Theoretical
The theoretical bond forward price at date \(t\) for a maturity \(T\) is:
\[BF_{(0, T)} = B_0 * DF_{(0,T)}^{-1} - \sum_{i=0}^{N} C_i * DF_{(t_i,T)}^{-1}\]
With:
- \(BF_{(0, T)}\) the forward price of bond \(B\) at date \(t\) for maturity \(T\),
- \(B_0\) the spot dirty price of the bond \(B\): \(B_0 = B_0^{clean} + A_0\) where \(B_0^{clean}\) is the clean price of \(B\) a date \(t=0\) and \(A_0\) is the accrued interest a date \(t=0\),
- \(DF_{(t_i,t_j)}\): the discount factor associated to the risk free rate for the period \((t_i,t_j)\),
- \(C_i\): the coupon at date \(t_i\).
No Arbitrage Opportunity
Replication Payoff
The pricing of a Forward on equity can be replicated using this strategy.
At \(t=0\)
- I have \(B_0\) in cash to invest and I use it to buy the bond \(B\) with dirty value \(B_0\),
At \(t=t_i\) where \(t_i\) are coupon payment dates
- I receive the coupon \(C_i\) in cash,
- I reinvest the coupon \(C_i\) at the risk free rate.
At \(t=T\)
- The bond \(B\) current value is now \(B_T\),
- I get back my coupons and the interest earned: \(\sum_{i=0}^{N} C_i + R_i\) with \(t_n < T\) and \(R_i\) the interest earned on coupon \(C_i\) with \(R_i = C_i * (e^{r_{(t_i, T)}} - 1)\) where \(r_{(t_i, T)}\) is the risk free rate from \(t_i\) to \(T\).
Finally at \(t=T\) I have:
- Bond \(B\) with value \(B_T\),
- The coupons \(C_i\) and their interest: \(\sum_{i=0}^{N} C_i + R_i\),
Total: \(B_T + \sum_{i=0}^{N} C_i + R_i\)
Forward Payoff
At \(t=0\)
- I have \(B_0\) in cash that I invest at the risk free rate,
- I enter a forward agreement on \(B_0\) with value \(B_0 * DF_{(0,T)}^{-1} - \sum_{i=0}^{N} C_i * DF_{(t_i,T)}^{-1}\)
At \(t=T\)
- I get back the \(B_0\) that I have invested at the risk free rate plus the risk free rate on the period \(R\),
- I buy the bond with value \(B_T\) using my forward agreement at price \(BF_{(0, T)} = B_0 + R - \sum_{i=0}^{N} C_i + R_i\)
Finally at \(t_T\) I have:
- Bond \(B\) with value \(B_T\),
- The difference of interests earned and forward payed: \((B_0 + R) - (B_0 + R - \sum_{i=0}^{N} C_i + R_i) = \sum_{i=0}^{N} C_i + R_i\)
Total: \(B_T + \sum_{i=0}^{N} C_i + R_i\)
Practical
In theory, the cost of financing is usually given as the risk free rate, although in the real world a forward’s implied financing cost can deviate from the market’s risk free rate.
In a market with particular supply/demand dynamics bond repo rates reach extreme levels and lending an owned bond is an investment opportunity.
The practical formula takes into account this investment opportunity rising from the lending of the bond owned.
The practical bond forward price at date \(t\) for a maturity \(T\) is:
\[BF_{(0, T)} = B_0 * \left(DF_{(0,T)}^{repo}\right)^{-1} - \sum_{i=0}^{N} C_i * DF_{(t_i,T)}^{-1}\]
With:
- \(BF_{(0, T)}\) the forward price of bond \(B\) at date \(t\) for maturity \(T\),
- \(B_0\) the spot dirty price of the bond \(B\): \(B_0 = B_0^{clean} + A_0\) where \(B_0^{clean}\) is the clean price of \(B\) a date \(t=0\) and \(A_0\) is the accrued interest a date \(t=0\),
- \(DF_{(t_i,t_j)}\): the discount factor associated to the risk free rate for the period \((t_i,t_j)\),
- \(DF_{(t_i,t_j)}^{repo}\): the discount factor associated to the repo rate for the period \((t_i,t_j)\),
- \(C_i\): the coupon at date \(t_i\).
\(DF_{(t_i,t_j)}^{repo}\) is the discount factor associated with the combination of the risk free rate on the period and the repo spread on the period (with the repo spread almost always negative - ie positive quotation as the quotation for repo is inverted).
No Arbitrage Opportunity
Replication Payoff
The pricing of a Forward on equity can be replicated using this strategy.
At \(t=0\)
- I have \(B_0\) in cash to invest and I use it to buy the bond \(B\) with value \(B_0\),
- I sell the bond \(B\) through a repurchase agreement contract and I get back \(B_0\) in cash,
- I invest \(B_0\) at the risk free rate (risk free rate \(\textit{almost always}\) greater than repo rate).
At \(t=t_i\) where \(t_i\) are coupon payment dates
- The buyer of the repo pays back immediately the coupon \(C_i\) in cash,
- I reinvest the coupon \(C_i\) at the risk free rate.
At \(t=T\)
- I get back the asset \(B\) with current value \(B_T\),
- I get back the \(B_0\) that I have invested at the risk free rate plus the risk free rate on the period \(R\),
- I pay back the \(B_0\) in cash to the buyer of the repo plus the repo rate on the period \(Re\) (\(Re\) being the sum of the risk free rate and the inverse of the repo spread: \(Re=R-S\) and \(S=R-Re\)),
Finally at \(t=T\) I have:
- Bond \(B\) with value \(B_T\),
- The coupons \(C_i\) and their interest: \(\sum_{i=0}^{N} C_i + R_i\),
- The difference of interests earned and payed: \(R-Re\) (with \(\textit{almost always}\) \(R-Re>0\))
Total: \(B_T + (R - Re) + \sum_{i=0}^{N} C_i + R_i\)
Forward Payoff
At \(t=0\)
- I have \(B_0\) in cash that I invest at the risk free rate,
- I enter a forward agreement on \(B_0\) with value \(BF_{(0, T)} = B_0 * \left(DF_{(0,T)}^{repo}\right)^{-1} - \sum_{i=0}^{N} C_i * DF_{(t_i,T)}^{-1}\)
At \(t=T\)
- I get back the \(B_0\) that I have invested at the risk free rate plus the risk free rate on the period \(R\),
- I buy the bond with value \(B_T\) using my forward agreement at price \(BF_{(0, T)} = B_0 + Re - \left(\sum_{i=0}^{N} C_i + R_i \right)\)
Finally at \(t_T\) I have:
- Bond \(B\) with value \(B_T\),
- The difference of interests earned and forward payed: \((B_0 + R) - \left(B_0 + Re - \left(\sum_{i=0}^{N} C_i + R_i \right)\right) = (R - Re) + \left(\sum_{i=0}^{N} C_i + R_i\right)\)
Total: \(B_T + (R - Re) + \left(\sum_{i=0}^{N} C_i + R_i\right)\)
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