How Early Termination Creates Volatility Exposure in Share Plans

# Equity Derivatives Question


# Equity Derivatives Question












I'm preparing to take a test on equity derivatives and have met with some difficulty, so I need some help on the following question:

> Please analyze the following share purchase plan. Assume client is committed to purchasing a fixed number of shares? Original payout : client receives the average of daily close prices over 10 periods a. Clearly there’s no optionality here Bank changes the payout such that bank has the right to choose to terminate the payout at any time from period 5 to period 10 a. Please explain why the following payout (2) has volatility exposure b. Bonus: if the client has a fixed dollar amount to spend, how does this affect the optionality? Client still receives the arithmetic average of daily prices until trade termination. If they don’t have enough cash they just get fewer shares than original target

## Answer by ZRH (score 1, accepted)

https://quant.stackexchange.com/a/44408

ad i) - of course you are right.

ad ii) If there is optionality or not, depends on how you read this:

A) If the bank is entitled to terminate, and e.g. does so after 5 days, and the price at which the shares are purchased, is the average of those 5 days, then there is NO optionality, as the bank would be indifferent to go for another 6th day, and then charge the average over the last 6 days.

B) If the price which is charged is the average over all ten days, regardless of termination or not, then there is clearly optionality. The bank would keep hedging for the full period of ten days, and buy shares pro-rata (say lots) each day. Then it would in hindsight evaluate all possible scenarios: Pass all 10 lots to customer (i.e. do not terminate); pass 9 lots to customer, keep one; pass 8 lots to customer, keep 2; down to pass 5 lots, keep 5. Out of these scenarios they would evaluate the one that generates the highest profit, where of course the "worst" outcome is not to terminate, in which case there is no cost to the bank. So there is a nonlinear payoff, i.e. either a profit or no loss, thus the bank holds optionality, and should indemnify the customer for granting this optionality. The higher the volatility, the more profit the bank can potentially reap, so there is vega.

Bonus: The fixed USD amount clause reduces optionality in the deal. Should prices go up, the deal will potentially terminate earlier, as the customer would be out of USD sooner in such scenario. This means the potential of the bank to benefit from price scenarios in their favour is reduced.