Question

1. Let’s say that a $50 strike call pays a 2.0% continuous dividend, r = 0.07, σ = 0.25, and the stock price is $48.00. What is the profit or loss, per share, for a short call position if the option expires in 60 days and the price rises to $50.00 after 5 days?

Correct Answer= $0.84 loss per share

Question- how do we get to this?

2.Assume S = $33.00, σ = 0.32, r = 0.06, div = 0.01. You short 100 $35 strike calls at 68 days until expiration. Under a delta hedge position, what is your overnight profit/loss if the stock rises

Correct Answer = $ 7.62 loss overnight

Question - how do we get to this?

Answer 1

1]

Selling price of call (premium received)

We use Black-Scholes Model to calculate the value of the call option.

The value of a call option is:

C = (S₀ * e^-qt * N(d₁))   - (Ke^-rt * N(d₂))

where :

S₀ = current spot price

K = strike price

N(x) is the cumulative normal distribution function

q = dividend yield

r = risk-free interest rate

t is the time to maturity in years

d₁ = (ln(S₀ / K) + (r + σ²/2)*T) / σ√T

d₂ = d₁ - σ√T

σ = standard deviation of underlying stock returns

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(48 / 50). We input the same formula into Excel, i.e. =LN(48/50)
• (r + σ²/2)*t = (0.07 + (0.25²/2)*(60/365)
• σ√t = 0.25 * √(60/365)

d₁ = -0.2385

d₂ = -0.3399

N(d₁) and N(d₂) are calculated in Excel using the NORMSDIST function and inputting the value of d₁ and d₂ into the function.

N(d₁) = 0.4057

N(d₂) = 0.3670

Now, we calculate the values of the call option as below:

C = (S₀ * e^-qt * N(d₁))   - (Ke^-rt * N(d₂)), which is (48 * e^{(-0.02 * (60/365))} * 0.4057) - (50 * e^{(-0.07 * (60/365))} * 0.3670)    ==> $1.2728

Buying price of call (premium paid)

We use Black-Scholes Model to calculate the value of the call option.

The value of a call option is:

C = (S₀ * e^-qt * N(d₁))   - (Ke^-rt * N(d₂))

where :

S₀ = current spot price

K = strike price

N(x) is the cumulative normal distribution function

q = dividend yield

r = risk-free interest rate

t is the time to maturity in years

d₁ = (ln(S₀ / K) + (r + σ²/2)*T) / σ√T

d₂ = d₁ - σ√T

σ = standard deviation of underlying stock returns

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(50 / 50). We input the same formula into Excel, i.e. =LN(50/50)
• (r + σ²/2)*t = (0.07 + (0.25²/2)*(55/365)
• σ√t = 0.25 * √(55/365)

d₁ = 0.1572

d₂ = 0.0602

N(d₁) and N(d₂) are calculated in Excel using the NORMSDIST function and inputting the value of d₁ and d₂ into the function.

N(d₁) = 0.5625

N(d₂) = 0.5240

Now, we calculate the values of the call option as below:

C = (S₀ * e^-qt * N(d₁))   - (Ke^-rt * N(d₂)), which is (50 * e^{(-0.02 * (55/365))} * 0.5625) - (50 * e^{(-0.07 * (55/365))} * 0.5240)    ==> $2.1139

Loss per share = option buying price (premium paid) - option selling price (premium received)

Loss per share = $2.1139 - $1.2728

Loss per share = $0.84