Actuarial finance : derivatives, quantitative models and risk management
Boudreault, Mathieu, Renaud, Jean-FrançoisThis book is targeted mainly toward undergraduate students in actuarial science and practitioners
with an actuarial science background seeking a solid but yet accessible introduction to
the quantitative aspects of modern finance. It covers pricing, replication and risk management
of derivatives and actuarial liabilities, which are of paramount importance to actuaries in areas
such as asset-and-liability management, liability-driven investments, banking, etc.
The book is divided into three parts:
Part 1: Introduction to actuarial finance
Part 2: Binomial and trinomial tree models
Part 3: Black-Scholes-Merton model
In each chapter, on top of the presented material, there is a set of specific objectives, numerical
examples, a point-form summary and end-of-chapter exercises. Additional complementary
information, such as historic notes ormathematical details, is presented in boxes. Finally, warning
icons appear in the margin when a given topic, a concept or a detail deserves extra care or
thought.
The content of the book is as follows.
Part 1: Introduction to actuarial finance
After an introductory chapter that puts into perspective the work of actuaries in the financial
world, more standard chapters on financial securities, forwards and futures, swaps and options
follow. Then, the next two chapters are devoted to the engineering of derivatives payoffs and
liabilities, i.e. to the analysis of the structure of payoffs/liabilities, which is at the core of noarbitrage
pricing. Finally, a whole chapter describes insurance products bearing financial risk,
namely equity-linked insurance and annuities (ELIAs).
In Chapter 1 –Actuaries and their environment, we put into context the role of the actuary
in an insurance company or a pension plan. We explain how to differentiate between (actuarial)
liabilities and (financial) assets, and between financial and insurance markets.We describe
the various insurance policies and financial securities available and we compare actuarial and
financial risks, short- and long-term risks, and diversifiable and systematic risks. Finally, we
analyze various risk management methods for systematic risks.
In Chapter 2 – Financial markets and their securities, we provide an introduction to financial
markets and financial securities, especially stocks, bonds and derivatives. We present the
term structure of interest rates, we calculate the present and future value of cash flows and we
explain the impact of dividends on stock prices.We also explain how actuaries can use derivatives
and why pricing in the financial market is different from pricing in the insurance market.
Finally, we look at price inconsistencies and how to create arbitrage opportunities.
In Chapter 3 – Forwards and futures, we provide an introduction to forwards and futures.
We look at situations where forward contracts and futures contracts can be used to manage
risks, and we explain the difference between a forward contract and a futures contract. We
explain how to replicate the cash flows of forward contracts and calculate the forward price
of stocks and of foreign currencies. Finally, we describe the margin balance on long and short
positions of futures contracts.
In Chapter 4 – Swaps, we provide an introduction to swaps with an emphasis on those
used in the insurance industry, namely interest rate swaps, currency swaps and credit
default swaps. We present their characteristics, explain their cash flows and compute their
values.
In Chapter 5 – Options, we give an introduction to standard options.We explain the differences
between options to buy (call) and options to sell (put), as well as the difference between
options and forward contracts. We explain when an option is used for hedging/risk management
or for speculating purposes. Finally, we describe various investment strategies using
options.
In Chapter 6 – Engineering basic options, we want to understand how to build and relate
simple payoffs and then use no-arbitrage arguments to derive parity relationships.We see how
to use simplemathematical functions to design simple payoffs and relate basic options and how
to create synthetic versions of basic options, including binary options and gap options. Finally,
we analyze when American options should be (early-)exercised.
In Chapter 7 – Engineering advanced derivatives, we provide an introduction to exotic/
path-dependent options and event-triggered derivatives. We describe the payoff of various
derivatives including barrier, Asian, lookback and exchange options, as well as weather, catastrophe
and longevity derivatives. We explain why complex derivatives exist and how they can
be used. Finally, we show how to use no-arbitrage arguments to identify relationships between
the prices of some of these derivatives.
