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