On the nature of jump risk premia

Agenda

Evidence of jump premia

• continuous-time models fitted to low-frequency data
• analysis of short-maturity options (still too long!)

What are jumps?

• intraday discontinuities?
• overnight returns (frictions, liquidity)?

Is there a premium for jumps in high-frequency returns?

• examine continuous trading during market hours

Where do option-implied jump premia come from?

• examine profits around non-trading periods
  • no options / only futures
  • no more pure jump interpretation
• aversion to hedging restrictions or mispricing?

Core results

Model-free strategy that pays the realized skew of jumps at settlement

• Dynamic option trading
• Fully non-parametric investigation of jump skew pricing
• How to interpret dynamic option trading

Pricing of jump skew risk:

• Not present in daytime option trading
  • large literature on jumps in high-frequency returns
• Present when trading options around non-trading periods
  • but is it mispricing?
• Different from the pricing of variance risk

Motivation

Measurement of jump skew risk

Price and premium for jump skew risk

Empirical results: trading jump skew risk

Motivation – theoretical

Skewness and preferences

• Kraus and Litzenberger (1976), Arditti (1967)
  • positive preference for skewness for reasonable utility functions
  • with systematic skewness: low returns for positively skewed payoffs.

Jumps and asset pricing

• Drechsler (2013), Schreindorfer (2014), Segal, Shaliastovich, and Yaron (2015)
  • uncertainty shifts and return jumps
• Harvey and Siddique (2002), Schneider, Wagner, and Zechner (2016)
  • conditional skewness explains cross-sectional anomalies
• Barro (2006), Barro and Ursúa (2008), Tsai and Wachter (2016), Barro and Liao (2019)
  • rare consumption disasters

Motivation – empirical

Tail risk \(\neq\) volatility risk

• Andersen, Todorov, and Fusari (2015), Andersen, Fusari, and Todorov (2017)
  • Variation in negative jump risk not spanned by volatility
  • Prices of OTM puts (relative to ATM) do not depend on diffusive volatility
  • New factor required to describe variation in price of left tail risk.

• Bollerslev and Todorov (2014):
  • \(\mathbb{P}\)-measure tail risks evolve over time.

Asymmetric pricing of tail risks

• Andersen, Fusari, and Todorov (2017):
  • Prices of negative jump risk can be recovered from prices of deep OTM short-maturity puts.
• Andersen, Todorov, and Fusari (2015):
  • The premium for negative jump risk is large, persistent
  • Little to no premium for positive jump risk.

Motivation – empirical

Jump risk \(=\) priced risk?

• Kelly and Jiang (2014), Bollerslev, Li, and Todorov (2016):
  • Jump risk (of S&P500) priced in cross-section of stock returns.
  • \(\alpha = 5.4\%\) p.a. wrt FF3.
• Bollerslev, Li, and Todorov (2016):
  • only jump component of S&P500 priced.

Other important facts

• Bollerslev, Todorov, and Xu (2015), Kelly and Jiang (2014):
  • Price of jump risk predicts S&P500 returns.

Motivation

Measurement of jump skew risk

Price and premium for jump skew risk

Empirical results: trading jumps kew risk

Measurement of jump skewness

Consider a semi-martingale forward price process (finite activity jumps):

\[ \frac{dF_t}{F_{t-}} = \mu_{t} dt + \sigma_{t-} dW_t + \int_{\mathbb{R}\setminus \{0\}}( e^x - 1 ) \, \nu_t(dx,dt) \]

Focus on skew of jumps in log-returns:

\[ \int_t^T \int_{\mathbb{R}\setminus \{0\}} \left( x^3 + O(x^4) \right) \, \nu_s(dx,ds) \]

Use divergence-based measures

• Family of realized variation measures defined in Schneider and Trojani (2019),
• Statistical inference for divergence in Khajavi (2017).

Few assumptions

• Trading results do not hinge on semi-martingale world,
• Only the “jump” interpretation does.

