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Q1 – Option pricing under lognormal distribution. ON 12/17/2021 Friday, SPX currently trades at 4620. March put has 3 months (63 trading days) to expiration. March 4275 strike PUT currently trades at 25% implied volatility. Assuming risk free rate of zero.
Q1a. What is the probability that the put will expire in the money (5 points)?
Q1b. What is the average underlying spx price when put expires ITM (5 points)?
Q1c. What is the average put payment conditional on put expiring ITM? (5 points)
Q1c. How much should the put be priced at today? (5 points)
Q2 – Underlying at 4620. 4275 put trades at 25% IV (same put as Q1). 4625 put trades at 19% IV. You set a put front ratio spread: long 1 put at 4625 strike and short 2 puts at 4275 strike. Q2a. What is the delta for the front ratio spread? (5 points)
Q2b. What is the gamma of the front ratio spread? (5 points)
Q2c. what is the one day theta value for the front ratio spread? (5 points)
Q2d. If SPX drops to 4500 on Monday (one trading day later), what is the PnL from delta, gamma, and theta respectively?