The authors suggest the following formula (Proposition 1, (5.1), p.14) for the upper bound of a call option:

Where, if I got it right,:

k_1, k_2 - stock buy and sell transaction costs,

t - valuation date,

T - option maturity date (terminal date),

S_t - stock price on valuation date (initial price),

S_T - the expected stock price on option maturity date,

K - option strike price,

The part

I thought about using historic EOD stock prices as a sample of the population that includes all past and future stock prices. From the sample, for a given call option duration and strike price, I would calculate the expected value of the call option payoff and the expected terminal value of the stock price, as % of the initial stock price. I assume that the historic record is sufficiently long so that the derived expected values can be considered as valid characteristics of the whole population. These would then be useful in forward-looking calculations of the call’s upper bound.

Below is how I thought my calculation would go in practice for given S_0, K, t and T using my stock historic data:

1) calculate strike values of the same moneyness as K for all past stock EOD prices.

2) compute realized call option payoffs (S_T – K), as % of initial stock prices, for all time intervals, including overlapping ones, equal to option duration (T – t) in days,

3) replace negative payoff values with 0 and calculate the mean of the distribution of all these historic payoffs as our expected option return in % of initial stock prices,

4) similarly calculate the expected stock terminal price, also as % of initial price,

5) convert these % values into actual values of the expected payoff and terminal stock price for the option and put them in the formula

Should I assume that the values of terminal stock prices and values of the call payoffs derived from my historic sample are lognormally distributed? And calculate the mean accordingly? Will it improve the result if I limit the distribution to only values with a short-term (for example 10 days?) historic volatility equivalent to the short-term stock historic volatility observed when making forward-looking estimates of the upper bound for a given call option?