4. The price of a European put is given by the Black-Scholes formula P. =-5,0(-d, )+ Ke="(T-1) (T-1)o(-d,) where, S, In+r+ (t K 2 d, = 1 +(+*)(1-1) o -t 1 2 + r- O*)(T- ) 1 S, In (T t) K de -= d, -OVT-t OVT-t and 0(.) is the standard normal cumulative distribution function. Please evaluate the following limits of put price. 1). When time is approaching the maturity date, t T, lim p,=? 2). When the maturity is far far away or the time-to-maturity is infinite, 1 -00, lim p,=? 3). When the volatility is approaching zero, o +0*, lim p, = ? 4). When the volatility is approaching infinity, o too, lim p, = ? S(1) 5). The price of an at-the-money put, -=1, p. = ? 2 00+ 0+0 = Ker(1-1) S, 6). The price of a deep-in-the-money put, 1 4. The price of a European put is given by the Black-Scholes formula P. =-5,0(-d, )+ Ke="(T-1) (T-1)o(-d,) where, S, In+r+ (t K 2 d, = 1 +(+*)(1-1) o -t 1 2 + r- O*)(T- ) 1 S, In (T t) K de -= d, -OVT-t OVT-t and 0(.) is the standard normal cumulative distribution function. Please evaluate the following limits of put price. 1). When time is approaching the maturity date, t T, lim p,=? 2). When the maturity is far far away or the time-to-maturity is infinite, 1 -00, lim p,=? 3). When the volatility is approaching zero, o +0*, lim p, = ? 4). When the volatility is approaching infinity, o too, lim p, = ? S(1) 5). The price of an at-the-money put, -=1, p. = ? 2 00+ 0+0 = Ker(1-1) S, 6). The price of a deep-in-the-money put, 1