金融数学入门(英文)
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– There is only one interest rate: All investors can borrow and lend at this (riskless) rate.
– The interest rate is constant over time.
– The same rate applies for all maturities.
• Financial Mathematics has been established as a separate academic discipline only since the late eighties, with a number of dedicated journals.
3
Structure of this talk
What is Financial Mathematics?
1
Introduction
• Financial Mathematics is a collection of mathematical techniques that find applications in finance, e.g. – Asset pricing: derivative securities. – Hedging and risk management – Portfolio optimization – Structured products
6
• Let r denote the continuously compounded interest rate, so that one unit of currency deposited in a (riskless) bank account grows to erT units in time T .
• Thus borrowing isn’t free: the borrower pays a premium to induce the lender to part with his/her money. This premium is the interest.
• We shall make the simplifying assumptions that
• There are two main approaches: – Partial Differential Equations – Probability and Stochastic Processes
2
Short History of Financial Mathematics
• 1900: Bachelier uses Brownian motion as underlying process to derive option prices.
• Modelling stock price behaviour • Naive stochastic calculus • PDE approach to finance • Martingale approach to finance • Numerical methods来自百度文库• Current Research
• Preliminary notions: Time value of money, financial securities, options.
• Arbitrage and risk–neutral valuation via a one–period, two–state toy model.
• Decision makers must therefore – be able to compare the values of cashflows at different dates – take a probabilistic view
5
Discounting
• The time value of money: R1.00 in the hand today is worth more than the expectation of receiving R1.00 at some future date.
• 1973: Black and Scholes publish their PDE-based option pricing formula.
• 1980: Harrison and Kreps introduce the martingale approach into mathematical finance.
• Thus an amount X at time T is the same as Xe−rT now.
• Discounting allows us to compare amounts of money at different times.
7
Returns
• The return on an investment S is defined by
R = ln ST S0
i.e. ST = S0eRT
The random variable R is essentially the
“interest” obtained on the investment,
and may be negative.
4
Preliminary Notions
Discounting and Financial Instruments
• Finance may be defined as the study of how people allocate scarce resources over time.
• The outcomes of financial decisions (costs and benefits) are – spread over time – not generally known with certainty ahead of time, i.e. subject to an element of risk
– The interest rate is constant over time.
– The same rate applies for all maturities.
• Financial Mathematics has been established as a separate academic discipline only since the late eighties, with a number of dedicated journals.
3
Structure of this talk
What is Financial Mathematics?
1
Introduction
• Financial Mathematics is a collection of mathematical techniques that find applications in finance, e.g. – Asset pricing: derivative securities. – Hedging and risk management – Portfolio optimization – Structured products
6
• Let r denote the continuously compounded interest rate, so that one unit of currency deposited in a (riskless) bank account grows to erT units in time T .
• Thus borrowing isn’t free: the borrower pays a premium to induce the lender to part with his/her money. This premium is the interest.
• We shall make the simplifying assumptions that
• There are two main approaches: – Partial Differential Equations – Probability and Stochastic Processes
2
Short History of Financial Mathematics
• 1900: Bachelier uses Brownian motion as underlying process to derive option prices.
• Modelling stock price behaviour • Naive stochastic calculus • PDE approach to finance • Martingale approach to finance • Numerical methods来自百度文库• Current Research
• Preliminary notions: Time value of money, financial securities, options.
• Arbitrage and risk–neutral valuation via a one–period, two–state toy model.
• Decision makers must therefore – be able to compare the values of cashflows at different dates – take a probabilistic view
5
Discounting
• The time value of money: R1.00 in the hand today is worth more than the expectation of receiving R1.00 at some future date.
• 1973: Black and Scholes publish their PDE-based option pricing formula.
• 1980: Harrison and Kreps introduce the martingale approach into mathematical finance.
• Thus an amount X at time T is the same as Xe−rT now.
• Discounting allows us to compare amounts of money at different times.
7
Returns
• The return on an investment S is defined by
R = ln ST S0
i.e. ST = S0eRT
The random variable R is essentially the
“interest” obtained on the investment,
and may be negative.
4
Preliminary Notions
Discounting and Financial Instruments
• Finance may be defined as the study of how people allocate scarce resources over time.
• The outcomes of financial decisions (costs and benefits) are – spread over time – not generally known with certainty ahead of time, i.e. subject to an element of risk