Why an American Put Can Be Worth More Than Its Continuation Value

# Binomial tree prices the American put


# Binomial tree prices the American put












When we use the binomial tree to price the American put, we should compare the discounted value from last nodes and the intrinsic value at each node.

But I confuse that, discounted value from last nodes is the value of European put at this node, and the value of European put is always greater than its intrinsic value, how does it occur the intrinsic value is greater than discounted value?

For the statement `the value of European put is always greater than its intrinsic value,` I get from the book `Problems and Solutions in Mathematical Finance Equity Derivatives. Volume 2` `page 74`

## Answer by Quantuple (score 2, accepted)

https://quant.stackexchange.com/a/36085

Assume deterministic interest rates and no dividends to keep notations uncluttered.

Because the discounted value of all self-financing portfolio is a martingale under $\Bbb{Q}$ one can express the European put price as a risk-neutral expectation as follows: $$ P(S_0;K,T)=\frac{1}{B_T} \Bbb{E}^\Bbb{Q}_0\left[\text{max}(K−S_T,0)\right] \tag{1}$$ Since $f : x \to \max(0, x)$ is a convex function, one can apply Jensen's inequality to the RHS of $(1)$. Further using the fact that $S_t/B_t$ is also a $\Bbb{Q}$ martingale under our working assumptions, one gets: \begin{align} P(S_0;K,T) &\geq \text{max}\left( \Bbb{E}^\Bbb{Q}_0\left[\frac{K}{B_T} −\frac{S_T}{B_T} \right], 0 \right) \\ &\geq {\color{green}{\max\left( \frac{K}{B_T} - S_0, 0 \right)}} \end{align} Hence we have an equality relating the put price (in blue below) to the RHS above (in green below), which is not the intrinsic value (in orange below). This is illustrated taking the example given in @Chris Taylor's comment

## Answer by Mark Joshi (score 2)

https://quant.stackexchange.com/a/36081

the result is simply not true. If it were true American puts would not be early exercised and they would be worth the same as Europeans.

The proof is flawed. The set -up of $\pi_t$ does not make sense. We cannot take $$ \max(S_t-K,0) $$ as part of a self-financing portfolio. We could take bonds worth that but they wouldn't be worth $$ \max(S_T-K,0) $$ at the end.