Hedge Fund Strategies：对冲基金策略.doc

Hedging StrategiesByMichael WhiteMGMT 6380Thesis Statement: Hedging strategies can help make the lay investors portfolios profitable.Abstract: There are various types of hedging strategies that traders employ on a daily basis. However, the ordinary investor may be afraid to add such strategies into their portfolio. This study is to examine what strategies the lay investor may undertake within their portfolios, and hopefully become comfortable with some these strategies. Traders may use complicated strategies very frequently; but there are some hedging strategies that the lay investor can use to have a successful portfolio, such as creating option spreads combinations, following trends, and investing in hedge-like mutual funds.Investors have taken an interest of late in hedge fund strategies; but can hedging strategies be truly more profitable than other investment strategies. Hedge fund strategies are always changing; they can spread different types of risks among different portfolios. Hedge Funds are normally seen as mysterious, and an exciting prospect for many investors, for those who can afford it. It is exciting for many investors to want to generate dramatic gains that hedging strategies can provide. Although, the strategies that hedge funds implement can be quite risky; profits can soar to all time highs, or fall to all time lows.There are several examples of hedging; there are long hedges, short hedges, reverse trading, cross hedging, and the hedge ratio. For implementing a long hedge, an investor may buy a contract at a set price to be later sold at a possible higher price in order to make a profit. A short hedge is a hedge where the hedger will sell a contract, the hedger can avoid the risk of prices falling by setting the price and selling in the future; the investor can hopefully expect more gains than they originally anticipated. Although, in most cases of hedging products such as commodities- long and short strategies do not always match up so nicely. In reality, the amount of the stock or the characteristic of a good will differ within a certain time span during a hedged strategy- in this case a cross hedge would be appropriate. A cross hedge is the case when the spot and the derivative position do not match up well. When the specifications of a position that is hedged do not match well, the hedger needs to ensure that he/she is trading the right number of contracts in order to control the risk involved in the hedged position.Of course as in any kind of investing it is important to minimize risk; and minimizing risk in hedging strategies in no different. One of the most important tools to use in derivative trading is the hedge ratio, it is as follows:HR= COMMODITIES POSITION CASH MARKET POSITION Kolb, Robert W. (2003). Futures, Options, and Swaps (4th ed.). United Kingdom. Blackwell Publishing. Pp 115It is also known as Delta and can also be written as:DELTA= CHANGE IN DERIVATIVE PRICE CHANGE IN UNDERLYING ASSET PRICE Cohen, Guy. (2005). Options Made Easy (2nd ed.). New Jersey. FT Prentice Hall. Pp.47As a commodity becomes in the money, the delta will increase; so the larger the delta, the faster the price movements of the derivative as compared to the underlying stock price. Conversely, buying the derivative out of the money will reduce hedgers chances to make a profit; because the change in the derivative price as compared to the stock price is much slower, so the probability of the derivative becoming in the money is much lower.We can use the hedge ratio model to determine how many contracts we will need to sell in order to hedge. For example, lets say an owner of 150 gasoline stations wants to hedge the risk of his gasoline inventory. At any one of the gasoline stations the company will have 1.125 million gallons of gasoline inventory. In order to minimize his risk the owner wants to sell 42,000 gallons of unleaded in futures contracts; he will first have to minimize his variance by choosing the model:HR= COVSFs2FIn order to calculate the risk-minimizing ratio in a simpler way he converts the model to:DS= a+bFt+etWhere,a= The constant regression parameter,b= The slope regression parameter,e= An error term with zero mean and a standard deviation of 1 Kolb, Robert W. (2003). Futures, Options, and Swaps (4th ed.). United Kingdom. Blackwell Publishing. Pp117The negative part of b will give us the risk minimizing ratio, because the b equals the same covariance between the variance of the spot (St) and the independent variance of the futures price (Ft). Through the regression estimation we can measure the effectiveness of the hedge and determine the coefficient of determination of R2, which is the portion of the total variance for the changes in the cash price statistically related to the changes in the futures price. R2 will always be between the numbers 0 and 1, the closer to 1 the better the degree of fit in the regression between cash and the futures position. Also, it will be better for obtaining a hedge we will want to use. Using the model, the owner determines the following results,a= 0.5231b= 0.9217R2= 0.88With 1.125 million gallons of gasoline inventory and a hedge ratio of -0.9217 gallons in futures for each gallon of inventory we get the results of:# of contracts= -0.9217(1,125,000/42,000) = -24.7 Kolb, Robert W. (2003). Solutions Manual for Futures, Options, and Swaps (4th ed.). United Kingdom. Blackwell Publishing. Pp. 26So according to our regression results we conclude that 88% of the variance of the spot price has changed during the sample period. So it is recommended by the model that in order to create a successful hedge, the gasoline owner should sell 25 contracts in order to hedge against price risk.Hedgers will utilize several option combination strategies