The market crash of 1987 revealed the failure of risk management systems. This stimulated the search for new approaches, which led to the development of the Value at Risk (VaR) concept. In the 1990s, this risk measure became standard, and in 1999 it received official and international status in the
Basel agreements.
In the course of time, VaR became a compulsory characteristic in the reports of most financial institutions. Step by step, market professionals came to believe that this indicator adequately described the risk of any investment portfolio regardless of structure or complexity. However, the financial crisis of 2008 revealed inconsistencies between forecasts based on VaR and actual losses incurred by market participants.
Such divergence between forecasts and reality were due to fundamental changes in the financial markets over the last 20 years. A vast quantity of complex financial products has been created recently, and sophisticated investors have shifted priorities from plain assets (stocks and bonds) to derivatives (futures, options, convertible bonds, credit swaps, etc.). In spite of these developments, risk evaluation mechanisms remained static. Today, a new financial era is evident, one that requires not only adaptation of risk evaluation methods but also the invention of fundamentally new approaches.
This implies a shift away from a one-sided system of risk estimation (such VaR) to the development of a whole set of evaluating algorithms based on different principles. Instead of fixing a new common standard, a multitude of alternative risk indicators will be necessary. Their application area should also be extended from assessment of existing investments to active utilization at the portfolio composition and structuring stage. Alternative indicators should be unique and independent (not correlated). Each should supplement the information contained in the other indicators, not duplicate it.
This article will examine four indicators, traditional VaR and three alternatives. The next part of this series will examine their interrelationship in assessing the risk of a complex option portfolio.
RISK CALCULATION
Two of the four indicators are calculated analytically using mathematical formulas (index delta and asymmetry coefficient), while the two others are estimated numerically via Monte Carlo simulation (loss probability and VaR).
Index delta. The risk of a complex portfolio containing options on different underlying assets cannot be evaluated by mere summation of corresponding deltas. This non-additivity problem can be overcome by applying the concept of index delta developed in “A better risk gauge for options portfolios” (July 2009):
where Δi is the delta of option i, Αi is the price of the underlying asset corresponding to this option, βi is beta of the underlying asset, xi is the number of options in the portfolio, and I is the index value at the moment of evaluation.
Asymmetry coefficient. This indicator expresses the skewness of the payoff function of the option portfolio. Most strategies selling naked options are based on the concept of market neutrality. If the portfolio is really market-neutral, its payoff function should be sufficiently symmetrical relative to the current value of a certain index. Such symmetry implies that the value of the portfolio will change roughly equally regardless of market direction. If neutrality is violated, the payoff function will be biased and the asymmetry coefficient would measure this bias.
Because the portfolio value P is the sum of options values it includes, the relationship between P and the changes of the index I can be expressed as:
where Οi is the value of option i, d is the index change (e.g., d = 0, 07 means that the index rose by 7%), other symbols are the same as in the index delta equation. The relationship Αi (I, d) can be established using beta bi, which is a slope coefficient in a linear regression between returns of the underlying asset and the index. Knowing beta we can roughly estimate the underlying asset value given that the index changes by a known amount:
Applying the Black-Scholes model, we can estimate values of all options contained in the portfolio provided that the prices of their underlying assets are equivalent to the values obtained using the third equation. Summing Οi values, we get the portfolio value P corresponding to the second equation. Two P values have to be calculated to estimate the degree of portfolio skewness — for the case of index growth by d x I and for the case of its decline by the same value (these values will be denoted as P (Ai (I, d)) and
P (Ai (I, - d)) correspondingly). Knowing these values, the portfolio asymmetry coefficient can be calculated as follows:
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.
