# Collateral replication argument

I'm trying to follow the replication argument in the first page of the following paper

One can however instead replicate directly with the holder of the zero coupon bond: Consider the collateralized zero coupon bond is valued as $V(t)$. As holder of the bond requires $V(t)$ as cash collateral, the issuer receives nothing net. Note the collateral grows at the rate of $c(t)$. Therefore, putting $e^{-c(T)}$ as collateral, one sees that at time $T$, holder of the bond will find the collateral account containing exist one unit of currency $i$ which is exactly the payoff of the bond.

I'm afraid I don't understand many things. The purchaser gives $V(t)$ to the issuer, and the issuer gives $V(t)$ to the purchaser as collateral. So neither the issuer nor purchaser receives anything net at the start of the contract. Finally, the collateral is paid from the holder of the bond from the issuer, so I don't see how the holder would stand to gain from the collateral rates - shouldn't it be the issuer instead?

Overall, if the risk-free rate was not the collateral interest rate, how could we create arbitrage? Could someone explain in more detail what is going on?

Let me know whether this helps, but the author mentions a paper from Fujii and Takahashi; I have been looking for it on the internet and I have found what seems to be a version of it: Collateral Posting and Choice of Collateral Currency. I think they give a relatively transparent explanation $-$ in terms of funding costs $-$ of why the discount rate for collateralized trades must be the collateral rate. I reproduce some excerpts here $-$ the author is referring to some derivative contract between two parties, A and B $-$ my emphasis:

[Uncollateralized trade]: "Let us consider the situation where the firm A has a positive present value (PV) in the contract with the firm B with high credit quality. From the view point of the firm A, it is equivalent to providing a loan to the counterparty B with the principal equal to its PV. Since the firm A has to wait for the payment from the firm B until the maturity of the contract, it is clear that A has to finance its loan and hence the funding cost should be reflected in the pricing of the contract. If the firm A has (and continues to be) Libor credit quality, the funding cost is given by the Libor of its funding currency since it makes the present value of ”funding” zero. This is the main reason why Libor (London inter-bank offered rate) has been widely used as a proxy of the discounting rate in the derivative pricing."

[...]

[Collateralized trade]: "The above situation drastically changes when the trade is collateralized [...]. Let us assume that the trade has been made with a CSA (credit support annex, a legal document regulating credit support for derivative transactions), requiring cash collateral with zero minimum transfer amount as well as threshold. In this case, there is no external funding need for the firm A, since the cash amount equal to the PV is posted from the firm B. Now, the firm A has to pay the counterparty B the margin whose interest rate is called ”collateral rate” applied to the posted collateral amount. This effectively makes the funding cost of the contract equal to the collateral rate."

Note that it is the party receiving the collateral that pays the collateral rate to the party posting the collateral.

[Edit 15/09/17]

"[...] the collateral is paid from the holder of the bond [did you meant to instead of from?] the issuer, so I don't see how the holder would stand to gain from the collateral rates - shouldn't it be the issuer instead?"

Note that $V(t) > 0$, hence it will always be the bond seller who will post collateral to the buyer $-$ as the bond has always positive value to the buyer. Hence the buyer holds at any time $t$ in $[0,T]$ a collateral amount equal to $V(t)$, for which he has to pay the (deterministic) collateral rate $c(t)$ to the seller.

The seller knows he has to deliver $1\$$at time T to the buyer, but he has to continuously post a collateral equal to the bond's value to the buyer, who will remunerate him continuously at rate c(t). Hence if he values the zero-coupon bond at 0 as - we keep the author's assumption of deterministic rates:$$ V(0) = e^{-\int_0^Tc(t)dt}$$He knows he will "hold" 1\$$ of collateral in the buyer's collateral account at$T$, because the latter is paying$c(t)$for the collateral. "Overall, if the risk-free rate was not the collateral interest rate, how could we create arbitrage? Could someone explain in more detail what is going on?" For simplicity we assume rates are constant, i.e.$r$and$c$are the "risk-free" and collateral rates for borrowing/lending over period$[0,T]$. Now let us consider two cases: •$r<c$: if the seller prices the zero-coupon bond as $$V(0) = e^{-rT}$$ By the same replication argument as before, the seller needs to post$V(0)$as collateral; the collateral will grow at rate$c$hence at maturity the collateral account$C(T)$will have the value: $$C(T) =e^{(c-r)T} > 1$$ The seller can then recover$e^{(c-r)T}-1$from the collateral account as riskless profit. •$r>c$: the previous strategy would yield a loss this time. In this case there isn't a clear strategy that would yield the seller a riskless profit without making the buyer pay a price$V(0)$higher than the$1\$$he is going to receive at T. He could always price:$$ V(0) = e^{(r-c)T} $$So that the collateral account will contain e^{rT}>1 at T, but the buyer is unlikely to enter that deal - except in the case of negative rates maybe. The problem is that at t=0 the price V(0) is immediately pledged as the collateral amount C(0). [Edit 2 15/09/17, modified on 09/10/17]^{\text{ [1]}} To show more formally that the collateral rate is the correct rate, let us consider a tradable asset X(t) which follows a generic Ito process and does not pay any cash-flow (dividends, coupons, etc.):$$ dX(t) = \mu_X(t,X(t))dt+\sigma_X(t,X(t))dW(t)$$We also assume that the collateral rate r_C(t) is stochastic. We can interpret the zero-coupon bond as a derivative written on the asset X(t) with payoff function h(T,X(T))=1\$$ at maturity $T$ (note there is no uncertainty as to what the final payoff will be). We will note its value at time $t$ as $V(t) = V(t,X(t))$. The asset $X(t)$ is merely instrumental and allows us to apply the Feynman-Kac Theorem in order to derive the correct discount rate $-$ see footnote $\text{[1]}$.

