# 三項モデルはBlack-Scholesに弱く収束する

Consider risk-neutral trinomial model with $N$ periods presented by $$S_{(k+1)\delta}H_{k+1}, \ \ \text{for} \ \ k=0,\ldots,N-1$$ where $\delta:=\frac{T}{N}$ and $\{H_k\}_{1}^{N}$ is a sequence of i.i.d random variables with distribution $$H_k = \begin{cases} e^{\delta(r-\sigma^2/2)+\sqrt{3\delta}\sigma} \ &\text{with probability} \ \hat{\pi} = \frac{1}{6}\\ e^{\delta(r-\sigma^2/2)} \ &\text{with probability} \ 1 - \hat{\pi} = \frac{2}{3}\\ e^{\delta(r-\sigma^2/2)-\sqrt{3\delta}\sigma} \ &\text{with probability} \ \hat{\pi} = \frac{1}{6}\\ \end{cases}$$ and $\hat{\pi}$ < 1/2. Show that as $\delta\rightarrow 0$, this trinomial model converges to the Black-Scholes model in the weak sense. Hint: Find $Z_k$ such that $\ln(H_k) = (r - \sigma^2/2)\delta + \sigma\sqrt{\delta}Z_k$. Then show (3.6)

（3.6）は、$\ hat {\ mathbb {E}} [Z_1] = o（\ delta）$と$\ hat {\ mathbb {E}} [Z_1 ^ 2] = 1 + o（1）$ $\ frac {1} {\ sqrt {N}} \ sum_ {k = 1} ^ {N} Z_k$は$\ mathcal {N}（0,1）$に弱く収束します。

Attempted solution: Let $\{Z_k\}_{1}^{N}$ be a sequence of i.i.d. random variables with the following distribution $$Z_k = \begin{cases} \alpha \ &\text{with probability} \ \hat{\pi}\\ -\beta \ &\text{with probability} \ 1-\hat{\pi} \end{cases}$$ such that $\ln(H_k) = (r - \sigma^2/2)\delta + \sigma\sqrt{\delta}Z_k$.

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## 1 答え

あなたの$Z_k$は3つの値を持つ必要があります。可能な値ごとに$\ ln（H_k）$の値を書くだけでいいです

$X$を$（x_1、x_2、...、x_n）$と$p_i = P（X = X_i）$とすると、$P（f（X）= f（x_i））= \ sum_ {j = 1} ^ n p_j \ mathbf {1} _ {f（x_j）= f（x_i）}$

$f$が1対1のマッピングであれば、$f（x_j）= f（x_i）\ Rightarrow i = j$と$P（f（X）= f（x_i））= p_i$となる。

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