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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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  • $\begingroup$ Are you hedging statically (by buying actual calls) or dynamically (replicating the calls by dynamic hedging) ? $\endgroup$ – noob2 May 15 at 9:36
  • $\begingroup$ I won't be buying any calls. I'm buying S(t) in quantity $N(d_1)$ and putting cash in a money market account in quantity $(1 - N(d_2))$. I'm unsure if these quantities are correct. I guess this means replicating the calls by dynamic hedging? $\endgroup$ – s5s May 15 at 14:07

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