【正文】 s the PutCall Parity for European calls on stocks with a continuous dividend yield q? E. PutCall Parity for American calls on dividend paying stocks: SPV(D)K?CP?SKB To prove this, we show that neither of the inequality can be violated, by considering two cases: If the second inequality is violated。 ., CPSKB, then we can have the following arbitrage: Question: If the written call is exercised against you early what should be the value of your portfolio? If the first inequality is violated。 ., for T2 T1, C(S,K,t,T2)?C(S,K,t,T1) P(S,K,t,T2)?P(S,K,t,T1) Note: This does not always hold for European options. (Why?) F. An American option is worth at least its exercise value (what you would get if you exercise today). C(S,K,t,T)?max[0,S(t)K] P(S,K,t,T)?max[0,KS(t)] Example: Do we have an arbitrage opportunity if, for Intel stock with S(t) = $100, a call option with K=$90 and 6month to maturity is trading at $9? Note: This rule does not always hold for European options. (Why?) ? More Arbitrage Bounds for Options on NonDividendPaying Stocks: Example: Same as on the previous page. Assume S(t)=$100, and the price of an Intel call with K=$90 and 6month to maturity is $11. Assume that Intel will not pay any dividend within the next 6month and assume that the risk free interest rate (.) is 10%. Is there an arbitrage? A. For a stock does not pay dividends: c(S,K,t,T)?max[0,S(t)KB(t,T)] C(S,K,t,T)?max[0,S(t)KB(t,T)] Proof: To prove this we only need to show (why?) c(S,K,t,T)?S(t)KB(t,T) We show this by contradiction. If c SKB, we have an arbitrage. This implies that American calls on nondividendpaying stocks will never be exercised earlier. (Intuition?) An arbitrage: Transaction Payoff (at t) Payoff (at T) c St KB Max[0,S(T)K] S(T) K SKBc Max[0,S(T)K] [S(T)K] B. For European puts on nondividendpaying stocks, a similar arbitrage argument shows that: Intuition?) p(S,K,t,T)?max[0,KB(t,T)S] C. Combining these rules implies that the value of a European call on a nondividendpaying stock