Interpreting the Delta-Hedged Black–Scholes Portfolio
Summary
The document clarifies the portfolio convention used in a standard Black–Scholes derivation. The portfolio is marked at its current value, rather than viewed as the trader’s initial costs. In the example, it is formed by shorting one option and holding a number of underlying shares equal to the option’s delta.
When the option value rises, the short option position loses value; the long share position can offset the change associated with the underlying’s movement. This explains why the stated positions belong to the hedging portfolio, rather than describing the option buyer’s standalone holdings. The excerpt provides a conceptual clarification, not a full derivation of the pricing equation. The hedge is intended to remove the portfolio’s instantaneous exposure to small underlying price changes, and the document does not discuss how often it must be rebalanced or the assumptions and risks of the model.
Key ideas
- The portfolio value is the combined mark-to-market value of its positions.
- The Black–Scholes hedge described shorts one option and holds the option delta in shares.
- A rise in option value reduces the value of the short option position.
- The share position offsets the option’s local sensitivity to the underlying price.
Tags
Full text
# Black-Scholes model portfolio position # Black-Scholes model portfolio position 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): 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 ## Answer by Patrick (score 1, accepted) https://quant.stackexchange.com/a/78445 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.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.