Sum of European put A stock price is currently priced at $\$25$. Every week it will either go up or down $5\%$. The risk-free rate is $4\%$ per annum with continuous compounding. What would a sum representing the price of an at-the-money European put expiring in one year look like?

we can use a binomial tree with risk-neutral pricing to model this, assuming a continuous dividend rate $\delta$ and letting the risk free interest rate be $r$. Although the stock does not pay dividends, we can still use this general formula.

Assuming 52 weeks in a year, we will let $$u= 1.05, d= 0.95, p^* = \frac{e^{(r-\delta)/52}-d}{u-d}$$ where $p^*$ is the risk-neutral probability of an up move.

Then the price of a put is equal to

$$P=e^{-r}\sum_{i=0}^{52}\binom{52}{i}(p^*)^i(1-p^*)^{52-i}\max(0, 25 - 25u^id^{52-i})$$