Risk Management

LJM RISK MANAGEMENT: OVERVIEW

LJM offers differentiated investment vehicles based on trading options on the S&P500 futures index. Since clients vary in their investment goals and tolerance for risk, LJM employs state-of-the-art, real-time risk management processes specifically tailored to each investment strategy. These risk management overlays are designed to maximize returns while managing risk exposure appropriate to the respective investment vehicle. Below is a brief introduction to market risk, generally, and the methodologies and tools that LJM employs within its risk management program.

Extreme Events

After the crash of October 1987, the financial industry experienced a period of realization that extreme stock market movements were more likely than the existing statistical models suggested. Financial engineers developed new models that more accurately reflected true market behavior. Large institutions and sophisticated boutique trading firms built the first generation of “non-Gaussian” models to value financial instruments, which recognized that large down movements are much more likely than previously assumed.

The crash of October 2008 again demonstrated that risk managers must base their strategies on probability distributions that capture the likelihood of extreme events. The recent crash also highlighted the need to use simulation techniques that describe realistic market conditions, such as liquidity squeezes and the fact that the correlations of large swaths of financial instruments move towards 100% during extreme events.

A good risk management system must realistically measure performance under stress events to allow the strategy manager better control over portfolio diversification.

LJM’s risk management program incorporates probability distributions that accurately estimate the likelihood of large market sell-offs and a simulation engine flexible enough to handle extreme market behavior when re-valuing a portfolio of financial instruments.

LJM RISK MANAGEMENT: METHODOLOGIES

Informed investors recognize that a certain amount of risk must be assumed when targeting returns higher than those of a fixed income portfolio. In fact, the recent financial crisis has demonstrated that the very definition of what constitutes a risk–free strategy should be thoroughly revised.

LJM’s goal is to translate our clients’ differing levels of risk appetite to reliable risk control processes that provide an optimal balance of risk and reward . As described in our Investment Products and Performance History pages, LJM offers three distinct strategies that span a wide range of risk/reward profiles:

1. LJM Aggressive (no hedging);
2. LJM Moderately Aggressive (moderate downside hedging);
3. LJM Preservation and Growth (strong downside hedging).

As noted before, market behavior is not symmetrical. The distribution of historical market performance displays skewness such that extreme tail events are more likely to the downside than to the upside. Reflecting this market reality, LJM offers hedge overlays in response to downside risk.

LJM strategies have exposure to sustained market rallies but generally will recuperate losses once markets stabilize. As such, they can be viewed as a natural hedge for a portfolio containing an allocation to long equities.

To hedge downside risk, LJM purchases put contracts on the S&P 500 futures. Since option values decay over time, hedging costs must be judiciously analyzed. LJM has developed state-of-the-art technology to identify put contracts offering the greatest protection over multiple time-horizons per dollar invested. This technology allows LJM to ameliorate risk while having the least impact on expected returns.

Additionally, LJM continually calculates the short options exposure of each strategy in order to ensure that the risk/reward profile is in line with market moves.

All LJM traders use identical tools to analyze risk. As opposed to large institutions with multiple layers between the Chief Risk Officer (“CRO”) and the trading desks, LJM risk reports are reviewed by its CRO (see LJM Organization page) and traders multiple times per day. All hedging trades are analyzed and agreed upon for same day execution.

Differentiation among firms trading options on S&P 500 futures stems from two factors: the accuracy of the estimations of probabilities of market moves and a reliable description of how implied volatilities change under these moves. As explained in the next sections, LJM uses state-of-the-art technology to calculate probabilities of market changes and to describe the behavior of implied volatilities.

Probability Distributions and Historical Volatility Clustering

Sound risk management relies heavily on a realistic description of market behavior. As noted above, many modeling technologies have under-estimated the likelihood of large downward moves (“crashes”). LJM employs its proprietary engine, LJM STORM System (see discussion below) to fit the historical behavior of the S&P 500 index with a probability distribution that accurately incorporates the likelihood of extreme events.

The graph below shows the natural logarithm of the probability density as a function of the natural logarithm of daily S&P 500 returns:

In the plot above, the red line is the probability distribution of S&P moves as calculated by the LJM STORM System. The blue dots depict the actual moves of the index and the green line is the Gaussian probability distribution that best fits the market historical behavior. Note that both the green and red lines fit the market well for small moves, but only the LJM STORM generated line captures the probability of large down moves (specifically, the market crash of October 1987

Another aspect that needs to be modeled is whether market moves today depend on market moves on previous days. Whereas there is a tendency by financial modeling to assume total independence between today’s and yesterday’s market moves, it has been observed that volatility tends to “cluster”, that is to say, periods of increased volatility more often than not are followed by days of increased volatility.

