A pricing model is presented for capped-accumulated-return-call (CARC) with multiple underlyings. At each reset date, a weighted stock price is calculated. These weighted stock prices are used to compute returns.
Let be N stocks in a given basket, be the price process of the jth stock and , and be a set of reset dates and be a payoff settlement date. The CARC with the multiple underlyings is a European type derivative security whose matured payoff at the settlement date is given by
(1)
where is the global floor (strike) of the return rate, N is the notional principal, and is capped-accumulated-return and defined as
(2)
where is the capped return-rate for each period explained as follows. Define the actual period return-rate as , (3)
Where
. (4)
Here are the weights and
. (5)
Then we define
, (6)
where c is the cap.
Let t be the current value date, then the current value of this CARC can be written as
(7)
where is the discounting factor at the value date. The above formula is in a world that is forward risk-neutral with respect to a specific currency .
As a result, the notional principal N is measured in the currency , and the discounting factor should be calculated by a zero curve (ref. https://finpricing.com/lib/IrCurveIntroduction.html) given at the value date. If the underlying asset is measured in another currency , assuming the option is a Quanto type transaction, the governing price dynamics of the underlying asset in the risk-neutral world of should be written as
(8)
where is the short rate of , q is the dividend yield of the asset, is the volatility of the asset price, is the volatility of the exchange rate between and , is correlation coefficient between the asset price and the exchange rate, and is the Wiener process. All these parameters are assumed deterministic.