Callable Asian Option
Asian options allow their underwriters to call the options back from investors at a specified time and with a specified amount prior to option maturities. A hybrid of Monte Carlo simulation and the closed form solution is employed in pricing.
The deal can be forward starting. In this case, the strike K is not pre-determined in the contract, but rather a stock price at a future start date .
Equation (1) indicates that the principal is protected. A payoff type without notional is also allowed, which removes the first part of equation (1) at maturity; but the owner of the option can still receive the call amount if the option is called on the call date (of course this call amount will be greatly reduced in the contract).
Pricing of callable Asian options involve two stages. The first stage is for the cases when the current value date is prior to the call date. This is our focus. The second is for the cases when the current value date is beyond the call date; in these cases, either the option was called and therefore deceased, or the remaining is just a regular Asian option, which is not in our scope.
A hybrid of Monte Carlo simulation and the Michael Curran’s closed-form solution can be employed to price callable Asian options. The former is used to propagate the stock price from the value date to the forward start date to determine the strike if the deal is forward starting, and to the call date to obtain a price to be compared with the call amount. The latter is used to price the Asian option value.
It should be noted that we allow a separate volatility of stock, , for pricing the Asian option other than the volatility of stock, , for propagating the stock price in Monte Carlo simulation. This separate volatility is actually the forward volatility of stock from the call date to the option maturity date.
We also implemented a crude Monte Carlo model for callable Asian options. In the testing, We use Quasi Monte Carlo (Sobol Sequence) with 20,000 simulations and 2,000 samples in stratified sampling, and also use crude Monte Carlo with 20,000 simulations. The testing results, shown in Table 1, indicate that the Quasi Monte Carlo model calculated callable Asian options within a tolerable range of the prices generated by the crude Monte Carlo model.
References: