Derivatives

Derivatives are securities where the future payoff relies on the underlying security.
- Examples include forwards, options, futures, swaps etc.

Valuing Deriviatives

A deriviative payoff can be seen as a function of the underlying asset

\[ f(S_T)\]

An example of a derivative is the forward contract:
- \(S_T\) is the price of the underlying at maturity
- \(K\) is the forward price, aka what is paid today

\[ f(S_T) = S_T-K\]

There are a few ways to set the forward price \(K\) at a time \(t<T\)
- Expectation pricing considers the expected price of the security at \(t=T\), aka \(\mathbb{E}[S_T]\)
  - For example if we expect the returns to be log normally distributed, where \(S_T = S_0e^{X}\),\(X\sim N(\mu T,\sigma^2T)\)
  \[\mathbb{E}[e^{-rT}(S_T-K)]=0,\quad\therefore K = \mathbb{E}[S_T],\quad\therefore K=S_0e^{\mu T+0.5\sigma^2T}\]
- Arbitrage pricing is the conventional way of pricing derivatives. We consider different combination of securities at the current time with matching future cash flows. For a forward contract consider 2 portfolios at \(t=0\)
  - Portfolio A: Forward contract, \(Ke^{-rT}\) cash
  - Portfolio B: One unit of underlying \(S_0\)
- At \(t=T\) both portfolios will have a cashflow of \(S_T\), therefore they should be priced equally at \(T=0\)
  
  \[ Ke^{-rT} = S_0,\quad\therefore\quad K=S_0e^{rT} \]
For a forward with continuous dividends, we consider 2 portfolios:
- Portfolio A: Forward contract, \(Ke^{-rT}\)
- Portfolio B: \(e^{-qT}\) units of underlying \(S_0\)
  - At time \(t=T\) both portfolios will have a cashflow of \(S_T\), therefore they should be priced equally at \(T=0\)
  \[ Ke^{-rT} = S_0e^{-qT},\quad\therefore\quad K = S_0e^{(r-q)T}\]

Options

European

European options give the right, but not obligation, to exercise at maturity.
Call Payoff:

\[ f(S_T) = \max(S_T-K, 0)\]

Put Payoff:

\[ f(S_T) = \max(K-S_T, 0) \]

American

American options give the right, but not obligation, to exercise at any time leading up to maturity.
Call Payoff:

\[ f(S_U)=\begin{cases}S_U-K,\quad\text{if}\quad U\leq T\\ 0,\quad\text{if}\quad U = \infty \end{cases} \]

Put Payoff:

\[f(S_U)=\begin{cases}K-S_U,\quad\text{if}\quad U\leq T\\ 0,\quad\text{if}\quad U = \infty \end{cases}\]

Position Diagram

A position diagram is a graphical display of profit/loss of an option strategy as a function of the underlying price.

Long Option Diagram

Bounds for Option Prices

All derivatives have general upper and lower bounds for pricing.
An example of the arguments required for a European call option is as follows:
- Consider Portfolio A: A European call \(c_t\), \(Ke^{-r(T-t)}\) cash
  - If \(S_T>K\) the portfolio will be worth:
  \[ \underbrace{Ke^{-r(T-t)}*e^{r(T-t)}}_{\text{Cash Payoff}}+\underbrace{S_T-K}_{\text{Call Payoff}} = S_T \]
  - If \(S_T<K\) the portfolio will be worth:
  \[ \underbrace{Ke^{-r(T-t)}*e^{r(T-t)}}_{\text{Cash Payoff}}+\underbrace{0}_{\text{Call Payoff}} = K \]
- Therefore an option + cash portfolio produces a value at least as great as \(S_T\), therefore the lower bound would be:
\[ c_t+Ke^{-r(T-t)}\geq S_t,\quad\therefore\quad c_t\geq S_t-Ke^{-r(T-t)}\]
- For the upper bound, note that the payoff at \(T\) for a European call is:
\[ \max(S_T-K,0)\leq S_T\] - Therefore \(c_t\leq S_t\) for any \(t\leq T\) as you could buy the stock itself otherwise.
General Bounds for other options can be seen in the table:

Option Bounds

Put-Call Parity

Considering European options
We consider 2 portfolios
- Portfolio A: 1 call \(c_t\), \(Ke^{-r(T-t)}\) cash
- Portfolio B: 1 put \(p_t\), One underlying \(S_0\)
\[ A = \begin{cases}S_t-K+K=S_T,\quad\text{if}\quad S_T>K\\ K,\quad\text{if}\quad S_T\leq K\end{cases} \] \[ B = \begin{cases}S_T,\quad\text{if}\quad S_T>K\\ K-S_T+S_T = K\quad\text{if}\quad S_T\leq K\end{cases}\]
Both portfolios have the same payoff at time \(T\), therefore they should be equally priced at time \(t\)

\[ c_t + Ke^{-r(T-t)}=p_t+S_t\]

