Convexity of Trend following & Variance arbitrage · Trend and Convexity Trend versus Risk Parity Variance arbitrage Conlusions ConvexityofTrendfollowing&Variancearbitrage Tung-Lam

Trend and ConvexityTrend versus Risk Parity

Variance arbitrageConlusions

Convexity of Trend following & Variance arbitrage

Tung-Lam Dao

in collaboration with Research team at CFM:

T.T.Nguyen, C. Deremble, Y. Lemperiere, J.P. Bouchaud, M. Potters

Metori Capital Management, December 2017




Convexity of CTA

W. Fung and D. A. Hsieh: Empirical Characteristics of Dynamic Trading Strategies: The Case of Hedge Funds, Review of Financial studies,10, 275-302 (1997)

W. Fung and D. A. Hsieh: The Risk in Hedge Fund Strategies: Theory and Evidence from Trend Followers, Review of Financial studies,Review of Financial studies, 14, 313-341 (2001)

M. Potters and J.-P. Bouchaud: Trend followers lose more often than they gain, Wilmott Magazine, 26, 58-63 (January 2006).

B. Bruder and N. Gaussel: Risk-Return Analysis of Dynamical investment Strategies, Lyxor White Paper, Issue 7 (June 2011).

T.-L. Dao et al: Tail Protection for Long Investors: Trend Convexity at Work Journal of Investment Strategies (May 2016)

P. Jusselin et al: Understanding the Momentum Risk Premium: An In-Depth Journey Through Trend-Following Strategies, Amundi WhitePaper (Oct 2017).




Outlook

1 Trend and Convexity

2 Trend versus Risk Parity

3 Variance arbitrage

4 Conlusions




How to observe the convexity of CTA?B

TO

P 5

0 I

ndex

S&P 500 Index

-10 %

-5 %

0 %

5 %

10 %

-20 % -15 % -10 % -5 % 0 % 5 % 10 % 15 % 20 %

Monthly returns of the Barclays BTOP 50 Index vs. monthlyreturns of the S&P 500 Index (symbols) and parabolic fit

SG

CT

A I

ndex

S&P 500 Index

-10 %

-5 %

0 %

5 %

10 %

-20 % -15 % -10 % -5 % 0 % 5 % 10 % 15 % 20 %

Monthly returns of the SG CTA Index vs. monthly returns of theS&P 500 Index (symbols) and parabolic fit




Approach: Volatility at different timescalePrice model

St = S0 +

t∑t′=1

Dt′ .

Price changes Dt′<t are stationary random variables with zero mean and covariancegiven by:

E[DuDv ] = C(|u − v|) ,

Uncorrelated random walks corresponds to C(u) = σ2δu,0. Trending random walks aresuch that C(u) > 0, while mean-reverting random walks are such that C(u) < 0

How does this translate in terms of the volatility of the walk? We define the volatility ofscale τ :

σ2(τ) :=

1τE[

(St+τ − St )2]

= σ2 +

2τ

τ∑u=1

(τ − u) C(u) ,

with single step volatility σ2(1) = σ2.

0.5

1

1.5

2

2.5

20 40 60 80 100 120 140 160 180 200

σ2(τ

) / σ

2

τ

TrendUncorrelated

Mean-reversion




Simple Trend = long variance - short varianceConsider a simple strategy such that the position Πt is proportional to the price difference between t and 0:

Πt := (St − S0) ,

P&L from t − 1 to t is given by:

Gt := Πt−1Dt = Dt

t−1∑t′=1

Dt′ , G1 := 0 .

Cummulative performance of from day 0 to day T :

GT =

T∑t=1

Gt =

T∑t=2

t−1∑t′=1

Dt Dt′

=12

(T∑

t=1

Dt

)2

−12

T∑t=1

D2t

⇒ GT =12

(ST − S0

)2︸︷︷︸Long-term Variance

−12

T∑t=1

D2t︸︷︷︸

Short-term Variance




Convexity with general trend

Trend estimator defined with EMA filter:

dπt = −2τπt dt +

2τ

dSt .

The solution of this SDE given by the following expression:

πt = Lτ (St ) =2τ

∫ t

−∞

e2(t−s)/τ dSs .

Daily profit and loss: dGt = φ(πt )× dSt

dGt = φ(πt )πt dt +τ

2φ(πt )dπt .

Let F (x) be such that F ′(x) = φ(x), then using Ito’s lemma we have:

dF (πt ) = φ(πt )dPt +2φ′(πt )τ2

dS2t .

Inserting this expression in the equation of the P&L, we obtain:

dGt = φ(πt )πt dt −φ′(πt )τ

dS2t︸︷︷︸

Drift term

+τ

2dF (πt )︸︷︷︸

Risk term

Re-arrange the different terms and introduce the filter LT :

LT [dGt ] =τ

TF (πt )︸︷︷︸

Payoff

−LT

[φ′(πt )τ

dS2t

]︸︷︷︸Volatility Cost

+LT

[φ(πt )πt −

τ

TF (πt )

]dt︸︷︷︸

Error term




Illustration of convexityLinear trend: Lτ/2[dGt ] = π2

t −1τLτ/2[dS2

t ].

P&

L o

f a

Tre

nd F

oll

ow

er

Trend Signal

-30 %

-20 %

-10 %

0 %

10 %

20 %

30 %

40 %

50 %

60 %

-3 -2 -1 0 1 2 3

Sign of the trend: E[Lτ [dGt ]|πt ] = |πt | −√

2πτσ.

