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The London Graduate School in Mathematical Finance is a consortium of the mathematical finance groups of
Birkbeck College, Brunel, Cass, Imperial College, King’s College, LSE and UCL. Its main purpose is to provide a
programme of advanced courses in mathematical financial, primarily but not exclusively for first year PhD
students in the various groups.
LGS PhD
Presentation Day
Friday 14 March 2014
09:15 – 17:30
Vera Anstey Room (VAR)
Old Building
London School of Economics and Political Science
Please enter the Old Building by the main entrance on Houghton Street. The Vera Anstey Room is up the first short flight of stairs on the right, past the reception desk and immediately before the lift.
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Session One
Chair: Shiju Liu
09:15 – 09:25
Welcome and introduction by Professor Pauline Barrieu
09:25 – 09:55
25mins + 5mins Q&A
Patrick Roome, Imperial
[email protected]A dynamic view of the Heston model
09:55 – 10:25
25mins + 5mins Q&A
Yavor Stoev, LSE
[email protected]Equilibrium with imbalance of the derivative market
10:25 – 10:55
25mins + 5mins Q&A
Cheng Li, LSE
[email protected]Asymptotic Glosten Milgrom equilibrium
Coffee break
Refreshments and biscuits will be providedSession Two
Chair: Cheng Li
11:15 – 11:45
25mins + 5mins Q&A
Raffaele Corvino, Cass
[email protected]Market value of leverage: quantitative estimation and economic applications
11:45 – 12:15
25mins + 5mins Q&A
Benoit Pham-Dang, Imperial
[email protected]Optimal default time of a Merton investor when borrowing is collateralised
12:15 – 12:45
25mins + 5mins Q&A
Gabriele Sarais, Imperial
[email protected]Inflation derivatives pricing with macroeconomic foundations
12:45 – 13:15
25mins + 5mins Q&A
Anna Zaremba, UCL
[email protected]Extracting causal relations in complex financial datasets
Lunch
A light sandwich lunch and refreshments will be providedSession Three
Chair: Ali Habibnia
14:15 – 14:45
25mins + 5mins Q&A
Mathieu Dubois, LSE
[email protected] Optimal diversification in the presence of parameter uncertainty for a risk averse investor14:45 – 15:15
25mins + 5mins Q&A
Pedro Vergel, Birkbeck
[email protected] Investing in fertilizer-mining companies in times of food scarcity15:15 – 15:45
25mins + 5mins Q&A
Peter Divos, UCL
[email protected]Valuation and hedging of in-play football bets
Coffee break
Refreshments and biscuits will be providedSession Four
Chair: Raffaele Corvino
16:05 – 16:35
25mins + 5mins Q&A
Alessandra Crisafi, UCL
[email protected]Optimal order execution in lit and dark pools
16:35 – 16:55
25mins + 5mins Q&A
Qing Liu, Imperial
[email protected]Consistent valuation of collateralized OTC deals under credit and risk banking
16:55 – 17:30
25mins + 5mins Q&A
Juraj Spilda, Cass
[email protected]3
Abstracts
Raffaele Corvino (Cass)
Market value of leverage: quantitative estimation and economic
applications
The market value of the leverage is extremely important for the capital structure of a company, but it is usually unobservable because of many potential factors. Our research combines approaches from two areas: the capital structure literature and the literature on default prediction. We model default as the first time the firm's asset value crosses a deterministic barrier. We produce quantitative measures of the actual dynamics of the firm's market leverage estimating a state-space model with the non-linear Kalman filter in conjunction with quasi-maximum likelihood, by using as observable variable the default probability extracted from CDS premia. Then, we use our estimates to construct an indicator of systemic risk which can be adopted by regulators as warning signal of systemic crisis.
