Introduction

• Logistics:

  • Homework

  • Midterm (Take-home)

  • Final (Take-home)

• No textbook as such

• Math tools:

  • Calculus: Solving First Order Linear Ordinary Differential Equations

  • Linear Algebra

• $X(t)$ is a function that changes with time

• $F = F(X, t)$ is a function with two parameters (Our derivative)

• $dF = \frac{\partial{F}}{\partial{X}}dX + \frac{\partial{F}}{\partial{t}}dt$ (Chain Rule)

• $\frac{dX}{dt}dt$

• Uncertainty is inherited by F from X. This means that if X is random, then F is non-deterministic.

• This is all stochastic calculus

• First-order linear ODE are solvable in general, regardless of form

• General form: $\frac{dX}{dT} + \alpha(t)X = \beta(t)$

• We will show in this class how derivatives are priced, as a function of the function of price of a security and time.

• General second-order linear differential equation: $\frac{d^2x}{dt^2}+\alpha(t)\frac{dX}{dt} + \beta(t)X = \gamma(t)$

• Can be solved when the coefficients are constant

• Solvable: $\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + x = 0$

• Not solvable (not constant coefficients): $\frac{d^2x}{dt^2} - tX = 0$ . This is the Airy Equation. You cannot write down an analytic solution, you can write down a power series solution.

• $dX = (\beta(t) - \alpha(t) ) dt$

• But in reality, our change in X (price) would be $dX = F(X, t) dt + R$, where R is a normal random variable sampled over an infinitely small period of time. Its mean is 0, its variance will be $dT$. Also, two different random variables R are independent. Hence, it is Markovian (not Non-Markovian) which means that it has no memory. If it had memory, the correlation between random variables would be embedded in the formula.

• The argument is that derivatives are forward-looking, and putting money based on expectation. They don't have to be theoretically accurate - it would just have to be the correct valuation of pricing the derivative.

• Returns on stocks are normally distributed, however prices are lognormal.

• Simple return over $[t_1, t_2] = \frac{S(t_1) - S(t_2}{S(t_1)}$

• Logarithmic return over $[t_1, t_2] = log \frac{S(t_1)}{S(t_2)}$

• X is the random variable, q(x) is the density function

• Then, $E[X] = \int_{-\infin}^{+\infin}{x q(x)dx)}$

• Variance $VAR[X] = E[X^2] - E[X]^2 = \int{x^2q(x)dx} - (\int{xq(x)dx})^2$

• $Y = g(x)$, then $E[Y] = \int{g(x)q(x)dx}$

• Everything depends on the expected value and the amount you are willing to pay today for the option security based on its expected return

• Kelly Criterion of risk management

• The goal is to maximize the expected return in the future, even though there is a random variable involved.

• Euler-Lagrange formula that is useful for solving optimization problems.

• Bellman equation

• "Regular" Call option payoff = $MAX(S(T) - K, 0)$, where S(T) is the price at future T and K is the strike price.

• Put option payoff = $MAX(K - S(T), 0)$

• The pricing of derivative cannot just be a function of today's price and time like $C(S, t)$. Because we do not live in a vacuum, we need to have a baseline to compare to. There are other factors that come into the model.

• "Average" Call option payoff = $MAX(\frac{S'(T_1) + S'(T_2)+S'(T_3)+\dots + S'(T_n)}{n}) - K, 0)$

• Variance Swap is also an option $MAX(VAR(log(\frac{S(T_2)}{S(T_1)} + \dots + log\frac{S(T_n)}{S(T_{n-1})}) - k, 0)$

• "Log" Call option payoff = $log(S(T))$

• Amazingly, variance swaps and log call option payoffs are very closely related.