Rather than assuming that daily market moves are independent of one another, LJM uses a GARCH(1,1) model, tying the volatility of a day’s move to that of its prior period.

In summary, LJM’s risk management program is predicated on analyzing multiple market scenarios using a probability distribution that accurately estimates the likelihood of large market moves while capturing the volatility clustering effect.

Modeling of Implied Volatilities

The second key component of good risk management is a realistic description of the behavior of options volatilities under changing market conditions.

As a result of the recognition that large down side market moves are more likely than gap moves to the upside, the implied volatilities of out-of-the-money puts are higher than those of at-the-money contracts. This is reflected in what market participants call the volatility-by-strike structure, as shown below:

In the graph above the y axis represents implied volatilities (in %) of 1 month options and the x axis represents the option strikes. The red dot is where the market was at the time this screenshot was taken.

The circles represent the market volatilities, whereas the solid line describes the volatility-by-strike functional form that best fits the market.

Accurately describing how this functional form behaves under changes in market levels and time to expiration is essential for a reliable risk management program.

Market practitioners have identified two distinct behaviors of the relationship between implied volatilities and market levels.  Each of these has been prevalent at different times, depending upon a few market characteristics.   LJM’s risk management system incorporates both descriptions of implied volatilities, thereby providing an accurate way of gauging the impact of each on the funds’ risk profiles.

Value-at-Risk Versus Expected Shortfall

Most financial institutions rely on Value-at-Risk (“VaR”) to measure their exposure to extreme events. As explained in the paper below, the VaR risk parameter fails to capture the true nature and exposure to large market moves (“tail events” also known as Black Swan events).

• “Uncommon” Risks: Using the Probabilities of Extreme Events to Enhance Returns

After the crash of October 2008, many financial managers realized that portfolio dynamics including correlation, liquidity and performance need to model exposure to tail events and that Expected Shortfall (“ES”) far exceeds the more traditional VaR metric.  In response, LJM now incorporates ES to define and measure downside risk exposure”

Monte Carlo Simulations

Besides using a large number of analytical models, most risk management systems simulatemarket scenarios by generating random paths to describe possible future moves. The general techniques used to generate these paths are called Monte Carlo (“MC”) simulations.

The essence of the methodology is to consider a very large number of future market scenarios, re-evaluate the positions for each case, and assign a probability to such scenario based on the distribution being used. Hence, these techniques live and die by the accuracy of the underlying probability distribution used to generate the given scenarios.

As explained in a recent WSJ article, except for large institutions and sophisticated firms, the probability distributions most widely used in MC simulations are based on Gaussian (i.e. normal or “bell shaped”) distributions. Unfortunately, relying on the Gaussian probabililty function results in a drastic under-estimation of extreme market moves. In other words, the Gaussain distribution assigns close to zero probability to crashes such as those of October 1987 and October 2008. To everyone who has now experienced two crashes within one generation, it is quite apparent that market crashes do occur somehwat more frquently.

In summary, a key component of LJM’s risk management program is utilization of the MC tool based on a probability distribution that accurately represents the likelihood of tail events.

LJM RISK MANAGEMENT: TOOLS/TECHNOLOGY

LJM uses a custom implementation of a “best of class” option analytics software called ProOpticus® (“PO”) developed by Prime Analytics, LLC. (“Prime”). Coupled to this cutting-edge, real-time risk management system, LJM uses proprietary LJM STORM System calculations to estimate the likelihood of extreme market moves

The decision to adopt the PO platform was mainly based on the system’s following strengths: 1) high level of flexibility in aggregating risk across multiple dimensions; 2) clear translation of advanced mathematical computations into trading and risk management tools; 3) ability to incorporate multiple assumptions about implied volatility behavior to ascertain their impact on trading strategies; 4) user friendly design.

Custom Risk Management System

LJM has worked with Prime to develop a custom implementation of their technology that integrates the proprietary LJM STORM System calculations to the PO risk management tools. This unique platform provides LJM a competitive advantage as a derivative trader, generally, and advanced capabilities relative to other CTAs trading S&P500 futures options.

Using LJM’s proprietary probability distributions and an accurate functional representation of the implied volatilities, the LJM risk platform values the impact of market moves on the funds’ values across 4 different time horizons, currently set at: intraday, 1, 2 and 3 weeks out.

The risk profile is then downloaded to an Excel file to facilitate discussion among traders and CRO. A screen shot of a report from 6/19/2009 is shown below:

This report is run multiple times a day to ascertain the evolution of the risk profile under market moves and as new trades are carried out.

The report above was generated on Friday, June 19th, 2009 immediately upon June expiration, when the LJM positions included July and August contracts.

The risk is broken up by expiration so as to allow traders and the CRO to ensure that it is not overly concentrated in a given expiration. The 4 time horizons mentioned before (intra-day, 1 week, 2 weeks and 3 weeks) appear all on the same page, with the time dimension increasing as one moves down the spreadsheet.

Pink backgrounds correspond to losses whereas green filled cells depict gains.

It should be emphasized that the reports depict static positions, i.e., they assume that LJM traders will not move contracts when the market levels change. Since LJM traders will quickly trade in and out of contracts to readjust the portfolio during big market moves, the reports offer guidance as to what the worst case scenario risk levels are.

Notice the rows in yellow displaying the probability of market moves. These are calculated using the LJM STORM methodology.

In addition to the matrix report LJM’s risk management system provides the traders with a pictorial description of the risk profile as a function of market moves. Below is a blowup of what LJM traders analyze during trading hours, each time a new report is produced:

This example was obtained from a risk report produced in mid-April 2009. The risk profile of each expiration is displayed separately, together with the aggregate values. The time horizon means that the calculations include 1 week of option decay. The vertical axis displays the expected profit/loss, while the horizontal axis corresponds to potential market moves.

LJM traders weigh a fund’s targeted returns versus this powerful visual depiction of the risk profile to choose the day’s trades.

GAP RISK

As discussed above, the first section of the daily risk profiles provide an estimate of the expected profit and loss should large intra-day market gaps occur.
By their very nature, the Aggressive and Moderately Aggressive strategies are exposed to market gaps, which are defined as almost instantaneous moves that preclude any possibility of readjusting the positions.  Since the P&G strategy has a higher level of down-side hedges, its risk profile is less steep than that exhibited by the Moderately Aggressive fund, but it should be stressed that it is still exposed to market gaps.

Since the implied volatilities explode during large, rapid sell offs, these strategies will see the short option premiums increase in value dramatically, leading to potentially large short term losses when the positions are marked to market at the end of the trading day – even if equities markets rapidly recover short term gap losses, option prices will retain elevated implied volatilities over an intermediate timeframe.  The reason for this is that the prices of out-of-the-money options depend not only on the market volatility but also on how quickly the volatility itself changes: when the market volatility increases very rapidly the volatility value of these options explodes at a much higher rate than that of near-the-money options. 

If the new prices of the positions exhibit a large imbalance between down-side and up-side risk, or should the sudden market move result in margin calls due to increased down-side risk, some contracts may need to be closed out, leading to real losses.

Our down-side hedging, as explained before, involves buying put contracts in order to reduce potential losses, but focuses primarily on the risk of sell-offs over multiple days, such as during the financial crisis of 2008, which we believe strikes the best balance between risk and rewards for a strategy based on capturing option decay.

Our analyses have shown that buying enough hedges substantially to reduce exposure to market gaps would also considerably reduce returns for all LJM investment products.  In addition, these strategies have demonstrated the ability to recover from large draw-downs over a reasonable timeframe.  It is therefore LJM’s focus for its hedging and risk modeling capabilities to manage exposure to the more typical market sell–off that evolves over a multi-day period.  LJM believes this approach strikes the optimal risk/reward balance for all of our investment strategies which are designed to outperform the market over an investment horizon of 3-5 years.

LJM STORM^sm System

The LJM STORM^sm System was developed over a two year period starting in 2006. Its computational techniques are currently being integrated into the Prime platform for exclusive use by LJM in its custom PO application. This proprietary technology allows LJM to calculate the funds’ risk profiles using probability distributions that accurately capture both the likelihood of extreme events and the volatility clustering effects shown by the market throughout the years.

As explained above, only by using probability distributions similar to the one generated by the LJM STORM System will the actual risk of extreme events be adequately measured: Distributions that underestimate the likelihood of tail events will greatly understate the risk such market moves represent to a portfolio of financial instruments.

LJM believes that coupling this methodology to its real-time risk management solution produces a system that is at the forefront of financial technology.