in order to profit from price movements. Combinations can be used to protect the underlying profits of a portfolio from losses due to stock volatility, avoiding long positions in securities that have severe liquidity or solvency problems, or benefiting from select companies whose industry position is strong and growing. Spread strategies are common when trying to hedge against the ups and downs of the market. A spread is when a hedger will simultaneously buy and sell an option on the same underlying stock, but with different strike prices or expiration dates, or both. A spread may be used with calls or puts. One such spread is the bull spread; it is designed to profit if the value of the underlying stock or good rises. A bull spread will require two calls with the same underlying stock, and the same expiration dates, but different exercise prices. The buyer of a bull spread strategy will buy a call with the exercise price below the stock price, and then sell a call option where the exercise price is above the stock price. The hedger will hope to profit from this strategy by having the stock price increase; this strategy helps to limit the risk the hedge trader may incur. However, this trading strategy may not be the most profitable as compared to other strategies. Furthermore, costs need to be accounted for when deciding to enter into a bull spread strategy; the cost of the bull spread option is the cost of the option purchased less the cost of the option the hedge trader sells. We can use the below model to determine the bull spread in factoring the cost, using the current price, expiration dates, underlying stock prices, and exercise prices.Ct*(St, X1, T)-Ct*(St, X2, T)Ct= Current price at t timeT=Expiration date of optionSt=price of underlying stock at t timeX1=Exercise price Kolb, Robert W. (2003). Futures, Options, and Swaps (4th ed.). United Kingdom. Blackwell Publishing. Pp. 353The below example illustrates a bull spread strategy,OutInBuy I call XYZ Jan 95 7700Sell 1 call XYZ Jan 105 3300Breakeven400If the stock were trading at $102 the long position will show a profit, but the short position will not show a profit if it exceeds $108. Profits can also be obtained if an investor would use a put bull spread strategy. A hedger would buy a put with a lower exercise price and then sell a put with a higher exercise price. Costs included in the put bull spread strategy will follow the following model: Pt*(St, X1, T)-Pt*(St, X2, T)Another common spread strategy is the bear spread, which profits from dropping prices. In order to profit from a bear strategy, it requires two options with the same underlying stock with the same expiration date. The difference in the options would be the exercise prices. In order to transact a bear spread strategy an investor would sell one option such as a call with a lower exercise price and buy another option such as a call with a higher exercise price. The value of the bear spread including costs would be:- Ct*(St, X1, T) +Ct*(St, X2, T) Kolb, Robert W. (2003). Futures, Options, and Swaps (4th ed.). United Kingdom. Blackwell Publishing. Pp. 355In a bear spread a trader gambles that the stock price would fall, but profits can be limited; one advantage about this strategy is that risk is limited.Combining the bull and bear strategies together will create a box spread; this strategy consists of combining a bull spread with calls plus a bear spread with puts along with the two having the same exercise prices. This hedging strategy helps to protect an investment while giving the investor a chance at profits no matter which way the market turns. The cost of a box spread can be valued as:Ct*(St, X1, T)-Ct*(St, X2, T) - Pt*(St, X1, T) +Pt*(St, X2, T) Kolb, Robert W. (2003). Futures, Options, and Swaps (4th ed.). United Kingdom. Blackwell Publishing. Pp. 358The following example represents what to expect from a box-spread strategy:OutInSell 1 call XYZ Sep 40 6600Buy 1 call XYZ Sep 35 2200Buy 1 put XYZ Sep 40 4400Sell 1 put XYZ Sep 35 1100Breakeven100Here we see that no matter how the price of the underlying stock reacts we are in the money for at least $100. According to our example if the stock were to trade at $38 per share, we can exercise the call at $40 to net an intrinsic value of $2/share, then exercise the buy call at $35/share for an intrinsic value of $3/share, next we exercise the put buy at $40/share for an intrinsic value of $2/share, and lastly we exercise the sell put at $35/share for an intrinsic value of $2/share. Our profit value for the entire transaction would be:200+300+200+200+100= $1000 profitProfits may be even greater with much larger transactions. In order for a hedger to remain profitable and not become exposed to a lot of risk, the hedger should keep a careful watch on one side of a box spread strategy. If the strategy has different expiration dates; the hedger could be exposed to the ups and downs of the market on the longer expiration date. Furthermore, it is best if a hedger closes both the long and the short positions to avoid unintended short positions that are not covered or hedged. Another hedge strategy that we will look at is trend following, this strategy has become popular, particularly with commodity traders. It is also used by some hedge fund traders. The model of trend following can produce returns that are uncorrelated with the standard equity, bond, currency, and commodity indices. However, trend following may be able to produce option-like returns; they tend to be large and positive during the best and worst performance months within the world equity markets. The comparison between trend followers and the relationship to the equity markets is nonlinear, and follows a framework of systematic risk. Returns from trend following usually have a low beta against