A non$-$dividend paying stock currently trades for $$30$ and has an annualised return standard deviation of $20\%.$

Given that the continuously compounded risk$-$free rate of return is $5\%$ p$.$a$.,$ complete the following:

Using a two$-$step binomial tree, price the European put option on the stock when the put has an exercise price of $$29$ and $6$ months to maturity.

I have just sold $100$ European call options written over shares in ABC Bank. The stock is currently trading for $$50$ a share and has a return standard deviation of $15\%$ p$.$a$.$ The options mature in $3$ months and have a strike price of $$48.$ The continuously compounded risk$-$free rate of return is $5\%$ p$.$a$.$

Explain exactly how I can trade today in the underlying shares to hedge my market risk.

How often should I rebalance my portfolio?

Discuss how an increase in an options time$-$to$-$maturity would affect its price if it was a call or a put, holding all other variables constant.

Explain how to construct a collar as a means of hedging downside risk of a stock you own.

State clearly what options you buy or write, and which have higher$/$lower strike prices.

Why would you choose this over simply buying a put option to the hedge risk?