B is correct. The delta ($\Delta$) of an option position can be calculated as:

$\Delta = \frac{f_u - f_d}{S_u - S_d}$

where:

• $S$ is the current price of the underlying stock,
• $S_u$ is the stock price after a move up in the binomial tree,
• $f_u$ is the option value given this stock price increase,
• $S_d$ is the stock price after a move down in the binomial tree, and
• $f_d$ is the option value given this stock price decrease.

Using the information in the given tree, the option delta is approximately:

$\Delta = \frac{9.25 - 1.02}{98.88 - 85.59} = 0.6193$

(As noted in the VRM text, this technically gives the delta at time $t$, not at time zero. However, when $t$ is small, there is little difference between the delta at time zero and the delta at time $t$.)

A correct interpretation of this result is that when the stock price increases by a small amount, the call option price increases by about 62% of that amount.

A is incorrect. The value of the portfolio at time $t$ is $S_u\Delta - f_u$ if the stock price increases and $S_d\Delta - f_d$ if the stock price decreases, which are equal for the value of delta in the equation above. Therefore, a portfolio with a short position in 100 options and a long position in 62 shares would have approximately the same value at nodes B and C. Option A incorrectly reverses the position sizes.

C is incorrect. This incorrectly calculates delta as $\Delta = \frac{S_d - f_d}{S_u - f_u}$, and also makes the incorrect reversal of position size described in the explanation for A above.

D is incorrect. This incorrectly calculates delta as $\Delta = \frac{S_d - f_d}{S_u - f_u}$.

Learning Objective: Define and calculate delta of a stock option.

Reference: Global Association of Risk Professionals, Valuation and Risk Models (New York, NY: Pearson, 2023). Chapter 14. Binomial Trees [VRM-14]