Option Greeks: Delta: Practice Problems and Solutions

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:

∆_put = ∆_call – e^-∂T = e^-∂TN(d₁) – e^-∂T = e^-∂T(N(d₁) -1) = ∆_put = -e^-∂TN(-d₁)

We note that delta changes with the stock price and so the replicating portfolio for an option must be adjusted every time the stock price changes.

Meaning of variables:

S = current stock price.

K = strike price of the option.

C = call option price.

P = put option price.

σ = annual stock price volatility.

r = annual continuously compounded risk-free interest rate.

T = time to expiration.

∂ = annual continuously compounded dividend yield.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 12, pp. 382-384.

Problem OGD1. The stock of Delta Corporation has a price of $506. Certain call options on Delta Corporation stock have a delta of 0.4 and trade for $33 per option. The stock price suddenly increases to $508. Assuming that this move did not substantially alter delta, what is the new price of the call option?

Solution OGD1. ∆_call = 0.4 means that the option price increases by $0.4 for every increase of $1 in the stock price. Since the stock price has increased by $2, the option price increased by $0.8, and the new option price is $33.8.

Problem OGD2. The Black-Scholes price for a certain call option on Vacuous LLC stock is $50. The stock currently trades for $1000 per share, and it is known that $452 must be borrowed in the replicating portfolio for this option. Find the delta of the option.

Solution OGD2. The Black-Scholes price for the option is C = 50 = Se^-∂TN(d₁) – Ke^-rTN(d₂). We are given that S = 1000 and Ke^-rTN(d₂) = 452. We are left to find ∆_call = e^-∂TN(d₁).

50 = 1000∆_call – 452, so 502 = 1000∆_call and ∆_call = 0.502.

Problem OGD3. The stock of Voracious Co. currently trades for $95 per share. The annual continuously compounded risk-free interest rate is 0.06, and the stock pays dividends with an annual continuously compounded yield of 0.03. The price volatility relevant for the Black-Scholes formula is 0.32. Find the delta of a call option on Voracious Co. stock with strike price of $101 and time to expiration of 3 years.

Solution OGD3. First, we find

d₁ = [ln(S/K) + (r – ∂ + 0.5σ²)T]/[σ√(T)] = [ln(95/101) + (0.06 – 0.03 + 0.5*0.32²)3]/[0.32√(3)] = d₁ = 0.3290109439

In MS Excel, using the input “=NormSDist(0.3290109439)”, we find that N(d₁) = 0.628926227

Now we use the formula

∆_call = e^-∂TN(d₁) = e^-0.03*30.628926227 = ∆_call = 0.5747952921

Problem OGD4. The stock of Voracious Co. currently trades for $95 per share. The annual continuously compounded risk-free interest rate is 0.06, and the stock pays dividends with an annual continuously compounded yield of 0.03. The price volatility relevant for the Black-Scholes formula is 0.32. Find the delta of a put option on Voracious Co. stock with strike price of $101 and time to expiration of 3 years.

Solution OGD4. Since the call delta is known from Solution OGD3, we use the put-call parity formula to find the put delta: ∆_put = ∆_call – e^-∂T = 0.5747952921 – e^-0.03*3 = ∆_put = -0.3391358932

Problem OGD5. For otherwise equivalent call options on a particular stock, for which of these values of strike price (K) and time to expiration (T) would you expect delta to be the highest? The stock price at both T = 0.3 and T = 0.2 is $50.

(a) K = $43, T = 0.3
(b) K = $43, T = 0.2
(c) K = $55, T = 0.3
(d) K = $55, T = 0.2
(e) K = $50, T = 0.3
(f) K = $50, T = 0.2

Solution OGD5. The more an option is in-the-money, the higher the value of delta. So, other things equal, a $43-strike option should have a higher delta than a $50-strike or a $55-strike option. As expiration time gets closer, delta becomes higher at low strike prices because there is less of a likelihood that an in-the-money option will become out-of-the-money prior to expiration. Thus, the highest delta should occur for the stock with the lowest strike price and lowest time to expiration, i.e., answer (b).

See other sections of The Actuary’s Free Study Guide for Exam 3F / Exam MFE.