Similarly to the Black and Scholes formula, I am looking to replicate Bachelier's caplet formula with two digital options: (1) asset-or-nothing (forward rate in this case) and (2) cash-or-nothing. For reference, Bachelier's caplet formula is: $$c(t,T_{i-1},T_i) = \delta*P(t,T_i)*\Bigl((F(t,T_{i-1},T_i)-K)*\Phi(D)+\sigma*\sqrt{T_{i-1}-T_i}*\phi(D)\Bigr)$$ $$where$$$\delta=$ frequence factor, $P(t,T_i)$ is the discount factor, $D={F(t,T_{i-1},T_i)-K}/{\sigma*\sqrt{T_{i-1}-T_i}}$, $\Phi$ is the cumulative distribution function and $\phi$ is the probability density function

For reference, in the BS formula, the part $$S*N(d_1)$$ is for the asset-or-nothing and the part $$K*e^{-rt}*N(d_2)$$ is for the cash-or-nothing.

My take is that, as $\Phi(D)$ represents the probability to be in-the-money, a digital caplet cash-or-nothing is value as: $$D_{cash}(t,T_{i-1},T_i) = \delta*P(t,T_i)*K*\Phi(D)$$ and a digital caplet asset-or-nothing is value as: $$D_{asset}(t,T_{i-1},T_i) = \delta*P(t,T_i)*\Bigl(F(t,T_{i-1},T_i)*\Phi(D)+\sigma*\sqrt{T_{i-1}-T_i}*\phi(D)\Bigr)$$

Many thanks!

  • $\begingroup$ This may be helpful for you. $\endgroup$
    – Gordon
    Jan 18 '19 at 19:03
  • $\begingroup$ Thanks for the link unfortunately, unless I am missing something, I don't see how this replies to my question. $\endgroup$
    – qbodart
    Jan 21 '19 at 17:06

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