Assume a cash collateral amount $C(t)$ needs to be posted continuously to the bond holder such that $C(t)=V(t)$ (perfect collateralization) and that this amount is remunerated at the collateral rate. To replicate the bond, the seller forms a portfolio $\Pi(t)$ made up of the asset and the cash account $\beta(t)$:

$$V(t) = \Pi(t) = \Delta_X(t)X(t)+\beta(t)$$

To be self-financing, the cash account needs to be made up of 1) the funding for the asset purchase $-$ opposite position to the asset's holding $-$ which we assume is done at a rate $r_X(t)$, and 2) the collateral amount which is equal to the derivative's value:

$$\beta(t) = C(t)-\Delta_X(t)X(t) = V(t)-\Delta_X(t)X(t)$$

Because the portfolio is self-financing, we have:

$$dV(t) = d\Pi(t) = \Delta_X(t)dX(t)+d\beta(t) \tag{1}$$

Given the collateral is remunerated at rate $r_C(t)$ and the financing of the asset purchase is done at rate $r_X(t)$, the cash account growths according to:

$$d\beta(t) = r_C(t)V(t)dt - r_X(t)\Delta_X(t)X(t)dt$$

On the other hand, by Ito's lemma:

$$dV(t) = \frac{\partial V}{\partial t}dt + \frac{\partial V}{\partial X}dX(t)+\frac{1}{2}\frac{\partial^2 V}{\partial X^2}dX(t)^2 \tag{2}$$

By the standard hedging arguments used in the Black-Scholes framework, we choose $\Delta_X(t)$ so as to eliminate risk $-$ i.e. cancel random terms in $dX(t)$ $-$ in $\text{(1)}$ and $\text{(2)}$, and equating both:

$$\frac{\partial V}{\partial t}dt + \frac{1}{2}\frac{\partial^2 V}{\partial X^2}dX(t)^2 = r_C(t)V(t)dt - r_X(t)\Delta_X(t)X(t)dt$$

This is the Black-Scholes equation with different collateral and asset financing rates and no "risk-free" rate.

Given the terminal condition of the bond $-$ $V(T)=h(T,X(T))=1$ $-$ we can apply the Feynman-Kac theorem and obtain:

$$V(t) = E^Q\left[e^{-\int_t^Tr_C(u)du}|\mathcal{F}_t\right]$$

where $Q$ is a risk-neutral measure under which $X(t)$ growths at rate $r_X(t)$. You see that the collateral rate is the correct discount rate in this case: the bond buyer funds the position through the collateral rate.

$\text{[1] }$This proof has been modified from an earlier version in order to rigorously apply the Feynman-Kac theorem. In the opinion of the writer, if we do not define an auxiliary tradable asset $X(t)$ and instead we make the bond price dependent on the collateral rate, then without (strongly) assuming either that 1) the rate $r_C(t)$ is traded in itself or 2) the 1st and 2nd derivatives of the bond price with respect to the collateral rate are null, it is not possible to derive a pricing PDE to which we can apply Feynman-Kac to find the price of the zero-coupon bond.

[Edit 08/10/17]

"Overall, if the risk-free rate was not the collateral interest rate, how could we create arbitrage? Could someone explain in more detail what is going on?"

For more information on arbitrage from different discount rates and collateral agreements, you can read Risk magazine's article "Goldman and the OIS Gold Rush".