Understanding the Stock and Cash Holdings in a Call-Based Insured Portfolio

Summary

The document examines how to interpret the stock and money-market components of a portfolio formed by holding a call and discounted cash equal to the strike. Under Black–Scholes assumptions, it combines the call price with the present value of the strike and rewrites the portfolio value using the Gaussian cumulative distribution function evaluated at the option’s two standard terms. The author asks whether those terms directly represent the proportions of capital to allocate to stock and cash.

This is a conceptual question about translating a pricing decomposition into portfolio holdings. The document supplies the pricing formula and its algebraic rearrangement, but no answer, worked allocation, or numerical evidence. It also does not distinguish portfolio value weights from the number of shares and cash units required to replicate the payoff. Its conclusions therefore remain unresolved, and the stated interpretation should not be treated as established guidance.

Key ideas

• A call plus discounted strike cash is presented as a candidate insured portfolio.
• The document rewrites the Black–Scholes call and cash value using normal cumulative distribution terms.
• It questions whether those terms directly give stock and money-market allocation proportions.
• The text does not provide a worked replication or resolve the distinction between holdings and capital weights.
• Its setup assumes no arbitrage and constant interest rates.

Tags

# Insured Portfolio via call + cash: how much cash?


# Insured Portfolio via call + cash: how much cash?












I am unsure about the quantities to keep in the risky asset, S, and the non-risky asset, M, when constructing an insured portfolio via Call + Cash (rather than Stock + Put). My understanding so far is below.

In a Black-Scholes framework trading 2 unrelated securities S (stock) and M (money market) and assuming no arbitrage and constant interest rates, we can construct an insured portfolio via

$$C(t) + K e^{-r(T-t)}$$

for a call maturing at $T$ with strike $K$.

Using the formula $C(t) = S(t)N(d_1) - K e^{-r(T-t)}N(d_2)$, when we substitute into the above, we get

$$ P(t) = C(t) + K e^{-r(T-t)} = S(t)N(d_1) + K e^{-r(T-t)}(1 - N(d_2)) $$

where N(x) is the Gaussian CDF. My understanding is that $N(d_1)$ is the amount of shares to buy (as a proportion of capital) and $(1 - N(d_2))$ is the amount to keep in M. Are these quantities correct? What are the quantities I need to keep in S and M?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.