# 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?

• Are you hedging statically (by buying actual calls) or dynamically (replicating the calls by dynamic hedging) ? – noob2 May 15 at 9:36
• 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? – s5s May 15 at 14:07