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Actuarial Economics CT8 > Syllabus Objectives > Flashcards

Syllabus Objectives Flashcards

1
Q

Explain the meaning of the term “utility function”

A

U(w) is a function representing an investor’s utility of wealth, w, at some
future date.

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2
Q

Expected utility theorem.

A

The expected utility theorem states that:
… a function, U(w) can be constructed
… representing an investor’s utility of wealth, w, at some future date.

Decisions are made on the basis of
… maximising the
expected value of utility
…. under the investor’s particular beliefs about the probability of different outcomes.

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3
Q

4 Axioms of the Expected Utility theorem

A
  • Comparability
  • Independence
  • Transitivity
  • Certainty Equivalence
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4
Q

Comparability

A

An investor can state a preference between all available certain outcomes.

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5
Q

Transitivity

A

If A is preferred to B and B is preferred to C, then A is preferred to C.

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6
Q

Independence

A

If an investor is indifferent between two certain outcomes, A and B, then he is also indifferent between the following two gambles:

(a) A with probability p and C with probability (1 − p); and
(b) B with probability p and C with probability (1 − p).

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7
Q

Certainty equivalence

A

Suppose that
… A is preferred to B
… and B is preferred to C.

Then there is a unique probability, p, such that:
the investor is indifferent between B and a gamble giving:
…. A with probability p
… and C with probability (1 − p).

B is known as the certainty equivalent of the above gamble.

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8
Q

Explain how it can be expressed mathematically in a utility function:
- non-satiation

A

U’(w) > 0

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9
Q

Explain it can be expressed mathematically in a utility function:
- risk aversion, risk neutrality and risk seeking

A

Risk Aversion:
U’‘(w) < 0

Risk Neutrality:
U’‘(w) = 0

Risk Seeking:
U’‘(w) > 0

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10
Q

State-dependent utility functions

A

Used to model the situation where there is a discontinuous change in the state of the investor at a certain level of wealth.

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11
Q

1st Order Stochastic Dominance (FSD)

A

Gamble A has first-order stochastic dominance over gamble B if, for any outcome x:
- A gives at least as high a probability of receiving at least x as does B,

and for some x,
- A gives a higher probability of receiving at least x.

P(A >= x) >= P(B >= x) for all x
and P(A >= x) > P(B >= x) for some x
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12
Q

Absolute dominance

A

When one investment portfolio provides a higher return than another in all possible circumstances.

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13
Q

Second-order stochastic dominance

A

Condition for A to dominate B is that:
int_a^x F_A(y) <= int_a^x F_B(y)

where a is the lowest return that the portfolios could possibly provide.

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14
Q

8 Key findings of behavioural finance

A

F - framing (and question wording)
A - anchoring and adjustment
M - myopic loss aversion
E - estimating probabilities

P - prospect theory
O - overconfidence
M - mental accounting
O - effect of options

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15
Q

7 Points under the “Effect of Options”

A

P - Primary effect
R - Recency effect
R - Regret aversion
I - People are more likely to select an intermediate option than one at either end.
S - Status-quo bias
M - More options tend to discourage decision-making.
A - Ambiguity aversion

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16
Q

2 Points under “Overconfidence”

A
  • Hindsight bias

- Confirmation bias

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17
Q

3 Points under “Estimating probabilities”

A
  • Dislike of negative events
  • Availability
  • Representative heuristics
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18
Q

Framing

A

The way in which a choice is presented (“framed”) and, particularly the working of a question in terms of gains and losses, can have an enormous impact on the answer given or the decision made.

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19
Q

Anchoring

A

Used to explain how people produce estimates.
They start with an initial idea of the answer (“the anchor”) and then adjust away from this initial anchor to arrive at their final judgement.

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20
Q

Myopic loss aversion

A

Similar to prospect theory but considers repeated choices rather than a single gamble.
Research suggests investors are less risk-averse when faced with a multi-period series of “gambles”, and that the frequency of choice or length of reporting will also be influential.

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21
Q

Prospect theory

A

Relates to how people make decisions when faced with risk and uncertainty.
It replaces the conventional risk-averse / risk-seeking decreasing marginal utility theory with a concept of value defined in terms of gains and losses relative to a reference point.
This generate utility curves with a point of inflexion at the chosen reference point.

