If we plot the portfolio payoff function with index values on the horizontal axis and portfolio values on the vertical axis, then Asym can be visualized as the slope of the line connecting two points with abscissas X = I × (1 + d) and X = I × (1 - d). The higher the absolute value of the slope, the more asymmetric the payoff function. If the coefficient is zero, the payoff function is perfectly symmetric.

“Asymmetry coefficient” (above) shows the interim data required to calculate the value for a portfolio of 10 short straddles. This portfolio was created on July 21, 2009. All options expire on Aug. 21, 2009, the risk-free rate is 1%, the quantity of each option is xi = 1. For example, the price of ED stock, provided that bi= 0.23 and the index rises by 10% (d = 0,1), is 37.92 × (1 + 0.23 × 0.1) = $38.79. Substituting this value into the Black-Scholes formula instead of the current stock price gives us the prices of call and put options ($1.89 and $3.06). When values of all options are determined in the similar way, they are summed up to obtain two P values. The table shows that if the index rises, the portfolio will be worth $46.66, and if it drops, $31.46. Substituting these data into the fourth equation above and taking into account that on the date of portfolio creation — the S&P 500 was 954.58 — we get:

Loss probability. A portfolio can yield a loss. The probability of this event can be estimated by Monte Carlo simulation. With this technique, a random price is generated for each underlying asset for a pre-defined future moment (for example, at expiration date). Then profits or losses of options are calculated for generated stock prices. The sum of these values represents an estimation of the portfolio’s profit/loss. The described cycle corresponds to one iteration. Performing it repeatedly for the same portfolio provides a reliable estimate for many useful characteristics.

“Monte Carlo” shows two iterations for the portfolio used in the previous example. Stock prices were generated using the lognormal distribution with historical volatility calculated on the basis of a historical period of 120 trading days (correlations of stock prices were taken into account). The first iteration for EIX stock generated the price of $31.04, which implies profit of 30 + 1.74 – 31.04 = $0.70, while the second iteration for the same stock generated a loss in the amount of $1.28. For the whole portfolio, the first iteration yielded a loss (-$2.74), while the second one was profitable ($5.87).

A full set of iterations for one portfolio represents a simulation. The ratio of the number of unprofitable iterations to their total quantity gives an estimate of the portfolio loss probability. For example, if 7,420 of the 20,000 iterations used are unprofitable, the loss probability is 0.37.

Value at Risk. VaR is an estimate of a maximum loss, which will not be exceeded with a given probability. In the next part of this series we, will use 95% probability to examine correlation analysis.

**Sergey Izraylevich is the chief investment officer of Hortan SARL and chairman of High Technology Invest Inc. Vadim Tsudikman is a director at Hortan. Contact the authors at izraylevich@htinvest.ru.**