The relationship between the volatility of returns and the number of jumps in financial markets

The contribution of this paper is two-fold. First we show how to estimate the volatility of high frequency log-returns where the estimates are not affected by microstructure noise and the presence of Lévy-type jumps in prices. The second contribution focuses on the relationship between the number of jumps and the volatility of log-returns of the SPY, which is the fund that tracks the S&P 500. We employ SPY high frequency data (minute-by-minute) to obtain estimates of the volatility of the SPY log-returns to show that: (i) The number of jumps in the SPY is an important variable in explaining the daily volatility of the SPY log-returns; (ii) The number of jumps in the SPY prices has more explanatory power with respect to daily volatility than other variables based on: volume, number of trades, open and close, and other jump activity measures based on Bipower Variation; (iii) The number of jumps in the SPY prices has a similar explanatory power to that of the VIX, and slightly less explanatory power than measures based on high and low prices, when it comes to explaining volatility; (iv) Forecasts of the average number of jumps are important variables when producing monthly volatility forecasts and, furthermore, they contain information that is not impounded in the VIX

Volatility and covariation of financial assets: a high-frequency analysis

Using high frequency data for the price dynamics of equities we measure the impact that market microstructure noise has on estimates of the: (i) volatility of returns; and (ii) variance-covariance matrix of n. assets. We propose a Kalman-filter-based methodology that allows us to deconstruct price series into the true effcient price and the microstructure noise. This approach allows us to employ volatility estimators that achieve very low Root Mean Squared Errors (RMSEs) compared to other estimators that have been proposed to deal with market microstructure noise at high frequencies. Furthermore, this price series decomposition allows us to estimate the variance covariance matrix of n assets in a more efficient way than the methods so far proposed in the literature. We illustrate our results by calculating how microstructre noise affects portfolio decisions and calculations of the equity beta in a CAPM setting

Pricing in Electricity Markets: a Mean Reverting Jump Diffusion Model with Seasonality

In this paper we present a mean-reverting jump diffusion model for the electricity spot price. We obtain a closed-form solution for forward contracts and calibrate it to market data from England and Wales. Finally, based on the calibrated forward curve we present months, quarters, and seasons-ahead forward surfaces.Energy derivatives, mean reversion, jump diffusion, electricity spot and forward.

Volatility and covariation of financial assets: a high-frequency analysis

Using high frequency data for the price dynamics of equities we measure the impact that market microstructure noise has on estimates of the: (i) volatility of returns; and (ii) variance-covariance matrix of n assets. We propose a Kalman-filter-based methodology that allows us to deconstruct price series into the true efficient price and the microstructure noise. This approach allows us to employ volatility estimators that achieve very low Root Mean Squared Errors (RMSEs) compared to other estimators that have been proposed to deal with market microstructure noise at high frequencies. Furthermore, this price series decomposition allows us to estimate the variance covariance matrix of $n$ assets in a more efficient way than the methods so far proposed in the literature. We illustrate our results by calculating how microstructure noise affects portfolio decisions and calculations of the equity beta in a CAPM setting.Volatility estimation, High-frequency data, Market microstructure theory, Covariation of assets, Matrix process, Kalman filter

The relationship between the volatility of returns and the number of jumps in financial markets

The contribution of this paper is two-fold. First we show how to estimate the volatility of high frequency log-returns where the estimates are not a affected by microstructure noise and the presence of Lévy-type jumps in prices. The second contribution focuses on the relationship between the number of jumps and the volatility of log-returns of the SPY, which is the fund that tracks the S&P 500. We employ SPY high frequency data (minute-by-minute) to obtain estimates of the volatility of the SPY log-returns to show that: (i) The number of jumps in the SPY is an important variable in explaining the daily volatility of the SPY log-returns; (ii) The number of jumps in the SPY prices has more explanatory power with respect to daily volatility than other variables based on: volume, number of trades, open and close, and other jump activity measures based on Bipower Variation; (iii) The number of jumps in the SPY prices has a similar explanatory power to that of the VIX, and slightly less explanatory power than measures based on high and low prices, when it comes to explaining volatility; (iv) Forecasts of the average number of jumps are important variables when producing monthly volatility forecasts and, furthermore, they contain information that is not impounded in the VIX.Volatility forecasts, High-frequency data, Implied volatility, VIX, Jumps, Microstructure noise

Dynamic hedging of financial instruments when the underlying follows a Non-Gaussian Process

