Black–Scholes Delta Hedging, Rebalancing Costs, and Portfolio Risk
Summary
The document raises two conceptual questions about the Black–Scholes replicating portfolio: whether changing the hedge ratio requires unlimited funding, and why risk is described through changes in portfolio value rather than cash flows into it. It introduces the setup of shorting a call and holding shares, with the share position chosen from the option’s sensitivity to the underlying price. Because that sensitivity changes over time, the hedge must be rebalanced to maintain local risk neutrality.
The text poses these questions but does not include answers or evidence resolving them. It is therefore useful as a statement of the distinction between delta hedging and a complete account of hedging costs: a locally riskless portfolio in the idealized model does not by itself explain the practical financing, transaction costs, or constraints involved. The discussion is introductory and leaves the reader to investigate the assumptions behind continuous rebalancing and the risk-free replication argument.
Key ideas
- A delta hedge offsets small changes in an option’s value using a position in the underlying asset.
- The hedge ratio can change as the underlying price and time to expiry change.
- The document asks whether rebalancing requires unlimited funding but does not answer the question.
- Portfolio value risk and the cash flows needed to maintain a hedge are distinct issues.
Tags
Full text
# Beginner question about Option pricing, Risk and Rational portfolios
# Beginner question about Option pricing, Risk and Rational portfolios
I'm a physics student currently reading "Econophysics and Physical Economics by Peter Richmond, J¨urgen Mimkes, and Stefan Hutzler" for the first time and this is my first touch with the world of economics so I want to apologize in advance becasue I feel my question is so dumb that I don't even know what am I confused about.
In page 80-81 the book starts laying down the Black-Scholes theory by introducing the "rational portfolio". The Idea is that you sell a call option at price $C(s,T)$ (where $s(t)$ is the price of the asset and $T$ is the expiry date) and immediately buy $n$ shares of the asset. So your portfolio now has the value:
$$\phi=-C(s,T)+ns(t)$$
Now it's said that the portfolio would be risk-free if its value doesnt change with time:
$$\Delta\phi =0=-\Delta C+n\Delta s \\ \Rightarrow n=\frac{\partial C}{\partial s}$$
And this means that, since in the general case $C$ would be a nonlinear function of $s$, then $\frac{\partial C}{\partial s}$ would depend on the time $t$, so the amount of assets in our portfolio would have to change in order to keep its value constant during the whole time. Here I have two questions which are somewhat related and need be answered together and again I'm sorry if they sound stupid or poorly formulated but I'm not sure how to ask them more corectly:
$1.$ Since $n$ has to change over time this means we need to constantly be selling and buying the assets, right? Then isn't there in theory possibility that we would need to be spending unlimited amount of money in order to satisfy $\Delta \phi=0$? If not, why?
$2.$ Why is the risk solely quantified by the changing of the portfolio but not also by the money going in the porfolio?
(I feel like I'm missing some super trivial point of view that would make me see this problem way simpler than it seems to me now)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.