Lesson 4 of 6 · 13 min

Arbitrage: forwards versus options, and option price bounds

A forward's symmetric payoff pins down one no-arbitrage price, but an option's one-sided payoff only lets arbitrage fix a range: a lower bound and an upper bound.

In short

  • No-arbitrage rests on the law of one price: identical future payoffs must cost the same today.
  • A forward costs nothing to enter and has a symmetric payoff ST−F0(T)S_T - F_0(T): the buyer can gain or lose without limit.
  • An option buyer pays a premium and exercises only when in the money, so profit = payoff − premium and the payoff is never negative.
  • European call: max⁡(0,St−PV(X))≤ct≤St\max(0, S_t - PV(X)) \le c_t \le S_t.
  • European put: max⁡(0,PV(X)−St)≤pt≤X\max(0, PV(X) - S_t) \le p_t \le X.
  • An option trading below its exercise value, or above its upper bound, offers riskless arbitrage.

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Arbitrage: forwards versus options, and option price bounds · Pricing and Valuation of Options