rvarb – Page 4 – Relative Value Arbitrage

A Volatility Term Structure Based Trading Strategy

In previous 2 articles, we explored a volatility trading strategy based on the volatility risk premium (VRP). The strategy performed well up until August 2015, and then it suffered a big loss during the August selloff.

In this article, we explore another volatility trading strategy, also discussed in Ref [1]. This strategy is based on the volatility term structure [2].

It is well known that volatilities exhibit a term structure which is similar to the yield curve in the interest rate market. The picture below depicts the volatility term structure for SP500 as at August 31 2016 [3].

volatility term structure trading strategy — SP500 Volatility Term Structure at Aug 31 2016

Most of the time the term structure is in contango. This means that the back months have higher implied volatilities than the front months. However, during a market stress, the volatility term structure curve usually inverts. In this case we say that the volatility term structure curve is in backwardation (a similar phenomenon exists in the interest rate market which is called inversion of the yield curve).

The basic idea of the trading strategy is to use the state (contango/backwardation) of the volatility term structure as a timing mechanism. Specifically, we go long if the volatility term structure is in backwardation and go short otherwise. To measure the slope of the term structure, we use the VIX and VXV volatility indices which represent the 1M and 3M implied volatilities of SP500 respectively.

The trading rules are as follows,

Buy (or Cover) VXX if 5-Day Moving Average of VIX/VXV >=1 (i.e. backwardation)

Sell (or Short) VXX if 5-Day Moving Average of VIX/VXV < 1 (i.e. contango)

The Table below presents the results

	All trades	Long trades	Short trades
Initial capital	10000	10000	10000
Ending capital	177387.15	19232.01	168155.15
Net Profit	167387.15	9232.01	158155.15
Net Profit %	1673.87%	92.32%	1581.55%
Exposure %	99.44%	6.64%	92.80%
Net Risk Adjusted Return %	1683.22%	1390.38%	1704.17%
Annual Return %	46.07%	9.00%	45.05%
Risk Adjusted Return %	46.33%	135.54%	48.54%

All trades	30	15 (50.00 %)	15 (50.00 %)
Avg. Profit/Loss	5579.57	615.47	10543.68
Avg. Profit/Loss %	13.29%	2.29%	24.28%
Avg. Bars Held	64.53	9.8	119.27

Winners	14 (46.67 %)	4 (13.33 %)	10 (33.33 %)
Total Profit	208153.11	36602.85	171550.26
Avg. Profit	14868.08	9150.71	17155.03
Avg. Profit %	36.58%	25.17%	41.14%
Avg. Bars Held	129.64	14.25	175.8
Max. Consecutive	3	1	4
Largest win	71040.59	18703.17	71040.59
# bars in largest win	157	35	157

Losers	16 (53.33 %)	11 (36.67 %)	5 (16.67 %)
Total Loss	-40765.96	-27370.84	-13395.12
Avg. Loss	-2547.87	-2488.26	-2679.02
Avg. Loss %	-7.09%	-6.02%	-9.45%
Avg. Bars Held	7.56	8.18	6.2
Max. Consecutive	5	6	2
Largest loss	-9062.89	-8222.29	-9062.89
# bars in largest loss	6	8	6

Max. trade drawdown	-28211.89	-15668.21	-28211.89
Max. trade % drawdown	-23.97%	-18.20%	-23.97%
Max. system drawdown	-32794.28	-26555.13	-37915.18
Max. system % drawdown	-50.07%	-90.85%	-34.31%
Recovery Factor	5.1	0.35	4.17
CAR/MaxDD	0.92	0.1	1.31
RAR/MaxDD	0.93	1.49	1.41
Profit Factor	5.11	1.34	12.81
Payoff Ratio	5.84	3.68	6.4
Standard Error	12109.91	6401.13	12526.9
Risk-Reward Ratio	1.49	0.15	1.36
Ulcer Index	11.25	42.48	8.24
Ulcer Performance Index	3.62	0.08	4.81
Sharpe Ratio of trades	0.8	0.53	0.97
K-Ratio	0.0745	0.0073	0.0683

The graph below shows the portfolio equity from 2009 up to August 2016.

