Structured products gained significant popularity in the retail market since the beginning of the second decade of the 21st century, due to the steady decline in interest rates across the global economies following the GFC. This led investors to the search for yield enhancement investments. One of them is the autocallable, a type of exotic option. The autocallable has the feature to be called by the issuer if the reference asset is at or above a pre-specified level on pre-specified futures dates. The principal guarantee disappears if the underlying asset price falls below a pre-specified level. A typical autocallable option is constructed using at least two barrier options. The option with lower barrier comes in the form of an at-the-money, down-and-in put option. The option with higher barrier has a knock-out feature, meaning the product matures early if the index rises above the upper barrier, this can be represented by a long up-and-out digital/binary call. Between these two barriers, investors earn an above-market coupon during the tenor of the deal. Autocallables are quite popular in the Asian market, especially in Japan and Korea. Assume the date is 25 February 2021. The Korean Composite Stock Price Index, or KOSPI, closed at 3100. The one-year risk-free rate for the Korean won is 0.8025%. KOSPI is expected to pay a dividend yield of 2% and the ex-dividend date is the 18 August 2021. The volatility of KOSPI on the same day is 27.92%. The head of structuring asks for your help with the knock-in put of the autocallable with a barrier of H. Assume a day count convention of 360 days, a cash rebate of 0 for exotics, and the expiration date for all exotics is 20 February 2022.

a. Explain what a down-and-in European barrier put option is. Can it be replicated by a set of more basic option contracts? If yes, how? If not, why not?

b. Using 10-step binomial tree and the Cox-Rubinstein-Ross parameterisation, price and at-the-money (ATM) European down-and-in put with barrier (H_L) set at the index level of 2900

c. Using the Binomial tree prices obtained in part b, plot and discuss the P&L of the short ATM European down-and-in put option with the barrier H_L = 2900

d. Using the Black-Scholes model, price the vanilla equivalent with the same contractual characteristics. What drives the differences between the two prices derived here and in part b? How can the two option price solutions converge?

Continue from Question 1: Assume that you sold an autocallable to a client. The structure has now 10 days left before expiry, with all the other parameters the same. There are fears of a correction in KOSPI level, and the head of structuring asks you to describe how the price of the European knock-in put in the structure behaves as a function of the underlying price.

a. Calculate the European knock-in put price as a function of the KOSPI from 2700 to 3100 with 10 index point increments.

b. Using basic numerical methods, calculate the delta of the European knock-in put

for the same KOSPI range as in part a

c. Discuss the behaviour of the delta around the barrier level. Does it behave the same way as plain vanilla puts? Why or why not? How would the behaviour of delta impact your hedging.