# Let W n (S; X) = lim e r Un(S, ; X), where U

## Question:

Let W_{n}^{∞} (S; X) = lim_{τ→∞ }e^{rτ}Un(S, τ ; X), where U_{n}(S, τ; X) is the value of the n-reset put option [see (5.4.20)]. For r n^{∞} (S) is given by (Dai, Kwok and Wu, 2003)

The auxiliary conditions are given by

where β_{n} = W^{∞}_{n}_{−1}(1; 1). Show that

where α = 2(q − r)/σ^{2}. The recurrence relation for β_{n} is deduced to be

Show that β_{1} = 1 and lim_{n→∞} β_{n} = 1 + 1/α . Also, find the first few values of S^{∗}_{n},_{∞}.

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