Derivatives: Option Markets and Contracts

Question 1

Delta Hedging

Answer 1

Number of shares in contract = Investment / Delta

Question 2

Gamma is Greatest for…

Answer 2

At the money options

Question 3

Put-Call Parity

Answer 3

Po = Co - So + x/e^(r)(t)

• • t should be in decimal form (90 days = .25)*
• • this formula assumes continuous compounding*

Question 4

Put-Call Parity for Futures and Forwards

Answer 4

P0 = Co + [(X - F(0,T) / 1+r^t]

• 1. F(0,T) = value of future now that expires in time T*
• 2. The numerator could be negative*

Question 5

Value of Call Option for one period (Binomial Model) including probability calculation

Answer 5

Step 1: Calculate max gain for up and down move

Step 2: prob = (1 + Rf% - d) / u - d

Step 3: Co = [prob(call+) + 1-prob(call-)] / (1 + Rf%)

• • d = 1 - the likely decline in value in percentage terms*
• • u = 1+ the likely rise in value in percentage terms*
• • call+ = The maximum value of the call assuming the positive forecast comes true at expiration*

Question 6

Value of Call (binomial model) - Two Periods

Answer 6

Step 1: Calculate max gain for up, down, and up/down-down/up moves - up and down move must be squared

Step 2: prob = (1 + Rf% - d⁻⁻) / u⁺⁺ - d⁻⁻

Step 3a: C⁺⁺ = [prob(call⁺⁺) + 1-prob(call⁺⁻)] / (1 + Rf%)

Step 3b: C⁻⁻ = [prob(call⁺⁻) + 1-prob(call⁻⁻)] / (1 + Rf%)

Step 4: C = [prob(C⁺⁺) + 1-prob(C⁻⁻)] / (1 + Rf%)

• d = 1 - the down move (or 1/1+upmove if no down move given)
• u = 1+ the up move
• call+ = The maximum value of the call assuming the positive forecast comes true at expiration

Question 7

Assumptions about Black Scholes Merton (BSM) Formula

Answer 7

1. Volatility of the return on the underlying stock is known and constant
2. Stock prices are lognormally distributed
3. Continuous Rf% is known and constant