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Outline how portfolio insurance is implemented and compare the strategy with delta hedging.

FRM® Part I · Valuation and Risk Models · Option Sensitivity Measures: The “Greeks” · FRM1.VR.16.i

The idea

Portfolio insurance protects an equity portfolio from falling below a floor by replicating a put option instead of buying one. The put's delta is negative, so the manager sells a fraction of the portfolio (or index futures) equal in size to the put's delta, and then sells more as the market falls and buys back as it rises. The trades are the same as those of a dealer who sold a put and delta hedges it. The difference is purpose: the insurer wants the floor, while the dealer wants to cancel a liability. The strategy relies on continuous trading and liquid markets. In a sudden price gap or market crash the sales cannot be done at planned prices, so the floor can fail, and mass selling by insurers can deepen the fall. The exam points are the direction of trading and its failure.

Formulas

Equity to sell to replicate a protective put

Sell = |Δp| × P

Δp
the delta of the put being replicated
P
the current portfolio value

Rebalance as the delta changes, selling after falls and buying after rises, and measure against the current portfolio value including cash already raised. Selling 1 − |Δp| of the portfolio is the trap.

A practice question

From the question bank, with the reason each wrong option is wrong.

How is portfolio insurance typically implemented for an equity portfolio?

  1. By buying the portfolio's stocks as the market falls and selling as it rises, to replicate a call option.Why not: Reverses the direction; a put replication cuts equity exposure as prices fall.
  2. By holding a fixed 50% equity weight at all times regardless of the market level.Why not: A constant weight is not portfolio insurance; the equity exposure must change with the put's delta.
  3. By buying a deep in-the-money call option on each stock in the portfolio.Why not: This is a static option purchase, whereas the dynamic strategy creates the put synthetically.
  4. By selling part of the portfolio (or index futures) as the market falls and buying back as it rises, so as to replicate a put option. - correct

A put gives a floor on portfolio value. Without buying one, the manager creates it synthetically: the put's delta is negative, so the equity exposure is reduced (by selling stock or index futures) as the market falls and rebuilt as it rises.

A harder one

A manager with a USD 200 million equity portfolio wants to replicate a put on it using the stock. The put's delta is -0.30. What should the manager do initially?

  1. Buy USD 60 million more stocksWhy not: Increases equity exposure; the put has negative delta, so exposure must fall.
  2. Sell USD 140 million of the stocksWhy not: Sells (1 - 0.30) of the portfolio; this reduces the equity exposure to 30% rather than to 70%.
  3. Sell USD 60 million of the stocks (or the equivalent in index futures) - correct
  4. Sell USD 30 million of the stocksWhy not: Takes half of the put delta (0.15) as the fraction of the portfolio to sell.

Equity to sell = |put delta| x portfolio value = 0.30 x 200 = USD 60 million. The remaining equity exposure is 70%, which replicates the delta of the portfolio plus the put.

The hardest tier

How do the trades of a portfolio insurer replicating a put compare with those of a dealer who has sold a put and delta hedges it?

  1. They are opposite: the dealer buys stock as the market falls, while the insurer sells.Why not: A short put has positive delta, so the dealer hedges by shorting stock, and sells as the market falls, as the insurer does.
  2. They are the same trades, so portfolio insurance and delta hedging are one strategy with one purpose.Why not: The trades match but the purposes differ: the insurer wants a floor on value, the dealer wants to neutralise a sold option.
  3. They differ because portfolio insurance requires no rebalancing while delta hedging does.Why not: Both strategies require continuous rebalancing as the put's delta changes.
  4. The direction is the same, selling as the market falls and buying as it rises; the insurer wants the put's floor, the dealer wants to offset a liability. - correct

Delta hedging a written put means shorting stock in proportion to the put's delta, and this holding is increased as the market falls. This is the same as creating a synthetic long put in portfolio insurance. The purpose differs: the insurer wants the floor, the dealer wants to neutralise a short-option risk.

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