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22
Q

Mental accounting

A

People show a tendency to seperate related events and decisions and find it difficult to aggregate events.
Rather than netting out all gains and losses, people set up a series of “mental accounts” and view individual decisions as relating to one or another of these accounts.

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23
Q

Primary effect

A

People are more likely to choose the first option presented.

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24
Q

Recency effect

A

In some instances the final option discussed may be preferred.
The gap in time between the presentation of the decision may influence this dichotomy.

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25
Q

regret aversion

A

by retaining existing arrangements, people minimise the possibility of regret.

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26
Q

Status quo bias

A

people have a marked preference for keeping things as they are.

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27
Q

Ambiguity aversion

A

people are prepared to pay a premium for rules.

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28
Q

Availability

A

People are influenced by the ease with which something can be brought to mind.
This can lead to biased judgements when examples of one event are inherently more difficult to imagine than examples of another.

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29
Q

Representative heuristics

A

People find more probable that which they find easier to imagine.
As the amount of detail increases, its apparent likelihood may increase.

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30
Q

Dislike of “negative” events

A

the “valence” of an outcome (the degree to which it

is considered as negative or positive) has an enormous influence on the probability estimates of its likely occurrence.

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31
Q

Hindsight bias

A

events that happen will be thought of as having been predictable
prior to the event, events that do not happen will be thought of as having been
unlikely prior to the event.

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32
Q

Confirmation bias

A 
people will tend to look for evidence that confirms their point
of view (and will tend to dismiss evidence that does not justify it).
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33
Q

4 Measures of investment risk

A

variance of return
downside semi-variance of return
shortfall probabilities
Value at Risk (VaR) / Tail VaR

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34
Q

4 Arguments against using semi-variance as a risk measure

A
  • not easy to handle mathematically
  • takes no account of variability above the mean
  • if returns are symmetrically distributed, semi-variance is proportional to variance, so it gives no extra information.
  • semi-variance measures downside relative to the mean rather than another benchmark that might be more relevant to the investor.
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35
Q

Argument for using semi-variance as a risk measure

A

Most investors do not dislike uncertainty of returns as such; rather, they dislike the possibility of low returns.

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36
Q

Value at risk

A

Var(X) = -t, where P(X

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37
Q

What can be deduced about and investor’s utility function if the investor makes decisions based on THE VARIANCE OF RETURNS

A

may imply that the investor has a quadratic utility function

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38
Q

What can be deduced about and investor’s utility function if the investor makes decisions based on the SHORTFALL PROBABILITY OF RETURNS?

A

This corresponds to a utility function which has a discontinuity at the minimum required return.

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39
Q

7 Assumptions of Mean-variance Portfolio Theory

A
  • All expected returns, variances and covariances of pairs of assets are known
  • Investors make their decisions purely on the basis of expected return and variance
  • Investors are non-satiated
  • Investors are risk-averse
  • fixed single-step time period
  • no taxes or transaction costs
  • Assets may be held in any amounts
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40
Q

State 2 conditions that need to be met in order for mean-variance portfolio theory to be consistent with utility theory.

A
  • Investors have quadratic utility functions

- Investment returns are normally distributed (or elliptically symmetrically distributed)

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41
Q

Explain the benefits of diversification using mean-variance portfolio theory.

A

As a portfolio is diversified, the return on the portfolio is less exposed to the specific risk of any one component.

This means that as portfolios are diversified the correlation components become less important, therefore variance of return is minimised.

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42
Q

3 Types of multifactor models of asset returns

A
  • Macroeconomic models
  • fundamental factor models
  • statistical factor models
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43
Q

Macroeconomic Multifactor model

A

A multifactor model where:

The factors are the main macroeconomic variables such as interest rates, inflation, economic growth and exchange rates.

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44
Q

Fundamental Multifactor model

A

A multifactor model where

The factors will be company specifics such as P/E ratios, liquidity ratios and gearing measurements.

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45
Q

Statistical Mutlifactor model

A

A multifactor model where:
The factors are not specific items initially.
The mehtod uses principal component analysis and historical returns on stocks to decide upon the factors.

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46
Q

Fundamental Factors

A
  • Level of gearing
  • Price earnings ratio
  • level of research and development spending
  • industry group to which the company belongs.
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47
Q

Discuss the single-index model of asset returns.

