Using Put-Call Parity to Bound an Index-Linked CD Return Multiplier
Summary
The document defines an index-linked certificate of deposit (CD) return multiplier as the CD’s return per invested dollar relative to the underlying index’s return per invested dollar. It asks whether the multiplier is below one for an at-the-money index call, given a riskless rate, the initial index level, and the call price.
It introduces put-call parity, equating the cost of a call plus a discounted strike payment with the cost of the index plus a put. The answer asserts that the riskless-rate factor exceeds the put price and uses this to claim the multiplier is below one. However, it does not show the derivation, clarify the units or normalization behind that comparison, or establish the assertion’s conditions. The parity equation alone is therefore not enough to verify the stated inequality; the argument is incomplete and should be checked against consistent pricing conventions.
Key ideas
- The return multiplier compares the CD’s return per dollar with the underlying index’s return per dollar.
- Put-call parity relates a call and discounted strike payment to the index and a put.
- The answer claims the multiplier is below one based on a comparison involving the riskless rate and put price.
- The document omits a derivation and does not explain the comparison’s units or assumptions.
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Full text
# How to prove with put-call parity: return multiplier ratio of ILCD (index linked certificate deposit ) < 1
# How to prove with put-call parity: return multiplier ratio of ILCD (index linked certificate deposit ) < 1
Index-linked CDs pay interests based on a specific index, and a guaranteed payment of all principal.
The return multiplier ratio (multiplier) is defined as the rate of return of the derivative versus the market rate of return of the underlying asset.
Return multiplier ratio is defined as: $\frac{r_f}{1+r_f}$$\frac{S_0}{C}$ It is a ratio: each dollar invested in the index-linked CD, gain riskless interest --- $\frac{r_f}{1+r_f}$$\frac{S-S_0}{C}$, divided by, the profit generated by each dollar invested in the index, which is $\frac{S-S_0}{S_0}$.
$r_f$ is the riskless interest rate, the current index is $S_0$, the index at maturity is S, S > $S_0$, the cost of at-the-money index call is C.
Question is: show $\frac{r_f}{1+r_f}$*$\frac{S_0}{C}$ <1 with put‐call parity
Put‐Call Parity says: payoff of the call‐plus bond portfolio is the same as the payoff of protective put position.
And by no arbitrage principle, the call‐plus‐bond portfolio (on left) must cost the same as the stock‐plus‐put portfolio (on right): C+ PV(K) = $S_0$ + P.
PV() is the present value, P is the put price, K is the exercise price.
## Answer by jimy (score -1)
https://quant.stackexchange.com/a/70674
Since $\frac{r_f}{1+r_f}$ is more than price of put, then $\frac{r_f}{1+r_f}$*$\frac{S-S_0}{C}$ is less than $\frac{S-S_0}{S_0}$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.