A non-dividend paying stock currently trades for $37.61 and has an annualised return standard deviation of 17.5%. Given that the continuously compounded risk-free rate of return is 2.737% p.a., complete the following:

1. Using a two-step binomial tree, price the European put option on the stock when the put has an exercise price of $35 and 6 months to maturity.
   
    
2. Use BSM to calculate the theoretical price of the put option.  
   
    
3. Discuss the advantages and disadvantages of buying verses selling put options.  
   
    
4. Explain the concept of implied volatility including how it is calculated.