Caplet and Floorlet Pricing under Backward-Looking Compounded Rates in a Discrete Hull-White Framework

This thesis prices backward-looking caplets and floorlets, non-linear derivatives written on compounded overnight risk-free rates (SOFR, SONIA, €STR) that have replaced LIBOR, in a discrete Hull-White model in which the short rate itself, and not only the compounding convention, is discrete. The short rate evolves by a first-order autoregressive recursion, the Euler-Maruyama discretization of the Hull-White equation, whose additive Gaussian noise preserves the distributional structure of the continuous-time model. Under the single-period convention p(k,k+1)=exp(-r_k), the compounded backward-looking rate telescopes exactly into the exponential of the sum of the short rates over the accrual period, reducing the payoff to a function of a single Gaussian sum. The pricing theory is then carried out entirely within the discrete affine term structure of the model, using only truncated univariate and bivariate Gaussian moments obtained by completing the square. No step uses stochastic calculus, and the derivation is, to the author's knowledge, the first fully self-contained discrete-time treatment of these prices. Closed-form prices are obtained both at the start of the accrual period and at any earlier valuation date, with the forward-looking prices following from the same argument as a special case. The two conventions share an identical Black-type formula and differ only through the variance entering it: the backward-looking total variance decomposes as nu_X = nu_W + nu_{m,N}, into a pre-accrual and an in-accrual component. This decomposition yields an exact integral representation of the backward-forward price gap, a closed-form expression at the money, and a simulation-free numerical study in which the single ratio nu_{m,N}/nu_W drives the main properties of the backward-looking premium.