Hedging Options on Asset Portfolios against Just One Underlying Asset in the Presence of Transaction Costs

This study investigates the optimal hedging strategy for options on a two-asset portfolio when transaction costs are present, demonstrating through simulation that hedging with a correlated but non-underlying asset can be preferable to hedging with the correct asset if the correlation is sufficiently high and transaction costs are low, as determined by risk-adjusted value metrics.

The Gist

Imagine you own a ticket (an option) that gives you the right to buy a specific house (Asset 1) at a fixed price in the future. The value of this ticket depends entirely on how the price of that house changes.

To protect yourself from losing money if the house price drops, you usually try to "hedge." In the perfect, frictionless world of math textbooks, you would constantly buy and sell the house itself to balance your risk. But in the real world, buying and selling houses is expensive. There are agent fees, taxes, and the hassle of moving (these are transaction costs). If you try to rebalance your portfolio every time the price moves a penny, the fees will eat up all your profits.

This paper asks a clever question: What if you hedge with a different house that is cheaper to trade, but moves in a very similar way?

The Setup: The "Right" House vs. The "Wrong" House

The authors set up a simulation with two houses:

1. The "Right" House (Asset 1): This is the actual house your ticket is based on. It's the perfect match, but imagine it's in a remote area where every time you try to buy or sell a share of it, it costs a lot of money (high transaction costs).
2. The "Wrong" House (Asset 2): This is a different house in a neighboring town. It's not the exact house your ticket is for, but it moves up and down in price almost exactly the same way as the "Right" house. Crucially, it's in a busy city where trading is cheap and easy (low transaction costs).

The researchers asked: Is it better to pay high fees to trade the "Right" house, or pay low fees to trade the "Wrong" house?

The Experiment: Simulating the Market

They used a computer to run 10,000 different "what-if" scenarios (simulations). They played with three main knobs:

• Correlation ($\rho$): How closely the two houses move together. If $\rho$ is 0.99, they are practically twins. If it's 0.2, they barely move together.
• Transaction Costs: How much it costs to trade each house (from 0% up to 10%).
• Risk Tolerance ($\lambda$): How much the investor cares about risk. A high number means they are very nervous and want to avoid risk at all costs. A low number means they are willing to take risks for potential gain.

They measured success using a metric called Risk-Adjusted Value (RAV). Think of this as a "score" that tells you: "Is the money I'm making worth the stress and risk I'm taking?"

The Findings: When to Trade the "Wrong" House

Here is what they discovered, translated into everyday logic:

1. The "Perfect Match" isn't always the winner
If the "Right" house is very expensive to trade, and the "Wrong" house is cheap to trade, you might be better off trading the "Wrong" house—but only if the two houses are practically identical in how they move.

• The Analogy: Imagine you need to cross a river. The "Right" bridge is the direct path, but the toll is \100. The "Wrong" bridge is a mile out of your way, but the toll is \1. If the "Wrong" bridge is 99% parallel to the "Right" bridge, taking the detour saves you money. But if the "Wrong" bridge goes in a completely different direction (low correlation), you'll end up lost, and the cheap toll won't help.

2. The "High Cost" Trap
If the fees to trade either house are huge (like 10%), the best strategy is often to do nothing at all.

• The Analogy: If the toll to cross any bridge is \1,000, and your ticket is only worth \500, you shouldn't cross at all. You're better off keeping your ticket and accepting the risk than paying the toll and losing money. The paper notes this often applies to very expensive assets like real estate or crypto, where trading fees are massive.

3. The "Risk Aversion" Factor
Your decision depends on how scared you are of losing money.

• If you are very risk-averse (high $\lambda$), you prefer the "Right" house even if it's expensive, because it's the perfect hedge.
• If you are less worried about risk (low $\lambda$) and the "Wrong" house is very cheap and very similar to the "Right" one, you might choose the "Wrong" house to save on fees.

4. The "Lockstep" Requirement
The paper found that to successfully trade the "Wrong" house, the correlation ($\rho$) needs to be extremely high (around 0.99).

• The Analogy: Even if two houses are in the same neighborhood, if one goes up 1% and the other goes up 0.5%, they aren't moving in "lockstep." To use the cheap "Wrong" house as a shield, they must move almost exactly together. If they drift apart even a little, the cheap hedge fails to protect you.

The Bottom Line

The paper concludes that hedging isn't just about picking the "correct" asset; it's about the cost of the trade.

• If trading costs are low: Stick to the "Right" asset (the one your option is actually based on). It's the safest bet.
• If trading costs are high: You have two choices. Either don't hedge at all (if costs are astronomical), or hedge with the "Wrong" asset (if it's cheap to trade and moves almost exactly like the real thing).
• The Catch: You can only use the "Wrong" asset if the two assets are practically twins (very high correlation). If they aren't, the savings on fees won't make up for the risk of the hedge failing.

In short: Sometimes, the "wrong" tool is actually the best tool, provided it's cheap to use and does the job almost as well as the "right" tool. But if the job is too expensive to do, sometimes it's better to just leave it alone.