(Jointly with Alessandro Beber and Gianluca Fusai)
Alessandra Crisafi (UCL)
Optimal order execution in lit and dark pools
We consider an optimal order execution problem over a finite period of time during which an investor has access to both a lit exchange and a dark pool. We take the exchange to be a limit order market and propose a continuous-time setup for the best bid and best ask prices, both modelled by explicit functions of incoming market and limit orders. In the situation where trades take place only in the exchange, we find that the optimal order execution strategy depends significantly on the book's resilience which we model by random arrivals of limit orders in the limit order book. We assume that the execution price in the dark pool is the mid-price taken from the exchange and that no fees are due for posting orders. We allow for partial trade executions in the dark pool, and we find the optimal order-size placement in both venues. Since the trading price in the dark pool is taken from the exchange, the resilience of the limit order book also affects the optimal allocation of shares in the dark pool. We propose a general objective function and we show that, subject to suitable technical conditions, the solution to the associated partial integro-differential equation can be characterised by the unique viscosity solution. We present a numerical example of which model parameters are analysed in detail
Peter Divos (UCL)
Valuation and hedging of in-play football bets
In-play football betting allows players to place bets on several events related to the outcome of a football game. In contrast to classical football betting where bets are placed before the beginning of the game, in-play football betting allows for placing bets during the game. The odds offered by bookmakers change dynamically as the game progresses and as the teams score goals. A stationary Poisson process is applied to model the values of these bets. It is shown that this model is in line with the first and second fundamental theorems of asset pricing, that is (1) assuming no arbitrage there exists a risk-neutral measure under which the bet values are martingales and (2) the market is complete, that is any bet can be replicated by taking a dynamic position in two other bets. Analytic formulas are developed for a wide range of different types of bets. Finally, the results are demonstrated using historical in-play data for the games of the 2012 UEFA European Championship. We find that, even though the stationary Poisson process is the simplest possible model with only two parameters, it gives a reasonably accurate explanation of in-play odds. (Joint work with Tomaso Aste and Sebastian del Bano Rollin)
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Mathieu Dubois (LSE)
Optimal diversification in the presence of parameter uncertainty for a risk
averse investor
We consider an investor who faces parameter uncertainty in a continuous- time financial market. We model the investor’s preference by a power utility function leading to constant relative risk aversion. We show that the loss in expected utility is large when using a simple plug-in strategy for unknown parameters. We also show that the loss due to estimation depends crucially on the coefficient of relative risk aversion. We provide theoretical results that show the trade-off between holding a well-diversified portfolio and a portfolio that is robust against estimation errors. To reduce the effect of estimation, we constrain the weights of the risky assets with an L1-norm leading to a sparse portfolio. We provide analytical results that show how the sparsity of the constrained portfolio depends on the coefficient of relative risk aversion. Based on a simulation study, we demonstrate the existence of an optimal bound on the L1-norm for each level of relative risk aversion.
Cheng LI (LSE)
Asymptotic Glosten Milgrom equilibrium
This paper studies the Glosten Milgrom model whose risky asset value admits an arbitrary discrete distribution. Contrast to existing results on insider’s models, the insider’s optimal strategy in this model, if exists, is not of feedback type. Therefore a weak formulation of equilibrium is proposed. In this weak formulation, the
inconspicuous trade theorem still holds, but the optimality for the insider’s strategy is not enforced. However, the insider can employ some feedback strategy whose associated expected profit is close to the optimal value, when the order size is small. Moreover this discrepancy converges to zero when the order size diminishes. The existence of such a weak equilibrium is established, in which the insider’s strategy converges to the Kyle optimal strategy when the order size goes to zero.
Qing Liu (Imperial)
Consistent valuation of collateralized OTC deals under credit and risk
banking
We develop an arbitrage-free framework for consistent valuation of OTC derivative trades with collateralization, counterparty credit risk, and funding costs. Based on the risk-neutral pricing principle, we derive a general pricing equation where CVA, DVA, and FVA are introduced by simply modifying the pay-out cash-flows of the deal. Funding risk breaks the bilateral nature of the deal price and makes the pricing problem a highly non-linear and recursive one. This means that FVA is not generally an additive adjustment as commonly assumed by market participants. Our
framework addresses common market practices of ISDA governed deals without restrictive assumptions on collateral margin payments and close-out netting rules. In particular, we allow for asymmetric collateral and funding rates. The pricing equation can be cast as a set of iterative equations that can be solved by least-squares Monte Carlo and we propose such a simulation algorithm. Our numerical results confirm that funding risk does have a non-trivial impact on the deal price.
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Benoit Pham-Dang (Imperial)
Optimal default time of a Merton investor when borrowing is collateralised
Patrick Roome (Imperial)
A dynamic view of the Heston model
The Heston stochastic volatility model was introduced over 20 years ago [1] and is arguably the most widely used stochastic volatility model in the industry. This is in large part due to analytical tractability and the existence of semi-closed formulae for European option pricing. Furthermore, asymptotics of the (spot) implied volatility smile have been thoroughly studied in Heston, giving insight into the behaviour of model-generated spot smiles. However, there are virtually no analytical results on the dynamics of model implied volatility smiles, a key model-risk metric for assessing the suitability of a model for exotic option pricing. In this talk we will first derive small and large-maturity asymptotics for the Heston forward implied volatility smile ([2], [3], [4]) using the theory of sharp large deviations (and refinements). We will then use these results to gain insight into some core dynamical properties of the model. We will provide a number of cases of degenerate large deviations behaviour and we will show that it is exactly the analysis of these pathological cases that gives the most insight into the dynamical features of the model. This is based on joint works with Antoine Jacquier (Imperial College London).
