【正文】 S(HKD/ bid USD) and S(HKD/ ask USD), can I conclude that the exchange rate of transaction cost is [ S(HKD/ bid USD) + S(HKD/ ask USD) ] / 2 ? In other words, x = z ? 35 Suppose HKD =1 USD, fixed by HKMA S(HKD/ bid USD) 1 USD ( x) HKD S(HKD/ ask USD) 1 USD ( + z) HKD 36 Nonzero transaction cost ?With nonzero transaction cost, S(j/ask k) S(j/bid k) in general ?S(j/ask k) = 1/S(k/bid j) ?S(j/bid k) = 1/S(k/ask j) ?Noarbitrage condition gives only ?S(1/ask 3) ? S(1/bid 2)S(2/bid 3) 37 S(HKD/USD) ? S(HKD/JPY) S(JPY/USD) without transaction cost A HKD AB JPY ABC USD S(JPY/HKD)=B S(USD/JPY)=C S(HKD/USD)=A 1 USD A HKD 38 Positive arbitrage profit if ? ABC 1 ? S(HKD/USD)*S(JPY/HKD)*S(USD/JPY) 1 ? S(JPY/HKD)*S(USD/JPY) 1/S(HKD/USD) ? S(JPY/HKD)*S(USD/JPY) S(USD/HKD) ? No arbitrage in this direction if ABC ? 1 Note that when transaction cost is zero 1/S(HKD/USD) =S(USD/HKD) 39 How do the arbitrage activities eliminate the arbitrage opportunities? ?Note that we have three markets: ? 1. USD HKD, 2. HKD JPY, 3. JPY USD ? arbitrage opportunities are present if ABC = S(HKD/USD)*S(JPY/HKD)*S(USD/JPY) 1 ?if ABC1 the route of arbitrage is USD ? HKD ? JPY ? USD ? HKD ? .... 40 If ABC1 the route of arbitrage is USD ? HKD ? JPY ? USD ? HKD ? .... ?In the USD HKD market, we sell (supply) USD and buy (demand) HKD. S(HKD/USD) = A will decrease. ?In the HKD JPY market, we sell (supply) HKD and buy (demand) JPY. S(JPY/HKD) = B will decrease. ?In the JPY USD market, we sell (supply) JPY and buy (demand) USD. S(USD/JPY) =C will decrease. ?Now ABC decreases because A, B and C decrease as arbitrage activities occur. ?The arbitrage activities continues until ABC = 1. 41 S(HKD/USD) ? S(HKD/JPY) S(JPY/USD) without transaction cost (reversing the previous strategy) A HKD AB JPY ABC USD S(HKD/JPY)=1/B S(JPY/USD)=1/C S(USD/HKD)=1/A 1 USD A HKD 42 Positive arbitrage profit if ? ABC 1。 k=. dollars ?S(HK/US)= approximately. 21 S(HKD/USD) vs S(HKD/JPY) S(JPY/USD) ?Suppose no transaction cost ?To get HKD, a holder of USD can use the rate S(HKD/USD) ?Or, he/she can buy JPY at the rate S(JPY/USD) and then use JPY to buy HKD at the rate S(HKD/JPY). In effect he/she is exchanging at a rate of S(HKD/JPY)S(JPY/USD) 22 S(1/3) vs S(1/2) S(2/3) 1 = HKD。 ., 1/(ABC)1 ? S(HKD/USD)*S(JPY/HKD)*S(USD/JPY) 1 ? S(JPY/HKD)*S(USD/JPY) 1/S(HKD/USD) ? S(JPY/HKD)*S(USD/JPY) S(USD/HKD) ? No arbitrage in this direction if ABC ? 1 Note that when transaction cost is zero 1/S(HKD/USD) =S(USD/HKD) 43 Thus, without transaction cost, we must have ? ABC = 1 ? S(HKD/USD)*S(JPY/HKD)*S(USD/JPY) = 1 ? S(JPY/HKD)*S(USD/JPY) = 1/S(HKD/USD) ? S(JPY/HKD)*S(USD/JPY) = S(USD/HKD) 44 with transaction cost A HKD AB JPY ABC USD S(JPY/bid H