A

The single-index model expresses the return on a security as:

Ri = αi + βiRM + εi

where:

  • Ri is the return on security i
  • αi and βi are constants
  • RM is the return on the market
  • The εi are independent, zero-mean random variables, uncorrelated with RM, representing the component of Ri not related to the market.
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48
Q

Specific risk

A

Risk that CAN be diversified away

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49
Q

Systematic risk

A

Risk that CANNOT be diversified away

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50
Q

Assumptions of the CAPM

A

ALL ASSUMPTIONS FROM MODERN PORTFOLIO THEORY PLUS:

All investors:

  • have the same 1-period horizon
  • can borrow/lend unlimited amounts at the same risk-free rate.
  • have the same estimates of the expected returns, standard deviations and covariances of securities over the one-period horizon.
  • measure in the same “currency” or in “real”/”money” terms.
  • The market for risky assets are perfect
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51
Q

4 Results of CAPM

A
  • All investors have same efficient frontier of risky assets
  • Efficient frontier collapses to a straight line in E-σ space in the presence of a risk-free asset.
  • All investors hold a combination of the risk free asset and the same portfolio of risky assets, M.
  • M is the market portfolio - it consists of all assets held in proportion to their market cap.
52
Q

Limitations of basic CAPM

A

Empirical studies do not provide strong support for the model.

It does not account for

  • taxes,
  • inflation or
  • where there is no riskless asset.

It does not consider

  • multiple time periods or
  • optimisation of consumption over time.
53
Q

Arbitrage Pricing Theory

A

Arbitrage pricing theory (APT) is an equilibrium market model that does not rely on the strong assumptions of the capital asset pricing model (CAPM).
APT requires that the returns on any stock be linearly related to a set of factor indices:

Ri = ai + bi,1 I1 + bi,2 I2 + … + bi,L IL + ci (*)

where
- Ri is the return on security i,
- ai and ci are the constant and random parts
respectively of the component of return unique to security i,
- I1 … IL are the returns on a set of L indices,
- bi,k is the sensitivity of security i to index k.

E[ci] = 0,
E[cicj] = 0 for all i, j where i ≠ j,
and Cov(ci,I) = 0 for all stocks and indices
54
Q

2 Weaknesses of APT

A

(1) In order to apply APT, we need to define a suitable multi-index model.

(2) We also need to come up with the correct factor forecasts. The hard part is the factor forecasts:
… finding the amount of expected excess return to
associate with each factor.
… The simplest approach is to calculate a history of factor returns and take their average.
This implicitly assumes an element of stationarity in the market.

55
Q

Strength of Arbitrage pricing theory

A

Based on no-arbitrage conditions.

- It allows us to describe equilibrium in terms of any multi-index model.

56
Q

2 Main assumptions underlying Arbitrage Pricing Theory

A
  • Principle of no arbitrage applied

- The returns on any stock can be linearly related to a set of factors.

57
Q

Principle of no arbitrage

A

It is not possible to make risk-free profits by exploiting anomalies in prices.
Any 2 assets that have identical payoff profiles must have the same price.

58
Q

3 Forms of market efficiency

A
  • Strong form
  • Semi-strong form
  • Weak form
59
Q

Efficient security market

A

One in which the price of every security fully reflects all available information; i.e. = “true” investment value.

60
Q

Strong form EMH

A

Market prices reflect all information; whether or not publicly available

61
Q

Semi-strong form EMH

A

Market prices reflect all publicly available information

62
Q

Weak form EMH

A

Market prices reflect all info from historical price data.

63
Q

Importance of market efficiency

A

If markets are inefficient, then investors with better information may be able to obtain higher investment returns.

64
Q

Outline the approach adopted by Shiller to test for excessive volatility

A

Shiller used a discounted cashflow model of equities going back to 1870.
A perfect foresight price was determined using actual dividends paid and a terminal value for the stock.
If markets are rational there would be no systematic forecast errors (i.e. error between the perfect foresight price and the actual price).
If markets are efficient, the perfect foresight price matches with share price.

Strong evidence was found that contradicted the EMH.

65
Q

Criticism of Shiller’s work

A
  • the choice of terminal value for the stock price
  • the use of a constant discount rate
  • bias in estimates of the variances because of autocorrelation
  • possible non-stationarity of the series, i.e. the series may have stochastic trends which invalidate the measurements obtained for the variance of the stock price
66
Q

State 2 difficulties of testing the strong form of the efficient market hypothesis and state the general conclusion of studies carried out on it.

