Suppose S = $ 37, r = 8 %, delta(the annualized dividend rate) is 3 %, sigma(the annualized standard deviation of the continously compounded stock returns) is 41 %. Consider the price of a $ 42 - strike put with 100 days to expiration.


a) Suppose that 4 days later the price of the underlying asset has risen to $ 37.95, using the Black-Scholes formula, compute the price of the $ 42 - strike put?



b) Suppose that 4 days later the price of the underlying asset has risen to $ 37.95, using a delta approximation, estimate the price of the $ 42 - strike put?



c) Suppose that 4 days later the price of the underlying asset has risen to $ 37.95, using a delta-gamma approximation, estimate the price of the $ 42 - strike put?



d) Suppose that 4 days later the price of the underlying asset has risen to $ 37.95, using a delta-gamma-theta approximation, estimate the price of the $ 42 - strike put?