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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.

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I've seen a Libor interest rate cap where the cap is a function of the change in RPI (UK inflation index) - stochastic strike. It definitely exists but don't ask me what the motivation is... – crunch Mar 28 '14 at 21:44
@crunch I will look into it - do you have any sources ? – Probilitator Mar 29 '14 at 9:01
Look up quanto options, that's where I'd start, for dealing with options with two sources of uncertainty and their correlation. – ghost.comet Mar 29 '14 at 19:17
@Probilitator Unfortunately nothing I can share, sorry. I've held the confirmation in my hand so I know it exists. Come to think of it, it was a euribor vs. fixed swap, however the float leg was capped at 2 * [CPI(n)/CPI(n-1)]. – crunch Mar 30 '14 at 22:13

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.

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I was not aware of the floating strike (or floating rate) Asian call option :) - but I will argue that knock-ins don't have a stochastic strike ;) - the strike itself can't change – Probilitator Mar 28 '14 at 19:49
I think one reason for having Asian Options, and other similar instruments, is that players with large positions in vanilla options have great incentive to "manipulate" share prices in order to avoid great losses or being able to exercise one's options, even if it means incurring losses from selling off/buying lots of shares in the underlying. With Asian options, however, this becomes much more difficult. – Good Guy Mike Mar 28 '14 at 20:44
@GoodGuyMike intersting perspective :) What is your opinion on Knock-Ins? Would you consider them as having a stochastic strike? I don't think that it is intended by design but mathematically $1_{\max S_t \geq B}(S_T-K)^+=(1_{\max S_t \geq B}S_T-1_{\max S_t \geq B}K)^+$ – Probilitator Mar 29 '14 at 6:18
Well, I wouldn't say that the strike is stochastic, because that is not entirely true. I mean the strike is always K, but the whole claim is either 0 or some value X depending on what happens up until maturity. On the other hand, one could see the strike as either K or $\pm \infty$ (depending on if it is a call or put). – Good Guy Mike Mar 29 '14 at 9:49

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.

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thank you - you are correct. When one starts with plain vanillas and spends most of one's time pricing derivatives with fixed strikes one easily forgets that the world is not so simple – Probilitator Apr 1 '14 at 8:13

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