Pricing and Hedging an American Call in a Dividend-Paying Tree
Summary
The document presents a finite-state price tree for an underlying asset, with zero interest and a cash dividend paid between observation dates. It sketches how to find risk-neutral transition probabilities by matching each node’s current value to the probability-weighted value of later stock prices, accounting for the dividend. It then describes valuing an American call by working backward and comparing immediate exercise value with the expected value of continuing.
The question asks how to construct a hedge for the option portfolio, including whether this means choosing stock holdings at each node. It does not provide a hedge solution, and the displayed calculations are part of the question rather than verified results. As presented, the exercise can illustrate backward induction and node-by-node hedging in a simple model, but the tree assumptions and dividend treatment should be checked before relying on its probabilities or option values.
Key ideas
- Risk-neutral transition probabilities can be derived by matching node prices to discounted expected future values, with dividends included.
- An American option’s value is found by comparing exercise and continuation values at each decision node.
- A hedge in a finite tree can involve adjusting the underlying position as the path unfolds.
- The document leaves the requested portfolio hedge unresolved and does not verify the proposed calculations.
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Full text
# Constructing a hedging strategy for an American option
# Constructing a hedging strategy for an American option
Question:
Consider the following model, where $r=0$, and a dividend of 1 unit of currency is paid at time 1.5. $$ \begin{array}{|c|c|c|c|} \hline & S(0,\omega) & S(1,\omega)^* & S(2,\omega)^* \\ \hline \omega_1 & 6& 9& 11\\ \hline \omega_2 & 6& 9& 7\\ \hline \omega_3 & 6& 4& 7\\ \hline \omega_4 & 6& 4& 1\\ \hline \end{array} $$ a) Calculate the risk-neutral probabilities at each node of the information tree.
b) Calculate the value at each node of an American call option with exercise price $K=5$
c) Construct a hedging strategy for the portfolio
My solution:
a) Risk neutral probabilities give expected value equal to the value. Make sure to use dividend in expected value.
$6=9\times p + 4 \times (1-p) \implies P(S(1)=9) = \frac{2}{5},\quad P(S(1)=4) = \frac{3}{5}$ $9=12\times p + 8 \times (1-p) \implies P(S(2)=11|S(1)=9) = \frac{1}{4}, \quad P(S(2)=7|S(1)=9) = \frac{3}{4}$ $4=8\times p + 2 \times (1-p) \implies P(S(2)=7|S(1)=4) = \frac{1}{3},\quad P(S(2)=1|S(1)=4) = \frac{2}{3}$
b) Work backwards from end, working out the max of value of exercising at that point or expected value of continuing. $$ \begin{array}{|c|c|c|c|} \hline & V_{amer}(0,\omega) & V_{amer}(1,\omega)^* & V_{amer}(2,\omega)^* \\ \hline \omega_1 & 2& 4 \text{ (stopping point)}& 6\\ \hline \omega_2 & 2& 4 \text{ (stopping point)}& 2\\ \hline \omega_3 & 2& \frac{2}{3}& 2\\ \hline \omega_4 & 2& \frac{2}{3}& 0\\ \hline \end{array} $$
c) ???
My question:
What is part C asking? What is the portfolio, is it the stock, is it the option, is it something else? Aren't there different ways to hedge something? Does it mean working out how many units of stock at each node to short to eliminate the risk at each step?
Also, are my answers correct for the other two parts? Is my notation correct?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.