5- Option pricing

Question 1

How many units of the underlying do you have to buy to price hedge an option?

Answer 1

Delta (Δ)

Question 2

What are the payoffs of a delta hedged position in the up (Cᵤ) and downside (Cd) scenarios?

Answer 2

• S₀uΔ - Cᵤ
• S₀dΔ - Cd

Question 3

What is the formula for Delta (Δ)?

Answer 3

Δ = (Cᵤ - Cd)/S₀(u-d)

Question 4

What is the present value of a delta hedged portfolio?

Answer 4

(S₀uΔ - Cᵤ)/Rբ = (S₀dΔ - Cd)/Rբ

Question 5

What is the long CALL option payoff at maturity (C)?

Answer 5

max(Sₜ - K, 0)

Question 6

What is the long PUT option payoff at maturity (P)?

Answer 6

max(K - Sₜ, 0)

Question 7

How do option payoffs at maturity change when short?

Answer 7

They are the same function but negative

Question 8

Briefly explain the intuition of the Black-Scholes formula

Answer 8

Discounted expected value based on risk-neutral probabilities

Question 9

What are the asset values being discounted and probability-weighted in the Black-Scholes formula?

Answer 9

-Forward value of asset at maturity Seʳᵀ
-Cash amount K, paid at maturity

Question 10

How does the PUT equation differ from that of a call?

Answer 10

All terms become negative, including the distributions [N(-d1)]

Question 11

What is the formula for a negative normal distribution N(-d1)?

Answer 11

N(-d1) = 1 - N(d1)

Question 12

How do you calculate option price using the binomial model?

Answer 12

-Draw a value tree and note the payoffs at each terminal node
-Use payoffs to calculate option value in prior period
-Iterate through to initial period t=0