Technical Summary

Technical Summary: Hedging Options on Asset Portfolios against Just One Underlying Asset in the Presence of Transaction Costs

Problem Statement
The paper addresses a practical limitation in standard option hedging theory: the assumption that the underlying asset of an option can be traded continuously and without friction. In reality, transaction costs (comprising fixed fees and proportional slippage) make continuous delta hedging prohibitively expensive, often leading to infinite costs in continuous-time models like Black-Scholes. Furthermore, the paper investigates a specific scenario where an investor holds a portfolio option written on a blend of two correlated assets ($S_{t1}$ and $S_{t2}$) but is constrained to hedge using only one of these assets. The core research question is determining the optimal hedging strategy under these constraints: should the investor hedge using the "right" asset (the one directly underlying the portfolio component), the "wrong" asset (the other correlated asset, which may be cheaper to trade), or abstain from hedging entirely (holding a naked position)?

Methodology
The authors employ a Monte Carlo simulation framework to analyze delta hedging strategies for equity options on assets following Geometric Brownian Motion (gBm).

• Portfolio Construction: The underlying portfolio is defined as $S_t = \alpha S_{t1} + (1-\alpha)S_{t2}$, where $\alpha$ represents the proportion of the first asset. The study focuses on cases where $\alpha=0$ or $\alpha=1$ (the portfolio is entirely on one asset) and considers hedging either $S_{t1}$ or $S_{t2}$.
• Hedging Mechanism: The study utilizes discrete-time delta hedging. The delta is calculated based on the Black-Scholes formula, adjusted for the specific asset being traded. The portfolio is rebalanced at fixed intervals ($\delta t$).
• Transaction Costs: Proportional transaction costs ($k$) are applied to the asset being traded. The cost structure includes a fixed component and a proportional component, modeled as slippage.
• Risk-Adjusted Value (RAV): To evaluate performance, the authors introduce the Risk-Adjusted Value (RAV) as the primary decision metric. RAV is defined as:
  $$RAV = \text{Average Simulated Value} - \lambda(\text{Standard Deviation of Simulated Value})$$
  where $\lambda$ represents the market price of risk. This metric balances expected return against volatility.
• Simulation Parameters: The study runs 10,000 simulations across varying parameters: correlation coefficients ($\rho$) ranging from 0.2 to 0.99, transaction costs ($k$) from 0% to 20%, rebalancing intervals (2 to 252 steps), and market prices of risk ($\lambda$) from 0.2 to 2.0.
• Leland's Number: The study references Leland's number ($A$) to determine the viability of modified Black-Scholes strategies. If $A > 1$, the paper notes that not hedging is theoretically superior to hedging under high transaction costs.

Key Results
The simulation results, analyzed through RAV, yield the following findings:

1. Impact of Transaction Costs: In the presence of transaction costs, frequent rebalancing leads to significant losses compared to unhedged positions or less frequent trading. When transaction costs are extremely high (e.g., 10% or 20%), the optimal strategy is consistently to not hedge at all, regardless of the correlation between assets.
2. Trading the "Wrong" Asset: The study identifies specific conditions where hedging with the "wrong" asset (the asset not constituting the portfolio's primary exposure) is superior to hedging the "right" asset or not hedging. This occurs when:
   • The correlation ($\rho$) between the two assets is very high (approaching 0.99).
   • The transaction costs on the "wrong" asset are significantly lower than those on the "right" asset.
   • The market price of risk ($\lambda$) is low (indicating lower risk aversion).
   • Specifically, with a 1% cost on the right asset and 0% on the wrong asset, trading the wrong asset becomes optimal at $\rho=0.99$ for high rebalancing frequencies.
3. Role of $\lambda$: The market price of risk ($\lambda$) is a critical determinant. High $\lambda$ values (high risk aversion) generally favor trading the "right" asset or not hedging, whereas lower $\lambda$ values make the "wrong" asset strategy more viable when correlations are high.
4. Blended Portfolios: For intermediate values of $\alpha$ (blended portfolios), simulated portfolio prices are generally lower than those for pure portfolios ($\alpha=0$ or $1$), and the risk (standard deviation) is higher when the proportion of the traded asset is low.

Significance and Claims
The paper claims to provide a quantitative framework for financial managers to decide whether to hedge a portfolio using a cheaper, correlated asset rather than the exact underlying asset when transaction costs are a factor.

• Managerial Insight: The authors assert that a portfolio manager can potentially achieve higher returns by trading a "wrong" but cheaper correlated asset, provided the correlation is sufficiently high and transaction costs on the "right" asset are prohibitive.
• Decision Thresholds: The study provides a decision matrix (summarized in Table 3.6) indicating when to trade the wrong asset, the right asset, or neither, based on the interplay of $\rho$, $\lambda$, and transaction costs.
• Limitations: The authors modestly note that their conclusions are based on simulated data using gBm and specific transaction cost models. They suggest that the strategy's effectiveness relies heavily on the assumption that the "wrong" asset is cheaper to trade and highly correlated. The paper does not claim to eliminate risk but rather to manage it more cost-effectively in frictional markets.

In conclusion, the study demonstrates that in markets with transaction costs, the rigid adherence to hedging the exact underlying asset is not always optimal. Under conditions of high correlation and disparate transaction costs, substituting the hedging instrument with a cheaper, correlated asset can be a rational, risk-adjusted strategy.