Pricing a Lookback Option in a Two-Period Binomial Tree
Summary
This note shows how to price a path-dependent lookback call in a two-period binomial model. A European payoff depends only on the terminal asset price, so the two paths that reach the same middle terminal node can be combined and their probabilities added. A lookback payoff instead depends on the maximum price observed along the path, so those paths may have different payoffs even when they end at the same price.
The method enumerates all four up/down paths, evaluates the maximum observed price on each, applies the payoff floor at zero, weights each payoff by its risk-neutral path probability, and discounts the expectation one period at a time (or over the full horizon). The response gives the risk-neutral up probability from the gross risk-free return and the up and down factors. The example is a small discrete model; it does not specify a numerical risk-free rate or calculate a final option value, and its result depends on the chosen tree inputs and discounting convention.
Key ideas
- A path-dependent payoff must retain the intermediate asset prices, not just the terminal price.
- A two-period binomial tree has four ordered paths, each with its own risk-neutral probability.
- For a lookback call, compute the maximum price along each path and apply the payoff floor.
- Paths with the same terminal price can be aggregated only when their payoff is also identical.
- The risk-neutral expected payoff is discounted at the risk-free rate to obtain the model price.
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Full text
# two-period binomial model, with price that is path-dependent
# two-period binomial model, with price that is path-dependent
Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.03$ and $l = 0.98$. How do you price a look-back option with payoff($\max_{t=0,1,2}S_t - 1)_{+}$? Hint: The price is path dependent and each path has a payoff.
I don't understand this question at all, any suggestions is greatly appreciated
## Answer by Quantuple (score 1, accepted)
https://quant.stackexchange.com/a/24930
For a standard European option (i.e. non path - dependent payoff):
$$V_0 = \frac {1}{1+R} E [ V (S_T) ] $$
Because, in a 2 period binomial tree, the terminal stock price $S_T$ can take 3 distinct values: $S_{uu}=S_0u^2$, $S_{ul}=S_0ul$ and $S_{ll}=S_0l^2$, you can write the expectation:
$$V_0 = \frac {1}{1+R} (q_{u}^2 V (S_{uu}) + {\color {red}{2}} q_u q_l V ( S_{ul} )+ q_l^2 V (S_{ll}))$$
With $q_u $ and $q_l $ figuring risk-neutral probabilities of an up/low move over a period:
$$q_u = \frac{(1+R) -l}{u-l}$$ $$q_l = 1-q_u $$
and the terminal option values $V (S_{uu})$, $V (S_{ul})$, $V (S_{ll})$ can be determined through the payoff function, for instance $$ V (S_T) = \max ( S_T - K , 0 ) $$ for a call option.
For a path dependent option it's almost the same, except $V (.) $ is now a function of the whole path $\{S_0,S_{T-1}, S_T\}$, not just $S_T $
$$V_0 = \frac {1}{1+R} E [ V (S_0,S_{T-1},S_T) ] $$
Because, in a 2 period binomial tree, you have 4 possible paths: up/up, up/low, low/up, low/low, you can write the expectation as:
$$V_0 = \frac {1}{1+R} (q_{u}^2 V (S_0, S_u, S_{uu}) + q_u q_l V ( S_0, S_u, S_{ul} ) + q_l q_u V ( S_0, S_l, S_{ul} ) + q_l^2 V (S_0, S_l, S_{ll}))$$
With, as given in the text: $$ V (S_0, S_{T-1}, S_T) = \max ( \max ( S_0, S_{T-1}, S_T) - K, 0) $$ for a call with lookback feature.
Note how in the non path - dependent case the paths up/low and low/up both finish with the same stock value, hence can be aggregated, hence the factor $ {\color {red}{2}}$.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.