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Answer: The option delta is approximately 0.62, therefore if the stock price increases by USD 1, the option price would increase by approximately USD 0.62.
## Explanation The delta (Δ) of an option position can be calculated using the formula: $$\Delta = \frac{f_u - f_d}{S_u - S_d}$$ Where: - $f_u$ = option value at the upper node (B) = 9.25 - $f_d$ = option value at the lower node (C) = 1.02 - $S_u$ = stock price at the upper node (B) = 98.88 - $S_d$ = stock price at the lower node (C) = 85.59 Calculating the delta: $$\Delta = \frac{9.25 - 1.02}{98.88 - 85.59} = \frac{8.23}{13.29} = 0.6193 \approx 0.62$$ **Why B is correct:** - Delta represents the sensitivity of the option price to changes in the underlying stock price - A delta of 0.62 means that for every $1 increase in the stock price, the option price increases by approximately $0.62 - This is the fundamental interpretation of delta in option pricing **Why A is incorrect:** - While the delta calculation is correct, the interpretation about the portfolio having the same value at nodes B and C is not accurate - Delta hedging creates a risk-free portfolio only when continuously rebalanced, not necessarily having the same value at different nodes **Why C and D are incorrect:** - The delta is approximately 0.62, not 0.94 - The calculation shows clearly that delta = 0.62 based on the given binomial tree data **Note:** As mentioned in the explanation, this technically gives the delta at time t (between nodes), but when the time step is very short, there is little difference between delta at time zero and delta at time t.
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A junior trader at an equity hedge fund is assessing the market risk of an option position. The option is a call option on stock MTP, which is currently trading at USD 92 per share. The trader has valued the option using a binomial tree, the three left-most nodes of which are shown below:
Stock price A
92.00
5.20
Option value
B
98.88
9.25
C
85.59
1.02
Stock price A
92.00
5.20
Option value
B
98.88
9.25
C
85.59
1.02
Given this information, and assuming the length of each time step in the tree is very short, which of the following conclusions would the trader be correct to reach?
A
The option delta is approximately 0.62, therefore a portfolio consisting of a short position in 62 options and a long position in 100 shares would have the same value at node B or node C.
B
The option delta is approximately 0.62, therefore if the stock price increases by USD 1, the option price would increase by approximately USD 0.62.
C
The option delta is approximately 0.94, therefore a portfolio consisting of a short position in 94 options and a long position in 100 shares would have the same value at node B or node C.
D
The option delta is approximately 0.94, therefore if the stock price increases by USD 1, the option price would increase by approximately USD 0.94.