The unit elasticity of options price with respect to maturity for far-out-of-the-money put options contrasts with the previously mentioned result from Brenner and Subrahmanyam (1988) that prices of at-the-money
put options in the Black-Scholes model are proportional to the square root of maturity.
Therefore, each of the 20 investors received the economic equivalent of 5-year "at-the-money
" put protection on their stock, and the amortized cost of that protection was only 1.38% per annum pre-tax or about 1% after-tax.
Consequently, the implied volatility measure is divided into three categories: at-the-money
implied volatility, near-the-money call implied volatility and near-the-money put implied volatility.
Time to expiration (years) Moneyness (S/K) 1/12 1/4 1/2 1 Deep-out-of-the-money 25.1459 5.9172 2.9991 2.0394 (0.90) 1.6812 0.3472 0.1720 0.1325 Out-of-the-money 2.8328 1.8461 1.6741 1.6500 (0.97) 0.1870 0.1166 0.1041 0.1054 At-the-money
1.4156 1.4219 1.4480 1.5203 (1.00) 0.0932 0.0817 0.0898 0.0964 In-the-money 0.9585 1.1688 1.2757 1.3936 (1.03) 0.0612 0.0710 0.0787 0.0862 Deep-in-the-money 0.1512 0.5227 0.7681 0.9774 (1.125) 0.0092 0.0320 0.0477 0.0628 Table 4: MAPE of Empirical Esscher Transform Estimates in a Heston World This table contains the prices for a European call option, where the underlying prices are simulated by stochastic volatility with [mu] = 0.1, [kappa] = 3.00, [theta] = 0.04, [sigma] = 0.40 and [rho] = -0.50, and they are compared to the true Heston call prices.
When this is not the case, the formula behaves well at-the-money
but the error increases far from the ATM value.
(2.) As firms usually grant at-the-money
options, the grant value is based on at-the-money
We define a metric to measure options returns from the seller point of view, write an algorithm that simulates the sale of options and collects premiums of at-the-money
european options at ex-ante market real prices (inflow of money to the seller of options), hold these contracts open until the original expiration, calculates the ex-post payoffs (outflow of money for the seller in case the option ends up in the money) and with that data calculating a realized return.
He, for the most part, reports strong feedback relationship between the options volume and expected future volatility, however results for at-the-money
(ATM) and out-of-the money (OTM) options are found to be more pronounced.
For a call option, in-the-money options refer to options with S/K being greater than 1, at-the-money
options refer to options with S/K being 1, and out-of-the-money options refer to options with S/K being less than 1.
Without going into technical details insubstantial for our purposes, we only mention that it is computed as an annualized, implied volatility averaged-out from at-the-money
call and put S&P500 options with ca.
option grants carried no expense so they were, predictably, wildly overused.
Fortunately, near maturity, at-the-money
options that have high trading volume also have high gamma, as illustrated in Table I.