# Financial and Actuarial Mathematics

# Sample PhD Projects in Financial and Actuarial Mathematics

## General Interests

Markets; insurance; pricing; risk; random processes; stochastic systems

**Project 1: Dynamic of agricultural prices**

The last decade has witnessed a surge in commodity prices and a widespread financialization of commodity products. The upward movements and the increased volatility of the commodity prices have been largely attributed to strong demand by China and other emerging markets as well as massive capital owes into the commodity markets by institutional investors, portfolio managers and speculators.

Agricultural markets constitute a great portion of the commodity market. Agricultural market emerges as the most strategic commodity market as it provides 40 percept of the worldwide employment and is responsible for 100 percent of the worldwide food-supply. Therefore, the well-being of the agricultural market has the highest priority for the government in any country and any effort to explain correctly the agricultural product prices is of a great importance.

In this project we aim to study new methods in order to model the dynamic of agricultural prices. Due to many frictions existing in the agricultural market, we have chosen to study the constant elasticity volatility model. This is a starting point for our project, and will be continued toward pricing different type of derivatives, in particular futures on agricultural commodities.

**Project 2: Pricing Agricultural Insurance**

Agriculture is a major source of livelihood in the world through farming the cultivation of animals, plants, fungi, fiber, bio-fuel, drugs and other products. A key challenge to farmers is access to appropriate insurance on their products, which is a key impediment in improving the efficiency in financing their business. Investment in agriculture sector is critical for driving global economic growth as agriculture sector contributes about 40 percent of worldwide employment.

Like other commodities, prices of the agricultural products do not follow a deterministic pattern. While in general the prices have to be set by the so-called demand-supply law, different shocks can affect prices. There are different types of shocks which change the direction of prices, among which the demand and supply shocks as well as shocks in inputs and harvest constitute the most important ones. Other important factors which have great impacts on agricultural prices are speculation and the government policies. That is why valuation of agricultural insurance contracts is a difficult task.

In this project we propose a new method of valuation of the agricultural prices, which helps for more accurate estimation of the agricultural insurance contracts. This project will evaluate the agricultural insurance prices and propose some new forms of modelling price dynamics to improve the pricing of insurance contracts. Our proposed methods are based on a rich literature developed on speculative storage models with harvest shocks. Our aim is to develop this approach in the literature of mathematical finance where we can adopt a stochastic analysis framework for further development.

Hence, the following research project will broadly cover:

a) Studying of different approaches in the literature on agricultural prices and insurance.

b) Proposing a new dynamic based on speculative storage model in a stochastic analysis framework for agricultural prices.

c) Identification of different parameters that can affect the prices and insurance contracts.

**Project 3: Dynamic Reserve Management of Insurance Companies**

The well-being of insurance-industry is one of the most important issues for the regulatory-sector in any country. Management of the reserve has a major role in keeping an insurance company solvent. Mismanaging the reserve could result in the ruin of an insurance company and consequently increasing the chance of systemic risk. That is why the regulatory-sector, among all other issues, gives a top priority to reserve-management of insurance companies. On the other hand, reserve-management is important from an individual company point of view, as it can offer better prices to its customers and pay better dividends to its share-holders.

In this project we aim to study the problem of reserve-management of an insurance company; first, to keep the company solvent, and second to maximize the profit of the company. Given the reserve’s law-of-motion we can set up a dynamical programing framework and derive the optimal policy for the reserve. The set-up of the problem can be either stochastic or deterministic. In this project we will extensively use the rich literatures of macro-economic and dynamic programing which have been developed for a similar problem, finding the optimal policies according to the capital’s law-of-motion.

Hence, the following research project will broadly cover:

a) Studying of different approaches in the literature on dynamic programing.

b) Proposing a new reserve-management framework for insurance companies.

c) Identification of different parameters that can affect the future reserve.

**Project 4: Hedging Insolvency of Insurance Companies**

Hedging is a common practice in banking, finance and insurance. In the literature of financial mathematics the main application of hedging is pricing a contingent claim (e.g. an option), by replicating the contingent claim's final value with a portfolio of viable financial positions. However, in practice the main use of hedging is not only for pricing, the main application is to form a portfolio which covers the risk of another position. With this idea in mind we propose a general theoretical framework to find capital reserve for an insurance company by hedging against the insolvency.