With Discrete Dividends, where \(D\) is the PV of dividends:

\[ c_t + Ke^{-r(T-t)} + D = p+S_t\]

With Continuous Dividends:

\[ c_t + Ke^{-r(T-t)} = p_t + S_te^{-q(T-t)}\]

Binomial Pricing

Setup

Assumptions

Can hold arbitrarily large amounts of bonds and cash
Securities market is arbitrage free
No trading costs
No min/max trading limits

Modelling Process

Consider the steps of length \(\delta t\) of 2 instruments
- Risky stock that pays no dividends. Price can either go up or down each period. The price of the stock is \(S(t)\)
- Risk free ZCB/Cash. Value of cash invested at \(t=0\) until \(t=t\) at the risk free rate. Risk free rate is often assumed to be continuously compounded. \(B(t) = B(0)e^{r\delta t}\), where \(B(0)=1\) typically.

Principle of No Arbitrage

Arbitrage is defined as risk-free trading profit
Arbitrage exists in capital markets if either
- Immediate profit can be made with no risk of future loss
- A non-zero probability of future profit exists with no upfront cash-flow and no risk of future loss
Any combination of securities that give the same future payments should have the same price.

Binomial Branch Model

Stock is currently priced at \(S(0)\) at node \(s_1\). For a discrete time length \(\delta t\):
- Stock can go up to \(s_3\) or down to \(s_2\)
- Bond starts off at \(B(0)\) and grows to \(B(1) = B(0)e^{r\delta t}\)
Derivative pricing is also done in node fashion, where it is a function of the price at that node. For example a call option with strike K:

\[ f_3 = \max(s_3-K,0)\] \[ f_2 = \max(s_2-K,0) \]

Graphical Representation of a branch:

Binomial Branch

Stock and Bond Strategy

We can replicate the payoff of the derivative by buying \(\phi\) of stock \(s_1\) and \(\psi\) of bond \(B(0)\)
Value of portfolio today:

\[ V(0)=\phi s_1+\psi B(0)\]

Value of portfolio after 1 step (time \(\delta t\)):

\[ V(1) = \begin{cases}\phi s_3+\psi B(0)e^{r\delta t}\\ \phi s_2+\psi B(0)e^{r\delta t}\end{cases}\]

We can form a strategy where this portfolio payoff replicates the derivative payoff at time \(\delta t\) by equating the portfolio payoff to the derivative payoff. This results in the following equations:

\[ \phi = \frac{f_3-f_2}{s_3-s_2},\quad\psi = \frac{1}{B(0)}e^{-r\delta t}(f_3-\phi s_3)\]

We can find a ‘risk neutral’ probability:

\[ q = \frac{s_1e^{r\delta t}-s_2}{s_3-s_2}\]

So that:

\[ V(0)=\phi s_1+\psi B(0) =e^{-r\delta t}(qf_3+(1-q)f_2) = \mathbb{E}_Q[e^{-r\delta t}X]\]

Binomial Tree Model

Generalization of the branch model where we combine branches to create a tree.
General form of branches:

Binomial Branch General

Derivative Pricing Using Trees

The general form for the replicating portfolio inputs at a specific node ‘now’ are:

\[ \phi_{now} = \frac{f_{up}-f_{down}}{s_{up}-s_{down}}\] \[ \psi_{now}=\frac{1}{B(now)}e^{-r\delta t}(f_{up}-\phi s_{up})\] \[ q_{now} = \frac{s_{now}e^{r\delta t}-s_{down}}{s_{up}-s_{down}}\] \[ V_{now} = f_{now}=\mathbb{E}_Q\left[\frac{B(0)}{B(T)}X\right]\] ### 2 Step Example

We have to work backwards through the tree to price the derivative at \(T=0\).

