FE Trivial Part 1 - Relationship between Caplet and Option on Zero Coupon Bond

Caplet Payoff:

$X_{t_i}= N.\tau_i.Max(0,L(t_{i-1},t_i)-K)$

Given that forward rate:

$\tau_i.L(T_{i-1},T_i) = \frac{1-P(T_{i-1},T_i)}{P(T_{i-1},T_i)}$

Caplet Payoff:
$X_{t_i}$
$= N.\tau_i.Max(0,L(T_{i-1},T_i)-K)$
$=N.Max(0,\tau_i.L(T_{i-1},T_i)-K.\tau_i)$
$=N.Max(0,\frac{1-P(T_{i-1},T_i)}{P(T_{i-1},T_i)}-K.\tau_i)$

Discounting to today:
Caplet Price:
$= P(T_{i-1},T_i).N.Max(0,\frac{1-P(T_{i-1},T_i)}{P(T_{i-1},T_i)}-K.\tau_i)$
$= N.Max(0,1-(P(T_{i-1},T_i))(1+K.\tau_i))$
$= (1+K.\tau_i).N.Max(0,\frac{1}{(1+K.\tau_i)} - P(T_{i-1},T_i))$
$= N_{zero}.Max(0,K_{zero} -S_{zero})$

This the pricing of a put option on a Zero Coupon Bond with

Notional: $= (1+K.\tau_i).N$
Strike:$=\frac{1}{(1+K.\tau_i)}$

This will be important when we compare with Hull White 1 Factor Zero Coupon Bond Put European Option. Under the Hull White 1 Factor model, the close form solution is:

$Put_{zero} (t_0,t_1,t_2,K_{zero}) = K_{zero}P(t_0,t_1)\phi(-h+\sigma_{zero})-P(t_0,t_2)\phi(-h)$

Where
$\sigma_{zero}= \sigma_{black}\sqrt{\frac{1-e^{-2a(t_1-t_0)}}{2a}}B(t_1,t_2)$
$h=\frac{1}{\sigma_{zero}}ln{\frac{P(t-0,t_2)}{P(t_0,t_2)K_{zero}}+\frac{\sigma_{zero}}{2}}$

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