American Option Bounds with Continuous Dividend Yield

Summary

The document gives simple bounds for American calls and puts on an underlying with continuous dividend yield. Because an American option can be exercised at maturity, its value is at least that of the corresponding European option, so the European lower bounds carry over. For the upper bounds, monotonicity in strike bounds a call by the value at zero strike, while under an exponential price model a put is bounded by its value when the underlying price is zero.

The resulting upper bounds are the spot price for the call and the strike for the put; the lower bounds use discounted spot and strike with dividend yield and risk-free rate. The argument assumes positive spot, strike, and maturity inputs, and the put upper-bound reasoning explicitly relies on an exponential model. These are broad valuation bounds, not estimates of option prices or guidance on early-exercise decisions.

Key ideas

• An American option is worth at least as much as its European counterpart, so European lower bounds also apply.
• The American call value is bounded above by the current spot price using monotonicity in strike.
• Under an exponential model, the American put value is bounded above by the strike.
• The bounds provide valuation limits and do not determine whether early exercise is optimal.

Tags

# American Option Bounds with Dividend Yield


# American Option Bounds with Dividend Yield












What are the upper and lower bound of American call and put options for an underlying with continuous dividend yield?

For European options, the bounds are known as

\begin{align*} [S_te^{-d\tau}-Ke^{-r\tau}]^+<C_t<S_te^{-d\tau}\\ [Ke^{-r\tau}-S_te^{-d\tau}]^+<P_t<Ke^{-r\tau} \end{align*}

However, I could not find a clear result regarding American options with dividend yield.

## Answer by M. Jeunesse (score 2)

https://quant.stackexchange.com/a/29901

For the lower bound, since american call option (resp. put) is bigger than european call option (resp. put). So your lower bounds for european options hold also for american options.

For the upper bound, there is a slight difference. (sorry for my too quick comment).

Here $S_0,T$ and $K$ are positive real numbers.

Let $C^A_T$ be the american call option of maturity $T$:

$$K<K' \Rightarrow (x-K)^+\geq (x-K')^+ \Rightarrow C^A_T(S_0,K)\geq C^A_T(S_0,K') $$

so $C^A_T(S_0,K)\leq C^A_T(S_0,0)=S_0$

Let $P^A_T$ be the american put option of maturity $T$:

Assuming you are in an exponential model (like BS) $x\to P^A_T(x,K)$ is non-increasing. Thus, $P^A_T(S_0,K)\leq P^A_T(0,K)=K$

So you get:

$$(S_0e^{-dT}-Ke^{-rT})^+\leq C^A_T(S_0,K) \leq C^A_T(S_0,K) \leq S_0$$ and $$(Ke^{-rT}-S_0e^{-dT})^+ \leq P^A_T(S_0,K)\leq P^A_T(S_0,K) \leq K$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.