Binomial Trees for Option Pricing

Options and Binomial Trees

The session covers using binomial trees to price options, which are derivatives. The binomial tree method is similar to probability trees.

Binomial Trees

Binomial trees represent possible stock price paths over time.
The current stock price is known, but future prices are uncertain.
The price can either go up or down.
Binomial trees help visualize potential price variations.
They consider both upward and downward price movements.

Key Assumptions of Binomial Trees

Random Walk: Future stock prices follow a random walk, with no guaranteed direction. We can only estimate probabilities.
No Dividends: Stocks do not pay dividends during the considered period.
Up and Down Moves: Prices either go up by a factor of $U$ or down by a factor of $D$ , relative to the initial stock price $S$ .
- If the price goes up: $S \times U$
- If the price goes down: $S \times D$
No Arbitrage: It's not possible to start with zero money and end up with a profit.
Solvency: A portfolio's value must always be non-negative, as stock prices cannot be negative.

Options as Derivative Contracts

A derivative is a financial instrument whose value is derived from an underlying asset.
The derivative is a document that represents an underlying asset which in this case we assume is a stock.
The value of the derivative is linked to the price of the underlying stock.
The derivative's price is derived from the underlying asset's price.
An option is a type of derivative.

Call Option

Gives the buyer the right, but not the obligation, to buy an underlying asset (e.g., stock $Z$ ) at a predetermined price (strike price, $X$ ) at a specified time in the future (maturity).
Two parties are involved: the buyer (Mister A) and the seller (Mister B).
Mister A has the right to buy the stock at the strike price.
The strike price ( $X$ ) is set at time $t_0$ .
The buying occurs at a later time, $t_6$ (e.g., in six months).
Mister B has the potential obligation to sell the stock at the strike price.
Example: Strike price ( $X$ ) = $100. If the stock price at maturity ( $ST$ ) is $150, Mister A will exercise the option and buy at $100. If $ST$ is $50, Mister A will not exercise the option and buy directly in the market for $50.
The call option protects against the stock price increasing. It allows you to buy at a set price regardless of how high the market value increases.

Put Option

Gives the buyer the right, but not the obligation, to sell an underlying asset at a predetermined price at a specified time in the future.
Mister A has the right to sell the stock at the strike price.
Mister B has the potential obligation to buy the stock at the strike price.
Example: If $ST$ is $150, Mister A will not exercise the option and sell directly in the market for $150. If $ST$ is $50, Mister A will exercise the option and sell at $100.
The put option is most beneficial when the stock price decreases because gives you the right to sell at a higher price.

European vs. American Options

European Option: Can only be exercised at maturity.
American Option: Can be exercised at any time up to maturity, giving it more value because it gives you more flexibility on when to use it.

Payoff Formulas

Call Option Payoff: $max(0, S_T - X)$
$S_T$ is the stock price at maturity.
$X$ is the strike price.
Put Option Payoff: $max(0, X - S_T)$

Profit Diagrams

Call option profits increase when the stock price at maturity is above the strike price ( $K$ or $X$ ).
Put option profits increase when the stock price is below the strike price.

Binomial Tree Illustration

Example: Stock Axis is priced at $20. It can go up or down by 10% at each time step.
Drawing the tree:
- Start with the initial stock price ( $S_0 = 20$ ) at time zero.
- At time one, the price can go up ( $Su$ ) or down ( $Sd$ ).
- $U = 1.1$ (10% increase).
- $D = 0.9$ (10% decrease).
- $Su = S0 \times U = 20 \times 1.1 = 22$
- $Sd = S0 \times D = 20 \times 0.9 = 18$
Two-time step tree:
- From each node in the one-time step tree, add two more branches (up and down).
- $S{uu} = S0 \times U \times U$
- $S{ud} = S0 \times U \times D$
- $S{dd} = S0 \times D \times D$
- Note: $S{ud} = S{du}$ , so they are merged into a single node.
Incrementing to a three time step tree just means to continue the branching from the two time step tree. The more the branches the further out you can predict the price in the future.

