Risk-Neutral Probabilities and Discounted Martingales in a Binomial Tree

Summary

The question considers a three-step binomial stock model and asks how to verify that the process is a martingale. It supplies an initial stock price, up and down multipliers, and an interest rate, and mentions candidate probabilities, but focuses on computing expected prices at later steps.

The answer clarifies that under an arbitrage-free pricing framework, it is the discounted stock price that is a martingale. Thus the terminal expected stock price must be discounted back to the initial value, and the tree can be evaluated recursively using the up and down moves. The response gives the governing principle but not the full calculation, and its wording uses continuous discounting without discussing alternative rate conventions.

Key ideas

• In an arbitrage-free binomial model, the discounted stock price is the martingale.
• The expected terminal stock value must be discounted to compare it with the initial price.
• Tree expectations can be computed recursively from each node’s up and down outcomes.
• The stated probability candidates require consistency with the model’s discounting convention.

Tags

# Martingale Binomial Tree Process


# Martingale Binomial Tree Process












3 step binomial tree process with $S_0=4,u=2,d=0.5,r=0.25.$ Determine the probability p and q such that the stock price process is a martingale (i.e. $E[S3]=S_0)$

I know P = 1/3 and Q = 2/3 but having trouble to get to $E[S_2]$ and $E[S_3]$ to prove it's the same as $S_0$

## Answer by user49802 (score 2)

https://quant.stackexchange.com/a/57710

Under Martingale framework you can admit ,without loss of generality, to be under an arbitrage free market. By the way the martingale process is the discounted spot, you then need to use $$\exp^{-3*0.25} E[S_3]=S_0 $$. Finally, remember that under Up event $$S_{t+1} = S_t * u$$. You'll be able to solve your tree recursively.

I may have made a mistake but still hope to be useful, good luck.

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.