In Chapter 8 – Equity-linked insurance and annuities, we give an introduction to a large
class of insurance products known as equity-linked insurance and annuities. First, we present
relationships and differences between ELIAs and other derivatives. Then, after defining three
indexing methods, we show how to compute the benefit(s) of typical guarantees included in
ELIAs.We explain how equity-indexed annuities and variable annuities are funded and analyze
the losses tied to these products. Finally, we explain how mortality is accounted for when risk
managing ELIAs.
Part 2: Binomial and trinomial tree models
In the second part, we focus on the binomial tree model and the trinomial tree model, two
discrete-time market models for the replication, hedging and pricing of financial derivatives
and equity-linked products. By keeping the level of mathematics low, the binomialmodel allows
greater emphasis on replication (as opposed to pricing), a concept of paramount importance for
actuaries in asset and liability management. The intuition gained from the binomial model will
be used repeatedly in the Black-Scholes-Merton model. Finally, as market incompleteness is a
crucial concept in insurancemarkets, the trinomial treemodel is treated in a chapter of its own.
This model is simple and yet powerful enough to illustrate the idea of market incompleteness
and its consequences for hedging and pricing.
In Chapter 9 – One-period binomial tree model, we first describe the basic assets available
and we identify the assumptions on which this model is based. Then, we explain how to
build a one-period binomial tree.Most of the chapter is devoted to the pricing of derivatives by
replication of their payoff, from which we obtain risk-neutral pricing formulas.
In Chapter 10 – Two-period binomial tree model, we consider again the replication and
the pricing of options and other derivatives, but now in a two-step tree. First, we explain how
to build a two-period binomial tree using three one-period binomial trees. Then, we build
dynamic replicating strategies to price options, from which we obtain risk-neutral pricing formulas.
Finally, we determine how to price options inmore complex situations: path-dependent
options, options on assets that pay dollar dividends, variable annuities or stochastic interest
rates.
In Chapter 11 –Multi-period binomial tree model, we see how to build a general binomial
tree. We relate the asset price observed at a given time step to the binomial distribution and
we highlight the differences between simple options and path-dependent options. We explain
how to set up the dynamic replicating strategy to price an option and then obtain risk-neutral
pricing formulas in a multi-period setup.
In Chapter 12 – Further topics in the binomial tree model, we extend the binomial tree to
more realistic situations.We determine replicating portfolios and derive risk-neutral formulas
for American-style options, options on stocks paying continuous dividends, currency options
and futures options.
In Chapter 13 –Market incompleteness and one-period trinomial treemodels, we define
market incompleteness and we present the one-period trinomial tree model. We build subreplicating
and super-replicating portfolios for derivatives and explain that, in incompletemarkets,
there is a range of prices that prevent arbitrage opportunities. Then, we derive bounds
on the admissible risk-neutral probabilities and relate the resulting prices to sub- and superreplicating
portfolios in a one-period trinomial tree model. Second, we show how to replicate
derivatives if the model has three traded assets. Also, we examine the risk management implications
of ignoring possible outcomes when replicating a derivative. Finally, we analyze how
actuaries cope with the incompleteness of insurance markets.
Part 3: Black-Scholes-Merton model
The third and last part of the book is devoted to the Black-Scholes-Merton model, the famous
Black-Scholes formula and its applications in insurance. Both the model and its main formula
are presented without the use of stochastic calculus; justifications are provided mainly by using
the detailed work done previously in the binomial model and taking the appropriate limits. For
the sake of completeness, a more classical treatment with stochastic calculus is also presented
in two starred chapters, which can be skipped. In the last chapters, we apply generalizations of
the Black-Scholes formula to price more advanced derivatives and equity-linked products, we
provide an introduction to simulation methods and, finally, we present several sensitivity-based
hedging strategies for equity risk, interest rate risk and volatility risk.