Solvency initially was studied in risk theory by managing the reserve within the control of initial investment. A mathematical tool used for this measurement is the probability of ruin which measures the likelihood of the reserve to be insufficient for the liabilities. However, in recent years the development of risk measures and their application in finance and actuarial science, and accordingly the use of the notion of the solvency set, have changed this traditional way of viewing to the problems in the ruin theory. The problem of an insurance company's solvency has been studied by using coherent risk measures in recent years. Indeed, a coherent risk measure and its associated solvency set can be used to find the capital reserve. In this project we go one step further and assume that the insurance company has access to the financial market and can hedge against the insolvency of the company by forming an appropriate portfolio.

Hence, the following research project will broadly cover:

- Studying of different approaches in the literature on solvency and hedging.
- Proposing a framework to study the optimal hedging the insolvency.
- Identification of different parameters that can affect the optimal policies to solvency.

**Project 5: Optimal reinsurance contracts**

Reinsurance is a contract which transfers part of the risk of an insurance company (cedent), to another insurance company (reinsurer). Optimal reinsurance contract is one of the most important issues that an insurance company has to take care of, which otherwise, will cause some irreversible losses or even will cause the insurance company to ruin. Managing a portfolio of reinsurance contracts is one of the insurance-companies top priorities. Different type of reinsurance contracts have been introduced in the reinsurance market, among which the stop-loss, stop-loss after quota-share and quota-share after stop-loss have been used on a regular basis. Usually in the literature the reinsurance premium is calculated according to the expected value premium principle, and the risk of a company is measured either in a ruin probability framework, or by a risk measures like VaR (Value at Risk) and CVaR (Conditional Value at Risk).

In this project we want to extend the study of the optimal reinsurance contracts to a general framework using general premiums, the Choquet premiums, and also a general risk measures, the spectral measures of risk. This framework covers the well-known Wang’s premium function as well as the celebrated risk measures VaR and CVaR.

Hence, the following research project will broadly cover:

- Studying of different approaches in the literature on optimal reinsurance.
- Proposing a framework to study the optimal reinsurance contracts.
- Identification of different parameters that can affect the optimal reinsurance contracts.

The above five projects require a strong mathematical background and a keen interest to work in the interface of financial mathematics and actuarial science. Applicants should have either a background in Mathematical or Actuarial/Financial Sciences or in relevant fields. A combined education in these fields would be an advantage.

### Dr Carmen Boado Penas

**Project 1: ****How has the financial and economic crisis affected retirement behaviour?**

Although the net effect of the financial and economic crisis on retirement is far from clear it is possible to identify the prospective channels through which the effects will be transmitted and highlight interaction between household decisions concerning stock market investments, housing wealth and pension provision. In particular, while the current decline in home values and stock prices would be expected to lead to a decrease in the number of retirements, the effect of the increase of the unemployment is just the opposite. The aim of this project is to measure how the economic crisis may affect the retirement decisions of the individuals. The dissertation will develop an extension of Coile and Levine (2011) and will address questions about how the stock market wealth, housing wealth and unemployment ratio could influence on the retirees’ decisions,

**Project 2: Valuing Public Obligations in the Face of Economic and Longevity Risk **

The traditional methods of valuing long-term public obligations rely on specific assumptions: (i) the value of future claims as well as the duration of future payments are assumed known at the time when the programme is introduced, (ii) the discount factor, which translates into present terms the value of future claims, is determined in an ad-hoc fashion. From this starting point, the literature adds realism to the analysis by allowing claims and duration to be random, the so called, the “stochastic valuation methods”. The question of determining the correct discount factor for time and risk is left unresolved.

In contributing to this important debate about the value of public obligations, this project proposes an integrated approach correctly predicting uncertainty and risk. With this in mind, the research combines advances from several areas: modern finance theory for discounting and valuing cash-flows, econometrics for modeling of a population of outcomes (e.g. earnings or labour market participation), actuarial science and demography for modeling longevity and health risks, and finally macroeconomics to account for economic risk in the future.