Traditional dynamic hedging strategies are based on local information (ie Delta and Gamma) of the financial instruments to be hedged. We propose a new dynamic hedging strategy that employs non-local information and compare the profit and loss (P&L) resulting from hedging vanilla options when the classical approach of Delta- and Gammaneutrality is employed, to the results delivered by what we label Delta- and Fractional-Gamma-hedging. For specific cases, such as the FMLS of Carr and Wu (2003a) and Merton’s Jump-Diffusion model, the volatility of the P&L is considerably lower (in some cases only 25%) than that resulting from Delta- and Gamma-neutrality

Hedging Non-Tradable Risks with Transaction Costs and Price Impact

A risk-averse agent hedges her exposure to a non-tradable risk factor $U$ using a correlated traded asset $S$ and accounts for the impact of her trades on both factors. The effect of the agent's trades on $U$ is referred to as cross-impact. By solving the agent's stochastic control problem, we obtain a closed-form expression for the optimal strategy when the agent holds a linear position in $U$. When the exposure to the non-tradable risk factor $\psi(U_T)$ is non-linear, we provide an approximation to the optimal strategy in closed-form, and prove that the value function is correctly approximated by this strategy when cross-impact and risk-aversion are small. We further prove that when $\psi(U_T)$ is non-linear, the approximate optimal strategy can be written in terms of the optimal strategy for a linear exposure with the size of the position changing dynamically according to the exposure's "Delta" under a particular probability measure.Comment: Originally posted to SSRN April 27, 2018. Forthcoming in Mathematical Financ

Trading Cointegrated Assets with Price Impact

Executing a basket of co-integrated assets is an important task facing investors. Here, we show how to do this accounting for the informational advantage gained from assets within and outside the basket, as well as for the permanent price impact of market orders (MOs) from all market participants, and the temporary impact that the agent's MOs have on prices. The execution problem is posed as an optimal stochastic control problem and we demonstrate that, under some mild conditions, the value function admits a closed-form solution, and prove a verification theorem. Furthermore, we use data of five stocks traded in the Nasdaq exchange to estimate the model parameters and use simulations to illustrate the performance of the strategy. As an example, the agent liquidates a portfolio consisting of shares in Intel Corporation (INTC) and Market Vectors Semiconductor ETF (SMH). We show that including the information provided by three additional assets, FARO Technologies (FARO), NetApp (NTAP) and Oracle Corporation (ORCL), considerably improves the strategy's performance; for the portfolio we execute, it outperforms the multi-asset version of Almgren-Chriss by approximately 4 to 4.5 basis points.Comment: 32 pages,4 figures, 11 tables. First appeared on SSRN Oct 2015 at http://ssrn.com/abstract=268130

Deep Attentive Survival Analysis in Limit Order Books: Estimating Fill Probabilities with Convolutional-Transformers

One of the key decisions in execution strategies is the choice between a passive (liquidity providing) or an aggressive (liquidity taking) order to execute a trade in a limit order book (LOB). Essential to this choice is the fill probability of a passive limit order placed in the LOB. This paper proposes a deep learning method to estimate the filltimes of limit orders posted in different levels of the LOB. We develop a novel model for survival analysis that maps time-varying features of the LOB to the distribution of filltimes of limit orders. Our method is based on a convolutional-Transformer encoder and a monotonic neural network decoder. We use proper scoring rules to compare our method with other approaches in survival analysis, and perform an interpretability analysis to understand the informativeness of features used to compute fill probabilities. Our method significantly outperforms those typically used in survival analysis literature. Finally, we carry out a statistical analysis of the fill probability of orders placed in the order book (e.g., within the bid-ask spread) for assets with different queue dynamics and trading activity

Foreign Exchange Markets with Last Look

We examine the Foreign Exchange (FX) spot price spreads with and without Last Look on the transaction. We assume that brokers are risk-neutral and they quote spreads so that losses to latency arbitrageurs (LAs) are recovered from other traders in the FX market. These losses are reduced if the broker can reject, ex-post, loss-making trades by enforcing the Last Look option which is a feature of some trading venues in FX markets. For a given rejection threshold the risk-neutral broker quotes a spread to the market so that her expected profits are zero. When there is only one venue, we find that the Last Look option reduces quoted spreads. If there are two venues we show that the market reaches an equilibrium where traders have no incentive to migrate. The equilibrium can be reached with both venues coexisting, or with only one venue surviving. Moreover, when one venue enforces Last Look and the other one does not, counterintuitively, it may be the case that the Last Look venue quotes larger spreads.Comment: 40 pages, 7 figure