Measurement of jump skewness

Realized variance (divergence for \(p=0\)):

• realized divergence (\(p=0\)) \(\approx\) 1/2 realized variance (\(\color{red}{=}\) if no jumps) + \(f(p,\text{jump})\)
• consider VIX index; VIX²: price of Itakura and Saito (1968) divergence
• VIX²-based variance swap with payoff \(R - 1 -\log{R} \approx O(\sigma^2)\).

Realized jump skew:

• sensitivity of realized divergence to choice of \(p\): \(\frac{\partial D_p(r)}{\partial p}\).

Take log return:

\[ r_i := \log{F_{t_i}} - \log{F_{t_{i-1}}}. \]

Jump skew variation:

\[ S_{t,T} := \sum_{i=1}^N \left( \underbrace{\color{#0072B2}e^{r_i}-1 - r_i}_{\text{VIX}^2\text{ payoff}} -\frac{r_i^2}{2} \right) \overset{\mathbb{P}}{\longrightarrow} \sum_{t \leq s \leq T} \frac{1}{6} \left(\Delta J_s\right)^3 + O(J_s)^4 \]

Motivation

Measurement of jump skew risk

Price and premium for jump skew risk

Empirical results: trading jumps kew risk

Central question: trading strategy

How to dynamically trade forwards and options so that their gross settlement payoff equals \(S_{t,T}\)?

Gross settlement payoff: before paid/received option premia

Like the floating leg of a variance swap

Trading strategy

Index option market:

• sparse maturities
• near continuum of strikes.

Non-linear trading:

Exposure to divergence of function of forward price \(\color{green}{\phi}(F_T)\) available through Breeden and Litzenberger (1978) and Carr and Madan (2001): \[ D_{\color{green}{\phi}}(F_T,F_t) := \color{green}{\phi}(F_T) \color{black}- \color{green}{\phi}(F_t) \color{black}- \color{green}{\phi}'(F_t)\color{black}(F_T- F_t) = \int_0^{\infty} \color{green}{\phi}''(K) \color{red}{O(F_T; K, F_t)} \color{black}dK \]

Existing literature: semi-static strategies

\[ D_{\phi}(F_T,F_t) + \sum_i \delta_i (F_T - F_i) \]

Replication of \(S_{t,T}\) not possible with semi-static strategies

There is no \(\phi\) and \(\{\delta\}_{i=1}^{N_T}\) such that \(S_{t,T} = D_{\phi}(F_T,F_t) + \sum_i \delta_i (F_T - F_i)\).

Trading strategy

Express \(S_{t,T}\) as sum of option payoffs between \(F_{t_i}\) and \(F_T\):

\[ S_{t,T}= \sum_{i=1}^N \left( e^{r_i}-1 - r_i -\frac{r_i^2}{2} \right) = \underbrace{\phi_s(F_T,F_{0})}_{\color{green}{\bigstar}} + \sum_i \underbrace{\gamma_{i}\phi_v(F_T,F_i)}_{\color{purple}{\bigstar}} + \underbrace{\delta_i (F_T - F_i)}_{\color{orange}{\bigstar}}. \]

\(\color{green}{\bigstar}\) at time \(t_0\) buy a skewness swap

\(\color{purple}{\bigstar}\) at time \(t_i > t_0\) trade \(\gamma_i\) variance swaps \(\color{red}//\) \(\color{orange}{\bigstar}\) at time \(t_i > t_0\) trade \(\delta_i\) forwards

Features of the trading strategy

Arbitrary trading frequency

• if options and underlying traded continuously
• jump interpretation at sufficiently high frequencies

Arbitrary trading horizon

• close positions appropriately, retain interpretation

Analytical convenience

• relatively easy to calculate moments in affine models

Features of the trading strategy

Arbitrary trading frequency

• if options and underlying traded continuously
• jump interpretation at sufficiently high frequencies

Arbitrary trading horizon

• close positions appropriately, retain interpretation

Analytical convenience

• relatively easy to calculate moments in affine models
  • anyone interested?

Another question: interpretation

How to interpret the resulting jump skew premium?

