Bond Future Option Valuation
A bond future option is an option contract that gives the holder the right but not the obligation to buy or sell a bond future at a predetermined price. The writer/seller receives a premium from the buyer for undertaking this obligation. Options are leveraged instruments that allow the owner to control a large amount of the underlying asset with a smaller amount of money. Bond future options offer significant advantages for reducing costs, enhancing returns and managing risk. They could be European style or American style.
Bond futures options are also exchange traded derivatives on treasury instruments. They provide market participants with the ability to adjust their interest rate exposures. A bond future option is also a good tool for hedging, income enhancement, duration adjustments, interest rate speculation and spread trading.
There are established global markets for bond futures and future options so they are very liquid. Investors use bond future options to hedge an existing bond portfolio against adverse interest rate movements or enhance the long-term performance of a portfolio of assets. Arbitrageurs profit from the price difference between the spot bonds and the bond futures. Speculators use bond future option in the hope of making a profit on short-term movements in prices. This presentation provides an overview of bond future option product and valuation.
The present value of a call bond future option is represented as:
PV(t)=N[F_T Φ(d_1 )-KΦ(d_2)] D_T
where d_1,2=[ln(F_T⁄K)±(σ^2 T)⁄2]⁄(σ√T) K the strike N the notional F_T=[(P-C_Σ ) exp(r_T T)-A ]/CF the forward clean price of the delivered bond (CTD) at t T the option maturity date D_T the discount factor the conversion factor for a bond to deliver in a bond futures contract C_Σ=∑(t_i≤T)〖Cexp(-r_i t_i)〗 the present value sum of all coupons of the underlying bond between t and T A the accrual interest before T. P the bond dirty price at t r_T the continuously compounded interest rate between t and T σ=αDyσ_y/CF the volatility of forward bond price. forward yield volatility of the CTD bond of the underlying futures. We use the swaption volatility implied volatility scaling factor y the forward yield that can be solved by P-C_Σ=∑(T≤t_i≤T_B)〖Ce^(-yt_i ) 〗 T_B the maturity of the underlying CTD bond D=(∑(T≤t_i≤T_B)〖t_i Ce^(-yt_i ) 〗)/(∑(T≤t_i≤T_B)〖Ce^(-yt_i ) 〗) the forward modified duration of the CTD bond of the underlying futures
The present value of a put bond future option is represented as:
PV(0)=N[KΦ(-d_2 )-F_T Φ(〖-d〗_1 )] D_T
Valuation: American Style
We use the Cox-Ross-Rubinstein (CRR) binomial tree to price American bond future option.
Build forward bond price tree.
F_0=[(P-C_Σ ) exp(r_T T)-A ]/CF
∆t=T/m
F_j^u=F_j e^(σ√∆t) with probability p=(1-e^(-σ√∆t))/(e^(σ√∆t)-e^(-σ√∆t) )
F_j^u=F_j e^(-σ√∆t) with probability 1-p
J = 1,…,m
σ=αDyσ_y/CF is the volatility described above
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