Option Valuation Methods

No-Arbitrage Valuation Method

A theoretical approach to determine the price of a call option.
Steps:
1. Draw the binomial tree and predict stock prices.
2. Find the option payoff at maturity.
  - Formula:
    - Call option max(0, ST - X)
    - Put option max(0, X - ST)
3. Set up a riskless portfolio:
  - Go long (buy) a certain number of shares (Delta shares).
  - Go short (sell) one call option.
4. Calculate the portfolio value at each node:
  - Value of shares, delta times stock price.
  - Subtract the value of the call option.
5. Equate portfolio values when the stock price goes up or down. Find what combination of stock to options makes the portfolio “riskless”
  - Set the equation up where the value of down equals the value of up and solve to find delta.
6. Determine delta (number of shares) to make the portfolio riskless.
7. Compute the value of the riskless portfolio.
8. Find the discounted PV of the portfolio to find its value at time zero.
9. Calculate the option price at time zero by looking at what the portfolio consisted of.
  - Long delta + short call option = know value. Now find value of the short call option.
Example:
- Stock Axis has a price of $20.
- Price can go up or down by 10%.
- Call option with strike price ( $X$ ) = $21.
- Maturity at three months.
- Find option payoff at the end of three months:
  - If the price goes up to $22, the payoff is max(0, 22 - 21) = $1
  - If the price goes down to $18, the payoff is max(0, 18 - 21) = $0.
- Formula for continuous compounding: $FV = PV\times e^{r \times t}$ . So $PV = FV \times e^{-r \times t}$

Risk-Neutral Valuation Method

Simpler, use probabilities by assuming a risk-neutral probability of an up and down move.
Formula for risk neutral probability from it going up:
- $p = \frac{e^{r \times t} - d}{u - d}$
  - r = risk-free rate.
  - t = one time step (expressed in years).
  - d = down move.
  - u = up move.
Formula for risk neutral probability from down move from up move. q = one - p
Steps:
1. Compute $p$
2. Option price at maturity must be found by the formula again.
3. We must now find F Naught, we will work backwards from the end to the beginning.
  - Expected F at maturity equals from it going up p times FU plus Fd q times.
  - Find out formula to be PV expected equal f(E) * ( e to the third power over rt
  - This requires concept of probability trees.

European Put Option

Two-year maturity.
Strike price = $52.
Current price = $50.
Each time step, stock goes up or down 20%.
Risk-free interest rate = 5%.
1. First step, find the values to the stock by considering two time steps.
2. Second find the probabilities of a risk neutral from up to down move.
  - P equals E to the r times t
  - Where T is delta T or just one time step
  - So be always careful of this step.
3. Find out payoffs of put options and these payoffs must be found max.
  - Max zero X minus $s^t$
Must work backwards to get P(Naught) or price of option.
P fu formula e to the power r to the t p f u u plus q for U to the power d. E to power R T = two branches .
Do the same to that of Fd find final f naught number p f u plus Q Fd times e to the negative power RT

American Put Option

Has the same structure and computations but must consider the time when you choose to exercise that choice that makes them most valuable.
Compute for the graph draw the tree from 50 to 6/4 and 2 and F -d equals 0 for U to do equals 4 + f -2 equals 20. We must figure out early exercise for t1 and t2.
To find the maximum is to find which is worth more, exercise early with the American or wait the f u + f -d do we consider the highest valley we will not consider FD. We will consider high value for FD.
Make FD a new FD =2 =0 because it's more valuable than doing anything else

Put-Call Parity

$C0 + X \times e^{-rt} = P0 + S_0$
$C_0$ = price of the call at time zero.
$P_0$ = price of the put at time zero.
This equation relates the call and the put, values have to be exactly equal otherwise there will be a possibility of arbitrage. Arbitrage should not happen normally, so put and call should be equivalent .

Volatility

When given the volatility (sigma) of the stock price, rather than percentages for upward or downward movements, use the following formulas:
- $U = e^{\sigma \sqrt{\Delta t}}$
- $D = e^{-\sigma \sqrt{\Delta t}} or D= \frac{1}{U}$