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I have been reading a finance paper that claims that the following function, which is a value for a financial derivative (1): $$V(s,t)=E_{Q} \left[\zeta\big(S(T)\big)e^{-\int_t^T r_F(\nu) d\nu}\\+\int_t^T C\big(S(u),u\big) e^{-\int_t^u r_F(\nu) d\nu}\big(r_F (u)-r_C (u)\big)du\ \middle\vert\ S(t)=s \right]$$

can be rearranged to be expressed as (2):

$$V(s,t)=E_{Q}\left[\zeta\big(S(T)\big)e^{-\int_t^T r_C(\nu) d\nu} \\-\int_t^T\big(V(S(u),u)-C(S(u),u)\big) e^{-\int_t^u r_C(\nu) d\nu}\big(r_F (u)-r_C (u)\big)du\ \middle\vert\ S(t)=s \right]$$

where $S(t)$ (which is the price of the underlying - eg a stock) follows a Wiener process with SDE:

$$\frac{dS(t)}{S(t)} =\big(r_R(t)-r_D(t)\big) dt+\sigma_S(t)dW_Q(t)$$

where $W_Q(t)$ is a Brownian Motion under measure $Q$.

The representation (1) is obtained by using the Feynman-Kac technique to solve the following PDE: $$ \frac{\partial V(S,t)}{\partial t}+\big(r_R(t)-r_D(t)\big)\frac{\partial V(S,t)}{\partial S}S(t)+\frac{\sigma_S(t)^2}{2} S(t)^2 \frac{\partial ^2 V(S,t)}{\partial S^2}\\=r_F V(S,t)-\big(r_F(t)-r_C(t)\big)C(S,t) $$

with the boundary condition: $$V(S(T),T)=\zeta\big(S(T)\big)$$

I can see no way to derive (2) from (1). I have tried several, but those that worked proved to have invalid steps. The paper itself just presents (2) as a fait accompli, with no attempt to justify it. On an intuitive level, (2) looks plausible, and seems like the sort of thing that may well be true. But a vague sense of plausibility is not the same as a rigorous demonstration of equivalence.

I am still working away at it, to see if it can be justified, but I was hoping that somebody might have seen something similar and be able to suggest a derivation, or alternatively point to a counterexample that invalidates the claimed equivalence.

I have the feeling that I may be overlooking something obvious, but I can't think what it is.

Thank you for any suggestions or comments anybody can make.

Background: the formula is intended as an adjustment to the classical derivative valuation formula, allowing for the trader's funding costs. $r_C(t), r_R(t),r_F(t)$ are respectively the interest rates for risk-free, repo and unsecured borrowing at time $t$ and $r_D(t)$ is the dividend rate. $C(s,t)$ is the collateral posted under the derivative agreement. $\zeta$ is the option payoff function and $V$ is the derivative value.

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