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Lectures on Financial Mathematics - Harald Lang.pdf
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Lectures on Financial Mathematics
Harald Lang
Preface
My main goal with this text is to present the mathematical modelling of financial markets in a mathematically rigorous way, yet avoiding mathematical technicalities that tends to deter people from trying to access it.
Trade takes place in discrete time; the continuous case is considered as the limiting case when the length of the time intervals tend to zero. However, the dynamics of asset values are modelled in continuous time as
in the usual Black-Scholes model. This avoids some mathematical technicalities that seem irrelevant to the reality we are modelling.
The text focuses on the price dynamics of
forward (or futures) prices rather than spot prices, which is more traditional. The rationale for this is that forward and futures prices for any good—also consumption goods—exhibit a Martingale property on an arbitrage free market, whereas this is not true in general for spot prices (other than for pure investment assets.) It also simplifies computations when derivatives on investment assets that pay dividends are studied.Another departure from more traditional texts is that I avoid the notion of “objective” probabilities or probability distributions. I think they are suspect constructs in this context. We can in a meaningful way assign probabilities to outcomes of experiments that can be repeated under similar circumstances, or where there are strong symmetries between possible
outcomes. But it is unclear to me what the “objective” probability distribution for the price of crude oil, say, at some future point in time would be. In fact, I don’t think this is a well defined concept.
The text presents the
mathematical modelling of financial markets. In order to get familiar with the workings of these markets in practice, the reader is encouraged to supplement this text with some text on financialeconomics. A good such text book is John C. Hull’s:
Options, Futures, & Other Derivatives (Prentice Hall,) which I will refer to in some places.Contents
I: Introduction to Present-, Forward- and Futures Prices . . . 1
Zero Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Money Market Account . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Relations between Present-, Forward- and Futures Prices . . . . . 3
Comparison of Forward- and Futures Prices . . . . . . . . . . . . . 4
Spot Prices, Storage Cost and Dividends . . . . . . . . . . . . . . . 6
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
II: Forwards, FRA:s and Swaps . . . . . . . . . . . . . . . . . . . 8
Forward Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Forward Rate Agreements . . . . . . . . . . . . . . . . . . . . . . . . 10
Plain Vanilla Interest Rate Swap . . . . . . . . . . . . . . . . . . . . 11
Exercises and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 12
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
III: Optimal Hedge Ratio . . . . . . . . . . . . . . . . . . . . . . 17
Exercises and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
IV: Conditions for No Arbitrage . . . . . . . . . . . . . . . . . 21
Theorem (The No Arbitrage Theorem) . . . . . . . . . . . . . . . . 22
The No Arbitrage Assumption . . . . . . . . . . . . . . . . . . . . . . 23
V: Pricing European Derivatives . . . . . . . . . . . . . . . . . 25
Black’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
The Black-Scholes Pricing Formula . . . . . . . . . . . . . . . . . . . 26
Put and Call Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
The Interpretation of
¾ and the Market Price of Risk . . . . . . . 26Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Exercises and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 28
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
VI: Yield and Duration . . . . . . . . . . . . . . . . . . . . . . . 32
Forward Yield and Forward Duration . . . . . . . . . . . . . . . . . 34
Black’s Model for Bond Options . . . . . . . . . . . . . . . . . . . . 36
Portfolio Immunising . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Exercises and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
VII: Risk Adjusted Probability Distributions . . . . . . . . . 42
An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Forward Distributions for Different Maturities . . . . . . . . . . . . 44
Exercises and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
VIII: Conditional Expectations and Martingales . . . . . . . 48
Martingale Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49IX: Asset Price Dynamics and Binomial Trees . . . . . . . . . 51
Black-Scholes Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Binomial Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 51
The Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Pricing an American Futures Option . . . . . . . . . . . . . . . . . . 53
American Call Option on a Share of a Stock . . . . . . . . . . . . . 54
Options on Assets Paying Dividends . . . . . . . . . . . . . . . . . . 54
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Exercises and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
X: Random Interest Rates: The Futures Distribution . . . . 66
XI: A Model of the Short Interest Rate: Ho-Lee . . . . . . . 69
The Price of a Zero Coupon Bond . . . . . . . . . . . . . . . . . . . 70
Forward and Futures on a Zero Coupon Bond . . . . . . . . . . . . 71
The Forward Distribution . . . . . . . . . . . . . . . . . . . . . . . . 71
Pricing a European Option on a Zero Coupon Bond . . . . . . . . 72
XII: Ho-Lee’s Binomial Interest Rate Model . . . . . . . . . . 73
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Exercises and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
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