\(\mathbb{E}^{\mathbb{P}}[S_{t,T}]\) : estimate from high-frequency data

\(\mathbb{E}^{\mathbb{Q}}[S_{t,T}]\) : we don’t know the future prices of options that we will have to trade

Jump skew premium

Let \(VS_t(\tau)\) be variance swap with \(\tau\) years to maturity

• swap leg involves a series of variance swaps

\[ \mathbb{E}^{\mathbb{P}}[S_{t,T}] - \mathbb{E}^{\mathbb{Q}}\left[ -\frac{1}{2}\log\left(\frac{F_T}{F_t}\right)^2 -\log\left(\frac{F_T}{F_t}\right)\right] - \underbrace{\int_t^{T} \mathbb{E}^{\mathbb{P}}\left[ \color{orange}\log{\frac{F_s}{F_t}}\color{black} d \color{blue} VS_s(T - s)\right]}_{\color{purple}{\bigstar}} \]

If we could trade forwards on VS: this term would have \(\mathbb{Q}_T\)-expectation 0.

But we can’t. Option rebalancing exposes investors to state variable risk.

However, reasonable definition of jump skew premium

Consider Merton (1976) -type models

• Theoretical premium: \(\mathbb{E}^{\mathbb{P}}[S_{t,T}] - \mathbb{E}^{\mathbb{Q}}[S_{t,T}] = \lambda (T-t) \color{blue} \mathbb{E}^{\mathbb{Q}}[J^3]\)
• Discrepancy between tradable and theoretical premium depends on maturity: \[ \color{BurntOrange}(T-t)^2 \color{black}\mathbb{E}^{\mathbb{Q}}[d \ln F_s]\left(\mathbb{E}^{\mathbb{\color{red}P}}[d \ln F_s] - \mathbb{E}^{\mathbb{\color{blue}Q}}[d \ln F_s] \right) \]

  details

Motivation

Measurement of jump skew risk

Price and premium for jump skew risk

Empirical results: trading jump skew risk

CBOE SPX Weeklys

Weeklys:

• \(> 1/3\) of open interest @ CBOE
• Sufficient trading activity since 2011
• Quotes are 99.7% of the data

Profits from trading return skew

During trading hours and around non-trading periods

• Lower profits during trading hours
• Profits over non-trading periods higher in turbulent markets
  • 2011: US debt downgrade / sovereign EU crisis
  • 2015: mini flash-crash in August

Magnitude: margin for $1m skew notional \(\sim\) $10k variance notional

Profits from trading return skew

First conclusions

• No compensation for jump skew risk in continuous trading,
• Positive premium for holding skew positions over non-trading periods
  • Same effect for shorting variance

Further questions

Long skew vs short variance trades

• Both strategies short put options
  • Skew shorts more OTM puts
• Are deeper OTM options sufficiently different to bring about a different risk exposure?

Known risk factors

• To what extent can we explain these profits with standard factors?
  • Tradable quantities, require \(\alpha=0\) in TS regression
• CAPM and Intermediary factor
  • Supply/demand models (Gârleanu, Pedersen, and Poteshman (2009)) give intermediaries a particular role
  • Important in empirical studies (He, Kelly, and Manela (2017))

Mispricing vs risk

• Is this strategy a good hedge of return skew?
• Are premia predictable by measures of return variation?

Spanning by known risk factors

Regress skew trading profits \(s_t\) across non-trading periods on:

• contemporary market return \(r_{mt}\),
• contemporary return on the primary dealer (intermediary) portfolio \(r_{dt}\),
• contemporary variance profit \(v_t\).

\[ s_t = \color{Maroon}\alpha\color{black} + \beta_m r_{mt} + \beta_d r_{dt} + \beta_v v_t + \eta_t \]

Interpretation

• all three proposed factors are tradable,
• if \(\color{Maroon}\alpha\color{black} = 0\), \(s_t\) profits are explained by known systematic factors.

Spanning by known risk factors: raw profits

• Exposure to market risk explains ~30% of after-hours skew profits,
• Positive association between profits from shorting variance and skew profits,
  • But profits from shorting variance subsumed by the market return in ON trading,
• Residual \(\color{Maroon}\alpha\) in ON but not WKND regressions with profits from shorting variance
• Little space for the intermediary risk factor.