A
  • Researchers require access to information that is not in the public domain.
  • Studies suggest that it is difficult to out perform with inside information.
67
Q

Explain how mean-reversion in the stock market can be consistent with an efficient market.

A

After a crash, many investors may have lost a significant proportion of their total wealth; it is not irrational for them to be more averse to the risk of losing what remains.
As a result, the prospective equity risk premium could be expected to rise

68
Q

Explain what is meant by “excessive volatility” of share prices.

A

Excessive volatility is when the change in market value of stocks (observed volatility), cannot be justified by the news arriving.

This is claimed to be evidence of market over-reaction which was not compatible with efficiency

69
Q

State 3 examples of empirical evidence of the “under-reaction” of share prices to events

A
  1. Stock prices continue to respond to earnings announcements up to a year after their announcement. An example of under-reaction to information which is slowly corrected.
  2. Abnormal excess returns for both the parent and subsidiary firms following a de-merger. Another example of the market being slow to recognise the benefits of an event.
  3. Abnormal negative returns following mergers (agreed takeovers leading to the poorest subsequent returns). The market appears to over-estimate the benefits from mergers, the stock price slowly reacts as its optimistic view is proved to be wrong.
70
Q

Advantages of the lognormal model

A
  • The mean and variance of return are proportional to the length of the time interval considered.
  • Returns over non-overlapping time intervals are independent of each other.
  • Cannot use past history to identify whether prices are cheap or dear implying weak form market efficiency consistent with empirical observations.
  • Does not permit negative share prices.
71
Q

disadvantages of the lognormal model

A
  • Estimates of volatility vary widely over time periods. This is supported by implied volatility from option prices.
  • The model is not mean reverting, which is contradicted by some evidence of momentum effects and reversion after market crashes.
  • Does not reflect jumps and discontinuities observed in the market.
72
Q

State how investors are assumed to make decisions in modern portfolio theory
(MPT).

A

Select on the basis of expected return and variance of return over a single time
horizon.

73
Q

Describe the lognormal model for security prices.

A

The lognormal model has independent, stationary normal increments for the log returns on the asset.

Thus, if Su denotes the stock price at time u, then

log(St /Ss) ~ N(μ(t − s), σ2(t − s))

where

  • μ is the drift
  • and σ is the volatility parameter.
74
Q

Outline the evidence against normality assumptions in models of market returns.

A
  • market crashes appear more often than one would expect from a normal distribution. While the random walk produces continuous price paths, jumps or discontinuities seem to be an important feature of real markets.
  • Furthermore, days with no change, or very small change, also happen more often than the normal distribution suggests. This would seem to justify the consideration of Levy processes.

• Q-Q plots of the observed changes in the FTSE All Share index against those which would be expected if the returns were lognormally distributed show substantial differences. This demonstrates that the actual returns have many more extreme events, both on the upside and downside, than is consistent with the
lognormal model.

• a quintic polynomial distribution whose parameters have been chosen to give the best fit to the data, clearly provides an improved description of the returns
observed, in particular more extreme events are observed than is the case with the lognormal model. The rolling volatilities of a simulation from the non-normal distribution show significant differences over different periods. This volatility process has the same characteristics as the observed volatility from the equity
market.

75
Q

Outline the nature of auto-regressive models of security prices and other economic
variables, including the economic justification for such models. (NO ANSWER) (NO THEORY FOUND)

A

…

76
Q

Discuss the main issues involved in estimating parameters for asset pricing models.

A

The estimation of parameters is one of the most time-consuming aspects of
stochastic asset modelling.
The simplest case is the purely statistical model, where parameters are
calibrated entirely to past time series. Provided the data is available, and
reasonably accurate, the calibration
can be a straightforward and mechanical process.
Of course, there may not always be as much data as we would like, and
the statistical error in estimating parameters may be substantial.
Furthermore, there is a difficulty in interpreting data which appears to
invalidate the model being fitted.
For example, what should be done when fitting a Gaussian model in the
presence of large outliers in the data?
Perhaps the obvious course of action is to reject the hypothesis of
normality, and to continue building the model under some alternative
hypothesis. After all, in many applications, the major financial risks lie in
the outliers, so it seems foolish to ignore them.
In practice, a more common approach to outliers is to exclude them from
the statistical analysis, and focus attention instead on the remaining residuals
which appear more normal.
The model standard deviation may be subjectively nudged upwards after [½]
the fitting process, in order to give some recognition to the outliers which
have been excluded.