[1] S. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6 (2): 327-342, 1993.
[2] A. Jacquier and P. Roome. Asymptotics of forward implied volatility. Submitted,
http://arxiv.org/abs/1212.0779. 2013.
[3] A. Jacquier and P. Roome. Large-maturity regimes of the Heston forward smile. In progress.
[4] A. Jacquier and P. Roome. The small-maturity Heston forward smile. SIAM J. Finan. Math., 4 (1): 831-856, 2013.
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Gabriele Sarais (Imperial)
Inflation derivative pricing with macroeconomic foundations
We develop a model to price inflation and interest rates derivatives using continuous-time dynamics that have some links with macroeconomic monetary DSGE models equipped with a Taylor rule: in particular, the reaction function of the central bank, the bond market liquidity, inflation and growth expectations play an important role. The model can explain the effects of non-standard monetary policies (like
quantitative easing or its tapering) and shed light on how central bank policy can affect the value of inflation and interest rates derivatives.
The model is built under standard no-arbitrage assumptions. Interestingly, the model yields short rate dynamics that are consistent with a time-varying Hull-White model, therefore making the calibration to the nominal interest curve and options
straightforward. Further, we obtain closed forms for both zero-coupon and year-on-year inflation breakevens and options. The calibration process is fully separable, which means that the calibration can be carried out in many simple steps that do not require heavy computation.
The advantages of such structural inflation modelling become apparent when one starts doing risk analysis on an inflation derivatives book: because the model
explicitly takes into account economic variables, a trader can easily assess the impact of a change in central bank policy or growth expectation on a complex book of fixed income instruments, which is normally not straightforward when using standard inflation pricing models.
Juraj Spilda (Cass)
Good deal bounds on variance swaps
In this presentation, we will investigate good-deal price bounds for variance swaps implied by utility maximization techniques. Our aim is to find bounds tighter than the already known no-arbitrage bounds in a setting with jumps modeled by a Levy process, excluding not only prices that would lead to arbitrage, but also to those that would offer very attractive risk-adjusted hedged portfolio returns (i.e. a good deal) to the buyer. We will compare results obtained from mean-variance preferences and exponential utility maximization.
Yavor Stoev (LSE)
Equilibrium with imbalance of the derivative market
This presentation investigates the impact of imbalanced derivative markets - markets in which not all agents hedge - on the underlying stock market. The availability of a closed-form representation for the equilibrium stock price in the context of a complete (imbalanced) market with terminal consumption allows us to study how this equilibrium outcome is affected by the risk aversion of agents and the degree of imbalance. In particular, it will be shown that the derivatives imbalance leads to significant changes in the equilibrium stock price process: volatility changes from constant to local, while risk premia decrease and become stochastic processes. Moreover the model produces implied volatility skew consistent with empirical observations.
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Pedro Vergel (Birkbeck)
Investing in fertilizer-mining companies in times of food scarcity
The primary goal of the paper is to show the validity of investing capital in fertilizer– mining companies, both from a market return perspective for individual or
institutional investors, or from a hedging standpoint for insurance companies and other economic actors exposed to inflation risk and high agricultural commodity prices. After providing some elements on the fertilizer market and describing the joint dynamics of corn, wheat and fertilizer prices over the last decade, we analyze an exhaustive sample of listed fertilizer producing companies over the years January 2004–December 2012. We show that their shares generated quite good returns over the whole period and extremely high ones during the years January 2004–December 2007, both in absolute terms and compared to their betas. We also exhibit that these returns display higher sensitivities to major agricultural indexes than to the World Bank Fertilizer Index, making the hedging argument quite compelling.
Anna Zaremba (UCL)
Extracting causal relations in complex financial datasets
The ability to properly quantify the causality structure in financial data is of great importance both for investors and regulators. We have compared various linear and non-linear causality measures based on methods from econometrics, machine learning and information theory. Specifically we discuss three generalizations of the concept of Granger causality: Geweke’s measure, kernelized Geweke’s measure, Hilbert Schmidt Normalized Conditional Independence Criterion (HSNCIC) and we compare them with transfer entropy. We analyse their properties and practical applicability, with special attention given to those aspects that are important to financial time series: non-linearity, noisiness, limited length of the time-series and non-stationarity. We report about extensive testing and validation of these measures on a range of simulated data and real data (S&P vs. exchange rates of carry trade currencies; inflation vs. interest rates).
