题目
Patty and Peter are risk analysts who are attempting to utilize the Black-Scholes-Merton option pricing model (BSM OPM) in order to price a call option on a publicly-traded stock. Their BSM OPM is a simple Excel model; of course they can modify inputs and perform calculations, but they are not prepared to make sophisticated adjustments to the model. Consider the following five issues: I. The both agree that the stock's volatility is not constant II. They both believe that continuously compounded returns on the stock exhibit leptokurtosis; i.e., are heavy-tailed III. They disagree on the stock's expected return: Patty believes E[return] is only +8.0% but Peter believes E[return] is at least +13.0% IV. The stock pays a dividend and they agree on the forecast, but the dividend assumption consists of multiple, quarterly (aka, lumpy) dollar payouts rather than a continuous dividend yield assumption V. They both agree that the stock will pay a continuous dividend yield of 5.0%, but the call option is an American style option (and they do require a convenient analytical solution per their simple Excel model) Which of the issues above creates a genuine theoretical problem that cannot be easily addressed such that their basic BSM model's output will not be highly dependable?
选项
A.None of these are real problems (each is easily addressed by the basic BSM OPM)
B.Only III. and IV. are real problems
C.Only I., II., and V. are real problems
D.All of these are real problems (none can be easily addressed by the basic BSM OPM)
答案
C
解析
They both agree that the stock's volatility is not constant: this is a real problem because the BSM assumes constant volatility and volatility is a critical input, unlike say the risk-free input. If the risk-free rate is stochastic, we are not as concerned They both believe that continuously compounded returns on the stock exhibit leptokurtosis; i.e., are heavy-tailed: this is a real problem because the critical assumption is that log returns exhibit a normal distribution; aka, prices exhibit a lognormal distribution. They disagree on the stock's expected return: Patty believes E[return] is only 8.0% but Peter believes E[return] is at least 13.0%: this is irrelevant because the model does not make an assumption about the stock's expected return The stock pays a dividend and they agree on the forecast, but the dividend assumption consists of multiple, quarterly (aka, lumpy) dollar payouts rather than a continuous dividend yield assumption: this is not even a problem, as the model easily handle dividends of either sort. In fact, lumpy dividend can be translated into their continuous equivalent. They both agree that the stock will pay a continuous dividend yield of 5.0%, but the call option is an American style option and they do require a convenient analytical Merton-like solution per their Excel model: this is a real problem because the adjustment requires more advanced numerical solutions 股票的波动率不是恒定的:这是一个真实的问题，因为BSM假设波动率是恒定的，而波动率是一个关键的输入，不像无风险的输入。如果无风险利率是随机的，我们就不那么担心了。股票支付红利和他们达成一致预测,但假设股息由多个季度(aka,块状)美元支出而不是一个连续的股息收益率假设:这不是一个问题,因为该模型容易处理的股息。事实上，波动股利可以转换成它们的连续等价物。他们都一致认为,股票将付出持续的股息收益率为5.0%,但看涨期权是一个美式期权,他们需要一个方便的分析Merton-like解决方案/Excel模型:这是一个真正的问题,因为调整需要更先进的数值解。