In Chapter 14 – Brownian motion, we provide the necessary background on Brownian
motion to understand the Black-Scholes-Merton model and how to price and manage (hedge)
options in that model. We also focus on simulation and estimation of this process, which are
very important in practice. First, we provide an introduction to the lognormal distribution and
compute truncated expectations and the stop-loss transform of a lognormally distributed random
variable. Then, we define standard Brownian motion as the limit of random walks and
present its basic properties. Linear and geometric Brownian motions are defined as transformations
of standard Brownian motion. Finally, we show how to simulate standard, linear and
geometric Brownianmotions to generate scenarios, and how to estimate a geometric Brownian
motion from a given data set.
In Chapter 15 – Introduction to stochastic calculus***, we provide a heuristic introduction
to stochastic calculus based on Brownian motion by defining Ito’s stochastic integral and
stochastic differential equations (SDEs). First, we define stochastic integrals and look at their
basic properties, including the computations of the mean and variance of a given stochastic
integral. Then, we show how to apply Ito’s lemma in simple situations. Next, we explain how
a stochastic process can be the solution to a stochastic differential equation. Finally, we study
the SDEs for linear and geometric Brownian motions, the Ornstein-Uhlenbeck process and the
square-root process, and understand the role played by their coefficients.
In Chapter 16 – Introduction to the Black-Scholes-Mertonmodel, we lay the foundations
of the famous Black-Scholes-Merton (BSM) market model and we provide a heuristic approach
to the Black-Scholes formula.More specifically, we present the main assumptions of the Black-
Scholes-Mertonmodel, including the dynamics of the risk-free and risky assets, and connect the
Black-Scholes-Merton model to the binomial model. We explain the difference between realworld
(actuarial) and risk-neutral probabilities. Then, we compute call and put options prices
with the Black-Scholes formula and price simple derivatives using risk-neutral probabilities.
Also, we analyze the impact of various determinants of the call or put option price. Finally, we
derive replicating portfolios for simple derivatives and show how to implement a delta-hedging
strategy over several periods.
In Chapter 17 – Rigorous derivations of the Black-Scholes formula***, we provide amore
advanced treatment of the BSM model. More precisely, we provide two rigorous derivations
of the Black-Scholes formula using either partial differential equations (PDEs) or changes of
probabilitymeasures. In the first part, we define PDEs and show a link with diffusion processes
as given by the Feynman-Ka?c formula. Then, we derive and solve the Black-Scholes PDE for
simple payoffs and we show how to price and replicate simple derivatives. In the second part,
we explain the effect of changing the probability measure on random variables and on Brownian
motions (Girsanov theorem). Then, we compute the price of simple and exotic derivatives using
the risk-neutral probability measure.
In Chapter 18 – Applications and extensions of the Black-Scholes formula, we analyze
the pricing of options and other derivatives such as options on dividend-paying assets, currency
options and futures options, but also insurance products such as investment guarantees, equityindexed
annuities and variable annuities, as well as exotic options (Asian, lookback and barrier
options). Also, we explain how to compute the break-even participation rate or annual fee for
common equity-linked insurance and annuities.
In Chapter 19 – Simulation methods, we apply simulation techniques to compute approximations
of the no-arbitrage price of derivatives under the BSM model. As the price of most
complex derivatives does not have a closed-form expression, we illustrate the techniques by
describe three variance reduction techniques, namely stratified sampling, antithetic and control
variates, to accelerate convergence of the price estimator.
In Chapter 20 – Hedging strategies in practice, we analyze various risk management practices,
mostly hedging strategies used for interest rate risk and equity riskmanagement. First, we
apply cash-flow matching or replication to manage interest rate risk and equity risk. Then, we
define the so-called Greeks.We explain how Taylor series expansions can be used for risk management
purposes and highlight the similarities between duration-(convexity) matching and
delta(-gamma) hedging.We show how to implement delta(-gamma) hedging, delta-rho hedging
and delta-vega hedging to assets and liabilities sensitive to changes in both the underlying
asset price and the other corresponding financial quantity. Finally, we compute the newhedging
portfolio (rebalancing) as conditions in the market evolve.