It has often been the practice in actuarial modelling to use the same data
set to specify the model structure, to fit the parameters, and to validate the
model choice.

A large number of possible model structures are tested, and testing stops [½]
when a model which is found which passes a suitable array of tests.
Unfortunately, in this framework, we may not be justified in accepting a
model simply because it passes the tests.
Many of these tests (for example, tests of stationarity) have notoriously
low power, and therefore may not reject incorrect models.
Indeed, even if the “true” model was not in the class of models being
fitted, we would still end up with an apparently acceptable fit, because
the rules say we keep generalizing until we find one.
This process of generalization tends to lead to models which wrap
themselves around the data, resulting in an understatement of future risk,
and optimism regarding the accuracy of out-of-sample forecasts.
In the context of economic models, the calibration becomes more
complex. The objective of such models is to simplify reality by imposing
certain stylised facts about how markets would behave in an ideal world.
This theory may impose constraints, for example on the relative
volatilities of bonds and currencies. Observed data may not fit these
constraints perfectly.
In these cases, it is important to prioritise the features of the economy
that are most important to calibrate accurately for a particular application.

77
Q

8 Properties of Brownian Motion

A
  • Independent increments
  • Stationary increments
  • Gaussian increments
  • Continuous sample paths
  • B_0 = 0
  • Cov(Bs, Bt) = min(s,t)
  • Brownian motion is a markov process (follows from independent increment property)
  • is a martingale
  • returns infinitely often to 0, or any other level
78
Q

Complete Market

A

In economics, a complete market is a market with two conditions:

1 Negligible transaction costs and therefore also perfect information,
2. there is a price for every asset in every possible state of the world

79
Q

Define an arbitrage opportunity

A

An arbitrage opportunity exists if we can set up a portfolio that:

a) has zero initial cost, implying long in some assets and short in others.
b) at some future time T: has 0 probability of loss, and greater than 0 probability of strictly positive profit.

80
Q

Principle of no arbitrage

A

States that arbitrage opportunities do not exist.

81
Q

Law of One price

A

States that any 2 securities or combination of securities that give exactly the same payments must have the same price.

82
Q

5 factors that effect the price of a European put option on a non dividend-paying share.

A

(1) The premium would decrease as the underlying share price increased.
(2) The premium would increase as the strike price increased.
(3) The premium would increase as the time to expiry increased.
(4) The premium would increase as the volatility of the underlying share
increased.
(5) The premium would decrease as interest rates increased.

83
Q

Define the Real-world probability measure P

A

Let A be some event contained in F.

Then P(A) is the actual probability that the event A will occur.

On a more intuitive level with m independent
realisations of the future instead of one we would find that the event A occurs on approximately a proportion P(A) occasions (with the approximation getting better as m gets larger and larger).

84
Q

Equivalent measures

A

Two measures P and Q which apply to the same sigma-algebra F are said to be equivalent if, for any event E in F :
P(E)>0 if and only if Q(E)>0,

where P(E) and Q(E) are the probabilities of E under P and Q respectively.

85
Q

Describe the advantages of the martingale approach to derivative valuation compared with an approach based on deriving an appropriate partial differential equation.

A

The martingale approach gives much more clarity in the valuation process. By providing an explicit expectation to evaluate.

It gives the replicating strategy for the derivative.

It can be applied to exotic options where the PDE approach cannot.

86
Q

State the risk-neutral approache to

valuing derivatives

A

Risk neutral pricing approach is the same as the martingale approach, i.e. values are derived from the risk neutral world.

87
Q

State the state price deflator approache to

valuing derivatives

A

Deflators values the derivative in a real world probability measure with a stochastic adjustment factor.

88
Q

Compare the risk-neutral and state price deflator approaches to valuing derivatives

A

The approach for deflators is the same as risk-neutral pricing, the only difference is that:
calculations are presented using the real world measure and a stochastic adjustment factor versus a risk neutral measure.

Intuitively the deflator approach can also give information about real world expected outcomes.

