Exercise: Vanilla and barrier options in discrete time

a) Stock BCD currently quotes 505. Compute the current value of the European Call (with 2 decimals) written on BCD, strike K = 510, using the binomial model (all intermediary variables you will use should have at least 5 decimals). You also know that :

• - The option maturity is T = 0.25 (91.25 days)
• - The yield curve is flat at 2% (annualized, discrete rate)
• - A dividend of 16 is paid out at date t = 50 days (between time nodes #1 and #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 0.57 and the expected volatility of BCD is 42% (annual).

b) Compute the delta, gamma, theta and (approximate) vega (each with 4 decimals).

c) Let a European Up-and-In Call be also written on BCD. Its strike K is 510, its maturity is T = 0.25 and the barrier is H = 565. Such a call becomes a vanilla European call if and only if BCDs price hits the 565 barrier (even though it may fall afterwards), but expires worthless if the 565 barrier has never been hit (so, beware of the path followed by BCDs price). Compute the current value of this barrier call using the tree you built for BCDs price in question a).