P&

L o

f a

Tre

nd F

oll

ow

er

Trend Signal

-40 %

-30 %

-20 %

-10 %

0 %

10 %

20 %

30 %

40 %

-3 -2 -1 0 1 2 3




Replication of Risk Parity and CTA

Commodities Stock Indices Foreign Exchange Rates Short term interest rates Government bondsWTI Crude Oil (CME) S&P 500 (CME) EUR/USD (CME) Euribor (ICE) 10Y U.S. Treasury Note (CME)

Gold (CME) EuroStoxx 50 (Eurex) JPY/USD (CME) Eurodollar (CME ) Bund (Eurex)Copper (CME) FTSE 100 (ICE) GBP/USD (CME) Short Sterling (ICE) Long Gilts (ICE)Soybean (CME) Nikkei 225 (JPX) AUD/USD (CME) JGB (JPX)

CHF/USD (CME)

Employ the most liquid futures in each sector.

This selection is stable across time.The time series are considered between January 2000 and October 2015.




0 %

20 %

40 %

60 %

80 %

100 %

0 50 100 150 200 250 300 350 400

Corr

ela

tion c

oeff

icie

nt

τ

Correlation between CTA replicator and the SG CTA Index as a function ofthe time-scale of the trend. maximum is around τ = 180 days.

0 %

20 %

40 %

60 %

80 %

100 %

120 %

140 %

2000 2002 2004 2006 2008 2010 2012 2014 2016

SG CTA IndexReplicator

Cumulated returns of trend replicator and the SG CTA Index. We seem tocapture all the alpha contained in the SG CTA Index.




Convexity of CTA versus S&P500

Plot aggregated performance over τ ′ days the SG CTA Index as afunction of a τ -day trend on S&P 500 index.

we need a careful choice of timescale to observe the convexity.Trend on S&P 500 is computed with τ = 180 while trend on SGCTA Index is computed with τ ′ ≈ 90.

Convexity feauture is much more significantly than the firstmonthly plot.

SG

CT

A I

nd

ex

Trend Signal

-10 %

-5 %

0 %

5 %

10 %

15 %

20 %

25 %

30 %

-3 -2 -1 0 1 2 3




Convexity of CTA versus Risk Parity

A simple trend following strategy applied on Risk Parity providesthe exact quadratic behavior of convexity.

CTA index provides also convexity to Riks Parity strategy

We can derive an inequality for lower bound of convexity:

E[GCTA|T RP] ≥ Υ(τ)((T RP)2− 1)

where T RP is the trend on Risk Parity index.

SG

CT

A I

nd

ex

Risk Parity Index Replicator

-10 %

-5 %

0 %

5 %

10 %

15 %

20 %

25 %

30 %

-20 % -10 % 0 % 10 % 20 %




Conclusion: trend as a hedge of Risk Parity

Trend following is a natural way to do stop-loss for long risk parity.

The hidden cost of for doing this hedge is the realized volatility.

Tuning the mixing between Trend and Long-only, one may obtainthe desired the convexity for the portfolio.

Both global trend and diversified trend can be used to hedge longrisk parity.




Straddle and collection of stranglesATM straddle payoff:

GstraddleT := |ST − S0| − (CS00,T + PS0

0,T )

= |ST − S0| −

√2Tπ

S0σa0

For a collection of strangles:

GstranglesT :=

∫ S0

0

(K − ST )+dK +

∫ ∞S0

(ST − K)+dK︸︷︷︸12 (ST−S0)2

−

∫ S0

0

PK0,T +

∫ ∞S0

CK0,T︸︷︷︸

12 T σ̄20,T

We obtain the payoff:

GstranglesT =12

(ST − S0)2 −12

T σ̄20,T

Buying straddle or strangles is a simple way to have long exposure on long-term variance by paying fixed implied volatility




Trend as Delta-hegde: Model-free approachWhen the price move, we need to trade a quantity of underlying tokeep neutral Delta. The Delta-hedge position is given simply by:

∆hdg = −(St − S0)

Total P&L of the hedge from t = 0 to the maturity is given by:

GhedgeT :=12

T∑t=1

D2t −

12

(ST − S0)2

Add the pay-off of strangles to this hedge P&L, we find:

GstranglesT + GhedgeT =12

T∑t=1

D2t −

T2σ̄20,T︸︷︷︸

Variance Swap payoff

Spot Price

Portfolio at time t

Spot Price Strike Price

Portfolio at time t'




Variance arbitrageSelling option will allow a risk premium. In fact, E[GstranglesT ] < 0 because the implied volatility is usually overpriced and more expensivethan the realized volatility.

GstranglesT =12

(ST − S0)2 −12

T σ̄20,T

Doing simple trend is also a simple way to exposure to long-variance. Indeed, trend anomaly has been proved empirically E[GtrendT ] > 0.

GtrendT =12

(ST − S0

)2−

12

T∑t=1

D2t

Trading volatility is interesting because we profit from both effects. Selling long-variance with high cost "implied variance" and buyinglong-variance with smaller cost "realized variance"⇒ Variance arbitrage

GtrendT − GstranglesT =12

T σ̄20,T −12

T∑t=1

D2t




ConclusionsTrend following and convexity:

Trend is an arbitrage between long-term variance and short-term varianceConvexity of trend is observed in association with a timescale (investment horizon or trend timescale)Timescale of trend defines a maturity (like call/put). Trend behaves as a hedge only for this maturity.

Trend following as a hedge of Risk Parity:Trend can be used as a hedging tool of long risk parityBoth diversified trend and global trend can be used for hedging the long risk

Variance arbitrage:Delta hedge of options is a trend. To harvest option premium, hedging frequency is important.Choosing hedging frequency is choosing timescale of volatility that one wants to be exposed.Best choice for hedging is to employ the trend anomaly with low frequency.


Documents

Convexity of Trend following & Variance arbitrage · Trend and Convexity Trend versus Risk Parity Variance arbitrage Conlusions ConvexityofTrendfollowing&Variancearbitrage Tung-Lam