1. Volatility Arbitrage Trading with Leverage SV
(stochastic volatility) Models via Particle Filtering
presented by
Mahsiul Khan, PhD
Quantum Filtering Algorithm LLC, NY
NYC Quant Research Group, Morgan Stanley Building
June 15, 2015
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 1 / 24
2. Outline
1 Modeling Volatility
2 Dynamic Asset Price Model
3 Realized volatility
4 Implied volatility
5 Leverage SV Models
6 Particle Filtering (PF)-A Sequential Bayesian Filtering
7 Volatility Trading
8 Variance and Volatility Swaps
9 PF to predict Volatility on SP500
10 Backtesting with VIX
11 Conclusions
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 2 / 24
3. Modeling Volatility
Volatility refers to uncertainty. It is the standard deviation (σt) of
asset price which is heteroscedastic & non-stationary.
rt = µ + σtvt (1)
yt = σt vt (2)
yt ∼ N(0, σ2
t ) (3)
σ2
t = α0 + α1y2
t−1 + β1σ2
t−1, GARCH(1, 1) (4)
where St = price, rt = ln(St /St−1) log-return, µ= expected return,
yt = rt − µ is excess return, vt is standard Gaussian noise process.
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 3 / 24
4. Dynamic Asset Pricing Models
Continuous Time Asset Pricing Model with SDE:
dSt
St
= µdt + σt dWt , GBM(Geometric Brownian Motion)(5)
d(ln St ) = (µ −
1
2
σ2
t )dt + σt dWt (with Ito′
s Lemma) (6)
V(0,T) = E[
1
T
T
0
σ2
t dt] (7)
ln(
st
s0
) ∼ N[(µ −
1
2
σ2
t )t, σ2
t t] (8)
where St = asset price with log-normal distribution, µ= expected
return, σt =volatility, and Wt is Wiener process (Brownian motion).
The V(0,T) is the expected future variance over [0, T].
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 4 / 24
5. Realized (Historical) Volatility
The annualized Realized Volatility is calculated with daily return
σrealized =
252
N − 1
N
t=1
r2
t , assume E[rt ] = ¯rt = 0 (9)
Where rt is log-return with log-normal distribution.
It is also called historical (observed) volatility
SV and GARCH family models are predictive models
Mean zero assumption has little effect on volatility estimation
Converting 1-Day vol to h-Day vol by scaling
√
h causes
overestimation on long-horizon vol (Diebold, 97)
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 5 / 24
6. Implied Volatility
Implied volatility parameter appear in Black-Scholes (BS) model.
It is the expected future volatility of underlying.
dSt
St
= µdt + σimplied dWt (10)
C(St , t) = f(St, K, T − t, r, σimplied ), (BS call option model)(11)
C(St , t) = e−r(T−t)
E[payoff(ST , K)] (12)
ˆσ¯C = f−1
( ¯C, · · · ) (13)
If ¯C(·) is the market-price of an option, then vol ˆσ¯C is implied
by the market price ¯C(·), and is called impled vol.
Objective: model theoretical vol with market-price of option
In Idealized world there is only one volatility:
σrealized = σimplied = σ
Mispricing due to Model uncertainty and the estimation
error creates arbitrage
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 6 / 24
7. Leverage SV Models with Correlated Noise
The standard SV model
xt = β1 + β2xt−1 + σuut (state) (14)
yt = ext /2
vt (obs) (ut , vt indep) (15)
The Leverage SV model
ut = ρvt−1 + 1 − ρ2u′
t (16)
xt = β1 + β2xt−1 + β3yt−1e−
xt−1
2 + ζu′
t (state) (17)
yt = ext /2
vt (obs) (18)
yt ∼ N(0, ext ) (19)
p(xt|y1:t) = sequential filtering (objective)
p(xt, θ|y1:t) = sequential filtering (Liu, & West) (20)
where xt = log(σ2
t ) is log-volatility, corr(ut , vt−1) = ρ, and ut , vt ,&
u′
t are standard Gaussian noise. θ is unknown static parameters.
