Pricing a Lookback Call in a Two-Period CRR Model
Summary
The document considers a lookback call in a two-period Cox–Ross–Rubinstein binomial model. Its payoff is the terminal asset price minus the lowest price reached across the life of the option, so the payoff depends on the path as well as the terminal price. The discussion contrasts this with a fixed-strike call, whose payoff is a function of the terminal price alone, and asks how to price the path-dependent contract in general.
The included response computes a discounted risk-neutral expectation by assigning probabilities to the four paths and calculating each path’s minimum. This illustrates that pricing can proceed by evaluating the payoff on each possible path and averaging under the risk-neutral measure. However, the response’s arithmetic and assumptions are offered for validation rather than independently established, and it does not give a general recursive method. In a larger tree, tracking the running minimum alongside the asset price provides a way to retain the path information needed for valuation.
Key ideas
- A lookback call’s payoff depends on the minimum asset price observed over the option’s life.
- A payoff depending on path history cannot generally be represented as a function of the terminal asset price alone.
- In a finite binomial tree, evaluate the payoff on each path and take its risk-neutral expected value, discounted to the present.
- For larger trees, the running minimum can be tracked as part of the state used for valuation.
- The numerical response is presented for validation and does not establish a general recursive pricing procedure.
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Full text
# What's the price of a lookback call option in the arbitrage-free CRR-model?
# What's the price of a lookback call option in the arbitrage-free CRR-model?
If we consider the CRR-model in two periods, i.e. T=2. Let $S^1$ be the risky asset with $S_0^1=100$ and $S^0$ the bond with $S_0^0=1$. Furthermore, we assume the model is arbitrage-free with $y_b=-0.1<r=0.05<y_g=0.2$. Therefore, an unique equivalent martingale measure $\mathbb{Q}$ exists with $$\mathbb{Q}(\lbrace\omega\rbrace) =0.5^{Z_1(\lbrace\omega\rbrace)+Z_2(\lbrace\omega\rbrace)}\cdot 0.5^{2-Z_1(\lbrace\omega\rbrace)-Z_2(\lbrace\omega\rbrace)}$$, where $\omega\in\lbrace 0,1\rbrace^2$. Furthermore, the lookback call option is given by $$\gamma (\omega)=S_2^1(\omega)-\min_{t\in\lbrace 0,1,2\rbrace}S_t^1(\omega)$$. I see that the price of the option depends on these numbers a lot, but I want a general approach. In some literature I have found out that the hedging price ( or arbitrage-free price) is given by $\mathbb{E}^\mathbb{Q}\left[\frac{f(S_2^1)}{(1+r)^2}\right]$, but how exactly do I determine the function $f$. If it was a call option with a fixed strike price $K$, i.e. $\gamma' (\omega)=(S_2^1(\omega)-K)^+$ I assume the $f$ would be given by $f(x)=\max(x-K,0)$. But how do I model the dependency of $\gamma$ on $\min_{t\in\lbrace 0,1,2\rbrace}S_t^1(\omega)$?
## Answer by stats19 (score 0)
https://quant.stackexchange.com/a/60590
I thought about it and think that in this case it does not really matter how $f$ looks exactly, because $f(S_2^1(\omega ))=\gamma (\omega)$ should be true. Then, with the formular for the equivalent risk-neutral measure $\mathbb{Q}$ given above we can compute the probabilities $Q((0,0))=Q((1,0))=Q((0,1))=Q((1,1))=0.25$. Furthermore, in the CRR-model it holds, that $$S_t^1(\omega )=S_0^1(1+y_g)^{D_t(\omega)}(1+y_b)^{t-D_t(\omega)}$$. So for the numbers above we get $$ \begin{split} \frac{1}{(1+r)^2}\mathbb{E}^\mathbb{Q}[\gamma]&=\frac{400}{441}\left(\mathbb{E}^\mathbb{Q}[S_2^1]-\mathbb{E}^\mathbb{Q}\left[\min_{t\in\lbrace 0,1,2\rbrace}S_t^1\right]\right)\\ &= \frac{400}{441}(0.25\cdot(81+108+108+144)-0.25\cdot(81+100+100+90))\\ &=15.87 \end{split} $$
Could someone validate that ? Furthermore, it does not really answer the question on how to proceed, if no specific numbers are given. How should one approach this problem in general ?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.