The annual rerun is 46% and the drawdown is 50%. There are 2 interesting observations

This strategy did not suffer a large loss like the VRP strategy during the August selloff of last year
Long volatility trades are profitable

In the next installment we will compare the 2 strategies, volatility risk premium and roll yield, in details.

References

[1] T Cooper, Easy Volatility Investing, SSRN, 2013

[2] Note that there is a so-called term structure risk premium in the options market that is not often discussed in the literature. The strategy discussed in this post, however, is not meant to exploit the term structure risk premium. It uses the term structure as a timing mechanism.

[3] The volatility term structure presented here is calculated based on VIX futures, which are the expectation values of 30-day forward implied volatility. Therefore, it is theoretically different from the term structure of spot volatilities which are calculated from SP500 options. Practically speaking, the 2 volatility term structures are highly correlated, and we use the futures curve in this article for illustration purposes.

Volatility Trading Strategy, a System Based on Volatility Risk Premium

Last year, we presented backtested results for a VXX trading strategy. The system’s logic is based upon the concept of volatility risk premium. In short, the trading rules are as follows:

Buy (or Cover) VXX if 5D average of (VIX index -10D HV of SP500) < 0

Sell (or Short) VXX if 5D average of (VIX index -10D HV of SP500)>0

The strategy performed well in backtest. In this follow-up post, we look at how it has performed since last year. The Table below summarizes the results

	All trades	Long trades	Short trades
Initial capital	10000	10000	10000
Ending capital	3870.55	9095.02	4775.53
Net Profit	-6129.45	-904.98	-5224.47
Net Profit %	-61.29%	-9.05%	-52.24%
Exposure %	100.00%	14.67%	85.33%
Net Risk Adjusted Return %	-61.29%	-61.68%	-61.23%
Annual Return %	-60.50%	-8.86%	-51.48%
Risk Adjusted Return %	-60.50%	-60.42%	-60.33%

All trades	11	5 (45.45 %)	6 (54.55 %)
Avg. Profit/Loss	-557.22	-181	-870.74
Avg. Profit/Loss %	-3.35%	-4.40%	-2.47%
Avg. Bars Held	24.55	8.6	37.83

Winners	6 (54.55 %)	2 (18.18 %)	4 (36.36 %)
Total Profit	2366.97	365.38	2001.59
Avg. Profit	394.49	182.69	500.4
Avg. Profit %	13.75%	6.46%	17.40%
Avg. Bars Held	32.5	7.5	45
Max. Consecutive	2	2	2
Largest win	1308.08	239.26	1308.08
# bars in largest win	102	8	102

Losers	5 (45.45 %)	3 (27.27 %)	2 (18.18 %)
Total Loss	-8496.42	-1270.36	-7226.06
Avg. Loss	-1699.28	-423.45	-3613.03
Avg. Loss %	-23.87%	-11.64%	-42.21%
Avg. Bars Held	15	9.33	23.5
Max. Consecutive	2	2	1
Largest loss	-6656.33	-625.51	-6656.33
# bars in largest loss	29	11	29

The strategy produced 11 trades with 6 trades (55%) being winners. However, it suffered a big loss during August. The graph below shows the portfolio equity since last August.

quantitative trading volatility — VRP volatility trading strategy

Large losses are typical of short volatility strategies. An interesting observation is that after the large drawdown, the strategy has recovered, as depicted by the upward trending equity line after August. This is usually the case for short volatility strategies.

Despite the big loss, the overall return (not shown) is still positive. This means that the strategy has a positive expectancy. Drawdown can be minimized by using a correct position size, stop losses, and a good portfolio allocation scheme. Another solution is to construct limited-loss positions using VXX options.

