Summary Valuation and Risk Management Part I

Tilburg University

Master Program


Summary Valuation and Risk
Management

Supervisor:
Author:
Hambel, C
Rick Smeets
Schweizer, N

December 9, 2024

,Table of Contents
1 Introduction to Financial Modeling 2
1.1 Discrete vs. Continuous Time Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Fundamentals from Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Generic State Space Model 7
2.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 No Arbitrage and the First FTAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 The Numéraire-dependent Pricing Formula . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Replication and the Second FTAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 The PDE Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Contingent Claim Pricing 18
3.1 Black Scholes Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 The Fastest Way to the Black-Scholes Formula . . . . . . . . . . . . . . . . . 19
3.1.2 A Double-Barrier Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Option Pricing in Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 The Heston Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 Calibration vs. Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Fixed Income Modeling 22
4.1 Bonds and Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Interest Rates and Interest Rate Derivatives . . . . . . . . . . . . . . . . . . . . . . 23

5 Short Rate Models for the TSIR 29
5.1 Benchmark: Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Affine Term Structure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Empirical Models 32
6.1 The Nelson-Siegel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.2 The Nelson-Siegel-Svensson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 LIBOR Market Model and Option Pricing 34

8 Credit Risk 36




1

,1 Introduction to Financial Modeling
1.1 Discrete vs. Continuous Time Modeling
When considering a discrete time setting with time horizon T , we denote

t ∈ {0, ∆t, 2∆t, . . . , (n − 1)∆t, |{z}
n∆t} = {i∆t | i = 0, . . . , n}
=T

Furthermore, we can define continuous time as a limit of discrete time, that is, ∆t → 0 as n → ∞:

t ∈ [0, T ]

A risk-free asset (a bond) paying a constant interest rate is given by

Bt+∆t = Bt (1 + r · ∆t)

with returns defined by
∆Bt+∆t
= r · ∆t
Bt
A risky asset (a stock) is modeled by
√ i.i.d.
St+∆t = St (1 + µ · ∆t + σ · νt+∆t · ∆t), νt+∆t ∼ N (0, 1)

Note that µ stands for the expected rate of return (drift) and σ defines volatility. It holds that
µ > r and hence µ − r > 0 defines the risk premium. Returns are defined as
∆St+∆t √
= µ · ∆t + σ · νt+∆t · ∆t
St
Since returns are not necessarily bounded from below by -1, this means that asset prices can be
negative. We solve this issue by modeling log-returns Lt , and take the exponential:

St+∆t = St e∆Lt+∆t

For the risk-free asset we now have that
 
r·∆t Bt+∆t
Bt+∆t = Bt e ⇔ r∆t = ln = ∆ ln Bt+∆t
Bt
and the risky asset is now modeled as

∆Lt+∆t = ln(St+∆t ) − ln(St )
√
 
1 2 i.i.d.
= µ − σ ∆t + σ · νt+∆t · ∆t, νt+∆t ∼ N (0, 1)
2
Now, we take the limit to continuous time, i.e., let n → ∞ while keeping the time horizon constant,
i.e., ∆t = Tn → 0. Then, the final stock price ST is modeled using a product of exponentials of log
returns, simplified through several steps:
n−1
Y
ST = S0 e∆L(i+1)∆t
i=0


2

, This expression can be expanded and simplified as
( n−1  )
√

X 1 2
ST = S0 exp µ − σ ∆t + σ · ν(i+1)·∆t ∆t
i=0
2

Now using the fact that T = n · ∆t we can write
( √ n
)
n √ X

1 2
ST = S0 exp µ − σ T + σ · √ · ∆t νi∆t
2 n i=1
( n
)
√

1 2 1 X
= S0 exp µ− σ T +σ· T × √ νi∆t
2 n i=1

Using the CLT,
n
1 X d
√ νi∆t → ZT ∼ N (0, 1) as n → ∞
n i=1
Thus, the stock price in the limit becomes:
√
  
d 1 2
ST → S0 exp µ − σ T + σ · T · ZT
2
In the limit, under i.i.d. returns, the log return is normally distributed
√
 
1 2
LT = L0 + µ − σ T + σ · T · ZT
2
Consequently, using the properties of a log-normal distribution, we have that in the limit, the stock
price ST is log-normally distributed with

mean: E[ST ] = S0 eµ·T
 
σ2 T
variance: Var(ST ) = S02 e2µ·T e −1

This means that any discrete-time model converges to a log normal distribution if and only if we
have i.i.d. innovations such that the CLT can be applied.

Regarding trading in discrete time, assume a frictionless financial market. This means the market
operates without any transactional hindrances like taxes, transaction costs, or regulatory con-
straints such as short-selling limits. We define vector notations:
(i) m: the number of basic assets.

(ii) Yt : an m-dimensional vector representing the prices of these assets at any time t.

(iii) ϕt : a vector denoting the number of units of assets held at time t.
The value of the portfolio Vt at any time t is calculated by taking the dot product of the quantity
of assets held (ϕt ) and the current asset prices (Yt ):
m
X
Vt = ϕ′t Yt = ϕi,t Yi,t
i=1


3