Multiple Periods

Options and Futures

Lorenzo Naranjo

Spring 2024

Binomial Trees with Multiple Periods

In the previous examples, going first up and then down is the same as going first down and then up.
A recombinant tree is obtained whenever $u$ and $d$ are kept constant in each node of the tree.
- Note that in a recombinant tree $u$ need not be equal to $d$.
When this happens we say that the tree recombines.
- Recombinant trees are very useful in modeling the stochastic behavior of financial assets because the number of nodes increases linearly with the number of periods, i.e after $n$ periods there are $n + 1$ possible nodes.
- If the tree does not recombine then the number of nodes increases exponentially, i.e. after $n$ periods there are $2^{n + 1}$ possible nodes.

Is it the same for an asset to go up by 80% and then down by 30%, compared to first go down by 30% and then go up by 80%?
Consider an asset whose current price is $100.

The tree recombines because \[ S_{ud} = 100 \times 1.80 \times 0.70 = 100 \times 0.70 \times 1.80 = S_{du} \]

We now extend the economy to two periods
The spot rate is given by $S$
Each period the spot rate goes up by $u$ or goes down by $d$ with risk-neutral probabilities $p$ and $1 - p$, respectively
Therefore:
- $S_{u} = S \times u$ and $S_{d} = S \times d$
- $S_{uu} = S_{u} \times u$, $S_{ud} = S_{u} \times d = S_{d} \times u = S_{du}$ and $S_{dd} = S_{d} \times d$
A call option expiring at $T$ and strike $K$ trades at $C$
The time-step is then $\Delta T = T / 2$

The call price at expiration is the intrinsic value of the option: \[ C_{i} = \max(S_{i} - K, 0) \] where $i$ represents all the nodes at expiration.
In all other nodes of the tree it must be the case that: \[ C_{i} = \left( p C_{iu} + (1 - p) C_{id} \right) e^{-r \Delta t} \]
We can work backwards to find the price of the option today.

Let us price a European call option written on a non-dividend paying stock using a two-step binomial model.
The current stock price is $100, and it can go up or down by 5% each period for two periods.
Each period represents 3-months, i.e. $\Delta t = 0.25$.
The risk-free rate is 6% per year (continuously compounded).
Compute the price of a European call option with maturity 6 months and strike $100.

The risk-neutral probability of an up-move is then: \[ p = \frac{e^{r \Delta t} - d}{u - d} = \frac{e^{0.06 \times 0.25} - 0.95}{1.05 - 0.95} = 0.6511 \]
The risk-neutral probability of a down-move is just $1 - p = 0.3489$

We then compute the price of the call in 3-months if the stock price moves up: \[ C_{u} = \left( 10.25 \times p + 0 \times (1 - p) \right) e^{-0.06 \times 0.25} = \$6.57 \]
Next, we compute the price of the call in 3-months if the stock price moves down: \[ C_{d} = \left( 0 \times p + 0 \times (1 - p) \right) e^{-0.06 \times 0.25} = \$0 \]
Finally, we compute the price of the call: \[ C = \left( 6.57 \times p + 0 \times (1 - p) \right) e^{-0.06 \times 0.25} = \$4.22 \]

We can use the risk-neutral probabilities to price a European put with the same characteristics. \[ \begin{align*} P_{u} & = \left( 0 \times p + 0.25 \times (1 - p) \right) e^{-0.06 \times 0.25} = \$0.08 \\ P_{d} & = \left( 0.25 \times p + 9.75 \times (1 - p) \right) e^{-0.06 \times 0.25} = \$3.51 \\ P & = \left( 0.08 \times p + 3.51 \times (1 - p) \right) e^{-0.06 \times 0.25} = \$1.26 \end{align*} \]

It is possible to relate the up and down movements to the risk-neutral volatility observed in the market.
It can be shown that over an interval $\Delta t$, the choice $u = e^{\sigma \sqrt{\Delta t}}$ and $d = 1 / u$ produce a binomial model consistent with the Black-Scholes model of a Geometric Brownian Motion (GBM).
Note that in this case $u \times d = 1$, so the horizontal center of the tree stays constant (every other period).
The risk-neutral drift, however, is incorporated into the risk-neutral probabilities.

In this example we price a 6-month European call option with strike price $135 written on a non-dividend paying stock that currently trades at $132 and whose volatility of stock returns is 35% per year.
The interest rate of 3% per year with continuous compounding.
We will use a 5-period binomial tree.
Therefore, we have that
- $T = 6/12 = 0.5$
- $\Delta t = T / 5 = 0.1$
- $u = e^{0.35 \sqrt{0.1}} = 1.1170$
- $d = 1/1.1170 = 0.8952$

It’s convenient and efficient in this case to write the tree as a lower triangle.

The risk-neutral probability of an up-move is given by: \[ \begin{align*} p = \frac{e^{0.03 \times 0.1} - 0.8952}{1.1170 - 0.8952} = 0.4859 \end{align*} \]
We can now price the call as follows.