To compute the price of a put option with a strike price of $4755 using the information given, we can use the put-call parity, which relates the price of a call option and a put option with the same strike price. The put-call parity can be expressed as:

[ C(K) + Ke^{-rT} = S + P(K) ]

Where: - ( C(K) is the price of a call option with strike K, - ( P(K) is the price of a put option with the same strike K, - ( S ) is the current index value (SO), - ( r ) is the annual risk-free interest rate, - ( T ) is the time to expiration.

Given that the put-call parity holds, we can rearrange the formula to solve for ( P(4755) ):

[ P(4755) = S + P(4755) - C(4755)e^{-rT} ]

We are given: - ( S = $4754 ) (index value on January 8, 2024), - ( r = 0.04 ) (annual interest rate), - ( T = 2/12 ) (time to expiration in years), - ( P(4750) = $78.25 ) (put option price with a strike of $4750, which is the closest available data point to $4755).

Now, we need to estimate ( C(4755) ). Since we don't have the exact price for a call with a strike of $4755, we can interpolate between the prices of ( P(4740) ) and ( P(4750) ) to get a better estimate for ( P(4755) ).

[ P(4755) \approx \frac{1}{2}(P(4740) + P(4750)) ]

[ P(4755) \approx \frac{1}{2}(75.15 + 78.25) ]

[ P(4755) \approx 76.70 ]

Now, applying the put-call parity:

[ P(4755) = 4754 + 76.70 - 4750e^{-0.04 \cdot 2/12} ]

[ P(4755) \approx 4754 + 76.70 - 4750e^{-0.04 \cdot 2/12} ]

[ P(4755) \approx 76.70 + 4754 - 4750e^{-0.04 \cdot 2/12} ]

[ P(4755) \approx 76.70 + 4754 - 4750 \cdot 0.983 ]

[ P(4755) \approx 76.70 + 4754 - 4732.6 ]

[ P(4755) \approx 447.1 ]

However, since we don't have the exact price for a call with a strike of $4755, we can interpolate between the prices of ( P(4740) ) and ( P(4750) ) to get a better estimate for ( P(4755) ).

[ P(4755) \approx \frac{1}{2}(P(4740) + P(4750)) ]

[ P(4755) \approx \frac{1}{2}(75.15 + 78.25) ]

[ P(4755) \approx 76.70 ]

Now, applying the put-call parity:

[ P(4755) = 4754 + 76.70 - 4750e^{-0.04 \cdot 2/12} ]

[ P(4755) \approx 447.1 ]

Therefore, the best estimate for ( P(4755) ) is approximately $447.10.

So, the correct answer is:

b. 79.90