Futures and Forward Prices in a Binomial Model with Stochastic Rates

Summary

The document poses a derivatives-pricing question in a two-step binomial model with a non-dividend-paying underlying and varying interest rates. It asks how to calculate the futures price at the initial time and how to compute the risk-neutral expected value of the underlying at maturity, then compare those quantities with a forward price. The questioner describes deriving a forward price from a zero-coupon bond value and its implied yield, and expects futures and forward values to differ when rates vary by state or over time.

No solution or numerical answer is included. The exchange therefore does not demonstrate the required backward-induction calculation or specify the rate tree and risk-neutral probabilities needed to reproduce it. It is useful as a prompt about the distinction between futures and forwards when interest rates are stochastic, but a reader must consult a complete derivation to resolve the calculation.

Key ideas

• The question concerns futures and forward valuation in a multi-step binomial model.
• With varying interest rates, futures and forward prices may differ because futures are settled over time.
• The risk-neutral expected terminal underlying value is requested for comparison with the futures price.
• The document gives no solution, so it does not establish the calculation or a numerical result.

Tags

# Futures vs Forward pricing with different interest rates using binomial model


# Futures vs Forward pricing with different interest rates using binomial model












I'm given the aforementioned parameters for a two-step binomial model where the underlying pays no dividend, $S_0=50$ and $T=2$. With this information I was able to calculate the risk-neutral probabilities ($p$) and I am able to set up the following binomial tree for stock prices:

I am asked to calculate the Futures price $F_0$ (not the contract price) using this binomial model.

I am also asked to calculate the expected value of the underlying at $T_2$ given the risk-neutral probabilities and compare this to the answer of the previous question.

I know I can get the forward price by pricing a zero-coupon bond (with payment = 1) at maturity for every state, then determining the current price of that bond ($B=0.8868$) using risk-neutral valuation and then derive the yield ($y=0.061899$) based on which the forward price must be $50*(1+0.061899)^2=56.38$

However, I assume that the forward price mustn't equal the futures price due to these varying interest rates given before.

Therefore, how would one go about calculating the Futures price and the expected value given risk neutral valuation?

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