2. (Black-Schole formula) Suppose an European call option is currently at the money, i.e. St = K. Suppose the risk-free rate r = 0 and both time to maturity T t and volatility are very small. Assume the underlying stock pays no dividends.

1. (a) Using a numerical analysis to demonstrate that the Black-Scholes European call price may be approximated by the following formula:

   T t ct=St 2 .

   ( Specify 5 group of different values of T, t, K, St, and compute the European call price with both Black-Scholes formula and the approximation formula. )

2. (b) Then try to prove this approximation formula according to the Black-Scholes pricing for-

   mula. (Hint: simplify Black-Scholes formula for St = K and note that ex2/2 1 when x is small. Note. Microsoft Word allows the use of Math formula.)

2. (Black-Schole formula) Suppose an European call option is currently at the money, i.e. St = K. Suppose the risk-free rate r = 0 and both time to maturity T-t and volatility o are very small. Assume the underlying stock pays no dividends. (a) Using a numerical analysis to demonstrate that the Black-Scholes European call price may be approximated by the following formula: T-t c =OStV 2013 (Specify 5 group of different values of T, T, K, St, and compute the European call price with both Black-Scholes formula and the approximation formula. ) (b) Then try to prove this approximation formula according to the Black-Scholes pricing for- mula. (Hint: simplify Black-Schole's formula for St = K and note that e-24/2 1 when is small. Note. Microsoft Word allows the use of Math formula.) 2. (Black-Schole formula) Suppose an European call option is currently at the money, i.e. St = K. Suppose the risk-free rate r = 0 and both time to maturity T-t and volatility o are very small. Assume the underlying stock pays no dividends. (a) Using a numerical analysis to demonstrate that the Black-Scholes European call price may be approximated by the following formula: T-t c =OStV 2013 (Specify 5 group of different values of T, T, K, St, and compute the European call price with both Black-Scholes formula and the approximation formula. ) (b) Then try to prove this approximation formula according to the Black-Scholes pricing for- mula. (Hint: simplify Black-Schole's formula for St = K and note that e-24/2 1 when is small. Note. Microsoft Word allows the use of Math formula.)