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 7 / 24
8. Particle Filtering(PF)-Sequential Bayesian Filtering
1 Kalman Filter-optimal when system is Linear & Gaussian
2 Extended Kalman Filter-when system is nonlinear &
non-Gaussian. Non-linearity too high, approx. is poor.
3 PF is suboptimal when systems are highly nonlinear &
non-Gaussian. It is a simulation based method under Bayes’
theorem for sequential realtime inferences.
Let x0:t ≡ (x0, ..., xt) and y1:t ≡ (y1, ..., yt ) as state and
observation sequences. Transition from prior to posterior as,
p(x0:t|y1:t, Ψ)
posterior
=
likelihood
p(y1:t |x0:t, Ψ)
prior
p(x0:t |Ψ)
p(y1:t|Ψ)
evidence
(21)
{xi
0:t, wi
t }N
i=1 ≈ p(x0:t|y1:t) (22)
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 8 / 24
9. Particle Filtering-continue
Posterior : p(x0:t|y1:t) = p(x0:t−1|y1:t−1)
p(yt |xt )p(xt |xt−1)
p(yt |y1:t−1)
Filtering : p(xt |y1:t) =
p(yt |xt )p(xt |y1:t−1)
p(yt |y1:t−1)
Prediction : p(xt |y1:t−1) = p(xt |xt−1)p(xt−1|y1:t−1)dxt−1
Update : p(xt |y1:t) ∝ p(yt |xt )p(xt |y1:t−1)
wi
t ∝ wi
t−1
p(yt |xi
t )p(xi
t |xi
t−1)
q(xi
t|xi
t−1, yt )
{xi
0:t, wi
t }N
i=1
a.s
−→ p(x0:t|y1:t) as N → ∞
{xi
0:t}N
i=1 are streams of particles (realizations) with associated
weights {wi
t }N
i=1 to approximate posterior PDF p(x0:t|y1:t).
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 9 / 24
10. PF:Pictorial Representation (from Van Der Merwe)
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 10 / 24
11. Volatility Trading
Most volatility arbitrage strategies take advantage of the
difference between the implied and realized vol of an asset.
The strategy often implemented through delta neutral
method consisting options and its underlying assets.
Long on vol ⇒ buy (call) option + sell underlying
Short on vol ⇒ sell option + buy underlying
A long vol position have positive gain if realized vol is higher
than implied vol at the time of the trade
Delta neutral is not neutral over time, as price changes
Dynamic re-hedging constitute transaction costs
Delta-hedging is not pure exposure to volatility, also
depends on underlying price
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 11 / 24
12. Variance and Volatility Swap-Pure Vol Exposure
Variance (volatility) swap is considered an asset class with
pure exposures to volatility
Variance swaps is a forward contracts on annualized
variance (vol) when trader A pays strike variance (vol) Kvar ,
and receives realized variance (vol) σ2
R at Maturity T.
A long (short) variance swap position will profit if the realized
variance(vol) of the underlying is greater (smaller) than the
strike Kvar at Maturity.
Variance swap trade on spread of Realized vs Implied vol
Variance swaps are actively traded on major equity indices:
S&P500, Nasdaq, Nikkei, EuroStoxx50, ...
Drawbacks: It is OTC products, lack of liquidity
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 12 / 24
13. Variance and Volatility Swap
What is the difference bet variance and volatility swap?
Kvar = strike variance=Expected future variance on [0,T]
Kvol = strike volatility=Expected future volatility on [0,T]
Variance Swap Payoff: Nvar x(σ2
R − Kvar ), Nvar = vega notional
2xKvar
Volatility Swap Payoff: Nvolx(σR − Kvol) ≈ Nvol
2Kvol
(σ2
R − K2
vol)
Kvar = E[1
T
T
0 σ2
t dt], Kvol = E[1
T
T
0 σtdt]
Kvol ≈
√
Kvar , Why approximation ?
Because, E[
√
variance] <= E[variance] (Jensen’s Ineq).
Most trade variance swap, pricing volatility swap is not linear.
