How to devise strategies to match target risk profiles
In what follows, I illustrate three different ways of generating a targeted risk profile (as represented by the payoff plots. Method (1) is based on eye-balling and in some simple cases is simply the definitional. Method (2) uses call options only. Method (3) uses put options only.
In principle, you can always replace a call with a put via put-call parity. Hence, there are many possible ways of generating the targeted payoff. For simple, commonly used payoff structures such as straddles, strangles, butterfly spreads, and risk reversals, it is useful to just memorize its mostly commonly used composition (Method (1)). However, for a more complicated payoff structure and for a structure that you are not familiar with (or not sure about), it is safer to follow the mechanical strategy of using calls only (Method (2)) or using puts only (Method (3)), as described in the lecture notes. Furthermore, when time permits, it is always to double check: Given the composition you have devised, derive its payoff structure and see if it matches the original target.
Straddle:
(1) Long one call and one put, both at K=100.
(2) Long $100 par zero-coupon bond, short a call (forward) at K=0, long 2 call at
K=100.
(3) No need for zero-coupon bond, long 1 forward at K=100, long 2 puts at K=100.
Strangle:
(1) Long call at 110, long put at 90
(2) Long $90 par zero-coupon bond, short a call (forward) at K=0, long 1 call at 90,
long 1 call at 110.
(3) No bond, long 1 forward at 110, long 1 put at 110, long 1 put at 90.
Butterfly spread:
(1) Long a call at 90, long a put at 110, short a call at 100, short a put at 100, short
$10 bond.
(2) No bond, long 1 call at 90, short 2 calls at 100, and long 1 call at 110.
(3) No bond, no forward, long 1 put at 110, short 2 puts at 100, and long 1 put at 90.
(4) You can also long a strangle (a put at 90 and a call at 110), short a straddle at 100,
and long $10 par zero-coupon bond to generate the payoff.
Risk reversal:
(1) Long a call at 110, short a put at 90.
(2) Short $90 par of zero-coupon bond, long 1 call/forward at K=0, short a call at 90,
and long a call at 110.
(3) No bond, long 1 forward at 110, long a put at 110, and short 1 put at 90.
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