89
Q

6 Assumptions underlying the Black-Scholes

A
  1. The price of the underlying share follows a geometric Brownian motion
  2. There are no risk-free arbitrage opportunities.
  3. The risk-free rate of interest is constant, the same for all maturities and the same for borrowing or lending.
  4. Unlimited short selling (i.e. negative holdings) is allowed.
  5. There are not taxes or transaction costs
  6. The underlying assets can be traded continuously and in infinitesimally small numbers of units.
90
Q

5 Ways in which the Black-scholes assumptions are not realistic

A
  1. Share prices can jump. (the geometric Brownian motion has continuous sample paths, no jumps)
  2. The risk-free rate of interest does vary and in an unpredictable way.
  3. Unlimited short selling may not be allowed, except perhaps at penal rates of interest. These problems can be mitigated by holding mixtures of derivatives which reduce the need for short selling.
  4. Share can normally only be dealt with in integer multiples of one unit, not continuously, and dealings attract transaction costs: invalidating assumptions 4, 5 and 6.
  5. Distributions of share returns tend to have fatter tails than suggested by the log-normal model, invalidating assumption 1.
91
Q

Define a risk-Neutral Probability Measure

A

A risk-neutral (or Equivalent Martingale) probability measure, Q, is a probability measure that is
…. equivalent to the real-world probability measure, P,
… under which the discounted asset price is a martingale.

92
Q

Explain what it means for a portfolio V(t) = ψt St + ϕt Bt to be self-financing

A

dV(t) = ψt dSt + ϕt dBt

At t + dt, there is no inflow or outflow of money necessary to make the value of the portfolio back up to V(t + dt).

93
Q

Describe the concept of delta hedging

A

Delta-hedging is the creation of an instantaneously risk-free portfolio, consisting of:

  • a short position in one option contract
  • a position in delta units of the underlying asset.
94
Q

Why is a delta-hedged portfolio risk-free (instantaneously)?

A

Since delta measures the rate of change in the option price wrt changes in the price of the underlying…
… movements in these 2 components of the portfolio will cancel out.

95
Q

Why is a delta-hedged portfolio ONLY INSTANTANEOUSLY risk-free?

A

Since delta changes over time as the underlying security price changes, dynamic delta hedging requires regular rebalancing of the portfolio in order to maintain delta-neutrality.

96
Q

Delta, Δ

A

df/dS

The change of the derivative price with the share price.

97
Q

Gamma, Γ

A

d²f/dS²

The change of delta with the share price.

98
Q

Theta, θ

A

df/dt

The change of the derivative price with time.

99
Q

Vega, v

A

df/dσ

The change of the derivative price with volatility.

100
Q

Rho, ρ

A

df/dr

The change of the derivative price with the risk-free rate

101
Q

Lambda, λ

A

df/dq

The change of the derivative price with the dividend rate.

102
Q

8 Desirable characteristics of a term structure model

A
  • The model should be arbitrage free
  • Interest rates should ideally be positive.
  • Interest rates should be mean-reverting
  • Bonds and derivative contracts should be easy to price.
  • It should produce realistic interest rate dynamics.
  • It should fit historical interest rate data adequately.
  • It should be easy to calibrate to current market data.
  • It should be flexible enough to cope with a range of derivatives
103
Q

Market price of risk (expression)

A

γ(t, T) = {m(t, T) - r(t)} / S(t, T)

γ(t, T) represents the excess expected return over the risk-free rate per unit of volatility in return for an investor taking on this volatility.

104
Q

Risk premium on a bond

A

γ(t, T) S(t, T) = m(t, T) - r(t)

105
Q

5 Desirable Properties of:

the one-factor Vasicek model

A
  • arbitrage free
  • does not prescribe the short rate process
  • is tractable (allows closed form analytical solutions to a wide range of derivatives)
  • encompasses mean reversion
  • allows for a wide range of yield curves
106
Q

6 Properties of:

Cox-Ingersoll-Ross model

A

Same as Vasicek:

  • Incorporates mean reversion
  • Arbitrage free
  • Time homogeneous

Different from Vasicek:

  • Does not allow negative interest rates
  • More involving to implement than Vasicek model (linked to the chi-squared distribution)
  • Volatility depends on the level of the rates: it is high/low when rates are high/low
  • It is a one factor model
107
Q

Vasicek Model: L

A

L = μ - σ²/α²

108
Q

Vasicek Model: β

A

β = σ² / 2α

109
Q

Value at redemption of a company according to the merton model

A

value at redemption = min(F(t), L)