### Dr Corina Constantinescu

**1. Some exact and asymptotic results for the ﬁrst passage-time distribution of renewal jump-diffusions**

The investigation of the connections between the analytic and probabilistic aspects of the theory of diffusions and Markovian processes has been initiated by Kolmogorov (forward and backward Kolmogorov equations). Recently, the interest was reawakened in the context of ﬁnancial modeling. The most famous case is the Merton-Black-Scholes diffusion, which due to its mathematical tractability led to a large variety of explicit formulas for the transition and ﬁrst passage probabilities necessary for the pricing of options. For insurance modeling, jump-diffusion or renewal jump-diffusion processes are considered (small continuous changes modeled by diffusions, and ”catastrophic” changes modeled by pure jump processes) and exact or asymptotic results for the ﬁrst passage-time distribution are thought after.

**2. A comparative study of pricing and risk management of equity-linked life insurance products**

Given the development of the equity-linked insurance market, there is a need for an overview of methods employed and models used in practice versus academia. The academic literature for variable annuities and similar products is to be compared with the methods currently used in practice for managing financial and mortality/longevity risks. The results should appease or raise warning flags in both life insurance and pension fund industries.

**3. Fractional differential equations in risk theory**

In the ruin theory context, (integro-)differential equations are widely used. For instance, when the distribution of the inter-claim times are gamma distributed with the shape parameter integer, the ruin probability satisfies high order (integro-)differential equations. If the shape parameter is not an integer anymore, one needs to appeal to fractional calculus: analyze fractional differential equations and their boundary conditions.

### Dr Amogh Deshpande

**1. Risk-sensitive stochastic control**

In risk-sensitive stochastic control, the optimal expected utility function depends on an external parameter say $\theta$. It is a generalization of the traditional stochastic control in the sense that now the degree of risk aversion of the investor is explicitly parameterized through $\theta$ rather than importing it in the problem via an externally defined utility function. PhD topics could focus on relaxing regularity of functions and/or asymptotic of portfolio performance.

Ref: *A. Deshpande, **On the role of Föllmer-Schweizer minimal martingale measure in Risk Sensitive control Asset Management, 2015, Journal of Applied Probability.*

**2. Numerical analysis of financial derivatives**

Financial derivatives traded these days are complicated products and usually inherit or built up on traditional derivative instruments. They can be shown to solve certain class of PDEs which need not guarantee explicit solution. In a PhD project on this topic, we price the derivatives by solving these PDEs using advanced numerical analysis and techniques.

For motivation and insights refer,

Ref: *The Mathematics of Financial Derivatives: A Student Introduction Paperback – 29 Sep 1995 **by **Jeff Dewynne** , Howinson and Wilmott, CUP.*

**3. Exact computation of losses in financial inter-bank network**

We aim to provide at least some quasi-analytic expression for the inter-bank lending (undirected) network in order to determine the probability and size of occurrence of small, finite size financial contagion and a large size contagion.

Refer:* Gai, P. and Kapadia, Contagion in financial networks, Proc. Roy. Soc. A, 2010.*

**4. Stochastic Analysis**

Many areas of stochastic analysis, for e.g. stochastic stability and semi-martingale modelling and their application to finance and economics.

Refer: *A. Deshpande, Asymptotic Stability of Semi-Markov Modulated Jump Diffusions, 2012, IJSA.*

### Dr Julia Eisenberg

**1. Stochastic Discounting via a Compound Poisson Jump Process**

In this project, we can investigate at first the expected value of discounted dividend threshold strategies. The discounting will be given by an exponential function of a Compound Poisson Jump Process. The second target is to search for an optimal strategy, maximising the expected discounted dividends.

**2. The impact of a random walk as a discounting factor on the optimal capital injections**

As far as the interest rate remains positive, a company will inject additional capital only if the surplus becomes negative in order to avoid unnecessary costs. However, it is possible (confer for example European Union) that an interest rate can become negative. We look at the problem of minimal capital injections under the assumption that the discounting factor is given by a random walk.

**3. When is it optimal to blow up the business?**

Switching between two different market states will impact an insurance company considerably. It could happen that periods with a negative economic environment last too long, so that companies cannot come out of red numbers for many years. The target of this project is to investigate when is it optimal to stop the business. It means in particular, that stopping times become a part of the considered business strategies. As risk measure one can take for instance dividends.

### Dr Paul Eisenberg

**Project 1: Local time, placement and density inequalities**

This project deals with finding inequalities for the expected occupation time and for the density for semimartingales at fixed times with uncertain parameters. These inequalities can be applied for stochastic optimal control problems to measure the difference in performance of a given control to the possibly unknown, or even non-existing, optimal control.

