Replicating a Capped FX Bond Payoff with Two Options
Summary
The document derives a static option representation for an exchange-rate-linked payoff that pays nothing below a lower threshold, rises over an intermediate range, and is capped at a fixed amount above an upper threshold. By rewriting the payoff in terms of the reciprocal exchange rate, it expresses the structure as a scaled difference between two put payoffs with distinct strikes. This provides a direct way to identify the options from the payoff’s piecewise shape.
The answer clarifies that a zero-coupon bond paying the cap amount is not needed for this replication, since the original payoff can fall to zero and therefore has no principal protection. The discussion gives an algebraic decomposition rather than pricing or empirical evidence, and it assumes the stated maturity payoff and exchange-rate convention. It also points toward static hedging as a general method, while leaving contract details such as market conventions and transaction costs outside the analysis.
Key ideas
- The payoff is zero below the lower exchange-rate threshold, increases between thresholds, and reaches a fixed cap above the upper threshold.
- Expressing the exchange rate as its reciprocal makes the payoff easier to decompose into option payoffs.
- The structure can be replicated with a short put at one strike and a larger long put at another strike, scaled by the stated notional factors.
- A zero-coupon bond is unnecessary because the payoff does not guarantee repayment of principal.
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Full text
# Show that a derivative is a combination of two options and a regular bond
# Show that a derivative is a combination of two options and a regular bond
Consider the bond decribed by the following formula:
$$ 1000-\text{max}\left[0,1000 \;\text{min}\left(\frac{169}{S_T} -1,1\right)\right] $$
where $S_T$ is a yen-USD exchange rate at maturity;
So if $S_T$ is larger than $169$ yen/per USD the holder receives $1000$ USD while if $S_T<84.5$ then the holder receives nothing.
(This was ICON (index currency option notes) issued in 1995, according to the book, if anyone is interested).
I'm asked to show that this is a combination of two options and a bond. Now looking at the profit-maturity rate graph it sort of gives me the idea, as there are two "flat" parts of the graph (which probably is a result of combining two options) and the middle part connecting them comes from regular bond yield.
But I'm not sure how I would magically cook up the formula (for the derivatives involved. Or is there other way than that?) from just intuition alone. Also, I'm not entirely sure what's meant by "regular bond here". I'm quite new to all these financial concepts, so it'd be great if someone could give a significant hints towards the solution.
## Answer by Daneel Olivaw (score 2, accepted)
https://quant.stackexchange.com/a/36893
Your payoff $\pi_T$ is the following:
$$ \pi_T = \left\{\begin{array} \\ 0 & & \text{if } S_T<84.5 \\ 2,000 - 169,000/S_T & & \text{if } 84.5 \leq S_T \leq 169 \\ 1,000 & & \text{if } S_T > 169 \end{array}\right.$$
Letting $N=1000$, $k_1=84.5$ and $k_2=169$, it can be written:
$$ \begin{align} \pi_T&=1_{\{k_1 \leq S_T \leq k_2\}}\left(2N-\frac{k_2N}{S_T}\right)+1_{\{k_2<S_T\}}N\\[6pt] &=k_2N\left(1_{\{k_1 \leq S_T \}}\left(\frac{1}{k_1}-\frac{1}{S_T}\right)+1_{\{k_2 \leq S_T \}}\left(\frac{1}{S_T}-\frac{1}{k_2}\right)\right) \\[6pt] &=k_2N\left(1_{\{\frac{1}{k_1} \geq S_T^{-1} \}}\left(\frac{1}{k_1}-S_T^{-1}\right)-1_{\{\frac{1}{k_2} \geq S_T^{-1} \}}\left(\frac{1}{k_2}-S_T^{-1}\right)\right) \\[6pt] &=k_2N\left(\max\left(0,\frac{1}{k_1}-S_T^{-1}\right)-\max\left(0,\frac{1}{k_2}-S_T^{-1}\right)\right) \end{align}$$
where $S_T^{-1}$ is the USDJPY FX rate. Therefore the payoff can be hedged by selling a put option on the USDJPY rate with strike $1/k_1$ and buying a put option on USDJPY with strike $1/k_2$ in quantities $k_2N$.
If you use a zero-coupon bond paying $1,000$ at maturity $T$ in your hedging strategy (as your questions specifies) then you still need two options similar to those above. But you do not really need it as there is no capital protection, i.e. under some scenarios $\pi_T=0$.
For a general methodology to address this type of static hedging questions, refer to the beginning of my answer to question "Finding arbitrage opportunity".
Note: an earlier comment I made on a digital call option was wrong.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.