Continuous Time Finance
Lecture 4
The Martingale Approach
I: Mathematics
(Ch 10-12)
Tomas Bjrk
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Tomas Bjrk, 2009
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Introduction
In order to understand and to apply the martingale
approach to derivative pricing and hedging we will
need to some basic conce
Continuous Time Finance
Lecture 3
Black-Scholes
(Ch 6-7)
Tomas Bjrk
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Tomas Bjrk, 2010
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Contents
1. Introduction.
2. Portfolio theory.
3. Deriving the Black-Scholes PDE
4. Risk neutral valuation
5. Appendices.
Tomas Bjrk, 2010
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1.
Introduction
Tomas B
Continuous Time Finance
Lecture 2
Stochastic Calculus
(Ch 4-5)
Tomas Bjrk
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Tomas Bjrk, 2010
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Stochastic Calculus
General Model:
dXt = tdt + t dWt
Let the function f (t, x) be given, and dene the
stochastic process Zt by
Zt = f (t, Xt)
Problem: What does
Continuous Time Finance
Lecture 1
Stochastic Integrals
(Ch 4-5)
Tomas Bjrk
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Tomas Bjrk, 2010
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Typical Setup
Take as given the market price process, S , of some
underlying asset.
St = price, at t, per unit of underlying asset
Consider a xed nancial deriv
Continuous Time Finance 2005
Exercise List
Seminar 1
Exercise 4.1
Exercise 4.3
Exercise 4.4
Extra Exercise: Suppose X has the stochastic dierential given in exercise
4.4, but now let (t) = X (t) and and be real numbers. Calculate
E [ ln(XT )| Ft ]
Ex
Continuous Time Finance
The Martingale Approach to Optimal
Investment Theory
Ch 20
Tomas Bjrk
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Tomas Bjrk, 2010
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Contents
Decoupling the wealth prole from the portfolio
choice.
Lagrange relaxation.
Solving the general wealth problem.
Example: Log ut
Continuous Time Finance
Stochastic Control Theory
Ch 19
Tomas Bjrk
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Tomas Bjrk, 2010
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Contents
1. Dynamic programming.
2. Investment theory.
Tomas Bjrk, 2010
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1. Dynamic Programming
The basic idea.
Deriving the HJB equation.
The verication theorem
Continuous Time Finance
Forward Rate Models
Ch 25
Tomas Bjrk
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Tomas Bjrk, 2010
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Recap
The instantaneous forward rate with maturity T ,
contracted at t is dened as
f (t, T ) =
ln p(t, T )
T
Bond prices are then given by
p(t, T ) =
Tomas Bjrk, 2010
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RT
Continuous Time Finance
Martingale Models for the Short Rate
Ch 24
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Tomas Bjrk, 2010
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Contents
1. Recap
2. Martingale modeling
3. Inverting the yield curve
Tomas Bjrk, 2010
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I. Recap
Tomas Bjrk, 2010
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2
Recap
P -dynamics for the short rat
Continuous Time Finance
Bonds and Short Rate Models
Ch 22-23
Tomas Bjrk
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Tomas Bjrk, 2010
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Denitions
A zero coupon bond with maturity T (a T -bond)
is a contract paying $1 at the date of maturity T .
p(t, T ) = price, at t, of a T -bond.
p(T, T ) = 1.
M
Continuous Time Finance
Incomplete Markets
Ch 15
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Tomas Bjrk, 2010
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Derivatives on Non Financial Underlying
Recall: The Black-Scholes theory assumes that the
market for the underlying asset has (among other
things) the following properties.
Continuous Time Finance
Change of Numeraire
Ch 26
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Tomas Bjrk, 2010
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Recap of General Theory
Consider a market with asset prices
0
1
N
St , S t , . . . , S t
Theorem: The market is arbitrage free
i
there exists an EMM, i.e. a measure Q such t
Continuous Time Finance
Currency Derivatives
Ch 17
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Tomas Bjrk, 2010
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Pure Currency Contracts
Consider two markets, domestic (England) and foreign
(USA).
rd = domestic short rate
rf
= foreign short rate
X
= exchange rate
NB! The exchange rate
Continuous Time Finance
Dividends,
Forwards, Futures, and Futures Options
Ch 16 & 26
Tomas Bjrk
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Tomas Bjrk, 2010
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Contents
1. Dividends
2. Forward and futures contracts
3. Futures options
Tomas Bjrk, 2010
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1. Dividends
Tomas Bjrk, 2010
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Dividends
Continuous Time Finance
Lecture 5
The Martingale Approach
II: Pricing and Hedging
(Ch 10-12)
Tomas Bjrk
o
Tomas Bjrk, 2010
o
Financial Markets
Price Processes:
0
N
St = St , ., St
Example: (Black-Scholes, S 0 := B, S 1 := S )
dSt = Stdt + StdWt,
dBt = rBt
Continuous Time Finance
Lecture 3B
Completeness and Hedging
(Ch 8-9)
Tomas Bjrk
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Tomas Bjrk, 2010
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Problems around Standard Black-Scholes
We assumed that the derivative was traded. How
do we price OTC products?
Why is the option price independent of t