Spanning by known risk factors: transaction costs

• Modest transaction costs erase the risk-adjusted profitability of ON trading,
• In WKND trading, no profiability beyond exposure to standard factors,
• Real extent of transaction costs hard to judge, lower than posted spreads (Muravyev and Pearson (2015)),
  • We observe many transactions within the spread.

Option mispricing vs risk-based explanations?

Puzzling patterns in option returns

• Options (puts!) are very expensive, deliver strongly negative returns
  • zero / positive returns during market hours
  • negative returns when held over non-trading periods

Quantity vs price of risk explanations?

• It’s not quantity
  • Returns in non-trading periods have lower variance than during market hours
  • We see NT period returns are less risky by all imaginable measures
• It’s not price
  • No patterns in returns on options with different features (and thus exposures)
• All findings in Jones and Shemesh (2018), Muravyev and Ni (2019)

Our evidence

• Mispricing explanation bounded by the extent to which CAPM explains profits,
• The skew strategy is a good hedge of return skewness,
• The skew profits are predictable with return variation measures.

Hedging overnight return skew

Payoff design

• The settlement of the options and forwards pays exactly return skew

Empirically

• The option premia component is much larger than settlement
  • Is it correlated with return skew?

    
    

Predicting overnight skew profits

Temporal variation

• Average profitability of after-hours skew profits varies over time
  • it’s higher following days with high volatility and negative returns

Summary

Model-free strategy that pays the realized skew of jumps at settlement

• Dynamic option trading
• Fully non-parametric investigation of jump skew pricing
• How to interpret dynamic option trading

Pricing of jump skew risk:

• Not present in daytime option trading
  • large literature on jumps in high-frequency returns
• Present when trading options around non-trading periods
  • but is it mispricing?
• Different from the pricing of variance risk

Conclusions

No premium for jumps in high frequency data

• … beyond premium for changes in variance swap rates
• result aligns with findings about intraday option returns.

Premia for skew positions over non-trading periods

• overnight: skew (but no variance) premia
  • not spanned by known factors
• holidays / weekends: variance (but no skew) premia
• premium for inability to hedge?
  • mispricing?
    • hedging
    • predictability

Path forward for us and for models?

• demand/supply shock impact on the option market
  • disentangle effects of mispricing
• differentiate between trading and non-trading periods,

Interpreting the jump skew premium – details

Semi-static strategies (e.g. variance swaps):

• floating leg — realized variation measure \(\equiv\) fwd + option settlement,
• fixed leg — swap rate — \(\mathbb{E}^{\mathbb{Q}_T}_t[\phi(F_T) - \phi(F_t)]\).

Dynamic strategies:

• replicating leg — two components
  • static skew swap rate (as if there was no extra hedging)
  • weight integrated wrt \(\Delta\) SR , \(\int_t^T \color{orange} \gamma_s \color{black} d \color{blue}\mathbb{E}^{\mathbb{Q}_T}_s[\phi(F_T) - \phi(F_s)]\),

With MGF of log-return \(\varphi_t(p,\tau- t) \equiv \mathbb{E}_t^{\mathbb{Q}}\left[ e^{p\log{F_{\tau}/F_t}} \right]\)

\[ \mathbb{E}^{\mathbb{P}}_t[S_{t,T}] - \left[\underbrace{-\frac{1}{2}\varphi''_t(0,T - t) - \varphi'_t(0, T - t)}_{\color{green}{\bigstar}} - \underbrace{\int_t^{T} \color{orange}\log{\frac{F_s}{F_t}}\color{black} d \color{blue} \varphi'_s(0,T - s)}_{\color{purple}{\bigstar}}\right] \]

Hedging exposure to jump skewness exposes trader to volatility price drivers.

Profits from trading return variance

Spanning of variance profits

Capital requirements

Realized jump skew is two orders of magnitude smaller than realized variance

Bring swap rates to common magnitude, check margin requirements

• bets on variance / skew of return
• compare CBOE margin requirements for positions:
  • variance of return on \(\$10k\) USD
  • skewness of return on \(\$1m\) USD
  • this does not mean such big total option notionals

Skew swap margin requirement as fraction of VS margin requirement