Financial Engineering:
Samenvatting Exam
FINANCIAL ENGINEERING
Comprehensive Practice Examination
CHAPTER 0: introduction
Outline of the course
Exercises
• homework and in class
• 3 challenging exercises for the application of the material
Course material
• slides
• extra reading material → not optional but the material will also be (partially)
covered in class
Course content
1. modelling of asset prices
2. simulation
3. options - valuation and hedging
4. equity linked structured products
5. interest rate modelling
6. credit risk modelling
7. hybrid securities
, 8. recent applications
Exam
Written exam (50/50)
• closed book: max. 1.5 hours
→ 4 questions with subquestions
• open book: remaining time (max. 3 hours in total)
→ 4 questions with subquestions
Image 2
CHAPTER 1: modelling asset prices
Introduction
1. Stochastic processes
Stochastic process = a variable that evolves over time in a way that is at least in part
random
There are multiple examples, but they can be clearly different in nature
• temperature in Antwerp
o stationary = constant properties over time
→ variation in temperature next week will be similar to variation in 3 years
• stock price of KBC
o non-stationary = the further you look into the future, the more uncertain
you are
→ variation in stock prices 3 years from now is wider than variation 1 year
from now
We can measure both in different processes
• continuous-time process = process whose value varies all the time
→ both temperature in Antwerp and stock price of KBC are good examples
• discrete-time process = process whose values can change only at discrete points
in time
, → well-suited to take as starting example: discrete-time discrete-state random
walk
Discrete-time processes
1. Discrete-time discrete-state random walk
𝑥𝑡 is a random variable
• that starts at a known value 𝑥0 at time 𝑡 = 0
• that takes a jump of 0.1 up or 0.1 down at times 𝑡 = 1,2,3 …
• with each jump up or down having a 50% probability
Formally
• set 𝑥0 = 0and 𝑥𝑡 = 𝑥𝑡−1 + 𝑒𝑡 with 𝑡 = 1,2,3 …
• with 𝑝𝑟𝑜𝑏(𝑒𝑡 = 0.1) = 𝑝𝑟𝑜𝑏(𝑒𝑡 = −0.1) = 50%
[Diagram showing a binomial tree starting at 0, branching to 0.1 (50%) and -0.1 (50%),
and continuing to branch out to 0.2, 0, -0.2, then 0.3, 0.1, -0.1, -0.3]
This is an example of a (discrete time) random walk
• recombining tree → going up and down is the same as going down and up
• random and non-stationary (more possibilities the more you toss)
Image 3
Let’s look at some properties of 𝑥𝑡
• non-stationary: the range of possible outcome increase in t
• Markov property: the future depends only on the present, not on the past
→ probability distribution of 𝑥𝑡+𝑠 depends only on 𝑥𝑡 and not additionally on what
happened before time t
• unpredictable: 𝐸0 [𝑥𝑡 ] = 𝑥0 for all t and 𝐸𝑡 [𝑥𝑇 ] = 𝑥𝑡 for all 𝑇 ≥ 𝑡
→ you know the future will change, but the expected value of the future is that it
will not change
1.1 Rational expectations
, Predictions for the future are uncertain and must be expressed in terms of probability
distributions. The (rational) expectation value of a random variable is calculated as the
probability-weighted average of the different possible realisations of the variable.
𝐸[𝑋 ∣ 𝐼] = 𝜋1 𝑋1 + ⋯ + 𝜋𝑛 𝑋𝑛
In this expression, 𝜋's are conditional on information set 𝐼. This is the statistically optimal
forecast. In our applications, we make this equation actionable by stating that “the price
of an asset today is equal to the probability weighted average of the likely outcomes in
the future”.
In the current example, we have
• expected value of 𝑥1 at time 0 = 𝐸[𝑥1 ∣ 𝑥0 = 0] = 0.5 ⋅ 0.1 + 0.5 ⋅ (−0.1) = 0 = 𝑥0
• 𝐸[𝑥2 ∣ 𝑥0 = 0] = 0.25 ⋅ 0.2 + 0.5 ⋅ 0 + 0.25 ⋅ (−0.2) = 0 = 𝑥0
• 𝐸[𝑥2 ∣ 𝑥1 = 0.1] = 0.5 ⋅ 0.2 + 0.5 ⋅ 0 = 0.1 = 𝑥1
1.2 Law of iterated expectations
The definition of rational expectations has powerful implications, one being the law of
iterated expectations (LIE). We will use LIE to understand the link between returns and
prices.
→ LIE states that we can not predict our own future mistakes
Consider two information sets H and I, where H is included in I but I is superior because
it contains some extra information. If one has limited information H, the best forecast
one can make of a random variable X is the same as the forecast one would make of X if
one had superior information I.
→ one cannot use limited information H to predict the forecast error one would make if
one had superior information I
𝐸[𝑋 ∣ 𝐻] = 𝐸[𝐸[𝑋 ∣ 𝐼] ∣ 𝐻]or 𝐸[𝐸[𝑋 ∣ 𝐼] − 𝑋 ∣ 𝐻] = 0
EXAMPLE
• H = {Facebook Q2 revenues between $1B-$2B}
• I = {exact Facebook Q2 revenues}
• X = a variable that might be affected by this (eg. stock price)
Now consider LIE by using time to represent different information sets. Assume time t
information is included in time t+1 information, which is included in time t+2
information and so on. Put differently, we never forget anything. This is the case in the
discrete-time random walk we consider.