Volatility Trading through VIX ETFs

It is well known that persistent biases exist in various markets. For example, in the tennis market, there exists a longshot bias. Similarly, financial markets exhibit a persistent bias called the risk premium. Formally, the risk premium is defined as

where E_Q denotes the expectation value of X, a stochastic variable, in the risk-neutral world, and E_P denotes the expectation value of X in the real world. X can be, for example, commodity prices, FX rates, etc. See reference 1 for a thorough discussion of various risk premia in financial markets.

Of our particular interest is the case where X is the realized volatility (RV) of a stock or stock index. The risk premium in this case is often called volatility (or variance) risk premium (VRP). If we substitute X in the above equation with the RV of SP500, then E_Q (RV) becomes the VIX index, and the VRP becomes VIX- E_P (RV).

Traders often try to exploit the VRP by trading listed options or OTC variance swaps. Another way to harvest the VRP is through trading VIX -based Exchange Trade Funds such as VXX. In this post we explore the latter possibility.

The most difficult problem when designing a VRP-based trading strategy is that E_P(RV), which is needed in order to calculate the VRP, is not observable. The best we can do is to use a quantitative method to estimate it.

Reference 2 tested various forms of E_P(RV) estimate: GARCH, historical volatilities (HV). The author found that 10-day HV is the most effective. They then further smoothed out the HV by using a 5-day moving average in order to avoid whipsaws. The trading rules are as follows:

Buy (or Cover) VXX if VIX index <= 5D average of 10D HV of SP500

Sell (or Short) VXX if VIX index > 5D average of 10D HV of SP500

The Table below summarizes the trading strategy’s statistics. The starting capital is $10000. It is fully invested in each trade

	All trades	Long trades	Short trades
Initial capital	10000	10000	10000
Ending capital	727774.91	-74688.89	812463.8
Net Profit	717774.91	-84688.89	802463.8
Net Profit %	7177.75%	-846.89%	8024.64%
Exposure %	99.02%	6.58%	92.44%
Net Risk Adjusted Return %	7248.42%	-12868.01%	8680.57%
Annual Return %	93.38%	N/A	96.68%
Risk Adjusted Return %	94.29%	N/A	104.58%

All trades	43	21 (48.84 %)	22 (51.16 %)
Avg. Profit/Loss	16692.44	-4032.8	36475.63
Avg. Profit/Loss %	12.04%	2.34%	21.29%
Avg. Bars Held	38.79	6.14	69.95

Winners	32 (74.42 %)	12 (27.91 %)	20 (46.51 %)
Total Profit	918340.82	94671.52	823669.31
Avg. Profit	28698.15	7889.29	41183.47
Avg. Profit %	18.13%	8.45%	23.93%
Avg. Bars Held	47.47	5.92	72.4
Max. Consecutive	9	5	8
Largest win	198505.33	36704.45	198505.33
# bars in largest win	78	7	78

Losers	11 (25.58 %)	9 (20.93 %)	2 (4.65 %)
Total Loss	-200565.92	-179360.41	-21205.5
Avg. Loss	-18233.27	-19928.93	-10602.75
Avg. Loss %	-5.69%	-5.82%	-5.10%
Avg. Bars Held	13.55	6.44	45.5
Max. Consecutive	3	4	1
Largest loss	-38701.76	-38701.76	-16516.3
# bars in largest loss	4	4	41
Max. trade drawdown	-300537.33	-38701.76	-300537.33
Max. trade % drawdown	-46.12%	-12.49%	-46.12%
Max. system drawdown	-331363.27	-164250.75	-300537.33
Max. system % drawdown	-52.21%	-95.16%	-64.93%
Recovery Factor	2.17	-0.52	2.67
CAR/MaxDD	1.79	N/A	1.49
RAR/MaxDD	1.81	N/A	1.61
Profit Factor	4.58	0.53	38.84
Payoff Ratio	1.57	0.4	3.88
Standard Error	56932.62	32093.21	72795.73
Risk-Reward Ratio	1.67	-0.46	1.51
Ulcer Index	12.97	111.97	11.45
Ulcer Performance Index	6.78	N/A	7.97
Sharpe Ratio of trades	1.46	1.36	1.71
K-Ratio	0.0775	-0.0214	0.07

