Using initial inputs:

$$p_{\text{up}} = \frac{(e^{(r \times \Delta t)} - d)}{(u - d)} = \frac{(e^{3\%} - e^{-0.3})}{(e^{0.3} - e^{-0.3})} = 0.4756$$

The figure below shows the stock price and the corresponding option’s intrinsic value at each node.

• Node A:

  • Index Price: 100
  • Option’s Strike price: 90
  • Option’s Value: ?

• Node B:

  • Index Price: 134.99
  • Option’s Strike price: 90
  • Option’s Value: 0

• Node C:

  • Index Price: 74.08
  • Option’s Strike price: 90
  • Option’s Value: 15.92

Price at A = $(0 \times 0.4756 + 15.92 \times (1 - 0.4756)) \times e^{(-3.0\%)} = 8.10$

Using corrected inputs (annual volatility = 50%):

$$u = e^{0.5} = 1.6487$$ $$d = e^{-0.5} = 0.6065$$ $$p_{\text{up}} = \frac{(e^{0.03} - 0.6065)}{(1.6487 - 0.6065)} = 0.4067$$

• Node B (Up):

  • Index Price: $100 \times 1.6487 = 164.87$
  • Option’s Value: 0

• Node C (Down):

  • Index Price: $100 \times 0.6065 = 60.65$
  • Option’s Value: $\max(90 - 60.65, 0) = 29.35$

Corrected Price at A = $(0 \times 0.4067 + 29.35 \times (1 - 0.4067)) \times e^{-0.03} = 16.90$

Change in price = $16.90 - 8.10 = +8.80$ USD.