Would really appreciate it if someone could provide some last minute guidance on this. q6 is looking at a binomial tree american style option and asks to calculate the value of this call. the below explanation for how they do this is confusing to me

An American-style put can be exercised early. At Time Step 1, for the up move, *p*^{+} is 0.2517 and the put is out of the money and should not be exercised early (*X* < *S*, 40 < 49.4). However, at Time Step 1, *p*^{–} is 8.4350 and the put is in the money by 9.60 (*X* – *S* = 40 – 30.40). So, the put is exercised early, and the value of early exercise (9.60) replaces the value of not exercising early (8.4350) in the binomial treeAn American-style put can be exercised early. At Time Step 1, for the up move, *p*^{+} is 0.2517 and the put is out of the money and should not be exercised early (*X* < *S*, 40 < 49.4). However, at Time Step 1, *p*^{–} is 8.4350 and the put is in the money by 9.60 (*X* – *S* = 40 – 30.40). So, the put is exercised early, and the value of early exercise (9.60) replaces the value of not exercising early (8.4350) in the binomial tree

My q is why do they include the 0.2517 if thats not in the money and should be exercised? totally confused here. any comments will be very appreciated.

It does not matter if the option is in the money or not at that particular node. Whatever the value is at that particular node (in your case 0.2517) reflects the expected present value of the option in the subsequent nodes during the next steps. Look at the middle node in Step 3. Your option is in the money at that particular node (0.48). This brings value to upper node in time step 1. So, since the option may be in the value in the future, it has value at upper node at Step 1 in amount of 0.2517, which should by all means be accounted for. If the option is not in the money, does not mean we should take its value at that particular node equal to 0.

An analog. You expect to get a positive amount of value of the put option at time step 0, but it is out of the money (exercise price is 40, while the stock price is 48).

Thanks very much for your comments. I understood everything u said in the first paragraph. Didn’t follow u in the 2nd though. The stock price in time t=0 is $38 not 48? So I would actually expect the put option value to be positive at time t =0.

yeah, my bad. Misremembered the stock price at time 0.

Whatever… imagine stock price is 40.5, a little bit higher than the exercise price. Even though the option is currently out of money, this option should have value, right? The less is its out-of-moneyness, the higher is its value.