average equities; the state dependent beta estimates are usually positive in up markets while negative in down markets. Trend following strategies can differ greatly among one another; that is why we will primarily examine the strategy the “primitive trend-following strategy;” it has the same payout as the structured option known as the “look-back straddle.” The look-back straddle strategy is transacted when the owner of a call option has the right to buy the underlying stock at the lowest price over the life of an option. Similarly, the owner of a put option is allowed to sell at the highest price; these two strategies together will deliver the ex post maximum payout for any trend following strategy. The look-back straddle is designed to resemble the relative characteristics of all types of trend following strategies; trend followers should normally deliver returns resembling that of a portfolio of bills and look-back straddles. The Primitive Trend Following Strategy attempts to capture the largest price movements for an initial asset during a time interval; where the optimal payout is Smax-Smin. Trend followers will trade only if they have observed certain price movements during a specific period of time. In order to better understand the primitive trend following strategy; it is helpful to compare it with another form of trader that uses another type of strategy, the “market timer.” The market timer can trade long and short on an underlying asset; at the beginning of a period if a market timer forecast stocks to outperform bills, then stocks will be the only ones held in a portfolio; this strategy assumes the presence of short sale constraints. Assuming that Z(t) is the return per dollar invested in the stock market, and R(t) is the return per dollar invested in Treasury bills in period t; then R(t)+Max0,Z(t)-R(t) will show the returns of a portfolio of bills and a call option on stocks. Without short sale constraints the return for a market timer will be modified to reflect the short sale alternative. If the perfect market timer had a portfolio consisting of bills and a straddle on stocks then the model he/she would use is, R(t)+Max0,Z(t)-R(t)+Max0,R(t)-Z(t). Market timers and trend followers alike will follow the price movements of the market, but they usually study them differently. A market timer will forecast the direction of an asset, trading long in order to gain an increase in price, and then trading short in order to obtain the gains from a downturn. Alternatively, a trend follower attempts to benefit from market trends; a trend follower will attempt to identify price patterns and trade in the direction of that trend. Market timers use the concept of the Primitive Market Timing Strategy; this strategy attempts to capture the profits from price movements of the initial asset prices (S and S). If S is anticipated to be higher or lower than S, a long or short position will be initiated. At the end of the period, the trade will be reversed; so the optimal payout for the trade strategy will be |S-S|. Alternatively, the optimal payout for the primitive trend following strategy is Smax-Smin. PMTS or PTFS are determined by comparing the payout of either strategy to the return of a risk-free asset. If market timers and trend followers were perfect, then they would capture the optimal payout of |S-S| and Smax-Smin without incurring any costs; but in reality these traders cannot perfectly anticipate price movements. In reality market timers and trend followers do incur costs when attempting to gain from their strategies. The PMTS is evaluated by a long position in a standard straddle, while the PTFS is examined through a long position in a look-back straddle. We can illustrate the differences in pricing strategies by examining their deltas. The delta of a standard straddle is given as:d= 2N (a1)-1Where N ( ) is the cumulative standard normal distribution and a1= ln (S/X) +(r+1/2s2) T/ (sT1/2)S is the current price of the underlying asset with an instantaneous variances, r the instantaneous interest rate, and T is the time to maturity of the option.Alternatively, the delta of the look-back straddle is:d= 1+1/2s2/r N (-b3) + (u/s) exp (-rT+2rb/s2) N (b2)-1+1/2s2/r N (d3)-(u/s) exp (-rT-2rd/s2) N (d2)Where,u= (r-1/2s2)d= ln(S/Q)d1= ln(S/Q) +(r-1/2s2) T/ (sT1/2)d2= - ln(S/Q) +(r-1/2s2) T/ (sT1/2)d3= - ln(S/Q)-(r-1/2s2) T/ (sT1/2)b= ln (M/S)b1= ln (M/S-)-(r-1/2s2) T/ (sT1/2)b2= -ln (M/S-)-(r-1/2s2) T/ (sT1/2)b3= ln (M/S-)-(r+1/2s2) T/ (sT1/2)Q and M are the minimum and maximum prices of the asset respectively since the creation of the look-back straddle. Within any investment horizon, the payouts of the PMTS will equal that of the PTFS only if Smax and Smin occur at the beginning and end of the period in any order. As the investment horizon diminishes the payouts of the two strategies will diverge, because the probability of Smax and Smin rise within the investment horizon. Furthermore, the look-back straddle strategy is a perfect comparison when relating it to trend following strategies; the look-back straddle strategy can be replicated by rolling standard straddles over the life of the option, the rollover process is very similar to buying breakouts and selling breakdowns related to trend following strategies. Fung, William; Hsieh, David A. (2001). The Risk in Hedge Fund Strategies: Theory and Evidence from Trend Followers. The Review of Financial Studies Summer 2001 Vol. 14, No. 2, pp. 313-341. In the final analysis, PTFSS replicates the key features related to trend following fund in their returns. Both have a positive skewness and also have positive returns during the up and down movements in market equity prices. Trend following funds can be examined through a combination of PTFS on currencies, commodities, three-month int