Pembahasan Ujian PAI: F10 – No. 07 – November 2025

• Model pohon binomial untuk proses pergerakan harga saham
• Harga saham pada langkah waktu ke-\(n\): \(\{x_{n} : 0 \leq n \leq 3\}\).
• \(S = x_{0} = \$100\) ;
• \(u = 1 + 10\% = 1{,}1\) ;
• \(d = 1 – 10\% = 0{,}9\) ;
• Tingkat suku bunga bebas risiko efektif tahunan: \(r = 5\%\);
• Strike price put option pada langkah ke-3: \(K_{p3}= \$110\)

• Harga saham di setiap kemungkinan (nodes pohon binomial):
  $$\begin{aligned}
  Su &= S \times u \\
  Sd &= S \times d
  \end{aligned}$$
• Payoff opsi put:
  $$P = \max\left[K – S,\, 0\right]$$
• Risk Neutral Probability:
  $$p^{*} = \frac{(1 + r) – d}{u – d}$$

Harga saham di setiap kemungkinan pada \(n = 3\):

$$\begin{aligned}
Su^{3} &= 100\,(1{,}1)^{3} = 133{,}1 \\
Su^{2}d &= 100\,(1{,}1)^{2}(0{,}9) = 108{,}9 \\
Sud^{2} &= 100\,(1{,}1)(0{,}9)^{2} = 89{,}1 \\
Sd^{3} &= 100\,(0{,}9)^{3} = 72{,}9
\end{aligned}$$

Payoff opsi put di setiap kemungkinan pada \(n = 3\):

$$\begin{aligned}
P_{uuu} &= \max\left[110 – 133{,}1,\; 0\right] = 0 \\
P_{uud} &= \max\left[110 – 108{,}9,\; 0\right] = 1{,}1 \\
P_{udd} &= \max\left[110 – 89{,}1,\; 0\right] = 20{,}9 \\
P_{ddd} &= \max\left[110 – 72{,}9,\; 0\right] = 37{,}1
\end{aligned}$$

Hitung peluang risk neutral:

$$\begin{aligned}
p^{*} &= \frac{(1 + r) – d}{u – d} \\
&= \frac{(1 + 5\%) – 0{,}9}{1{,}1 – 0{,}9} \\
&= \frac{0{,}15}{0{,}2} = 0{,}75
\end{aligned}$$

Hitung harga opsi put sekarang, yakni pada \(n = 0\) dengan backward steps:

$$\begin{aligned}
P &= (1 + r)^{-3}(Pu^{3}(p^{*})^{3} + Pu^{2}d(p^{*})^{2}(1 – p^{*}) + Pud^{2}(p^{*})(1 – p^{*})^{2} + Pd^{3}(1 – p^{*})^{3}) \\
&= (1{,}05)^{-3}\left[0 \times (0{,}75)^{3} + (1{,}1)(0{,}75)^{2}(0{,}25) + (20{,}9)(0{,}75)(0{,}25)^{2} + 37{,}1(0{,}25)^{3}\right] \\
&= 3{,}44
\end{aligned}$$

Jadi, harga saat ini dari opsi jual tersebut adalah 3,44.