2 Step Binomial Model

Suppose we are at node 3:

\[ \phi_3 = \frac{f_7-f_6}{s_7-s_6},\quad\psi_3 = \frac{1}{B(1)}e^{-r\delta t}(f_7-\phi s_7)\] \[ v_3 = \phi_3 s_3+\psi_3 B(1)\]

Similarly at node 2:

\[ v_2 = \phi_2 s_2+\psi_2 B(1)\]

Now stepping back to \(t=0\):
- We want to form a portfolio that pays
\[ f_3 = v_3,\quad f_2 = v_2\]
Therefore we use the inputs:

\[ \phi_1 = \frac{f_3-f_2}{s_3-s_2},\quad\psi_1=\frac{1}{B(0)}e^{-r\delta t}(f_3-\phi_1s_3)\]

\[ v_1 = \phi_1 s_1+\psi_1 B(0) \]

As this payoff matches the derivative, the replicating portfolio will be able to finance the weight rebalancing at the next node.
Note that:

\[ f_3 = e^{-r\delta t}(q_3f_7 +(1-q_3)f_6),\quad f_2 = e^{-r\delta t}(q_2f_5 +(1-q_2)f_4)\]

Therefore \(f_1\):

\[ f_1 = e^{-r\delta t}(q_1f_3+(1-q_1)f_2)\]

\[ f_1 = e^{-2r\delta t}(q_1q_3f_7 + q_1(1-q_3)f_6+(1-q_1)q_2f_5+(1-q_1)(1-q_3)f_4) = \mathbb{E}_Q\left[\frac{B(0)}{B(2)}X\right] \]

Binomial Model Martingale Representation

Mathematical Definitions

Stock Price Process
- We consider a binomial tree for the possible states of the world, with associated stock price process \(S(t)\).
Filtration
- The filtration \(\mathcal{F}(t)\) can be thought of as the history of the stock price up until time t
  - At time \(0\) the filtration is node \((1)\)
  - At time \(1\), if the stock price went up the filtration is nodes \((1,3)\)
  - etc.
Probability Measure
- We have a set of probabilities \(\mathbf{P}\) for real world probabilities and \(\mathbf{Q}\) for risk neutral probabilities
Claim X
- The claim \(X\) as a function of the nodes at maturity \(T\). This can also be seen as a function of the filtration \(\mathcal{F}(t)\).
- Represents the payoff of the derivative.
Preversible Process
- A preversible process \(\phi(t)\) is a process on our tree whose value is only dependent on \(\mathcal{f}(t-1)\) (one step earlier)
Conditional Expectation Operator
- The conditional expectation operator is defined as \(\mathbb{E}_Q[.|\mathcal{F}(t)]\)
Martingales
- A process \(M(.)\) is a martingale with regares to the measure \(\mathbf{Q}\) and filtration \(\mathcal{F}\) if
\[ \mathbb{E}_Q[M(u)|\mathcal{F}(t)] = M(t),\quad\text{all }t\leq u\]
- Examples:
  - A process of constant value is a martingale under any measure
  - The process \(Z(.)\) is a measure under measure \(\mathbf{Q}\)
  \[ \mathbb{E}\left[\frac{X}{B(T)}|\mathcal{F}(t)\right]\]
  - The conditional expectation process is always by definition a martingale under \(\mathbf{Q}\)

Representation Theorem

Consider a binomial tree with probability measure \(\mathbf{Q}\), and two non-constant \(\mathbf{Q}\) martingales \(Z(.), Y(.)\)
The Binomial Martingale Representation Theorem says there exists a preversible process \(\phi(.)\) such that:

\[ Y(t) = Y(0)+\sum^t_{k=1}\phi(k)(Z(k)-Z(k-1))\]

Where

\[ Y(t) = \mathbb{E}_Q[B(T)^{-1}X|\mathcal{F}(t)],\quad Z(t)=\frac{S(t)}{B(t)}\]

Self-Financing Strategy

Consider the following portfolio rebalancing strategy:
- At time \(t\), buy portfolio consisting of \(\phi(t+1)\) of stock \(S(t)\) and \(\psi(t+1)=Y(t)-\phi(t+1)B(t)^{-1}S(t)\) units of bond
Considering a portfolio at time 0:

\[ V(0)= \phi(1)S(0)+\psi(1)B(0)\]

At time \(t\) the portfolio is worth \(B(t)Y(t)\), therefore at maturity this portfolio will be worth \(B(T)Y(T)\) where:

\[ B(T)Y(T) = B(T)\mathbb{E}_Q[B(T)^{-1}X|\mathcal{F}(T)] = \mathbb{E}_Q[X|\mathcal{F}(T)] = X\]

Therefore for no abitrage the cost at time 0 would be the discounted expected payout under the \(\mathbf{Q}\) measure:

\[ V(0) = \phi(1)S(0)+\psi (1)B(0)=B(0)Y(0)=Y(0) =\mathbb{E}_Q[B(T)^{-1}X]\]

Discrete Derivatives

Jake

06/11/2021