0
$\begingroup$

Question:

I am a physicist currently learning about the Black-Scholes model in a statistical mechanics course. I have been teaching myself financial terminology and was reading the "Derivation of the Black–Scholes PDE" of the wikipedia page attached.

The wikipedia article makes the following statement (also Hull's "Options, Futures, and Other Derivatives.", 8th edition, pp 309-310):

enter image description here

My understanding is as follows. The portfolio Π appears to be the buyer's portfolio for a call option. They have payed a "fee"/"premium" V for a call option. At the same time they have sold dV/dS shares of the underlying, S, of the call option. Therefore I would conclude that the buyer is in a "long position" on the call option (or rather the underlying?) and a "short position" on the underlying stock S, with such a delta to make their portfolio deterministic in time evolution.

My confusion lies in the wikipedia article's statement that the current portfolio is "...consisting of being short one option and long dV/dS shares ...". To me this seems to be the counter-party /seller's position in this trade?

I think my misunderstanding originates from my confusion as to what the portfolio Π is representing. Any clarification would be appreciated, thanks!

https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation#Derivation_of_the_Black%E2%80%93Scholes_PDE

$\endgroup$

1 Answer 1

1
$\begingroup$

Yes I think it was my misunderstanding of what the portfolio means. The portfolio is the value of my investments (rather than their costs which I mistakenly thought previously).

Certainly, if the value V of the option increases by dV then my portfolio P = -V + aS has fallen in value due to this increase in V, so I am in a short position V. At the same time, the value V of the option changed as a function of the underlying S, so I would expect my long position on S to cover this loss in my short position.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.