Options with a stochastic strike
Do options where the strike itself is a stochastic process exist? If they do - what are the motivations for such a product and where is it used ?
Example: Call-Option with stochastic strike:
$$ (S_T(\omega) - K_T(\omega))^+ $$ where $ K_t $ is a stochastic process. $ K_T $ could for example be of the form $ K_T=f(X_T) $ where $ f $ is a measurable function.
Asian options: strike is average of underlying over tenor. Underlying is stochastic.
Options with kock-ins/knock-outs: Underlying is stochastic and may cross the kock threshold as it evolves. Option value depends on this cross or lack thereof (boolean).
Options on Options, too.
Motivations for Asian options you can google. Kock-ins and knock-outs lower the cost of an option so that a buyer who needs them for hedging purposes can more cheaply acquire downside protection. Generally applies to Currency/Interest Rate hedging.
Options on options are more of a theoretical construct, although I’m sure they exist in practice. Generally useful for valuing complex projects with optionality when standard DCF approaches would not fully capture the stochasticity of the project’s value. Dixit has a good chapter on this, I believe.
I believe your example describes the payoff of a simple spread option. Some may argue that in reality this spread option has zero strike:
$$ (S_T(\omega) - K_T(\omega)-0)^+ $$ Which leads us to the question: What exactly strike is anyway? Is it uniquely identifiable term in each payoff function? No It isn’t.