At time t$,$ Titan stock is priced at $$27\mathrm{.}50$ per share. A trader uses a binomial

tree to calculate the price of a put option on Titan stock. In the upper node at time t$+\backslash$tau $,$ the stock price is $$30\mathrm{.}39$ per share, the put price is $$0\mathrm{.}935$ per share, and the put delta is $0\mathrm{.}345.$ In the lower node at time t$+\backslash$tau $,$ the stock price is $$24\mathrm{.}88$ per share, the put price is $$5\mathrm{.}278$ per share, and the put delta is $0\mathrm{.}961.$ Use this information to answer the following questions:

a$.$ At time t$,$ what is the delta of the put option?

b$.$ At time t$,$ what is the gamma of the put option?

c$.$ Suppose the price of Titan stock decreases from $$27\mathrm{.}50$ per share to $$26\mathrm{.}85$ per share. Using only delta from part a$,$ what is the estimated change in the put option's price per share? Using both delta and gamma from parts a and b$,$ what is the estimated change in the put option$$s price per share?

d$.$ Suppose the price of Titan stock decreases from $$27\mathrm{.}50$ per share to $$26\mathrm{.}85$ per share. Using your answers from parts a and b$,$ what is the estimated new delta?