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Financial Mathematics代写

MSc in Financial Mathematics, FM50/2019

Negative rates and portfolio risk management

Financial Mathematics代写 This document describes one of the available topics for the MSc-project in Financial Mathematics.

Cristin Buescu and Teemu Pennanen 

Department of Mathematics

King’s  College London

Financial Mathematics代写
Financial Mathematics代写

This document describes one of the available topics for the MSc-project in Financial Mathematics. The focus is on how to model negative rates in order compute and interpret certain risk measures of an investment portfolio.

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, identifying which real market data to be used for param- eter estimation and how long the historical time series should be.Financial Mathematics代写

To facilitate the analysis 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.

Part 1: Literature review Financial Mathematics代写

The first part is a literature review that should include a description of the con- tracts 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.

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. 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, including the EMMI ref- erence (for EONIA and EONIA based contracts), the ECB reference (for ECB deposit rate). 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.Financial Mathematics代写

It is particularly important that the information gathered from these sources is syntesized and presented as a flowing story that is consistent both in terms of notation and mathematical and financial content.

Part 2: Numerical analysis Financial Mathematics代写

This part applies the theoretical notions from Part 1 on an analysis of a specific portfolio with assets:

  • Equity: 1 share of the Deutsche Boerse AG German Stock Index (Bloomberg ticker DAX INDEX)
  • 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 = 0. 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.Financial Mathematics代写
  • European call option (long position) with a counterparty that may default. 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.

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 Financial Mathematics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 = (AXtt + 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).

Estimation of model parameters Financial Mathematics代写

At each stage write your estimated values as in Table 1.

  • For estimation we use two years of historical daily data from Bloomberg with t0= 0 being 27th of April 2017 (see, for instance, the packages lm,
1
1

Table 1: Table with estimated parameters

dynlm in R). The data has spikes at the end of most months (colloquially called the beat/the pulse); assume they are caused by expired regulatory requirements, so 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.
  • Identify b so that the log equity has an annual drift of 5%, and the long- term median of EONIA is its last observed value. Assume the central bank deposit rate remains constant at its last observed level at t = 0.
  • The covariance matrix Σ is derived as the cross-product of the residuals, normalized appropriately with respect to the number of observations.
Computational results Financial Mathematics代写
  • Analyse 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 (consider the percentage change in the value of the portfolio:(V0 V1)/V0). Analyse the numerical accuracy of the results.
  • 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.
  • 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.
Portfolio Expected

returns

Median

returns

[email protected] [email protected]
credit risk
no credit risk

Table 2: Format to use for the display of the results

Summarize your answers by completing a table in the format shown in Table 2.

Part 3: Advanced issues (to be done after part 2) Financial Mathematics代写

 This part should include any pertinent analysis that would contribute to enhanc- ing the understanding of the topic. Ideally this would be focused on negative rates and portfolio risk management. Among the possible extensions that could be studied in relation to the proposed topic we mention (but these are just suggestions, and the list is not comprehensive):

  • Analyse the portfolio impact of negative rates, and explain your reasoning. Give histograms of the residuals and perform statistical tests to check normality.
  • Change the weights of the assets, starting with same initial portfolio value, to find a better portfolio (i.e. higher expected return and smaller CVaR).
  • Comment on the model and investigatealternative(s).

References Financial Mathematics代写

  • European Money Markets Institute(EMMI):https://www.emmi-benchmarks.eu
  • European Central Bank(ECB):http://www.ecb.europa.eu
  • ”Getting started guide”, Bloomberg, 2012 (available at:https://www.kcl.ac.uk/nms/depts/mathematics/research/finmath/bloomberg/do education-userguide-a4.pdf
  • D.Montgomery, E. Peck, G. Vining, “Introduction to linear regression analysis”, 5th Ed, Wiley, 2012.
  • D.Filipovic, ”Term structure models: a graduate course”, Springer,2009.
  • M.Crouhy, D. Galai, and R. Mark, “A Comparative Analysis of Current CreditRiskModels”, available at http://www.defaultrisk.com/pp model 12.htm
  • CreditMetrics Technical Document, available atcom
  • D.Brigo, F. Mercurio, “Interest Rate Models: Theory and Practice”,2nd Edition, 2006, SpringerVerlag.
  • T.Bielecki and M. Rutkowski, ”Credit risk: Modeling, Valuation and Hedging”, Springer Verlag, 2002.
  • P.Jorion, “Value at Risk”, 3rd ed, McGrawHill.
  • For models of default see the numerous papers by M. Jeanblanc; or the paper by Jarrow and Protter “Structural versus reduced form models: a new information based perspective”, J of Inv. Manag., 2(2), 1-10, 2004.
Financial Mathematics代写
Financial Mathematics代写

 

 

 

 

 

 

 

 

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