**Project 2: Mathematical modelling of energy futures markets**

Energy markets have been liberalised across Europe; in 2000 for Britain. Electricity is mostly traded over the so called spot market (confer EPEX SPOT/Auction in UK) and to varying degree over futures markets. This project deals with the challenging mathematical side of modelling of electricity futures markets in a coherent and consistent way. Electricity markets have different features as stock markets which are mainly due to the difficulty of storing/transporting electricity. This has severe consequences for short termed markets (like spot and auction), and has also effects on the long term markets.

**Project 3:Boundary behaviour of polynomial processes**

This project deals with the local behaviour of polynomial processes on the boundary of their state spaces. Polynomial processes are Markov processes where the generator maps polynomials to polynomials of the same or lower degree. Naturally, these processes have interesting mathematical features. However, it is not known if the law of continuous polynomial processes is determined by the generator restricted to the polynomials. So called, local uniqueness in the interior of the domain is known which leads to the question: Can things go wrong on the boundary? For discontinuous polynomial processes it is known that their law does not need to be determined by the generators action on the polynomials. Even if the state space is the real line the law needs not to be specified.

### Dr Bujar Gashi

**1. Rough path theory with applications**

Rough path theory has been developed extensively in recent years. It gives a meaning to differential equations driven by rough signals, and in particular it gives a pathwise meaning to stochastic differential equations. This project aims at further study of this theory and its application. In particular we are interested in its application to optimal control, optimal stopping, stochastic differential equations, and mathematical finance.

**2. Forward-backward stochastic control with applications**

Forward-backward stochastic differential equations (FBSDE’s) appear in various applications. In particular they appear naturally in mathematical finance when dealing with pricing and hedging, and optimal investment and consumption. However, the control theory for such equations is still in infancy. The aim of this project is to study control problems for FBSDE’s, and to apply such results to problems in mathematical finance. In particular, we will study the pricing of options in a market with large investors, and optimal investment with stochastic interest rate.

**3. Optimal investment with stochastic interest rate**

Most of the known methods for optimal investment assume that the interest rate is either constant or it changes according to a known law. However, the finance practice has shown that such an assumption is not realistic. As a result, various different models for interest rates have become available in recent years. The aim of this project is to study the optimal investment problem under the assumption that the interest rate changes randomly.

### Dr Olivier Menoukeu Pamen

**Project 1: Optimal dividend policy of an insurance firm. **** **

This project deals with the problem of finding an optimal dividend policy for a class of diffusion processes under some constraints (for e.g., transaction cost). More precisely, properties of the value function of this control problem would be studied and convergence of numerical schemes for the optimal value function would be looked at.

**Project 2: Derivative pricing under Markov regime switching.**

Random structural changes in the movements of stock (or spot) indices are not generally caused by internal events of the market but are related to the global socioeconomic and political environment. Markov modulated models (or regime switching) have been widely used in financial mathematics and economics to capture this exogenous cycles. The aim of this project is to price Bermudan swaptions under Markov regime switching model and to check the impact of the regime in the derivative pricing.

**Project 3: Risk sensitive control for jump-diffusion model under Markov regime switching.**

The studies of risk-sensitive optimal control have attracted much attention of researchers in the past decades. The aim of this project is to investigate a risk-sensitive optimal control problem for a Markov modulated Lévy process. As an application, we shall analyse the optimal wealth, investment and consumption for a risk sensitive control problem of an investor. A Malliavin calculus approach to such problem would also be explored.

### Dr Apostolos Papaioannou

**1. Markov Arrival Process risk models in the presence of dividends strategies**

In this project, we can investigate the probability of ruin for a Markov Arrival risk model under various dividend strategies (constant, threshold, etc). This kind of risk processes can be used to model insurance portfolios with dependent risks .

**2. Absolute ruin probabilities in time – dependent risk processes**

When the surplus process of an insurance portfolio becomes negative or the insurer is in deficit, the insurer could borrow money at a constant debit interest rate to repay the claims.

In this project, we can investigate the absolute ruin probabilities for a general class of dependent risk models.

**3. Defective renewal equations for a class of dependent risk models**

In this project, we can investigate the derivation of defective renewal equations for many risk measures in a class of non –renewal risk models. Then, these defective renewal equations can be used to derive bounds for the corresponding risk measures.