We observe that the short trades are profitable while the long ones lost money. The CARG is 93.4%, which is high, but so is the drawdown of -52%. The graph below shows the portfolio equity

Volatility portfolio equity — VRP Volatility Trading Strategy

In summary, the VRP can be harvested through VIX ETF. However the drawdown is high. This strategy is viable if it is part of an asset allocation scheme. It would enhance the portfolio risk-adjusted return if we allocate, for example, 10% of our portfolio to this strategy, and the rest to equity and fixed income investments.

References

[1 ] A. Ilmanen, Expected Returns: An Investor’s Guide to Harvesting Market Rewards, John Wiley & Sons , 2011

[2] T Cooper, Easy Volatility Investing, SSRN, 2013

Pricing Convertible Bonds-An Example

In the previous post, we outlined the main steps in pricing a convertible bond using the Binomial Tree approach. In this follow-up post, we provide a hypothetical example of a straight convertible bond.

The specifics of the hypothetical convertible bond are as follows,

INPUTS
Stock price	100
Volatility	0.25
Risk Free Rate	0.02
Risky Rate (risk free+credit spread)	0.08
Coupon	0.06
Maturity (in years)	5
Conversion ratio	10

Using the formula provided in the previous post, we calculated the up and down moves, and probability of the up move. The results are

MODEL PARAMETERS
dt	1.00
u	1.28
d	0.78
Probability Up	0.48

Once the tree parameters are calculated, we next build the tree, and then work backward from the end nodes in order to obtain the convertible bond’s price at time zero. As the final result, the bond price is $1319 (per $1000 notional)

To see the details of the calculations of the tree, follow the instruction at the bottom of this page. Note that this is a simplified example. In real life, convertible bonds are usually more complex. They often include features such as call, put, contingency conversion options. The call and put options can be implemented using the formula given in the previous post.

I hope that this and previous posts demystified a little bit the complexities of convertible bonds. Let us know if you have any questions.

Follow this link for more derivative pricing models in Excel

Pricing Convertible Bonds and Preferred Shares

A convertible bond (or preferred share) is a hybrid security, part debt and part equity. Its valuation is derived from both the level of interest rates and the price of the underlying equity. Several modeling approaches are available to value these complex hybrid securities such as Binomial Tree, Partial Differential Equation and Monte Carlo simulation. One of the earliest pricing convertible bond approaches was the Binomial Tree model originally developed by Goldman Sachs [1,2] and this model allows for an efficient implementation with high accuracy. The Binomial Tree model is flexible enough to support the implementation of bespoke exotic features such as redemption and conversion by the issuer, lockout periods, conversion and retraction by the share owner etc.

In this post, we will summarize the key steps in pricing convertible bond method using the Binomial Tree approach. Detailed description of the method and examples are provided in references [1,2].

Generally, the value of a convertible bond with embedded features depends on:

The underlying common stock price
Volatility of the common stock
Dividend yield on the common stock
The risk free interest rate
The credit worthiness of the preferred share issuer

Within the binomial tree framework, the common stock price at each node is described as

pricing convertible bond

where S₀ is the stock price at the valuation date; u and d are the up and down jump magnitudes. The superscript j refers to the time step and i to the jump. The up and down moves are calculated as

and

where is the stock volatility, and is the time step.

The risk neutral probability of the up move, u, is

and the probability the down move is 1-p

After building a binomial tree for the common stock price, the convertible bond price is then determined by starting at the end of the stock price tree where the payoff is known with certainty and going backward until the time zero (valuation date). At each node, Pj,i the value of the convertible is

where m denotes the conversion ratio.

