wilmott.com

Apologies for not including formulae; I am waiting for answers on a separate post on how to include formulae.

Question:

I have a payoff that seems to me to be very vanilla in that it is European and that it is non path dependent. I have tried my best (happy to share workings) to start with Expectations and to work through to Call Option price using the FTAP. I am almost there, but there is a slight twist.

The payoff is:

MIN ( [MAX(S(T) - S(0),0] - N, 0 ]

Where:

• S(T) is the stock price at maturity
• S(0) stock price today
• N some notional fixed amount

So the only stochastic part is S(T) and assume constant/deterministic interest rates.

The inner part (the “MAX” part) on its own is just a vanilla Call, but I don’t have the technical skill to evaluate the outer “MIN” under the risk-neutral expectation. I know that Jenson’s inequality tells me that you can’t simply “take the Expectation into the min/max operands”, but that is as far as I got.

Thank you in advance.

\[ \min ( \max(S(T)-S(0), 0) - N , 0) \]

Thank you, I am however looking for two things here, (a) the derivation and (b) the closed form / analytical formulae for the price/value of the derivative. So starting with expectations of the above payoff, reducing this to integrals and then to nice and simple terms like S, N(d1), etc.

I can be wrong, but my assumption is that the payoff looks like a derivative (pardon the pun) form of a BS Call/Put payoff, from whence the desire to have some closed form formula.

Thanks in advance

Plot the payoff!

Oh wow that was pretty cool!

So plotting it, the left part of the payoff graph is a long call connected to a short put on the right (as the underlying increases)

Thanks so much, that was fun!

Something like that!

After a bit of digging I see my payoff mirrors that of a “bull call spread” or a “put call spread”, to be replicated by a long call + short call or long put + short put, respectively.

Case closed I suppose, no fun integration and Green formulae unfortunately!

Thanks Paul! I used the Additional Notes from CQF and relearnt a bunch. That course really was amazing.