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Pricing a Call in a Two-Stock Binomial Model

Article Quant Q&A · Author: Imean H
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Summary

The document outlines a risk-neutral valuation approach for a call option in a two-stock, two-period binomial model. Because there is no risk-free asset initially specified, the response uses the requirement that traded assets share the same expected value under the risk-neutral probabilities to solve for an up-move probability and the common expected value. It then recommends repeating the calculation at subsequent nodes to obtain the transition probabilities.

Once the probabilities are known, the option value is found by weighting its terminal payoffs across in-the-money states and discounting at the risk-free rate inferred from the model. The example derives a one-period rate from the common expected value relative to the initial stock price, under discrete annual compounding. The explanation is brief and assumes the stated tree is consistent and that the probability equations determine a valid pricing measure; it does not spell out every node or terminal payoff calculation.

Key ideas

  • Solve for risk-neutral transition probabilities using the requirement that traded assets share a common expected value at each step.
  • Repeat the probability calculation at each node in the tree.
  • Calculate the call value by taking the risk-neutral expected terminal payoff and discounting it.
  • Infer the risk-free rate from the common expected value and the initial asset price.

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OptionsDerivatives pricingStatistics
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# What is the call price in a two-stock two-period model (binomial)?


# What is the call price in a two-stock two-period model (binomial)?












I am trying to wrap my head around the binomial model. In particular, how would the calculation go for deriving the price of a call option for one of the stocks expiring at time 2? You can pick any K.

I am usually used to just one stock, so this has me a bit confused.

(Each cell is a price, the arrows indicate change of time into new states, and the upper cell contains prices of stock 1, and lower cell prices of stock 2. No risk-free asset, no dividends).

## Answer by NSZ (score 2, accepted)

https://quant.stackexchange.com/a/34866

You have to calculate the Risk-Neutral probability of the upmoves and downmoves. The key point to bear in mind is that every stock (and risk-free bond) have the same expected value at each time step.

Therefore calling $q_{01}$ the probability of an upmove from time 0 to 1, you would have:

$$120q_{01}+90(1-q_{01})=X $$ $$130q_{01}+80(1-q_{01})=X $$

Now solve for $q_{01}$ and $X$, which gives you:

$$q_{01}=0.5$$ $$X=105$$

Do the same at each node and you should find the probabilities. Finally to find the value of a call option just multiply the $q$s for the payoff at the in-the-money nodes (where the stock price is higher than $K$).

This payoff has to be discounted using the risk-free rate which you can easily find from the calculations above. For example at the first node: $$1+r_{01}=\frac{X}{100} \quad \Rightarrow \quad r_{01}=0.05$$ assuming a time step of one year and discrete compounding.

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.