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Capped Accumulated Return Call Volatility Surface A pricing model for capped-accumulated-return-call (CARC) with volatility surface is presented. Proprietary approaches to interpreting volatility surface are employed during pricing. To accelerate the convergence when low discrepancy sequences are used in Monte Carlo simulation (Quasi-Monte Carlo simulation), the Brownian Bridge Path Construction has been employed in some CARC transactions. As is known, quasi-Monte Carlo methods provides a way to improve the accuracy and reliability of Monte Carlo simulation by using deterministic sequences known as quasi-random sequences. This results in better convergence and deterministic error bounds (Joy, Boyle and Tan, 1996). There are a few techniques aimed at speeding up quasi-Monte Carlo, and the Brownian bridge path construction is one of them. It attempts to use the best coordinates of each point to determine most of the structure of a path. A new representation of the volatility skew is provided. To use this new representation, the user must input “STRIKE_REPRESENTATION PERCENTAGE” in the token file. Then, the ATM strike is always assumed to be 100. The first quantity in the volatility skew still signifies the stock price when the volatility skew was built. The model will interpret this skew as follows. At the time 1 year from the date when the volatility skew was built, the 10% ITM volatility is 0.4, with the strike level being 90*1050/100=945. The ATM volatility is 0.38 with the strike level being 1050*100/100=1050. The 10% OTM volatility is 0.36 with the strike level being 110*1050/100=1155. At the time 2.0 years, a similar interpretation can be obtained. As to the interpolation of the volatility surface in pricing CARC, for each reset period, the model needs vol_spot and vol_strike to determine the volatility to use for that period. For the current period, the vol_spot = current_stock_price at the value date, vol_strike = (1.0+cap) *l ast_reset_price. For all future periods, the vol_spo t= base_spot of the volatility skew, and the vol_strike = (1.0+cap) * base_spot. The base_spot is the stock price when the volatility skew was built, such as 1050 in the above example. Based on the value of vol_strike/vol_spot for each reset period, the volatility for that period is obtained by linear interpolating (or flat extrapolating) time in the volatility skew representation, followed by moneyness interpolation (or flat extrapolation).
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