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Explain what the delta of a stock option is and calculate it.

FRM® Part I · Valuation and Risk Models · Binomial Trees · FRM1.VR.14.d

The idea

The delta of a stock option is the rate of change of the option price with respect to the stock price, the slope of option value against stock price. In a one-step tree, delta = (value in up state − value in down state) ÷ (stock price up − stock price down). A call has a delta between 0 and 1, and a put between −1 and 0. For a European call and put with the same strike and expiry on a stock paying no dividends, put delta = call delta − 1. A dealer short a call with delta 0.6 buys 0.6 shares per option to be delta neutral, and a holder of a put with delta −0.4 also buys 0.4 shares per option. Delta changes from node to node, so in a multi-step tree each node has its own delta and the hedge must be rebalanced. Common mistakes are the wrong sign and dividing by the wrong price spread.

Formulas

Delta of an option in a one-step binomial tree

Δ = (fu − fd) ÷ (Su − Sd)

fu
the option value in the up state
fd
the option value in the down state
Su
the stock price in the up state
Sd
the stock price in the down state

Negative for a put. Divide by the spread of the two successor prices, not by the starting price, and recompute it at each node of a multi-step tree.

A practice question

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

What is the delta of a stock option?

  1. The change in the option price per unit change in the volatility of the stockWhy not: Describes vega, not delta.
  2. The change in the option price as one day of time passesWhy not: Describes theta, not delta.
  3. The change in the option price per unit change in the stock price, the slope of the option value against the stock price - correct
  4. The risk-neutral probability that the option finishes in the moneyWhy not: Mixes delta with an exercise probability; delta is a hedge ratio, not a probability.

Delta is the sensitivity of the option price to the underlying stock price, the slope of the value curve. In a tree it is the number of shares in the replicating portfolio, which is also the hedge ratio.

A harder one

A stock trades at 60 and will be either 69 or 51 in six months. A European put has strike 62. What is the put's delta from the one-step tree?

  1. +0.61Why not: Gets the sign wrong; the value falls when the stock rises, so a put delta is negative.
  2. +0.39Why not: Reports the delta of the corresponding call, which is the put delta plus 1.
  3. -0.18Why not: Divides the change in option value by the starting price 60 rather than by the stock price spread 18.
  4. -0.61 - correct

Put payoffs are 0 at 69 and 62 - 51 = 11 at 51. Delta = (0 - 11) / (69 - 51) = -0.6111. The call delta is 0.389, and put delta = call delta - 1 holds here as well.

The hardest tier

A stock at 80 moves up 10% or down 10% in each of two six-month steps. The risk-free rate is 5% continuously compounded. A European call has strike 82 and one year to expiry. After the first step the stock has risen to 88. What delta must the call hedge hold for the second step?

  1. 0.57Why not: Keeps the time-zero delta, although delta must be recomputed at each node.
  2. 0.46Why not: Uses the full spread of terminal prices, 96.8 and 64.8, instead of the two successors of the 88 node.
  3. 0.84 - correct
  4. 0.93Why not: Divides the second-step payoff change by the first-step price spread of 16 instead of 17.6.

At the 88 node the successors are 96.8 and 79.2 with call payoffs 14.8 and 0, so delta = 14.8 / (96.8 - 79.2) = 0.8409. The time-zero delta is 0.5653, so the hedge is rebalanced after the first step.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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