The Black-Scholes European option pricing formula can be derived in several ways. In this dissertation we present several methods that can be used to derive this formula, including partial differential equation method, the risk-neutral pricing method, the martingale measure method, and the change of numeraire technique

Policyholder exercise behavior presents an important risk factor for life insurance companies. Yet, most approaches presented in the academic literature – building on value maximizing strategies akin to the valuation of American options – do not square well with observed prices and exercise patterns. Following a recent strand of literature, in order to gain insights on what drives policyholder behavior, I first develop a life-cycle model for variable annuities (VA) with withdrawal guarantees. However, I explicitly allow for outside savings and investments, which considerably affects the results. Specifically, I find that withdrawal patterns after all are primarily motivated by value maximization – but with the important asterisk that the value maximization should be taken out from the policyholders’ perspective accounting for individual tax benefits. To this effect, I develop a risk-neutral valuation methodology that takes these different tax structures into consideration, and apply it to our example contract as well as a representative empirical VA. The results are in line with corresponding outcomes from the life cycle model, and I find that the withdrawal guarantee fee from the empirical product roughly accords with its marginal price to the insurer. I further consider the implications of policyholder behavior on product design. In particular – due to differential tax treatments and contrary to option pricing theory – the marginal value of such guarantees can become negative, even when the holder is a value maximizer. For instance, as I illustrate with both a simple two-period model and an empirical VA, a common death benefit guarantee may indeed yield a negative marginal value to the insurer.

In this thesis we use nonstandard methods for the valuation of derivatives on electricity. We model the dynamics of electricity spot price as mean reverting processes on the hyperfinite binomial tree and by switching to the risk-neutral world we derive analytical formulas for the price of forward contracts. Both of our models are fitted to the German electricity market and forward price predictions are compared with forward products traded on the exchange. We conclude that both the Ornstein-Uhlenbeck and the Schwartz one factor model fit long-term forward contracts well while our prediction results for short-term forward prod- ucts are not conclusive due to low liquidity and alternative approaches might be suitable. 1

Quanto EIA是一種具有選擇權特性且能連結至外幣投資的保險年金商品.以往針對權益連動年金所做的文獻中,均未考慮Quanto的特性.本文利用風險中立評價法求算出六種具有Quanto特性的鎖高型權益連動年金商品的評價公式,並進一步利用數值分析來探討各個契約及市場參數對契約價值的影響. / Quanto Ratchet EIAs link to foreign investments and provide options-like properties. The literature covers the pricing of the EIAs that are not quantos. This paper intends to fill the hole. To derive the pricing formulas, we added an exchange rate model as well as a foreign risk-free rate model to the pricing framework of Black and Scholes. Our formulas cover quanto ratchet EIA products for both compound and simple versions that may have a return cap and employ two types of geometric return averaging. We further provide numerical analyses on how contract features and market parameters affect the contract value.

Build-Operate-Transfer (BOT) is a form of Public-Private Partnerships that is commonly used to close the growing gap between the cost of developing and modernizing transportation infrastructure systems and the financial resources available to governments. When assessing the feasibility of a BOT project, private investors consider revenue risk - which is stemmed from the uncertainty about future traffic demand - as a critical factor. A potential approach to mitigating the revenue risk is the offering of revenue risk sharing mechanisms such as Minimum Revenue Guarantee options by the government. In addition to Minimum Revenue Guarantee options, a mechanism known as Traffic Revenue Cap options may also be negotiated, which makes the government entitled to a share of revenue when it grows beyond a specified threshold. Financial valuation of investments in BOT projects should take into account uncertainty about future traffic demand, as well as Minimum Revenue Guarantee and Traffic Revenue Cap options. The conventional valuation methods including Net Present Value (NPV) analysis are not capable of integrating the uncertainty about future traffic demand in the valuation of BOT projects and properly pricing Minimum Revenue Guarantee and Traffic Revenue Cap options. Real options analysis can be used as an alternative approach to valuation of investments in transportation projects under uncertainties. However, the appropriate application of real options analysis to valuation of investments in transportation projects is conditioned upon overcoming specific theoretical challenges. Current real options models do not provide a systematic method for estimating the project volatility, which measures the variability of investment value. Existing models do not provide a method for calculating the market value of Minimum Revenue Guarantee and Traffic Revenue Cap options. Also, current models are not able to characterize the impact of Minimum Revenue Guarantee and Traffic Revenue Cap options on private investors' financial risk profile. The overarching objective of this research is to apply the real options theory in order to price Minimum Revenue Guarantee and Traffic Revenue Cap options under the uncertainty about future traffic demand. To achieve this objective, a real options model is created that characterizes the long-term traffic demand uncertainty in BOT projects and determines investors' financial risk profile under uncertainty about future traffic demand. This model presents a novel method for estimating the project volatility for real options analysis. This model devises a market-based option pricing approach to determine the correct value of Minimum Revenue Guarantee and Traffic Revenue Cap options. An appropriate procedure is created for characterizing the impact of Minimum Revenue Guarantee and Traffic Revenue Cap options on the investors' financial risk profile. The proposed real options model is applied to a BOT project to illustrate the valuation process. The limitations of the proposed real options model, as well as the barriers to its implementation, are identified and recommendations for future research are offered. This research contributes to the state of knowledge by presenting a new method for estimating the project volatility, which is required for the real options analysis of transportation investments. It also introduces a risk-neutral valuation method for pricing the market value of Minimum Revenue Guarantee and Traffic Revenue Cap options in BOT projects. The research also contributes to the state of practice by introducing a novel class of assessment tools for decision makers that characterize the investors' financial risk profile under uncertainty about future traffic demand. Proper methods for pricing of Minimum Revenue Guarantee and Traffic Revenue Cap options are useful to public and private investors, in order to avoid wasting capital in transportation projects.

Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal.

Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal

Topic of the master thesis is practice of interest rate models. Literature dedicated to the interest rate models usually presents theory in very general form. Theory presented in general form leads to a gap between theory and practice. Author tries to fill this gap. Thesis describes basic theory and presents practical computations, which are relevant to generating interest rate scenarios. Contribution is given by derivation of formulas and computational methods in form directly applicable for implementation of presented models. It is common practice to validate quality of interest rate scenarios. Author presents several tests and implements them in programming language Python. Tests are implemented as application with graphical user interface.