• Global Frameworks: CFA Level 2 (Advanced Derivatives), GARP FRM Part 1.

1. Fundamental Boundary Constraints for European Options
Option values cannot drop below zero, nor can they exceed certain structural ceilings:
  • The absolute upper bound for a call option is the asset price itself (C ≤ S).
  • The absolute upper bound for a European put option is the present value of its strike price
    P ≤ X × e^(−r × T)

2. Put-Call Parity Mechanics
Put-Call Parity establishes a definitive, arbitrage-free relationship between the prices of European call options, put options, the underlying spot stock, and a risk-free bond with identical strike prices and expiration dates.
The relationship is derived by comparing two distinct investment strategies:
  • Fiduciary Call: Buying a European call option plus a zero-coupon risk-free bond that matures to match the strike price
    C + X × e^(−r × T)
  • Protective Put: Buying the underlying stock plus a European put option (S + P).

At expiration, both portfolios are guaranteed to be worth exactly the higher of the asset price (\(S_{T}\)) or the strike price (X). Because their future values match, their upfront costs must be identical, defining the Put-Call Parity formula:
C + X × e^(−r × T) = S + P
3. Synthetic Financial Engineering
Using Put-Call Parity, analysts can synthetically replicate any of the four asset components by rearranging the algebra. For example, a synthetic long stock position can be engineered without buying actual shares:

S = C − P + X × e^(−r × T)
 
PreviousNext