Replicating Strategies

In this lesson, we looked at replicating strategies in the multi-period binomial model.

We let S(t) denote the stock price at time t.
We let B(t) denote the value of the cash account that is invested in R at time t.
We let x(t) denote the number of shares held between times t-1 and t for times t=1...n.
We let y(t) denote the units of cash held between times t-1 and t for times t=1...n.
We let theta(t) = (x(t), y(t)) be the portfolio held at time t.
Theta(t) is known as a trading strategy.

The value associated with the trading strategy, V(t), is defined as:

The trading strategy theta(t) is a self-financing trading strategy.
A self-financing trading strategy is a strategy in which changes in V(t) are due entirely to gains or losses in trading, not due to deposits or withdrawals of funds in the stock account or the cash account.  This effectively creates a closed system.
Since the trading strategy is self-financed, then the change of V(t) over time periods is due to changes in stock price or cash account changes:

We can remember that when pricing options for the multi-period binomial model, we found the prices in 1-period starting from times t=3 and working backwards.  We will use this same technique to find our replicating strategy amounts x(t) and y(t).  However, in this case instead of equating x and y to the possible stock prices, we equated x and y to the possible option prices.

For example in the following 1-period binomial model:

we would use the following equations if this was a 1-period binomial model:
107x + 1.01y    = 107 and
93.46x + 1.01y = 93.46

However, if this was a multi-period binomial model with the final values at t=n on the right and working backwards, we would use the following equations:
107x + 1.01y    = 10.23 and
93.46x + 1.01y = 2.13

then solve for x and y as simultaneous equations.


The following is the full multi-period binomial tree for the European call option we have been using in previous posts.