4
$\begingroup$

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:

enter image description here

So I thought this was easy to price:

  1. discount \$$2,000$ by a swap curve and issuer's Z-spread;
  2. 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$;
  3. price a $4.8$Y Call struck @ $2,000$ using $S^{*}_{t}$;
  4. price a $4.8$Y Call struck @ $2,920$ using $S^{*}_{t}$;
  5. 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

  1. Do you think my reasoning is correct? If not, why?
  2. 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.

$\endgroup$
4
  • $\begingroup$ Are you sure it's an offered price and not a bid price? I did not look deeply into this but I get a mid price similar to yours $\endgroup$
    – Quantuple
    Mar 16, 2017 at 11:47
  • $\begingroup$ Maybe they apply a huge funding spread. $\endgroup$
    – Quantuple
    Mar 16, 2017 at 12:00
  • $\begingroup$ About 350 bps to get that bid... it belongs to Intesa Sanpaolo and I am assuming it's a senior note (maybe I am wrong there?), it cannot have such a huge credit spread. Even 5Y CDS on subordinated debt are @ 286 bps, there's nothing @ 350 bps out there. Ok, I am using EUR-denominated CDS, but quanto adjustments can't justify that. $\endgroup$
    – Lisa Ann
    Mar 16, 2017 at 12:06
  • $\begingroup$ Maybe I've found the issue, and of course it's my bad: I've discounted the ZC bond by using EUR swap curve instead of USD's one, 'cause I'm so used to discount by using it that I forgot to switch... now I get the same market maker's mid price, could you please check? $\endgroup$
    – Lisa Ann
    Mar 16, 2017 at 13:55

1 Answer 1

4
$\begingroup$

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 $K_2=3034.4056$, $\tau=4.8Y$
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.