For Binomial Interest Rate Trees, we discount future bond cashflows at time “t” using “Up/Down” Future 1Y Spot Rates at time “t-1”. Under this model, why are we able to discount cashflows using multiple (products) short-term rates?
E.g cashflow at t=2 is discounted by the R_u and R_d 1Y Spot Rates 1 Year from now, and then discounted again by Current 1Y Spot Rates. Intuitively, there is no mismatch if we use Current 2Y Spot Rates on that time=2 cashflow. I understand we want interest rates to vary, but can help but feel there is a timing perspective mismatch!
Is it because in the model we explicitly say/assume that future ST Spot Rates will take on certain values, and thus we can discount cashflows using multiple series of expected 1Y Spot Rates?
I understand that discounting a cashflow arriving at t=2 by the 2year spot rate is equivalent to discounting it by (1Y Spot Rate) * f(1,1). This is because they are derived from each other by arbitrage.
However, for Binomial Interest Trees, we seem to be discounting such said t=2 cashflow by (Current 1Y Spot Rate) * (1Y Ahead 1Y Up/Down Spot Rates), which to me registers differently. Unfortunately, I’m not sure where my understanding has broken down, I’m clearly missing something simple, sorry if this question seems trivial!
One way I see it is if we calculate the value of the bond at each node from Right to Left (in a tree), then one would use prevailing/expected future 1Y Spot Rates for each node timeframe, is that an appropriate interpratation?