Exercise: Vanilla and barrier options (discrete-time)

a) Stock ABC currently quotes 310. Compute the current value of the American call (with 2 decimals) written on ABC, strike K = 300, using the binomial model (all intermediary variables should have 5 decimals). You also know that:

• - The option maturity is T = 0.25 (91days)
• - The yield curve is flat at 3% (annual, discrete rate)
• - A dividend of 13 is paid out at date t = 61 days (time node #2 in a 3-period tree): Use the proportional dividend approach.
• - The tree has 3 periods (4 dates), the true probability of a positive jump is 0.65 and the expected volatility of ABC is 40% (annual).

b) Compute the delta (2 decimals), gamma, vega, and theta (4 decimals).

c) Let a European Up-and-In call be also written on ABC. Its strike K is 300, its maturity is T = 0.25 and the barrier is H= 392. Such a call becomes a vanilla European call if and only if ABCs price hits the 392 barrier (even though it may fall afterwards), but expires worthless if the 392 barrier has never been hit (so, beware of the path followed by ABCs price). Compute the current value of this barrier call using the tree you built for ABC in question a).