Where F(t) is the gross value of the company at time t and L is the outstanding debt/bonds

110
Q

3 limitations of 1-factor term structure models

A
  • Not perfect correlation across maturities
  • Different volatility phases
  • Pricing complex derivatives
111
Q

6 Deficiencies of the Vasicek Term Structure Model

A
  • Does not prevent negative interest rates
  • Is difficult to use to obtain humped yield curves
  • Has a lack of time dependence of parameters which is not compatible with empirical evidence
  • Gives, in the long run, spot rates normally distributed - which is not compatible with empirical evidence
  • Implies perfect instantaneous correlation of bond prices, which is not compatible with empirical evidence
  • Will need to be re-parameterised as the yield curve evolves.
112
Q

Compare the properties of the Vasicek and the Cox-Ingersoll-Ross models for interest rates

A
  • Vasicek and CIR both encompass mean reverting short rates
  • Both generate arbitrage-free yield curves
  • In both models, the parameters are time invariant
  • Vasicek permits negative rates, CIR does not
  • Vasicek is more mathematically tractable
  • CIR enforces a non-negative lower bound on yields.
113
Q

Properties of the Hull White model

A
  • Arbitrage Free
  • Mean reversion of rates
  • Ease of calculation of bonds and certain derivative contracts
  • Goodness of fit to historical data
  • Ease of calibration to current market data

NOT:

  • Positive interest rates
  • Realistic dynamics
  • Flexible enough to cope with a range of derivative contracts.
114
Q

Credit event

A

An event that will trigger the default of a bond.

115
Q

Recovery rate

A

In the event of a default, the fraction δ of the defaulted amount that can be recovered through bankruptcy proceedings or some other form of settlement is known as the recovery rate.

116
Q

Structural models

A

Models for a company issuing both shares and bonds, which aim to link default events explicitly to the fortunes of the issuing company.

117
Q

Reduced-form models

A

Statistical models that use observed market statistics such as credit ratings, as opposed to specific data relating to the company.

118
Q

Intensity-based models

A

A particular type of continuous-time reduced-form model.

They typically model the “jumps” between different states using transition intensities.

119
Q

Merton model for risk: Type

A

The Merton model is a structural model for credit risk.

120
Q

Merton model for risk: Assumption

A

It assumes that the shareholders are entitled to net assets of the company after redemption of the loan

121
Q

Merton model for risk: How are Gross assets modelled

A

Gross assets are modelled as the share price in a Black-Scholes market

Thus, if the loan matures at time T:

  • Lt is the loan value at time t,
  • Ft is the gross asset value,
  • Et is the equity value at time t ,

Then Lt = min(L, Ft) where L is the nominal amount of the loan.

It follows that Et is the value of a call option on the gross assets with strike L and Ft = Et + Lt

122
Q

3 Types of credit risk models

A
  • structural
  • reduced form
  • intensity-based
123
Q

Describe how the Merton model can be used to estimate the risk-neutral probability of default.

A

In the Merton model, the company is modelled as having
… a fixed debt, L
… and variable assets Ft.

This means the equity holders can be regarded as holding a
….European call on the assets with a strike of L.

It follows from the Black- Scholes model that we can deduce the (risk-neutral) default probability from the share price.

124
Q

Describe the 2-state model for credit derivatives

A

The company defaults at time-dependent rate λ(t) if it hasn’t previously defaulted.

Once it defaults, it remains permanently in the default state. It is assumed that after default, all bond payments will be reduced by a known factor, (1 − δ), where δ is the recovery rate.

125
Q

Default-free bond

A

A bond is default-free if the stream of payments due from the bond will definitely be paid in full and on time

126
Q

4 Possible Outcomes of a default

A

The outcome of a default may be that the contracted payment stream is:

  • Rescheduled
  • Cancelled by the payment of an amount which is less than the default-free value of the original contract
  • Continued but at a reduced rate
  • Totally wiped out
127
Q

4 examples of a credit event

A
  • Failure to pay either capital or a coupon
  • Loss event (when it becomes clear the borrower is not going to make a full payment on time)
  • Bankruptcy
  • Ratings downgrade of the bond by a rating agency such as S&P or Moody’s

Actuarial Economics CT8 (18 decks)

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