Time 0 value of an American Put in Cox-Ross-Rubinstein model

Question

This is a question from a problem sheet which I have handed in and have solutions for. The only examples of this in class I have seen are examples where the interest rate is 0.

"Consider a Cox-Ross-Rubinstein model with $T = 3$ periods, $S_0 = 100$, $u = 1.6$ and $d = 0.6$. The interest rate is $r = 0.1$. What is the time 0 price of an American put that has exercise price $e = 90$?"

So, we have a claim, $X_t = (e - S_t)_+$.

The solutions go as follows:

At time 3 the claim has values $(0,0,32.4,68.4)$ in the different states.

At time 2 values of the claim when immediately exercising are $(0,0,54)$.

The time 2 Snell envelope gives values $(0, \frac{32.4 \times \frac{1}{2}} {1 + r},\frac{(32.4 + 68.4) \times \frac{1}{2} } {1 + r})$ in the different states.

My question is, why is there a denominator of $1 + r$ included? I'm thinking we're perhaps calculating the Snell envelope for the discounted process, but surely then the Snell envelope at time 2 would be $\max\{\frac{X_2}{(1 + r)^2},\mathbb{E}(Z_3 | \mathcal{F}_2)\}$, and $Z_3 = \frac{X_3}{(1 + r)^3}$.

Answer 1

1. $S_0$: The stock price today.
2. $p$: The probability of a price rise.
3. $u$:The factor by which the price rises.
4. $d$: The factor by which the price falls.

Three equations are required to be able to uniquely specify values for the three parameters of the binomial model. Two of these equations arise from the expectation that over a small period of time the binomial model should behave in the same way as an asset in a risk neutral world.This leads to the equation \begin{align} p\,u+(1-p)d=e^{r\,\Delta t} \end{align} which ensures that over the small period of time $\Delta t$ the expected return of the binomial model matches the expected return in a risk-neutral world, and the equation, \begin{align} p\,u^2+(1-p)d^2=\sigma^2\Delta t \end{align} which ensures that the variance matches.

Cox-Ross-Rubinstein

Cox, Ross and Rubinstein proposed the third equation \begin{align} u=\frac{1}{d} \end{align} Rearranging the above three equations to solve for parameters p, u and d leads to, \begin{align} &p=\frac{e^{r\,\Delta t}-d}{u-d}\\ &u=e^{\sigma\,\sqrt{\Delta t}}\\ &d=e^{-\sigma\,\sqrt{\Delta t}} \end{align} The unique solution for parameters p, u and d given in above Equation ensures that over a short period of time the binomial model matches the mean and variance of an asset in a risk free world, and as will be seen shortly, ensures that for a multi-step model the price of the underlying asset is symmetric around the starting price $S_0$ .

In general the time period between today and expiry of the option is sliced into many small time periods. A tree of potential future asset prices is then calculated. Each point in the tree is refer to as a node. The tree contains potential future asset prices for each time period from today through to expiry.

Discounting the Payoff for American put option.

\begin{align} V_n=max\{K-S_n\,,\, e^{-\sigma\,\sqrt{\Delta t}}\left(p\,V_u+(1-p)V_d\right)\} \end{align}

where

1. $n$ designates a node prior to expiry.
2. $V_n$ is the option value.
3. $K$ is the strike.
4. $S_n$ is the price of the underlying asset.
5. $V_u$ is the option value from node upper node at $n+1$.
6. $V_d$ is the option value from the lower node at $n+1$.

This example shows how to price an American put option with an exercise price of $\$ 90$ that matures in 1 year. The current asset price is $\$100$, the risk-free interest rate is $ 10\%$, and the volatility is $92\%$. There is no dividend payment.

[Price, Option]=binprice(Price, Strike, Rate, Time, Increment, Volatility, Flag)

where

Flag:Specifies whether the option is a call (Flag = 1) or a put (Flag = 0).

Answer 2

Generally when doing trees, people discount one step at a time. So this is the value discounted from the end to that node in the tree.