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
1
,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
○ stationary = constant properties over time
→ variation in temperature next week will be similar to variation in 3 years
● stock price of KBC
○ 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 = 0 and 𝑥𝑡 = 𝑥𝑡−1 + 𝑒𝑡 with 𝑡 = 1, 2, 3 ...
( ) (
● with 𝑝𝑟𝑜𝑏 𝑒𝑡 = 0. 1 = 𝑝𝑟𝑜𝑏 𝑒𝑡 =− 0. 1 = 50% )
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)
2
,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+1 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] = π1𝑋1 + ... + π𝑛𝑋𝑛
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 = 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.
● use 𝐸𝑡[𝑋] to denote 𝐸[𝑋 | 𝑡𝑖𝑚𝑒 𝑡 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛]
● let H = time t information and I = time t+1 information
[ ] [
● then 𝐸𝑡[𝑋] = 𝐸𝑡 𝐸𝑡+1[𝑋] or 𝐸𝑡 𝐸𝑡+1[𝑋] − 𝑋 = 0 ]
3
, If you make a prediction of a stock price today, you should not change your mind if next week you learn things that could
have been anticipated already today. If you know something will happen next week, you should already take it into
account in your expectation today.
● we can not predict our own future mistake if we are a rational forecaster
● only things that will change your expectation are things that you did not anticipate (unexpected events)
EXAMPLES
‘1. Modelling asset prices’ slide 14-21
The LIE shows how the discounted present-value model of a security price is entirely consistent with randomness in
security returns.
→ LIE applied to equity valuation: dividend discount model and Gordon growth model
1.3 Changing assumptions
Let us now return to the discrete-time random walk. We make the process more general with respect to the probabilities
and the outcomes. This will help to understand and control the key properties of the process.
● the coin is not fair: probabilities are not 50/50
● instead of 10 cents, we add or subtract a certain amount h (still symmetrical)
1
① Assume ℎ = σ ∆𝑡 and 𝑝 = 2
(fair coin)
We can compute that
1 1
● 𝐸[𝑑𝑥] = 2
ℎ + 2
(− ℎ) = 0
2 2
[
● 𝑣𝑎𝑟[𝑑𝑥] = 𝐸 (∆𝑥) − (𝐸[∆𝑥]) = ] 1
2
ℎ² +
1
2
(− ℎ)² − 0² = ℎ² = σ²∆𝑡
Do this 𝑛 times to cover a period 𝑇 = 𝑛∆𝑡
● 𝐸[∆𝑥] = 𝑛 { 1
2
ℎ +
1
2 }
(− ℎ) = 0
2 2
{[
● 𝑣𝑎𝑟[∆𝑥] = 𝑛 𝐸 (∆𝑥) − (𝐸[∆𝑥]) = ] } 𝑇
∆𝑡
σ²∆𝑡 = σ²𝑇
The specific relation between ℎ and ∆𝑡 is the only one to make the variance of ∆𝑥 depend on 𝑇 and not on the number of
steps. This implies that predicting every year for 2 years is the same as making 24 monthly predictions (smaller steps).
● still no predictability (expected value = 0)
● the variance is now linked to the passage of time (non-stationary)
4
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
1
,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
○ stationary = constant properties over time
→ variation in temperature next week will be similar to variation in 3 years
● stock price of KBC
○ 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 = 0 and 𝑥𝑡 = 𝑥𝑡−1 + 𝑒𝑡 with 𝑡 = 1, 2, 3 ...
( ) (
● with 𝑝𝑟𝑜𝑏 𝑒𝑡 = 0. 1 = 𝑝𝑟𝑜𝑏 𝑒𝑡 =− 0. 1 = 50% )
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)
2
,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+1 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] = π1𝑋1 + ... + π𝑛𝑋𝑛
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 = 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.
● use 𝐸𝑡[𝑋] to denote 𝐸[𝑋 | 𝑡𝑖𝑚𝑒 𝑡 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛]
● let H = time t information and I = time t+1 information
[ ] [
● then 𝐸𝑡[𝑋] = 𝐸𝑡 𝐸𝑡+1[𝑋] or 𝐸𝑡 𝐸𝑡+1[𝑋] − 𝑋 = 0 ]
3
, If you make a prediction of a stock price today, you should not change your mind if next week you learn things that could
have been anticipated already today. If you know something will happen next week, you should already take it into
account in your expectation today.
● we can not predict our own future mistake if we are a rational forecaster
● only things that will change your expectation are things that you did not anticipate (unexpected events)
EXAMPLES
‘1. Modelling asset prices’ slide 14-21
The LIE shows how the discounted present-value model of a security price is entirely consistent with randomness in
security returns.
→ LIE applied to equity valuation: dividend discount model and Gordon growth model
1.3 Changing assumptions
Let us now return to the discrete-time random walk. We make the process more general with respect to the probabilities
and the outcomes. This will help to understand and control the key properties of the process.
● the coin is not fair: probabilities are not 50/50
● instead of 10 cents, we add or subtract a certain amount h (still symmetrical)
1
① Assume ℎ = σ ∆𝑡 and 𝑝 = 2
(fair coin)
We can compute that
1 1
● 𝐸[𝑑𝑥] = 2
ℎ + 2
(− ℎ) = 0
2 2
[
● 𝑣𝑎𝑟[𝑑𝑥] = 𝐸 (∆𝑥) − (𝐸[∆𝑥]) = ] 1
2
ℎ² +
1
2
(− ℎ)² − 0² = ℎ² = σ²∆𝑡
Do this 𝑛 times to cover a period 𝑇 = 𝑛∆𝑡
● 𝐸[∆𝑥] = 𝑛 { 1
2
ℎ +
1
2 }
(− ℎ) = 0
2 2
{[
● 𝑣𝑎𝑟[∆𝑥] = 𝑛 𝐸 (∆𝑥) − (𝐸[∆𝑥]) = ] } 𝑇
∆𝑡
σ²∆𝑡 = σ²𝑇
The specific relation between ℎ and ∆𝑡 is the only one to make the variance of ∆𝑥 depend on 𝑇 and not on the number of
steps. This implies that predicting every year for 2 years is the same as making 24 monthly predictions (smaller steps).
● still no predictability (expected value = 0)
● the variance is now linked to the passage of time (non-stationary)
4
