Problem $1$: Use the following parameter values to price a call and put option by applying

the Delta Hedging, Replicating Portfolio, and Risk Neutral Valuation approaches:

$S=$ current price of $($non$-$dividend paying$)$ underlying asset $=$$$50$;

$K=$ exercise price $=$$$50$;

$r=$ annualized riskless rate of interest $=3\%$;

$=$ annualized volatility $($standard deviation of return$)$ for the underlying asset $=\mathrm{.}1$;

$t=$ length of time$-$step in years $=1$;

$u=$ one plus the rate of return on the underlying asset after one up move $=e^{\sqrt[\phantom{2}]{t}}=$

$e^{\mathrm{.}1\sqrt[\phantom{2}]{1}}=1\mathrm{.}1052$; and

$d=$ one plus the rate of return on the underlying asset after one down move $=e^{-\sqrt[\phantom{2}]{t}}=$

$e^{-\mathrm{.}1\sqrt[\phantom{2}]{1}}=\mathrm{.}9048.$

Problem $2$: Now suppose there are $2$ timesteps, but the length of each timestep is $6$ months

rather than $1$ year. What are the arbitrage$-$free values for a call and put, based on these

parameter values?