# 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.

• It appears fine to me. For (c), I do not know what does it mean "running maximum". – Gordon Apr 20 '16 at 13:41
• @Gordon The running maximum at time $t$ is the maximum of the price until or at time $t$, That is how my professor defined it to be – Wolfy Apr 20 '16 at 16:28
• @Gordon did you see my last comment? – Wolfy Apr 21 '16 at 0:37
• I don't yet have any good idea for (c). – Gordon Apr 21 '16 at 2:07
• @Gordon Ok, could you possibly look at my recent post I am not sure how to replicate American options – Wolfy Apr 21 '16 at 19:40