Financial Engineering:
samenvatting EXAM
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
samenvatting EXAM
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
