OPT-5 Binomial Trees

Major Approaches to Derivatives Valuation

Price is the PV of expected risk-neutral future value

Trees or Lattices
Monte Carlo Simulation

Implementation of PDEs:

Finite Difference methods

One Step Tree

Consider a stock with price and an option with price . At time , the price can either move up to or down to where and . On the upside, the option is worth and on downside, the option is worth .

We construct a portfolio of long shares and short 1 call

The two payoffs must be equal for the portfolio to be riskless. Therefore,

The present value of the portfolio is

The cost of setting up the portfolio is . Therefore,

Substituting for , we get

where

We can consider as the (risk-neutral) probability of an upward movement and as the probability of a downward movement.

The expected stock price, will be:

To value an option, we can assume the portfolio earns the risk-free rate and discount at the risk-free rate.

For example,

then

Given above probability, we can compute the value of the option

Multiple Steps Trees

Example: European Call

Example: A European call option. u=1.1, d=0.9, step=3 month, K=$21, rf=0.12

Value at node B is

thus the resuling value at node A is

Example: American Put

Example: An American put option. u=1.2, d=0.8, step=1 year, K=$52, rf=0.05

Delta Hedging

Delta is the ratio of change in the price of the stock option to the change in the price of the underlying stock. Delta of a call is positive and delta of a put is negative.

In Binomial Trees, delta is defined as above

The delta of a portfolio changes after each time period. Therefore, to maintain a riskless portfolio, the stock holding (i.e. delta) has to be adjusted after each period.

For example,

Girsanov’s Theorem

(Girsanov’s Theorem): Volatility is the same in the real world and the risk-neutral world.

In above examples, we assume as some values. One way of matching the volatility is to set

where is the volatility and is the length of the time step.

We can measure volatility in the real word and use it to build a tree for the asset in the risk-neutral world.

Binomial Trees of Options on Other Assets

The same concept of pricing options using binomial trees with risk-free valuation can be extended to other assets. The generalized form of (up probability) is

where,

for a non-dividend paying stock

for a stock paying dividend yield

for a currency where is the foreign risk-free rate

for a futures contract

5_Example Binomial Model.xlsx

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