MSc in Financial Mathematics, FM50/2018Negative rates代写 This document describes one of the available topics for the MSc-project in Financial Mathematics. The focusNegative rates and portfolio risk managementCristin Buescu and Teemu Pennanen Department of MathematicsKing’s  College LondonThis document describes one of the available topics for the MSc-project in Financial Mathematics. The focus is on an investor who holds a portfolio of assets and who wants to compute and interpret certain risk measures in order to guide future actions in an environment with negative rates.Negative rates代写The project has three parts: Negative rates代写first part is a literature review that would de- scribe and explain the contracts in the portfolio, the models used for equity/ interest rates, the methods available for the modelling of the default, the meth- ods used to estimate the parameters, and the most common risk measures. The second part consists of a numerical analysis of a specified portfolio, where the models would be implemented on a practical level. The third part treats more advanced issues related to this topic, with a view of enhancing the understanding of the problem, the model and the results.Negative rates代写Implementing such a project in real life would require at a minimum identi- fying the risk factors and appropriate models for them, checking if counterparty credit risk is present, and how to model it if necessary, identifying which real market data to be used for parameter estimation and how long the historical   time series should be.To facilitate the analysis Negative rates代写we provide guidance for some of the steps mentioned above. The risk factors are modelled with stochastic models that have been introduced in previous modules, and the parameters of the models are estimated using real data from Bloomberg over the specified time horizon. Future paths are generated according to these models, and the possible future values are incorporated in a risk analysis through the computation of risk measures for the portfolio.The investor is subject to credit risk, where the counterparty of a certain contract in the portfolio can default before the maturity of the contract, thereby affecting the payoff of the contract. For our portfolio we consider a reduced-form model with constant intensity of default that will model the occurrence of the default.  The parameters for this model of default can be calibrated to market  data (e.g. CDS market prices), but we will assume specific values for them.Part 1: Literature review Negative rates代写In the first part the student is asked to write a literature review that should include a description of the contracts in the portfolio, particularly the EONIA- based interest rate swap, a brief outline of the models used for equity/ interest rates,  the methods available for the modelling of the default (i.e. structural  vs reduced form models, advantages and disadvantages of each class), a brief outline of the methods used to estimate the parameters, and a review of the most common risk measures.Negative rates代写The student is invited to consult a number of publications on EONIA/ECB rates, Credit Risk modeling, and on Value at Risk and risk measures in general.We present some suggestions below as a starting point:For EONIA and EONIA based contracts see the European Money Markets Institute (EMMI) website https://www.emmi-benchmarks.euForECB deposit rate see the European Central Bank (ECB) website”Getting started guide”, Bloomberg, 2012 (available at: https://www.kcl.ac.uk/nms/depts/mathematics/research/finmath/bloomberg/docs/education-userguide-a4.pdf)The references at the end of this document are classical books on risk man- agement, interest rate models, least squares parameter estimation and related topics, and give good starting points to the literature. The student should be proactive in researching the literature, which involves published journal papers and books. Working papers should be used mostly for orientation, given that their content has not been peer reviewed.Negative rates代写It is particularly important that the student synthesizes the information gathered from these sources and presents it as a flowing story that is consistent both in terms of notation and mathematical and financial content.Negative rates代写Part 2: Numerical analysis Negative rates代写This part applies the theoretical notions from Part 1 on an analysis of a specific portfolio. The assets in the portfolio are:Equity: 1 share of the Deutsche Boerse AG German Stock Index (Bloomberg ticker DAXINDEX)EONIA based interest rate swap with a counterparty that is default free. The maturity is one month, the notional amount is 10 million, and the swap rate can be obtained from Bloomberg market data on the day t = The investor is the swap payer, i.e. pays the fixed rate and receives the floating rate pegged to daily EONIA values. The formula for the EONIA swap variable rate can be found in EMMI publications.Europeancall option (long position) with a counterparty that may  The underlying is the equity above (DAX), the strike is 12,450 and the maturity is 50 days from t = 0. Initial price of 235.1 at t = 0 comes from Bloomberg. For later prices we use a pricing measure where the equity follows a Black-Scholes model with drift equal to the simulated EONIA rate at 30 days from t = 0, and the volatility is the square root of the element of Σ corresponding to equity. The counterparty of the option can default with zero recovery rate (in case of default the entire option becomes worthless). The default is modelled by a reduced form model with constant annual intensity of default 0.12.Negative rates代写The goal of the project is to analyse the risk and return characteristics of the portfolio using a stochastic model for the underlying risk factors.Stochastic model Negative rates代写 Consider the risk factors to be the equity (DAX) and the EONIA spread over the ECB deposit rate:Xt = (log Yt log St)j,and assume they follow under the subjective measure P a discretized version of  a stochastic differential equation (SDE) of the type:∆Xt = (AXt−∆t + b)∆t + ε, ε ∼ N (0, Σ),where ∆t = 1 day (for tractability make the simplifying assumption that week- ends or holidays are equivalent to 1 day periods).Negative rates代写Estimate model parameters: A, b, Σ using two years of historical daily data from Bloomberg (see, for instance, the packages lm, dynlm in R). Fix the date of the analysis (t0 = 0) as 27/04/2017. Assume the data spikes at the end of most months (colloquially called the beat/the pulse) are caused by expired regulatory requirements, so we exclude all end of the month observations from the data. Plot the data with and without the beats.If a parameter has a significance level above 5%, then temporarily set it    to zero (we will use alternative information to historical data to estimate them). Write the resulting discrete dynamics of log Yt and log St.Identifyb so that the log equity has an annual drift of 5%, and log EONIA spread over ECB satisfies E(log St0 +1 log St0 ) = 0. Assume the central bank deposit rate remains constant at its last observed level at t = 0The covariance matrix Σ is derived as the cross-product of the residuals, normalized appropriately with respect to the number of observations.Negative rates代写Computational resultsAnalyse the distribution of the potential losses incurred by the portfolio over a 30 days holding period, and use VaR and ES (CVaR) to quantify them with confidence level 99%. The relative losses are defined wrt to the value V1 of the portfolio in 30 days from t = 0 and the value V0 at time t = 0 (you can consider the percentage change in the value of the portfolio for instance). Analyse the numerical accuracy of the results.Negative rates代写To compute the value V0 find the equity price, the option price and the swap rate on the date t = 0 in the Bloomberg terminal, and explain how you obtained them.For V1 simulate 30 days for the process Y ,  and simulate also the default  of the counterparty of the call. Plot one future sample path together with  the historical path for each component of the process X.Negative rates代写Include in your analysis the histogram of the relative losses, a plot of the return distribution of the portfolio, computation of VaR and ES, and the expected and median returns of the portfolio.Analyse the impact of credit risk by repeating the calculations without credit risk (zero intensity) and comparing the results. Discuss.Summarize your answers by completing a table in the format shown in Table 1.

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