Mathematics for Finance, Capinski
Textbook serving as an introduction to “financial engineering”, the use of mathematical techniques to solve financial problems.
Authors: Marek Capinski, Tomasz Zastawniak
Introduction: A Simple Market Model
- Set of assumptions:
- Future stock prices are a random variable
- Future prices of a risk-free security is a known number
- (a security is any tradable financial asset, e.g. stocks or bonds)
- All stock and bond prices are strictly positive
- An investor may hold $x$ stock shares and $y$ bonds, where $x, y \in \mathbb R$.
- Being able to hold fractions of either is referred to as divisibility
- Being able to hold arbitrary quantities (including negative) is referred to as liquidity. This means any asset can be bought or sold on demand at the market price in arbitrary quantities.
- The wealth of an investor must be non-negative at all times, a portfolio satisfying this condition is called admissible
- The future price of a share of stock is a random variable taking only finitely many values
- No-arbitrage principle: There is no admissible portfolio with initial value $V(0) = 0$ such that $V(1) > 0$ with non-zero probability. In other words, it’s impossible to construct a portfolio which starts from nothing and ends up with anything.
- If an investor holds a positive amount of a particular security, this is a “long position”
- If an investor holds a negative amount of a particular security, this is a “short position”
- A short position in a stock can be realised by short selling: an investor borrows the stock, sells it, and uses the proceeds to make another investment. If the original owner of the stock wants to sell it, the investor has to re-purchase the stock (hopefully at a lower price for what they sold it for) and return it to the owner
- More on the no-arbitrage principle: If a portfolio did exist that violated the no-arbitrage principle, then an arbitrage opportunity is available
- These are opportunities are pursued by investors called arbitrageurs. Their activities make the market effectively free of arbitrage opportunities
- The expected return of a portfolio is just $\mathbb E[\text{return}]$
- The risk is the standard deviation of the return
- Forward contracts: A forward contract is an agreement to buy or sell a risky asset at a specified future time, called the delivery date. The investor agreeing to buy the asset is said to be entering into a “long forward position”, and the seller is said to enter into a “short forward position”
- Call options: A call option is a contract giving the contract holder the right, but not the obligation, to purchase a share of stock at the strike price at a specified future time
- Investors will have to pay to purchase a call option, otherwise a risk-free profit could be made (with non-zero probability)
- You can price call options in a binomial tree model by working backwards from the no-arbitrage assumption. This is done by finding an equivalent portfolio in stocks and bonds of the option (“replicating” the option), where equivalent means that the value of the portfolio is the same as the value of the option at the specified expiration date, and then calculating the initial value of this portfolio
- Put options: This is the other side of the contract, giving the contract holder the right, but not the obligation, to sell a share of stock at a given price and expiration date
- Options can be used to reduce risk
Risk-Free Assets
- The initial deposit in a bank account is called the principal $P$
- Simple interest is another name for non-compound interest
- Given the value of investment at time $t$, $V(t)$, the value $V(0)$ is called the present/discounted value and can be calculated from the interest rate
- A perpetuity is a sequence of payments of a fixed amount to made at equal time intervals and continuing indefinitely into the future
- Perpetuities can have finite value because of discounting
- An annuity is a sequence of finitely many payments of a fixed amounts at equal time intervals
- Can generalise to compounding and continuous compounding
- Two compounding methods are equivalent if the corresponding growth factors over a period of a year are the same
- Any compounding method with interest rate $r$ can be converted to an equivalent annual compounding with a modified growth factor $r _ e$
- The money market consists of risk-free securities, such as bonds
- Bonds are financial securities promising the holder a sequence of guaranteed future payments
- Zero-coupon bonds involves just a single payment of the face value at the maturity date
- Bonds are freely traded and their prices are determined by market forces, this implies an interest rate
- Coupon bonds promise a sequence of payments
- Investments in the money market can be realised by a financial intermediary who buys and sells bonds on behalf of its customers (to reduce transaction costs).
- Under the assumption that the interest rate is constant, the value of a money market account does not depend on the way the money market account is run