Backward Induction for American Puts in a Binomial Tree
Summary
The document addresses why a one-period discount factor appears when valuing an American put in a Cox–Ross–Rubinstein tree. At each node before expiry, the holder compares the payoff from exercising immediately with the discounted risk-neutral expected value of continuing. The discount applies over the single step to the next layer of the tree, so the value is built backward one period at a time. Discounting each future payoff all the way to time zero is unnecessary when applying this node-by-node recursion.
The example gives a three-period tree with specified stock moves, interest rate, and strike, and lists terminal payoffs and intermediate values. It does not provide a complete numerical time-zero price. The answer also briefly describes how CRR parameters are commonly tied to the risk-free return and volatility, but the quoted variance equation and a later discount expression appear questionable or inconsistent with standard formulas. The useful point is the one-step discounting logic; the document is not a reliable, fully worked derivation of every model parameter.
Key ideas
- At each pre-expiry node, compare immediate exercise value with continuation value.
- Continuation value is the risk-neutral expected next-step option value discounted for one period.
- Backward induction applies the discount step by step through the tree.
- The document's parameter discussion contains formulas that may be misstated, so it should not be treated as a complete derivation.
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# Time 0 value of an American Put in Cox-Ross-Rubinstein model
# Time 0 value of an American Put in Cox-Ross-Rubinstein model
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 by user16651 (score 6, accepted)
https://quant.stackexchange.com/a/18717
- $S_0$: The stock price today.
- $p$: The probability of a price rise.
- $u$:The factor by which the price rises.
- $d$: 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
- $n$ designates a node prior to expiry.
- $V_n$ is the option value.
- $K$ is the strike.
- $S_n$ is the price of the underlying asset.
- $V_u$ is the option value from node upper node at $n+1$.
- $V_d$ 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 by Mark Joshi (score 0)
https://quant.stackexchange.com/a/17272
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.