You are pricing a derivative using the risk-neutral approach on a Binomial tree.The length of each branch in the tree (delta t) is 5 months.The riskfree rate of interest is 6% per annum.The proportional up movement in stock price is u = 1.3. Calculate the risk-neutral probability (p*) that share price will move up. Note: if the probability is (say) 51.23%, enter 0.5123

Question

BlackTom AI · Accepted Answer

Risk-neutral probability p* satisfies the condition that the expected gross return of the stock under p* equals the risk-free gross return over the period. With possible up factor u and down factor d, and continuous compounding at rate r for time Δt, the equation is:
p*·u + (1 − p*)·d = e^{rΔt}.

Step 1 — relevant parameters and model choice
- Δt = 5 months = 5/12 year = 0.4166667 year.
- r = 6% = 0.06 per annum (use continuous compounding).
- u = 1.3. For a Cox–Ross–Rubinstein style binomial, take d = 1/u = 1/1.3 = 0.7692307692.

Step 2 — compute risk-free gross factor
rΔt = 0.06 × 5/12 = 0.025.
e^{rΔt} = e^{0.025} = 1.0253151205 (intermediate).

Step 3 — solve for p*
Start from p*u + (1 − p)*d = e^{rΔt}.
Rearrange: p*(u − d) = e^{rΔt} − d.
So p* = (e^{rΔt} − d) / (u − d).

Insert numbers:
d = 0.7692307692
u − d = 1.3 − 0.7692307692 = 0.5307692308
Numerator = e^{rΔt} − d = 1.0253151205 − 0.7692307692 = 0.2560843513
Therefore p* = 0.2560843513 / 0.5307692308 = 0.482547... ≈ 0.4825.

Why this is correct
This p* makes the expected discounted stock price equal to the current price under the risk-neutral measure, which is the required condition for arbitrage-free pricing.

Why other computations might be wrong
- Using a simple interest factor (1 + rΔt = 1.025000) instead of e^{rΔt} gives a slightly different p* ≈ 0.4820; that is an approximation but not the continuous-compounding CRR standard.
- Using any other value for d (not 1/u) breaks the usual CRR symmetric tree assumption and will change p*.

Conclusion
The correct risk-neutral up probability is p* ≈ 0.4825, which matches the provided answer 0.4825.

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