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 13 / 24
14. Variance Swap:Mark-to-Market
At inception, swap value is zero: E[e−r(T−t0)(σ2
R − Kvar )] = 0
Mark-to-Market(MTM) value computed dynamically on [t0, T]
Decompose variance V(t0, T) at time t
V(t0, T)(T − t0) = V(t0, t)(t − t0)
realized
+ V(t, T)(T − t)
future
V(t0, T) = λV(t0, t) + (1 − λ)V(t, T), λ = t−t0
T−t0
MTM(t) = Ne−r(T−t)[λ(V(t0, t) − Kvar ) + (1 − λ)(Kt − Kvar )]
λ= proportion of time elapsed by t
Kt=new strike var available at t
MTM(t) is used to adjust the position
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 14 / 24
15. Pricing Variance/Volatility Swaps
Estimating Kvar is the most important and difficult, closed
form, numerical and MC methods are used.
Javaheri, Wilmott, Haug (2004) proposed closed form
solution with mean-reverting OU process
Demeterfi et al (1999b) also proposed closed form solution
replicating portfolio of options under BS model
Heston’s (1993) SV model also used for estimating Kvar
Demeterfi rule of thumb, Kvar = σATMF 1 + 3Txskew2,
σATMF =at-the-money-forward vol, T=maturity in years
CBOE vol index VIX is fair value strike (Kvar ) for variance swap
on S&P500 index
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 15 / 24
16. Variance Swap-A Hypothetical Trade
Buyer=Trader A, Seller=Trader B, Underlying=S&P500
Start date=t0, Maturity=t0 + 3mo, Strike (Kvar ) = .30, N = $1
Buyer receives Realized vol & Seller receives Implied vol (Kvar )
Payoff Calculation:
Scenario-1
realized volatility=.20
payoff=$1x(.202
− .302
) = $ − .05
Trader A pays 5 cents to Trader B
Scenario-2
realized volatility=.40
payoff=$1x(.402
− .302
) = $.07
Trader A receives 7 cents from Trader B
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 16 / 24
21. Backtesting with VIX as Strike Variance (Kvar)
We did concept model backtesting with VIX as strike
variance, trading position based on PF-predicted vol.
We compared the performance of PF-algo with Actual
Realized vol & Short Only position
PF-SV Model is 1-step predicting model
Every day we take a new 21-day variance swap position
Payout is realized at 21-day, assume zero transaction cost
Signal: position(t)=sign(pfvol(t) − vix(t))
Daily PnL(t+21)=position(t)[N(t)(vol2
realized (t + 21) − Kvar (t))]
Kvar (t) = (vix(t) − 1)2, N(t) = Vega
2Kvar (t) , Vega = .30
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 21 / 24
22. Backtesting with VIX as Strike Variance:1990-2014
1993
1997
2001
2005
2009
2013
date
0
50
100
150
200
250
%
Cumulative Return
pf
always_short
actual
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 22 / 24
23. Backtesting Output
Table: Backtesting Output
Features PF Always Short Actual Realized
sharpe 5.93 5.43 5.05
max drawdown (%) -4 -18 -6
return (%) 5 4 4
volatility(%) 0.83 .83 .86
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 23 / 24
24. Conclusion
1 Variance/Volatility swap perfect for trading and hedging risk
2 S&P500 volatility arbitrage index consistently outperforms
S&P500 by 3% since 1990
3 PF-SV model outperforms existing methods
4 PF-signal correctly predicted volatility 2008-09 market crash
5 PF-SV model is ideal for high frequency trading
6 Backtesting is done on concept, refinement in the trading
frequency and algorithm will improve performance
7 Backtesting with S&P500 variance future: next assignment
8 Thanks to Dr. David Chan for his collaboration on
backtesting strategy
9 Thank you Quant Group
presented byMahsiul Khan, PhD (Quantum Filtering Algorithm LLC, NY)Volatility Arbitrage Trading with Leverage SV (stochastic volatility) Models via ParticleQuant Research Group 24 / 24
