Stochastic Processes · 免费预览
布莱克-斯科尔斯公式
令二项式步长趋于零，中心极限定理把对数价格化为带漂移的布朗运动；风险中性方法在漂移为 r 减去 sigma 平方再除以 2 的情形下对期望进行贴现，由此得到的布莱克-斯科尔斯公式对一只真实的「Google」看涨期权给出的价格，与其市场价格的偏差不超过 0.17 美元。
1. Shrinking the steps
Shrinking the steps
2. The risk-neutral drift
The risk-neutral drift
3. The formula in practice
The formula in practice
4. Why the kink smooths out
Why the kink smooths out
5. What the limit theorem does
What role does the central limit theorem play in passing from the binomial tree to continuous time?
6. Three defining properties
Which triple defines a standard Brownian motion?
7. Reading the volatility
In the model S t equal to S 0 e to the mu t plus sigma B t, what does sigma measure?
8. Choosing the pricing drift
What drift must the stock use under the risk-neutral measure in continuous time?
9. The Google sanity check
The Black-Scholes formula predicted 32.93 for the Google call while the market traded it at 33.10. What does the gap of 0.17 suggest?
10. What the PDE encodes
What is the financial meaning of the Black-Scholes partial differential equation?
11. The risk-neutral growth rate
A stock has volatility sigma equal to 0.4 and the bank pays r equal to 0.05. Compute the risk-neutral drift mu star equal to r minus sigma squared over 2, and write the answer as a decimal.
12. From d1 to d2
A call has d1 equal to 0.5 with sigma equal to 0.3 and t equal to 0.25 years. Using d2 equal to d1 minus sigma times the square root of t, compute d2 and write the answer as a decimal.
13. Gap to the market
The Black-Scholes prediction for the Google call was 32.93 and the market price was 33.10. Compute the absolute gap and write the number of dollars.