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Explain the dynamic nature of delta hedging an option position.

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

The idea

A delta hedge makes a position insensitive to small price moves by holding the underlying in proportion to the option's delta. A trader who is short calls buys delta shares per option, while one who is short puts sells shares, because a short put has positive delta. The hedge is dynamic because delta is not constant. It changes with the stock price (gamma) and with the passage of time, even if the price does not move. The position must therefore be rebalanced, either continuously or at discrete intervals, and a short option hedger tends to buy after rises and sell after falls. The common mistake is to treat the hedge as set and forget. The exam point is to compute the incremental trade, which is the new required holding minus the current holding.

Formulas

Shares to buy when rebalancing a delta hedge of short calls

Trade = n × (Δ₁ − Δ₀)

n
the number of calls written
Δ₁
the call's new delta
Δ₀
the call's delta when the hedge was last set

A negative result is a sale. For short puts the hedge is short stock, so the signs reverse. Trading the whole new hedge instead of the increment is the trap.

A practice question

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

A trader has written 10,000 call options on a stock; each call has a delta of 0.60. To make the position delta neutral she should:

  1. Sell 6,000 sharesWhy not: A short call has negative delta, so the hedge needs positive delta from long shares, not short shares.
  2. Buy 10,000 sharesWhy not: Ignores the delta: it hedges one share per option, which over-hedges a call with delta 0.60.
  3. Buy 6,000 shares - correct
  4. Buy 4,000 sharesWhy not: Uses (1 - delta) instead of delta, which is the put-call adjustment, not the hedge ratio.

The short calls have position delta of -0.60 x 10,000 = -6,000. Buying 6,000 shares adds +6,000 and makes the total delta zero.

A harder one

A dealer has written 50,000 European puts, each with a delta of -0.40. What delta-neutral trade in the stock is required?

  1. Buy 20,000 sharesWhy not: Treats the position like short calls; a short put has positive delta, so it needs short stock.
  2. Sell 20,000 shares - correct
  3. Sell 50,000 sharesWhy not: Ignores the delta of 0.40 and hedges one share per option.
  4. Sell 30,000 sharesWhy not: Uses (1 - 0.40) instead of the delta of the put.

Short puts: delta = -50,000 x (-0.40) = +20,000. Selling 20,000 shares gives -20,000 and brings the total to zero.

The hardest tier

A trader is short 100,000 calls and starts delta hedged at a stock price of USD 50 with delta 0.50. The stock rises to USD 52 (delta 0.62), the hedge is reset, then it falls to USD 49 (delta 0.44) and the hedge is reset again. After the two rebalancing trades, which result is correct?

  1. She holds 44,000 shares, and the two rebalancing trades produced a net cash inflow of USD 258,000. - correct
  2. She holds 44,000 shares, and the two rebalancing trades produced a net cash outflow of USD 258,000.Why not: Reverses the sign: she buys 12,000 at 52 (outflow 624,000) and sells 18,000 at 49 (inflow 882,000), a net inflow.
  3. She holds 62,000 shares, and the first rebalancing trade cost USD 624,000.Why not: Ignores the second rebalance: after the fall delta is 0.44, so holdings must be cut to 44,000.
  4. She holds 44,000 shares, and the rebalancing trades produced a net cash inflow of USD 1,506,000.Why not: Treats the purchase as a cash inflow too, adding 624,000 and 882,000 instead of netting them.

Start 50,000 shares. At 52 she needs 62,000: buy 12,000 for 624,000. At 49 she needs 44,000: sell 18,000 for 882,000. Net cash = 882,000 - 624,000 = +258,000, holding 44,000 shares.

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