# Calculation of the upper stochastic dominance bound of an option

#### Lenin

I’d like to calculate, for a call option on a stock, the upper stochastic dominance bound as proposed by Constantinides et al. in their 2002 paper 'Stochastic dominance bounds on derivatives prices in a multiperiod economy with proportional transaction costs'. Could you advise how this can be done in practice?
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 expected stock return on terminal date. The authors initially define R, the expected stock return, for a single period economy as a ratio of the expected terminal stock price S to the initial stock price (S_0), with accounting for transaction costs:
.

The part
in the formula I interpreted as the expected value of a distribution of possible option payoffs (S_T - K), given the initial stock price of S_t, and where negative payoff values are replaced by 0. Thus the whole formula seems to be a ratio of the expected option payoff scaled down by the expected stock return (as a ratio of the expected terminal stock price to the initial stock price). Dividends are ignored. Did I get it correctly so far?

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?

Replies
7
Views
1K
Replies
0
Views
774
Replies
8
Views
4K
Replies
0
Views
1K
Replies
0
Views
730