Option Greeks: Delta: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 39

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov.

This is Section 39 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10

here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here.

The option Greek delta (∆) "measures the option price change when the stock price increases by $1" (McDonald 2006, p. 382).

Delta is also the number of shares in the replicating portfolio for an option - otherwise known as the share-equivalent of the option.

An option that is in-the-money will be more sensitive to price changes than an option that is out-of-the-money. The more deeply an option is in-the-money, the more likely it is to be exercised, and delta approaches 1 in that case.

For an out-of-the-money option that is unlikely to be exercised, delta approaches 0.

As time to expiration increases, delta is smaller at high stock prices and greater at low stock prices. (McDonald 2006, p. 383).

The formula for a call option's Delta is

∆_call = e^-∂TN(d₁)

where d₁ = [ln(S/K) + (r - ∂ + 0.5σ²)T]/[σ√(T)] and d₂ = d₁ - σ√(T)

A replicating portfolio for a call option involves holding ∆ shares and borrowing B dollars.

Here, B = Ke^-rTN(d₂), so the cost of the replicating portfolio is the Black-Scholes price of the call option:

C = Se^-∂TN(d₁) - Ke^-rTN(d₂)

The formula for a put option's delta can be derived via put-call parity: