Following this post, let \( D(t) := e^{-rt} \) be the discount factor, \( D_i := D(t_i) \).

Given the relationship

Lemma

Let \( p_n^{Ber} \) be a n-Bermudan put with strike vector \( \mathbf{ K } \). Let \( c_n \) be the n-installment call with the same exercise dates, \( q_n = K_n \), and \[ \tag{1} q_i = K_i – \sum_{j = i+1}^n q_j \frac{D_j}{D_i}, \] then the following identity holds: \[ c_n + \sum_{i=1}^{n} D_i q_i = S + p_n^{\text{Ber}} \]

and the formula for the installment call from [GWK07]

Theorem (Griebsch/Wystup/Kühn)

The value of the \(n\)-installment call \(c_n\) is

\[ c_n = S N_n^+ – \sum_{i=1}^n D_i q_i N_i^- \]

according to the abbreviations here, we would like to derive the formula for the Bermudan put.

The recursive condition (1) yields

\[
q_i = K_i – K_{i+1} D_{i+1} / D_i
\]

with \( K_{n + 1} := D_{n+1} := 0 \).

Then, the formula for the Bermudan put becomes
\[
\begin {eqnarray}
p_n^{Ber} & = & c_n + \sum_{i=1}^n D_i q_i – S \\
& = & S(N_n^+ – 1) + \sum_{i=1}^n D_i q_i – \sum_{i=1}^n D_i q_i N_i^- \\
& = & S(N_n^+ – 1) + \sum_{i=1}^n D_i (1-N_i^-) (K_i – K_{i+1} D_{i+1} / D_i) \\
& = & S(N_n^+ – 1) + \sum_{i=1}^n D_i (1-N_i^-) K_i – \sum_{i=1}^n D_{i+1} K_{i+1} (1-N_i^-) \\
& = & S(N_n^+ – 1) + \sum_{i=1}^n D_i (1-N_i^-) K_i – \sum_{i=2}^n D_{i} K_{i} (1-N_{i-1}^-) \\
& = & S(N_n^+ – 1) + \sum_{i=2}^n D_i (N_{i-1}^- – N_i^-) K_i + D_1 (1-N_1^-) K_1 \\
\end{eqnarray}
\]

Fortunately, it follows from [SH87] that for \( k \geq 2 \)
\[
N_{i-1}^- – N_i^- = \overline{N_i^-},
\]

and \( 1 – N_1^- = \overline{N_1^-} \).
Hence,

Corollary (Geske/Johnson)

\[ p_n^{Ber} = S(N_n^+ – 1) + \sum_{i=1}^n D_i K_i \overline{N_{i}^-} \]

which coincides with [GJ84]’s formula for constant strike, since [SH87] also derive that:

\[
N_n^+ = 1 – \sum_{i = 1}^n \overline{N_i^+},
\]

only that this formula allows for variable strikes, too.

Notation and Definitions

Let \( N_k( \mathbf{y_k}; \mathbf{R_k}) \) be the \( k \)-dimensional cumulative normal distribution with upper bounds \( \mathbf{y_k} \in \mathbb{R}^k \) and covariance matrix \( \mathbf{ R_k } = ( \rho_{ij} ) \) with \( \rho_{ij} = \sqrt{t_i / t_j} \).

\( \mathbf{d_k^{\pm}} := (d^{\pm}(\overline{ S_1 }, t_1) , \ldots, d^{\pm}(\overline{ S_k }, t_k ) ) \), where \( d^{\pm}( strike, maturity ) \) are the usual Black-Scholes-Merton parameters.

\( \overline{ S_i } \) is the discrete stopping boundary which is recursively defined as in [GWK07] or [GJ84], respectively.


Let \( \mathbf{D_k} := diag(1, \cdots, 1, -1) \in \mathbb{R}^{k \times k} \), \( \mathbf{ \overline{ R_k } } := \mathbf{ D_k R_k D_k } \), and \( \mathbf{ \overline{ y_k } } := \mathbf{ D_k y_k } \).

Let’s also simplify the notation by \( N_k^{\pm} := N_k( \mathbf{ d_k^{\pm} }; \mathbf{ R_k }) \).
Analogously, define \( \overline{ N_k^{\pm}} := N_k( \mathbf{ \overline{ d_k^{\pm} }}; \mathbf{ \overline { R_k }} ) \).

References

  • [GJ84] Robert Geske and H. E. Johnson (1984). The American Put Option Valued Analytically, The Journal of Finance 39.5 (1984), 1511-1524.
  • [GWK07] Susanne Griebsch, Christoph Kühn, and Uwe Wystup (2007). Instalment options: a closed-form solution and the limiting case, CPQF Working Paper Series 5 (2007).
  • [SH87] Michael Selby, Stewart Hodges (1987). On the evaluation of compound options, Management Science 33.3 (1987), 347-355