# Duality between constant rebalanced portfolio (CRP) and corresponding derivative

One of the greatest achievements of modern option pricing theory is finding corresponding dynamical trading strategies in linear instruments with which you can replicate and by that price derivative instruments; of course the most prominent example being Black Scholes and dynamic hedging.

I wonder concerning another trading strategy, the constant rebalanced portfolio (CRP), what a corresponding derivative would look like:

• How would the payoff diagram and resulting return distribution look like?
• How to price it (in a BS world for a start)?
• How to statically hedge it with plain vanilla options?

I think this will be especially interesting since it is a well known fact that (under appropriate assumptions) a CRP uses the volatility component to shift the drift upwards (unlike most other trading strategies which leave the drift unchanged but only form the return distribution differently). Therefore volatility is not only an element of risk but of chance too. I want to fully understand this phenomenon, also known as volatility pumping, by using the machinery of option theory.

Do you have any literature concerning this topic (I found none) or any ideas and hints where to start?

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Fascinating question! I'd be very interested to know the answer as well, and will hopefully try to find an answer myself (if no one else does first) when I have some time. Do you have a source for the "well known fact" regarding volatility pumping? Also, I doubt you could achieve this with a static options position, since options expire and a CRP is meant to be held indefinitely. – Tal Fishman Sep 5 '11 at 14:53
– vonjd Sep 5 '11 at 16:12
– vonjd Sep 5 '11 at 18:27
– vonjd Sep 5 '11 at 18:40

## 2 Answers

Actually there are more than just ideas and hints concerning this topic. There is an intuitive model and solution to your question already using machinery of option theory. But don't worry, it's not a surprise that you didn't find any useful literature in your search because the proposed solution actually comes from a very different topic.

In addition to the references recommendation, here are some analogies and highlights that hopefully will bridge the proposed solution and your interests:

• Note that in volatility pumping, which is the easiest version of constant rebalanced portfolio (CRP), we always invest less than 100% capitals in risky assets. What if now, I propose to invest more than 100% capitals in risky assets? Will you buy it? :)

• For whatever reasons, many investors buy it! And that's why we have a fast growing leveraged exchange-traded funds (Leveraged ETFs) market. Yeah, >100% capital investment is 'leveraged'. As the business grow, there thus arise greater demands for better understanding and modeling of these leveraged services/products.

• There you go! The answer you are looking for, for a 'de-leveraged' trading strategy, actually lies in the research of modeling leveraged assets. All you need to do is to plug in your favored leverage (x1/2 for volatility pumping) and double think about the new formula and consequences (mostly reversed). Here is a list of the recommended references:

[1] Leveraged ETFs: All You Wanted to Know but Were Afraid to Ask

[2] A Dynamic Model for Leveraged Funds

[3] The Dynamics of Leveraged and Inverse Exchange-Traded Funds

[4] Path-Dependence Properties of Leveraged Exchange-Traded Funds: Compounding, Volatility and Option Pricing

• There is nothing magic after you read it: your volatility pumping (upward drift) will become volatility 'dragging' (downward) for leveraged ETFs. If you can fully understand dragging in leveraged ETFs (which is exactly the goal these references try to help you), you gain a complete understanding of pumping in CRP at the same time. Even better, these references indeed use the machinery of option theory :)

• You can also prove to yourself that x1/2 leverage will maximize your volatility pumping (not in paper discussion, but it's very obvious once you understand it).

Unfortunately, there are not much discussions regarding derivatives on leveraged assets in these references and others except for the last paper. But honestly once you master the underlying dynamics, solution for derivatives pricing isn't too far away :)

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Oh? The bounty reward was given away while I am posting my answer? I remember there are still few hours to go before the bounty deadline? :) Too bad I just back from a vocation! – 楊祝昇 Sep 15 '11 at 20:32
Do any of these papers discuss replicating a leveraged ETF with options? If so, then it may be applicable to this problem, but if not, then I'm not sure you've offered anything besides a useful framework (not irrelevant, but, like my answer, not a complete answer). BTW, if vonjd feels your answer is best you can still get 25 points with an up-vote and check mark. If Louis also feels your answer is much better than mine, I'd be willing to re-offer the bounty and award it to you. – Tal Fishman Sep 15 '11 at 20:44
Hello Tal, please keep your bounty. I am teasing myself :) As for your question, in my opinion, once you have sound underlying dynamics, replicating is trivial. To myself, my answer is complete enough, i.e. the information I suggest here is enough to solve vonjd's questions. – 楊祝昇 Sep 15 '11 at 21:21

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Thank you Chu-Sheng: I am accepting this answer since it opens up a whole new line of research relevant to the topic + it connects different quant topics impressively. – vonjd Sep 16 '11 at 6:15

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Thank you for accepting the answer, vonjd :) And thanks for your inspiring question. I always believe it's the smart questions and curiosity that lead us to great answers and ideas! I wish I could answer you more but I really favor short and inspiring answer in public. However, I am more than happy to discuss further in private :) Please feel free to contact me (Google/Linked) if you are interested in more discussions. – 楊祝昇 Sep 16 '11 at 18:35

While not really an answer, here are my thoughts on the problem.

For starters, I would approach the problem as one of whether a portfolio which is constantly rebalanced over some horizon can be replicated using vanilla options. Obviously the portfolio will have to be rolled over when the options reach expiration. Then the rest, such as payoff diagram and price in a B-S world can be easily derived.

Suppose you have a \$100 portfolio to be invested equally in two assets, X and Y, both of which have liquid options over a closely-spaced range of strikes. Let's say both assets start out at a price of \$50, so that initially you want to hold 1 share of each. In a traditional CRP, as the asset prices evolve, you will want to sell some of whichever asset outperforms and buy the underperformer. With options, however, an alternative strategy is to simply try to keep the exposure to each asset fixed at \$50. In order for the P&L from the options portfolio to match the hypothetical CRP, though, one must then "rebalance" by buying and selling the entire replicating portfolio. For example, if both assets increase by 1%, then the replicating portfolio should show a P&L of \$1. However, since the exposures will remain \$50/\$50, and additional \\$1 unit of the options portfolio must be purchased (essentially reinvesting the P&L).

Proceeding under these assumptions, the replicating portfolio of options for asset X must satisfy \begin{equation} \sum_i\Delta_{t,i}^X=\frac{p_0^X}{p_t^X} \end{equation} where the sum of $i$ is over all option contracts held for asset X, and $p_t^X$ is the price of X at time $t$. A similar equation holds for asset Y, which is henceforth interchangeable with X (so I drop the superscript).

In order to keep the exposures constant, the sum of deltas must not depend on volatility, interest rates, or other extraneous factors besides price. I'm not sure what the ultimate dependence between sum of deltas and price will be, so perhaps some additional dynamic trading will be necessary because of this factor. What remains now is to solve for the portfolio of options with this property. Perhaps with this first step someone else can take up the mantle, but I believe this will take me more time than I have to spend on this problem at the moment.

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This is really interesting - Thank you! I hope that somebody will take up the mantle - or that you will find the time to continue your work. Give me a shout if you would be interested in publishing a (white) paper on this, perhaps we can work together here... I would very much appreciate that! Thank you again. – vonjd Sep 16 '11 at 6:17
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