链接
http://www.columbia.edu/~mh2078/FoundationsFE/Assign7.pdf
(The Deflated Gains Process is a Q-martingale)
已知 \[dS_t = (r-q)S_tdt + \sigma S_tdW_t^Q\] 求证 \[S_0 = E_0^Q \left[ \int_0^T e^{-rt}qS_tdt + e^{-rT}S_T \right]\]
观察形式，就是含息债券的定价公式而已，用 \(r\) 贴现。
Proof：
可以交换期望和积分顺序。
\[\frac{d(e^{-rt}S_t)} {dt} = -re^{-rt}S_t + e^{-rt} \frac{dS_t}{dt}\]
套上期望式时，布朗运动项消去，用换元的技巧
\[E[dS_t] = E[(r-q)S_tdt]\]
\[E\left[\frac{d(e^{-rt}S_t)} {dt} \right] =E \left[ -re^{-rt}S_t + e^{-rt} \frac{dS_t}{dt} \right] = E\left[ -qe^{-rt}S_tdt \right]\]
\[\therefore E\left[\int_0^T e^{-rt}qS_tdt\right] = E\left[ \int_0^T \frac{e^{-rt}qS_tdt}{-qe^{-rt}S_tdt} d(e^{-rt}S_t)\right] = E\left[ \int_0^T - d(e^{-rt}S_t)\right] = E\left[ S_0 - e^{-rT}S_T\right] = S_0- E\left[e^{-rT}S_T\right]\]
代入结论的表达式得证。
(Deriving Black-Scholes When the Underlying Pays a Dividend Yield)
有付息的定价在 B-S-Model-Revisited 中给出了推导，是将股价的GBM的参数 \(\mu\) 从 \(r\) 调整到 \(r-q\)。这样做显然是因为股票持有期的收益 = 分红 + 股价上涨，那么股价上涨的漂移系数肯定要扣除 \(q\)。
\[\begin{aligned} C_t &= e^{r(t-T)}E[\max{(S_T-K, 0)}] \\ &= e^{r(t-T)} \big(E[S_T|S_T>K] - KP(S_T>K) \big) \\ &= e^{r(t-T)} \big(S_0 e^{\mu T}N(d_1) - KN(d_2) \big) \qquad \text{By } \mu=r - q \\ &= S_0e^{rt-qT} N(d_1) - e^{r(t-T)}KN(d_2) \\ &= S_t e^{rt-qT - (r-q)t} N(d_1) - e^{r(t-T)}KN(d_2) \\ &= S_t e^{q(t-T)} N(d_1) - e^{r(t-T)}KN(d_2) \end{aligned}\]
\(d_1, d_2\) 也将 \(\mu = r-q\)
\[d_2 =\frac{\ln{(S_0/K)} + (r-q - \sigma^2 /2)T}{\sigma \sqrt T}\]
\[d_1 = \sigma \sqrt T + d_2\]
(Futures Contracts and Black-Scholes)
- 用鞅定价理论推导期货 \(F_t^{(T)}\) 的价格，标的物为 \(T\) 时的股票。假设 B-S 成立、股票分红率 \(q\)。
类似 european call 的推导，仍然假设标的资产 \(S\) 满足 \(dS = \mu Sdt + \sigma S dW_t\)
根据鞅定价理论
\[F_t = e^{r(t-T)} E[S_T]\]
\(S_T\) 已知
\[S_T = S_0 e^{(r-q-\frac{\sigma^2}{2})T + \sigma \sqrt T Z}\]
考虑 \(E[e^{X}]\)，\(X\sim N(0, \sigma^2)\)
\[\begin{aligned} E[e^{X}] &= \frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{\infty} e^{ x} e^{-\frac{x^2}{2\sigma^2}}dx \\ &= \frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{\infty} e^{x-\frac{x^2}{2\sigma^2}}dx \qquad \text{Note that } x-\frac{x^2}{2\sigma^2} = \frac1{2\sigma^2}\left(2\sigma^2 x - x^2\right)= \frac1{2\sigma^2}\left(-(\sigma^2-x)^2 +\sigma^4 \right) = \frac{\sigma^2}{2} - \frac12 \left(\sigma - \frac x \sigma\right)^2\\ &= \frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{\infty} e^{\frac{\sigma^2}{2} - \frac12 \left(\sigma - \frac x \sigma\right)^2}dx \\ &= \frac{1}{\sqrt{2\pi}\sigma} e^{\frac{\sigma^2}{2}} \int_{-\infty}^{\infty} e^{ - \frac12 \left(\sigma - \frac x \sigma\right)^2}dx \\ &= \frac{1}{\sqrt{2\pi}\sigma} e^{\frac{\sigma^2}{2}} \int_{-\infty}^{\infty} e^{ - \frac12 u^2}\sigma du \qquad \text{Substitute } u= \frac x\sigma - \sigma \\ &= \frac{1}{\sqrt{2\pi}} e^{\frac{\sigma^2}{2}} \int_{-\infty}^{\infty} e^{ - \frac12 u^2} du \\ &= \frac{1}{\sqrt{\pi}} e^{\frac{\sigma^2}{2}} \int_{-\infty}^{\infty} e^{ - v^2} dv \qquad \text{Substitute } v = \frac{u}{\sqrt{2}} \\ &= e^{\frac{\sigma^2}{2}} \qquad \text {Gaussian Integral equals } \sqrt\pi \end{aligned}\]
P.S. 其实有对数正态分布的期望公式可以直接使用。
所以
\[\begin{aligned} E[S_T] &= S_0 e^{(r-q-\frac{\sigma^2}{2})T} E[e^{\sigma \sqrt T Z}] \\ &= S_0e^{(r-q-\frac{\sigma^2}{2})T} e^{\frac{\sigma^2 T}{2}} \\ &= S_0e^{(r-q)T} \\ &= S_te^{(r-q)(T-t)} \qquad \text{Change the starting time to }t \end{aligned}\]
\[\begin{aligned} F_t &= e^{r(t-T)} E[S_T] \\ &= e^{r(t-T)} (S_te^{(r-q)(T-t)})\\ &=S_t e^{q(t-T)} \end{aligned}\]
或者用 \(S_0\) 表示
\[F_t = e^{r(t-T)}S_0e^{(r-q)T} = S_0e^{rt-qT}\]
和带 dividend 的股票的 european call 对比
\[\begin{aligned} C_t &= S_t e^{q(t-T)} N(d_1) - Ke^{r(t-T)}N(d_2) \end{aligned}\]
- 以该期货为标的的期权 (option on the futures) 定价。假设期货在 \(T\) 到期而期权在 \(\tau < T\) 到期。
如果假设股价服从几何布朗运动，则 \(S_t = S_0e^{(r-q - \sigma^2/2)t+\sigma\sqrt tZ}\)，根据上文导出的期货定价，期货价格过程是
\[F_t = S_0e^{(r- \sigma^2/2)t -qT +\sigma\sqrt tZ} = S_0e^{-qT} e^{(r- \sigma^2/2)t+\sigma\sqrt tZ}\]
注意到
\[F_0 = S_0e^{-qT}\]
\[F_t = F_0e^{(r- \sigma^2/2)t+\sigma\sqrt tZ}\]
和几何布朗运动的定义一致。所以 \(F\) 就等价于一个不付息的 Stock，漂移系数就是 \(r\)。
既然如此，可以直接套用 BS 给出的期权定价公式 (不含息) 了。注意，期权
\[C_t = F_tN(d_1) - e^{-r(\tau - t)K}N(d_2)\]
其中
\[d_1 = \frac{\ln{(F_t/K)}+(r+\sigma^2/2)(\tau-t)}{\sigma\sqrt{\tau-t}}\]
\[d_2 = d_1 - \sigma\sqrt{\tau -t}\]
- 确定该期权价格满足问题2的 BS PDE。
该期权的定价其实和 no dividend 时的 BS 推导结果一致，因为仅仅看它的 underlying \(F_t\) 是 no dividend 的。