and $N=2000$

Thus taking a risk-neutral discounted expectation of the payoff yields the price at time $t$: $$ V_t = N B(t,T) + \frac{N}{S_0} C(K_1, T-t) - \frac{N}{S_0}C(K_2, T-t) $$ hence

- $N$ times the price of a zero coupon bond with time to expiry $\tau=T-t=4.8Y$

- plus, $N/S_0$ times the price of a call on the S&P struck at $K_1=S_0=2078.36$, $\tau=4.8Y$

- minus, $N/S_0$ times the price of a call on the S&P struck at $K_2=3034.4056$, $\tau=4.8Y$