Pricing Hedge Incompleteness Against Transaction Costs

Summary

The document considers how to balance hedge quality against bid–ask costs when hedging an exotic instrument with liquid instruments. It proposes measuring the cost of an imperfect hedge as a high percentile of losses over a chosen risk horizon, so the objective can account for adverse outcomes even when expected mis-hedge cost is near zero. This measure can be combined with trading costs to compare candidate hedges.

As alternatives, the response mentions deriving an optimal strategy under specified spread and utility assumptions, or using a utility function to translate P&L variance or another risk measure into a cost. The discussion offers a conceptual framework rather than a complete optimization procedure: it does not specify how to estimate the loss distribution, choose the percentile or horizon, or account for changing liquidity and model risk. Any resulting hedge depends on those choices and the institution’s risk preferences.

Key ideas

• A hedge objective can trade off transaction costs against losses from residual exposure.
• A loss percentile over a chosen horizon can represent the cost of hedge incompleteness.
• Utility functions can convert a measure of mis-hedge risk into a monetary cost.
• Optimal hedge ratios depend on assumptions about spreads, risk, and preferences.

Tags

# Cost function for hedging portfolio


# Cost function for hedging portfolio












Let's say I am hedging an exotic instrument $E$ with $N$ liquid instruments $L_i$, each of which has an associated hedging ratio $R_i$ and a bid-ask spread $\delta_i$ (per dollar of notional). What would you recommend as a cost function to balance the completeness of the hedge and minimize the hedging cost?

## Answer by Brian B (score 6, accepted)

https://quant.stackexchange.com/a/820

I would assign the cost of incompleteness as the 90th percentile of N-period losses expected on the mis-hedged portfolio (where N is perhaps 5 trading days -- enough for a trader to get hit by a bus and someone else to catch up on his book). This is nicely compatible with VaR computations, corrects for the fact that expected cost of a mis-hedge is usually zero, and doesn't involve any tricky utility function theory.

You sometimes see more precise measurements made. For example some papers in the 1990s calculated the exact optimal hedging strategy for European options given a particular bid-offer spread on the underlying and (I seem to recall) utility function assumptions.

If you like utility functions a lot, clearly you can assign one to the variance (or other metric) of P&L arising from mis-hedging, and use the utility function to turn that directly into a cost. You can approximate the "right" function parameters by, say, looking at your firm's recent returns and Sharpe ratio.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.