When a Derivative Hedge Has a Predictable Cash Balance Sign
Summary
The cash balance in a delta hedge is the derivative’s value minus the value of its asset holdings. In general, the document does not claim that this quantity has a fixed sign; it describes conditions under which sign information can be obtained. In a one-asset diffusion setting, a result attributed to Bergman says the sign of the option’s cash balance follows the sign of the corresponding terminal payoff expression, provided the relevant pricing assumptions hold.
For options whose prices are homogeneous in the asset price and strike, Euler’s theorem expresses the cash balance through the strike sensitivity. The answer then connects that sensitivity to risk-neutral probabilities using Breeden-Litzenberger relationships, which can yield a known sign for suitable contracts. It also cites work extending the analysis to repo financing and collateral. These conclusions depend on model structure, payoff properties, and homogeneity; they are not a general sign rule for every derivative, funding arrangement, or multidimensional underlying.
Key ideas
- A derivative’s hedge cash balance is its value minus the value of its delta holdings.
- In a one-asset diffusion model, the cash balance sign can inherit a sign property from the terminal payoff expression.
- For prices homogeneous in asset price and strike, Euler’s theorem relates the cash balance to strike sensitivity.
- Breeden-Litzenberger relationships connect strike sensitivity to risk-neutral probabilities.
- The sign conclusions rely on assumptions about the pricing model, payoff, and financing setup.
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Full text
# Cash balance sign in hedging portfolio
# Cash balance sign in hedging portfolio
Consider a derivative which depends on $n$ assets with price vector $X=(S^1,\dots,S^n)$. The derivative value $V_t$ is given by the function $v(t,S)$, so that the hedge ratios for the hedging portfolio are given by $\partial_iv(t,S)$ for each asset $S^i$ for $i=1,\dots,n$.
Is there anything we can say in general about the sign of $V_t-\sum_i\partial_iv(t,S)S^i_t$? If nothing, what additional structure do we need to equip the problem with in order to do so?
The motivation is that the quantity of interest is usually the value of the cash balance in a hedging portfolio. If its sign was known to be constant throughout, a potential differential between deposit and funding rates would not matter because we would only be paying either.
## Answer by Daneel Olivaw (score 0, accepted)
https://quant.stackexchange.com/a/71973
Bergman (1995) and Korn (1995) analyse the question in a diffusion setting, with Mercurio (2014) expanding their results.
For example, Proposition 3 in Bergman (1995) establishes that in a diffusion setting à la Black-Scholes where the derivative contract has a terminal payoff function $h$ and is written on some unidimensional asset price $S$, then $v(t,s)-sv_s(t,s)$ has same sign that $h(s)-sh^\prime(s)$ where $v_s$ is the partial derivative w.r.t. to $s$. This is based on a result from Bergman (1994, equation 24) which establishes the Black-Scholes $v$ function inherits properties from its boundary function $h$: $$v(t,S_t)-S_tv_s(t,S_t)=e^{-r(T-t)}\mathbb{E}(h(S_T)-S_Th^\prime(S_T)|\mathscr{F}_t)$$ where $h^\prime$ is a generalized function. Mercurio (2014) expands this result to a model with repo financing and collateral.
Assuming the derivative contract depends on some strike $k$, such as an option, it is possible to derive similar results for valuation functions $v$ which are homogeneous of degree 1 in both $s$ and $k$. This is because homogeneity implies that $v(t,s,k)=sv_s(t,s,k)+kv_k(t,s,k)$ by Euler's theorem for homogeneous functions, thus the cash balance is equal to $kv_k(t,s,k)$, then we can use Breeden-Litzenberger formulas which establish a link between the partial derivative $v_k$ and risk-neutral probabilities - for which the sign is known. Merton (1973) establishes conditions under which a derivative price is homogeneous, see e.g. Theorem 9.
References
Bergman, Yaacov (1994). "General Restrictions on Contingent Claim Prices When Those Solve a PDE", working paper, Hebrew University.
Bergman, Yaacov (1995). "Option Pricing with Differential Rates", Review of Financial Studies, Vol. 8, No. 2, pp. 475-500.
Korn, Ralf (1995). "Contingent Claim Valuation in a Market with Different Interest Rates", Mathematical Methods of Operations Research, Vol. 42, pp. 255-274.
Mercurio, Fabio (2014). "Differential Rates, Differential Prices", Risk, Vol. 27, No. 1, pp. 100-105.
Merton, Robert (1973). "Theory of Rational Option Pricing", Bell Journal of Economics and Management Science, Vol. 4, No. 1, pp. 141-183.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.