题目
A market maker writes 100 at-the-money call option contracts and delta hedges dynamically by daily rebalancing of a long position in the underlying shares. The delta hedge is based on an implied volatility assumption of approximately 10% per annum. However, at the end of the month, the realized (actual subsequent) volatility of the stock was over 20%. However, the stock fluctuated both up and down roughly evenly. If borrowing occurs at the constant risk-free rate, and transaction costs are ignored, what is the net profit (loss) to the market maker at the end of the month?
选项
A.Net loss due to gamma exposure
B.Net loss due to theta (time decay)
C.Approximately break-even due to the almost continuous delta hedge and roughly even up\/down movements
D.Net gain due to the gamma exposure
答案
A
解析
If the realized volatility matched the assumed volatility (which informs the delta!), then the market maker should be roughly break-even due to the delta-hedge; this is another way of viewing Hull's statement that the discounted cost of hedging should roughly equal the Black- Scholes price.However, the market marker, who is short options, remains "short gamma" (aka, "long short convexity"); i.e., the long shares have per share delta of 1.0 but gamma of zero (do you see why? the delta of 1.0 does not vary). The market maker's option counterparty, on the other hand, is "long gamma."In this case, a realized volatility higher than assumed implies losses to the short gamma position.In regard to (B), while time decay has a second-order impact on gamma (i.e., for ATM options, gamma increases as time to maturity decreases; so this effect only compounds the short gamma), the shorter maturity is already re-computed in each daily re-balancing.如果实现的波动率与假设的波动率相匹配（已知delta），则做市商应因delta对冲而大致达到收支平衡。 这是查看Hull关于对冲折现成本应大致等于BSM价格的说法的另一种方式。然而，做市商卖出期权，仍然是空头gamma；也就是说多头股票的差额为1.0，但是gamma为0. 另一方面，做市商的对手方的期权是多头gamma