Samenvatting Exam
FINANCIAL ENGINEERING
Comprehensive Practice Examination
CHAPTER 0: introduction
Outline of the course
Exercises
• homework and in class
• 3 challenging exercises for the application of the material
Course material
• slides
• extra reading material → not optional but the material will also be (partially)
covered in class
Course content
1. modelling of asset prices
2. simulation
3. options - valuation and hedging
4. equity linked structured products
5. interest rate modelling
6. credit risk modelling
7. hybrid securities
, 8. recent applications
Exam
Written exam (50/50)
• closed book: max. 1.5 hours
→ 4 questions with subquestions
• open book: remaining time (max. 3 hours in total)
→ 4 questions with subquestions
Image 2
CHAPTER 1: modelling asset prices
Introduction
1. Stochastic processes
Stochastic process = a variable that evolves over time in a way that is at least in part
random
There are multiple examples, but they can be clearly different in nature
• temperature in Antwerp
o stationary = constant properties over time
→ variation in temperature next week will be similar to variation in 3 years
• stock price of KBC
o non-stationary = the further you look into the future, the more uncertain
you are
→ variation in stock prices 3 years from now is wider than variation 1 year
from now
We can measure both in different processes
• continuous-time process = process whose value varies all the time
→ both temperature in Antwerp and stock price of KBC are good examples
• discrete-time process = process whose values can change only at discrete points
in time
, → well-suited to take as starting example: discrete-time discrete-state random
walk
Discrete-time processes
1. Discrete-time discrete-state random walk
𝑥𝑡 is a random variable
• that starts at a known value 𝑥0 at time 𝑡 = 0
• that takes a jump of 0.1 up or 0.1 down at times 𝑡 = 1,2,3 …
• with each jump up or down having a 50% probability
Formally
• set 𝑥0 = 0and 𝑥𝑡 = 𝑥𝑡−1 + 𝑒𝑡 with 𝑡 = 1,2,3 …
• with 𝑝𝑟𝑜𝑏(𝑒𝑡 = 0.1) = 𝑝𝑟𝑜𝑏(𝑒𝑡 = −0.1) = 50%
[Diagram showing a binomial tree starting at 0, branching to 0.1 (50%) and -0.1 (50%),
and continuing to branch out to 0.2, 0, -0.2, then 0.3, 0.1, -0.1, -0.3]
This is an example of a (discrete time) random walk
• recombining tree → going up and down is the same as going down and up
• random and non-stationary (more possibilities the more you toss)
Image 3
Let’s look at some properties of 𝑥𝑡
• non-stationary: the range of possible outcome increase in t
• Markov property: the future depends only on the present, not on the past
→ probability distribution of 𝑥𝑡+𝑠 depends only on 𝑥𝑡 and not additionally on what
happened before time t
• unpredictable: 𝐸0 [𝑥𝑡 ] = 𝑥0 for all t and 𝐸𝑡 [𝑥𝑇 ] = 𝑥𝑡 for all 𝑇 ≥ 𝑡
→ you know the future will change, but the expected value of the future is that it
will not change
1.1 Rational expectations
, Predictions for the future are uncertain and must be expressed in terms of probability
distributions. The (rational) expectation value of a random variable is calculated as the
probability-weighted average of the different possible realisations of the variable.
𝐸[𝑋 ∣ 𝐼] = 𝜋1 𝑋1 + ⋯ + 𝜋𝑛 𝑋𝑛
In this expression, 𝜋's are conditional on information set 𝐼. This is the statistically optimal
forecast. In our applications, we make this equation actionable by stating that “the price
of an asset today is equal to the probability weighted average of the likely outcomes in
the future”.
In the current example, we have
• expected value of 𝑥1 at time 0 = 𝐸[𝑥1 ∣ 𝑥0 = 0] = 0.5 ⋅ 0.1 + 0.5 ⋅ (−0.1) = 0 = 𝑥0
• 𝐸[𝑥2 ∣ 𝑥0 = 0] = 0.25 ⋅ 0.2 + 0.5 ⋅ 0 + 0.25 ⋅ (−0.2) = 0 = 𝑥0
• 𝐸[𝑥2 ∣ 𝑥1 = 0.1] = 0.5 ⋅ 0.2 + 0.5 ⋅ 0 = 0.1 = 𝑥1
1.2 Law of iterated expectations
The definition of rational expectations has powerful implications, one being the law of
iterated expectations (LIE). We will use LIE to understand the link between returns and
prices.
→ LIE states that we can not predict our own future mistakes
Consider two information sets H and I, where H is included in I but I is superior because
it contains some extra information. If one has limited information H, the best forecast
one can make of a random variable X is the same as the forecast one would make of X if
one had superior information I.
→ one cannot use limited information H to predict the forecast error one would make if
one had superior information I
𝐸[𝑋 ∣ 𝐻] = 𝐸[𝐸[𝑋 ∣ 𝐼] ∣ 𝐻]or 𝐸[𝐸[𝑋 ∣ 𝐼] − 𝑋 ∣ 𝐻] = 0
EXAMPLE
• H = {Facebook Q2 revenues between $1B-$2B}
• I = {exact Facebook Q2 revenues}
• X = a variable that might be affected by this (eg. stock price)
Now consider LIE by using time to represent different information sets. Assume time t
information is included in time t+1 information, which is included in time t+2
information and so on. Put differently, we never forget anything. This is the case in the
discrete-time random walk we consider.
