6 Options II

# 57 c as the time intervals get shorter ie as n and

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Unformatted text preview: bound, start with put- call parity for European options. 1) c + Ke−rT = p + (S0 – D0) ⇒ c – p = (S0 − D0) – Ke−rT. 2) In general, P ≥ p. 3) To start, let dividends D0 = 0. • In this case, we expect C = c • If C − p = S0 – Ke−rT, then C − P ≤ S0 – Ke−rT 4) If D0 rises (D &gt; 0), C falls and P rises, and we still conclude that C − P ≤ S0 – Ke−rT. 3. Arbitrage Opportunities when European Put- Call Parity is Violated: a) Setting. 1) Assume the following: rf = 10%/yr, T = 6 months, S0 = \$25, K = \$20, D0 = 0. 2) Parity implies that c + 20e−0.10(1/2) = p + 25, or c = p + 5.98. b) Example #5 – European call undervalued (or put overvalued). 1) Suppose that c = \$7 when p = \$2, so c &lt; p + 5.98 = 7.98. 2) Arbitrage opportunity. • At time t = 0, short sell the stock to raise \$25 in cash, write/sell put for \$2, buy call for \$7, and invest the remaining \$20 of cash (25 + 2 − 7) at rf = 10%. • At time t = T, cash is worth 20e0.10(1/2) = \$21.03: o If ST = \$21 &gt; K = \$20, put expires. Exercise call, pay \$20 to buy stock and close out short position. Profit = \$1.03. o If ST = \$19 &lt; K = \$20, call expires and put is exercised. Pay \$20 to buy stock from put holder and close out short position. Profit = \$1.03. 3) Arbitrageurs buying this call and selling this put will drive c up and p down. c) Example #6 – European put undervalued (or call overvalued). 1) Suppose that c = \$9 when p = \$2, so c &gt; p + 5.98 = 7.98. 2) Arbitrage opportunity. • At time t = 0, write/sell call for \$9 and borrow S0 + p − c = \$18. Buy put for \$2 and stock for \$25. • At time t = T, loan debt is 18e0.10(1/2) = \$18.92: o If ST = \$21 &gt; K = \$20, put expires and call is exercised. Sell stock to call holder for \$20, pay off loan for \$18.92, and pocket \$1.08. o If ST = \$19 &lt; K = \$20, call expires. Exercise put and sell stock for \$20. Pay off loan for \$18.92, and pocket \$1.08. 3) Arbitrageurs buying this put and selling this call will drive p up and c down. Ec 174...
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