# replicate option by dynamic hedging

I've just started working for a company with a decent commodity exposure. They manage this by as they call it dynamically hedging it. Basically when they start the hedging they identify a market price and a budget price x% above the market. This budget price is the maximum they are willing to pay on average for the commodity. To achieve this they hedge a percentage of the portfolio and increase this if prices go up and decrease this when prices go down. I'm pretty convinced by this strategy they replicate buying an option at the strike of the budget price but I'm struggling to mathematically prove this. Any help?

• This concept is known as Portfolio insurance, originally developed by Mark Rubinstein and Hayne Leland. Two of their articles on the subject are: Leland, H.E. and M.Rubinstein (1981) Replicating options with positions in stock and cash; and Leland, H.E. and M.Rubinstein (1976) The evolution of portfolio insurance. I do not think the original articles are freely available online, nor can I recommend any other specific resources on the subject, but I hope this will help you in finding more information. – MGL Jan 29 at 16:21
• I think this gets me going, do you know if I'm correct in that the cost of this dynamic hedging should be the same as a buying an option prized against the volatility that is realized in the hedging process? – bramvs Jan 29 at 16:25
• I have very little knowledge on the subject, but that is the goal of the process. However, as I understand, there are a few differentiating factors. Firstly, of course, portfolio insurance does not have any vega exposure, so contrary to buying options, it offers no protection from rising implied volatility. Secondly, the return of portfolio insurance is path dependent (price changes during the holding period will affect the final value of the portfolio), whereas options' final value at expiry is not. Portfolio insurance also assumes continuous trading, so price jumps can have adverse effects. – MGL Jan 29 at 16:41
• My notion is that portfolio insurance can be thought of as hedging a given amount of total price moves for a given budget (the time-span of the insurance is not pre-determined), whereas options do not have a limit in terms of total volatility they can hedge, but the time-span of the protection is pre-determined. – MGL Jan 29 at 16:45
• @MGL - an option is a finite time-to-expiry instrument, where $\Gamma$ varies (increases), as it draws closer to maturity, and so does $\theta$. For a predefined scheme, as stipulated in the original post, $\Gamma$ would by definition by time-independent, therefore there would be a constant cost $\theta$ per unit time to run this hedging strategy (assuming that positions are unwound, if prices drop) – ZRH Jan 31 at 0:31