In a binomial pricing model, the risk-neutral up probability ( p^ ) can be calculated using the relationship between the risk-free rate, the up factor, and the down factor. Given that ( r = 0.1 ) (the risk-free interest rate), ( u = 1.2 ) is the up factor (the asset price increases by 20% when the underlying asset moves up, and ( d = 0.9 ) is the down factor (the asset price decreases by 10% when it moves down), we can find ( p^ ) using the following equation:

[ p^* = \frac{(u / (u e^r) - d}{(u - d)(e^r - 1)} ]

Plugging in the values: [ p^ = \frac{(1.2 * e^{0.1}) - 0.9]}{(1.2 - 0.9)(e^{0.1} - 1)} ] [ p^ = \frac{1.32 - 0.81}{0.3(e^{0.1} - 1)} ] [ p^* = \frac{0.51}{0.3(e^{0.1} - 1)} ]

Since we don't know the value of ( e^{0.1} ), we cannot calculate an exact numerical value for ( p^* ). However, without knowing the actual value of ( e^{0.1} ), we cannot definitively determine if the correct answer is a, b, c, or d. Therefore, the correct answer to this multiple-choice question is:

d. ( p^* ) cannot be calculated as SO is not given.