Show that a derivative is a combination of two options and a regular bond

Question

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 1

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.