Stochastic Processes · 免费预览
二叉模型
在二叉模型中，股票每期乘以 u 或 d，而现金按 1 加 r 的因子增长；风险中性概率 p* 等于 1 加 r 减 d 与 u 减 d 之比，它将任何期权定价为贴现后的期望值，而对冲比率与期权变动和股票变动之比相匹配。
1. One period with interest
One period with interest
2. A trade that cannot lose
A trade that cannot lose
3. Many periods, backwards
Many periods, backwards
4. The martingale shortcut
The martingale shortcut
5. Keeping the model honest
Why must the model parameters satisfy 0 < d < 1 + r < u?
6. The pricing probability
For a one-period binomial model with multipliers u and d and interest rate r, what is the risk-neutral probability of the up move?
7. Reading the hedge ratio
The hedge ratio Delta 0 equals (V1(H) - V1(T)) / (S1(H) - S1(T)). What does it tell a trader?
8. Checking the worked probability
In the 60-stock example with u = 4/3, d = 5/6 and r = 1/18, the risk-neutral probability is computed as which value?
9. Why 5,000 shares
A trader sells 10,000 calls struck at 65 on the 60-stock and computes Delta 0 = 1/2. Why buy exactly 5,000 shares?
10. The shortcut and its meaning
Why is V0 equal to the discounted expected terminal payoff under the risk-neutral probabilities?
11. Pricing the traded call
For the 60-stock with u = 4/3, d = 5/6, r = 1/18 and a call struck at 65 paying 15 or 0, compute the no-arbitrage value V0 as an exact fraction or a decimal.
12. Shares to hold
For the same call, the two option values are 15 and 0 while the two stock values are 80 and 50. Compute the hedge ratio Delta 0 as a fraction or decimal.
13. A tree value by recursion
In the convenient model u = 2, d = 1/2, r = 1/4 the pricing recursion is V_n = 0.4 (V_up + V_down). At the node where the stock is 8 at time 2, the put struck at 10 has children worth 0 (at 16) and 6 (at 4). Compute this node's value and write the number.