Deriving Risk-Neutral Probability in a Binomial Option Model

Summary

This note derives the risk-neutral up-move probability for a one-period binomial stock model. Under the risk-neutral measure, the expected stock price grows at the risk-free rate over the period. Equating the probability-weighted up and down stock values to that growth, then rearranging, gives the probability in terms of the up and down growth factors and the risk-free growth factor. The numerical example uses the stated stock price, strike, up and down factors, and rate; it identifies that the questioner's calculation mistakenly used the strike where the initial stock price belonged.

The note also distinguishes compounding conventions: the risk-free growth factor may be expressed with simple per-period growth, continuous compounding, or a rate compounded over the period. The relevant form depends on how the model's rate and time step are defined. The derivation assumes a standard binomial setup with an appropriate risk-free asset and does not discuss dividends, transaction costs, or extensions such as multi-period trees.

Key ideas

• Risk-neutral valuation makes the expected stock growth match the risk-free asset's growth over a tree period.
• The one-period probability follows by equating the weighted up and down outcomes to risk-free growth from the initial stock price.
• The strike price does not enter the risk-neutral probability calculation.
• Use a growth factor consistent with the model's simple, discrete, or continuous compounding convention.
• The derivation assumes a standard binomial model and does not address dividends or trading frictions.

Tags

# Option pricing: Risk neutral probability calculation


# Option pricing: Risk neutral probability calculation












Let $u=1.3$ $d=0.9$ $r=.05$ $S(0)=50, X = \text{strike} = 60$. Assume binomial model

Why isn't the risk neutral probability found by solving the following for $p$: $$E[S(T)]=p65+(1-p)45=S(0)(1+r)^T=60(1.05)$$

Because risk neutral probabilities should be the same in all time steps, I just took $T=1$

The correct $p=0.375$

## Answer by Quantuple (score 2, accepted)

https://quant.stackexchange.com/a/25208

[Short Answer]

You write $E [S_T]=S_0(1+r)^T $ but you actually compute the RHS as $X (1+r)^T$ in your numerical application.

[Long Answer]

The stock price is a martingale in an equivalent measure using the risk-free asset as numeraire i.e.

$$ E [S(T)] = (S_0 u) q + (S_0 d) (1-q) = S_0 (1 + r ) \Delta t $$

In that case, dividing each member by $S_0$ and re-arranging the terms yields

$$ q ( u - d ) + d = (1 + r ) \Delta t $$ $$ q = \frac {(1 + r ) \Delta t - d}{u - d} $$

Doing the computations with your input data (assuming a tree period covers $\Delta t=1$) $$ q = \frac {(1+0.05) - 0.9}{1.3-0.9} = 0.375 $$

Note that this expression for $q $ is exactly the one given in any textbook provided $u $ (resp. $d $) figures the upward (resp. downward) growth rate of the stock over a binomial tree period $\Delta t $, while $1 + r $ represents the growth rate of the risk free asset.

Note that sometimes continuous compounding is used, in which case:

$$ q = \frac {e^{r \Delta t} - d}{u - d} $$

Depending on the compounding convention you could also have

$$ q = \frac {(1 + r)^{ \Delta t} - d}{u - d} $$

of course.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.