We have a 3 period binomial tree with values:
59.65 (C33 = 7.65)
56.24 (C22 = ?)
53.03 53.03 (C32 = 1.03)
50 50 (C21 = ?)
47.14 47.14 (C31 = 0)
44.45 (C20 = ?)
41.91 (C30 = 0)
W want to calculate a European call option, no arbitrage, with properties K = 52, u = 1.0606, d = 1/u = 0.943, maturity in 9 weeks, r = 0.001 per week. The value for a call option is given by $max[S_t -K, 0]$. We can calculate $C_2^{2}$ given the risk neutral formula from the literature (Bjork 3ed, 2.1.4):
$C_2^{2} = \frac{1}{1+R} (q*C_3^{3} + (1-q)*C_3^{2})$, $\frac{1}{1+R}$, given by Bjork Proposition 2.6, but as we have multiple nodes I assume we need to discount it, which gives the formula $e^{r-(T-t)}= e^{0.053348-(9/52)} = 1.009276$
$R = 1.001^{52} = 5.3348pct = 0.053348 ,$
$q = \frac{(1+R)-d}{u-d} = \frac{1.009276 - 0.943}{1.0606 - 0.943} = 0.5636$, if we then plug the values into the formula:
$C_2^{2} = \frac{1}{1+R} (q*C_3^{3} + (1-q)*C_3^{2}) = 1.009276*(0.5636*7.65 + 0.4363*1.03) = 4.8051$,
My questions are:
a) is the value of $C_2^{2}$ correct?
b) is there a faster way to calculate the option value of the tree because this takes a lot of time (yes you can write a program but I am following the theory and I believe I have to learn it by hand as well).