If the bond is callable, the payoff at each node is

The payoff of a putable bond is

Here C and P are the call and put values respectively; r denotes the risk-free rate.

The above equations are the key algorithms in the binomial tree approach. However, there are several considerations that should be addressed due to the complexities of the derivative features

Credit spreads (credit risk) of the issuers which usually are not constant.
Interest rates can be stochastic.
Discount rate ri,j depends on the conversion probability at each node. This is due to the fact that when the common share price is well below the strike, the preferred share behaves like a corporate bond and hence we need to discount with a risky curve. If the share is well above the strike then the preferred behaves like a common stock and the riskless curve need to be used.
The notice period: the issuer tends to call the bond if the stock price is far enough above the conversion price such that a move below it is unlikely during the notice period. For most accurate results, the valuation would require a call adjustment factor. This factor is empirical and its value could be determined by calibration to stock historical data.

This approach in pricing convertible bond can be implemented in scripting languages such as VBA and Matlab. In the next installment, we will provide a concrete example of pricing a convertible bond. If you have a convertible bond that you want us to use as example, send it to us.

References

[1] Valuing Convertible Bonds as Derivatives, Quantitative Strategies Research Notes, Goldman Sachs, November 1994.

[2] Pricing Convertible Bonds, Kevin B. Connolly, Wiley, 1998.

Brazil Beats Germany

The World Cup is over and Germany won the much coveted trophy. This World Cup will be remembered for its beautiful attacking-style games as well as one of the most crushing defeats in football history: the Host lost 1-7 to Germany in the semi final.

However, the Brazilians do not have to wait for another 4 years in order to have a chance to revenge against Germany. Their stock market has already beaten the German’s one. The chart below shows the ratio of the Brazil ETF (EWZ) v.s. Germany ETF (EWG) and we can see that the ratio is in an uptrend.

Statistically, the Brazilian market has underperformed over the last 3 years. It lost -29%, while the German market gained 45%. However, EWZ started bouncing at the beginning of this year from an oversold condition.

The bounce is also supported by fundamental factors: Brazil is an exporter of commodities and the uptrend in base metals lends support to its market recovery. Additionally, the upcoming Summer Olympics in Rio in 2016 will give a boost to the local economy.

Germany, on the other hand, is suffering from problems in Europe: deflation threat, a weak euro, negative interest rates, geopolitical tension in Ukraine and Russia, and a possible bank failure in Portugal, just to name a few. This is a good opportunity for pair traders who want to take advantage of the divergence.

Brazil, Emerging Markets and the World Cup

There are now less than 2 weeks until the start of the 2014 FIFA World Cup of Soccer, which is the biggest sport event in the world. The event is being organized in Brazil. From an economic point of view, Brazil is one of the BRIC countries; it has underperformed the overall emerging market during the last 4-5 years. The chart below shows the relative strength of Brazil ETF (EWZ) with respect to the emerging market ETF (EEM).

The ratio has been in a down trend for more than 4 years. We can observe, however, a rebound taking place in early 2014. Some analysts said that this rebound is supported in part by the preparation of the 2014 World Cup of Soccer and the 2016 summer Olympics.

Interestingly, Brazil is the favorite for winning the World Cup this year. It has the highest chance of winning the World Cup, followed by Argentina, Germany and Spain.

The implied probability of winning calculated from the various bookmakers odds is in the range of 20%-25%. A World Cup win can boost consumer confidence and hence the local economy in general. (We saw a similar situation before in 1998 when France won its first ever World Cup at home).

To play a potential recovery in Brazil, one can go long EWZ and hedge the downside with EMM. If you worry about the negative impact of the host nation’s not winning the World Cup, you can hedge by laying against Brazil on a sport exchange.

Good luck and enjoy the Game!