Bond option pricing with Forward price dynamic

This function is available in R-code (here). See also its upper level page (here).

In this below, we explain Bond option pricing with forward price BS dynamics.

Functions to load: Bl(), ZBOFwd()
Usage: ZBOFwd(K, T, U, PT, PU, nu)
Arguments:
K: strike
T: maturity of option
U: maturity of the underlying zero bond
PT: spot price of T-Bond
PU: spot price of U-Bond
nu: standard deviation of log forward price of U-Bond at time {T}
Value: 2-D vector of call and put price.

Denote by {P(t,T)} the price of {T}-bond at {t}. Then, for {t< T< U}, the forward price of {U}-bond delivered at {T} is given by

\displaystyle F_{P}(t, U, T) = P(t, U)/ P(t, T).

Since the dynamic {t\mapsto F_{P}(t, U, T)} is a non-negative martingale with respect to the {T}-forward measure {\mathbb P_{T}}, one can assume

\displaystyle d F_{P}(t, U, T) = F_{P}(t, U, T) \eta(t, U, T) \cdot d W^{T}(t),

where {W^{T}} is {n}-dim {\mathbb P_{T}-}BM and {t\mapsto \eta(t, U, T)} is {n}-dim deterministic process. Now, we want to compute

  1. zero bond call price at {t}, which has payoff at {T} with amount {(P(T, U) - K)^{+}}, denoted by {ZBC(t, T, U, K)}
  2. zero bond put price at {t}, which has payoff at {T} with amount {(P(T, U) - K)^{-}}, denoted by {ZBP(t, T, U, K)}.

R-code is written according to the following fact:

Proposition 1

\displaystyle ZBC_{t} = P(t, T) Bl(K, F_{P}(t, U, T), \nu, +1),

and

\displaystyle ZBP_{t} = P(t, T) Bl(K, F_{P}(t, U, T), \nu, -1),

where {\nu^{2} = \int_{t}^{T} |\eta(s, U, T)|^{2}ds}.

Ex. Spot prices for 2-year zero-coupon bond is 0.90, and 5-year zero-coupon bond is 0.72. Volatility of the forward price {F_{P}(t, 5, 2)} is given as constant {0.2.} For strike {K = .8} and maturity {T = 2} years, we can compute call and put prices underlying {5}-year bond as follows. With given parameters {PT = .9; PU = .72; eta = .2; T= 2; U = 5; K = .8}, one can compute {nu = eta \sqrt T}. Then, option price is

\displaystyle [ZBC, ZBP] = ZBOFwd(K, T, U, PT, PU, nu) = [0.0809733, 0.0809733].

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