Derivatives #49 - Option Markets and Contracts

Question 1

Put-Call Parity

Answer 1

LOS 49.a

put-call parity equation for European options:

C₀ + PV(X) = P₀ + S₀

C<sub>0</sub> = call value today
PV(X) = present value of strike price
P<sub>0</sub> = put value today
S<sub>0</sub> = stock value today
PV(X) = X / (1+R<sub>f</sub>)<sup>T</sup>, R<sub>f</sub> = risk-free rate, T = exercise & maturity

Question 2

synthetic securities

Answer 2

LOS 49.a

Derive from put-call parity equation (e.g. synthetic call):

C₀ = P₀ + S₀ - PV(X)

”+” ⇒ “long”

”-“ ⇒ “short”

Question 3

reasons for creating synthetic securities

Answer 3

LOS 49.a

1. to price options by using combinations of other instruments with known prices
2. to earn arbitrage profits by exploiting mispricings among the instruments, i.e.

Question 4

how to arbitrage using put-call parity

Answer 4

LOS 49.a

1. Compare security price to synthetic version of security.
2. Short the overpriced one, long the underpriced one

profit is realized immediately; CF’s offset at time T

Question 5

binomial model: equations for components

Answer 5

LOS 49.b

For stock price binomial tree:

U = size of up move

D = 1 / U = size of down move

For option price binomial tree:

π_U = (1 + R_f) - D / (U - D) = risk-neutral prob of up move

π_{<span>D</span>} = 1 - π_U = risk-neutral prob of down move

Question 6

binomial model: option value calculation steps

Answer 6

LOS 49.b

1. Calculate U and D; build stock price tree to get stock prices at time T.
2. Calculate option payoffs at time T e.g. C_T⁺⁺ = max(0, S_T⁺⁺- X)
3. Calculate π_U and π_D and add to the tree
4. Find C₀ by working backwards from C_T; e.g.:

C_T-1 = (π_{<span>U</span>}C_T⁺⁺ + π_DC_T^+-) / (1 + R_f)

Question 7

binomial model: interest rate caps & floors

Answer 7

LOS 49.b

• interest rate cap (floor) = bundles of European call (put) options on interest rates
• Value of the cap (floor) is the sum of the values of its component caplets (floorlets)
• caplet value = max[0, (r_t - r_cap)*notional principal] / (1+r_t)
• floorlet value = max[0, (r_cap - r_t)*notional principal] / (1+r_t​)

Question 8

assumptions of the Black-Scholes-Merton (BSM) model

Answer 8

LOS 49.c

1. price of underlying follows a lognormal distribution
2. R_f^c is constant and known (bad for bonds and int rate valuation)
3. σ_S is constant and known (not real world)
4. markets are “frictionless” (not real world)
5. underlying has no CFs (BSM can be altered for this)
6. European options only (American options can be priced using binomial option pricing models)

Question 9

inputs to BSM

Answer 9

LOS 49.c

Question 10

delta

Answer 10

LOS 49.e

delta - slope of option value

• Call: 0 <= delta_C <= 1
  
  • deep out-of-money: delta aproaches 0
  • deep-in-money: delta approaches 1
• Put: -1 <= delta_C <= 0
  
  • deep out-of-money: delta aproaches 0
  • deep-in-money: delta approaches -1
• delta (discrete time) = dC / dS
• delta (continuous time) = N(d₁)
  
  • dC ≈ N(d₁) * dS
  • dP ≈ [N(d₁) - 1] * dS

Question 11

dynamic hedging

Answer 11

LOS 49.e

• delta-neutral hedge - combo of long stock and short calls sych that the portfolio value doesn’t change with changing stock price
• # of shorts needed = # shares hedged / delta_call
• HINT: Delta < 1, so we always need more calls than shares

Question 12

gamma

Answer 12

LOS 49.f

• Gamma = 1st derivative of slope (2nd derivative of option & stock price function)
  
  • largest when option is at the money and close to expiration
  • small for deep out-of-money and deep-_in_-money and far from expiration