Backward Induction for American Options in a Two-Period Binomial Model
Summary
The document presents a two-period binomial option-pricing exercise with a specified initial asset value, up and down factors, interest rate, and strike. It asks for prices of an American put, an American call, and an American claim whose payoff is the maximum asset price reached along the path. The attempted put solution computes risk-neutral probabilities and terminal asset values, then compares immediate exercise with continuation at intermediate nodes. This illustrates the backward-induction framework used for American options and highlights that a path-dependent payoff must track the running maximum.
The put calculation is incomplete: its time-zero continuation calculation uses one-step terminal payoffs rather than the option values at the next-period nodes, so it does not implement the full two-period recursion. The document provides no completed call or running-maximum valuation, and the stated exercise is therefore not a validated worked solution. It is useful as a setup for practicing node-by-node valuation, but the arithmetic and recurrence need correction before relying on a price.
Key ideas
- A binomial model prices American options by comparing exercise value with discounted continuation value at each node.
- Risk-neutral up and down probabilities are determined from the asset moves and risk-free return.
- A running-maximum payoff depends on the asset path, so the pricing state must retain the maximum reached so far.
- The attempted put calculation does not complete the two-period backward recursion and should not be treated as a verified solution.
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Full text
# Two-period binomial model for American option
# Two-period binomial model for American option
> Consider a two-period binomial model for a risk asset with each period equal to a year and take $S_0 = 1$, $u = 1.5$, and $l = 0.6$. The interest rate for both periods is $R = .1$. a.) Price an American put option with $K = .8$ b.) Price an American call option with $K = .8$ c.) Price an American option with path dependent payoff which pays the running maximum of the path.
Note: The running maximum at time $t$ is the maximum of the price until or at time $t$
To clarify when it says "pays the running maximum", it means that the payoff is the running maximum, i.e. max(S_0,S_1,S_2).
Attempted solution a.) We have $S_0 = 1$, $S_0u = 1.5$, $S_0l = 0.6$, $S_0u^2 = 2.25$, $S_0ul = .9$, and $S_0l^2 = .36$. The risk neutral probabilities are $$\hat{\pi}_u = \frac{1+R-l}{u-l} = .5556 \ \ \ \ \ \ \ \hat{\pi}_l = \frac{u-R-1}{u-l} = .4444$$ Now we need to calculate the continuation values at nodes $t = 0, t = 1$ price-down, and $t = 1$ price up which we denote $C_0,C_{1,1},C_{1,2}$ respectively. We will then compare the continuation values and exercise value at each node in a backward manner. At time $t = 1$, the continuation value is $$C_{1,2} = \frac{1}{1+R}\hat{\mathbb{E}}\left[(K-S_T)_{+}|S_1 = 1.5\right] = \frac{1}{1.1}\left(.5556(.8 - 2.25)_{+} + .4444(.8-.9)_{+}\right) = 0$$ $$C_{1,1} = \frac{1}{1+R}\hat{\mathbb{E}}\left[(K-S_T)_{+}|S_1 = 0.6\right] = \frac{1}{1.1}\left(.5556(.8 - .9)_{+} + .4444(.8-..36)_{+}\right) = 0.1778$$ $$C_{0} = \frac{1}{1+R}\hat{\mathbb{E}}\left[(K-S_T)_{+}|S_0 = 1\right] = \frac{1}{1.1}\left(.5556(.8 - 1.5)_{+} + .4444(.8-.6)_{+}\right) = 0.0808$$ Therefore, the price of the option at these nodes are $$V_{1,2} = \max\{\frac{1}{1.1}\left(.5556(.8 - 2.25)_{+} + .4444(.8 - .9)\right),(.8-1.5)_{+}\} = 0$$ $$V_{1,1} = \max\{\frac{1}{1.1}\left(.5556(.8 - .9)_{+} + .4444(.8 - .36)\right),(.8-.6)_{+}\} = 0.2$$ $$V_{0} = \max\{\frac{1}{1.1}\left(.5556(.8 - 1.5)_{+} + .4444(.8 - .6)\right),(.8-1)_{+}\} = 0.0808$$ At time $t = 0$, we need to see if it is optimal to exercise or optimal to continue. Since the the exercise value $E := (K-1)_{+}$ clearly since $K = .8$ in this problem it is optimal to continue.
If this solution is correct then part b.) will be pretty straight forward. Although I am not totally sure I am right here I may have made some mistakes, and I also have no idea how to do part c.). Any suggestions is greatly appreciated.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.