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Black–Scholes Put Replication and the Value of the Hedge Portfolio

Article Quant Q&A · Author: emcor
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Summary

The document clarifies how a Black–Scholes put can be replicated when interest rates are zero. The put has a negative delta, so the hedge involves shorting shares and placing the corresponding funds in bonds. The bond holding can exceed the put’s price without implying that the replicating portfolio is worth more than the option: the short stock position has negative value and must be included in the portfolio total.

Using the standard put price and delta expressions, the response shows that the bond position equals the strike-weighted probability term, while the portfolio value combines that bond amount with the liability from the short shares. At inception, the combined value matches the put price. The explanation addresses the apparent mismatch between bond investment and option price, but assumes zero interest and does not discuss rebalancing through time or practical trading costs.

Key ideas

  • A Black–Scholes put has negative delta, leading to a short stock position in its replicating hedge.
  • The hedge also holds bonds, and the bond amount can be greater than the put premium.
  • The short shares contribute negative value to the combined hedge portfolio.
  • At inception, the bond and stock positions together equal the put price under the stated zero-interest assumption.

Tags

OptionsDerivatives pricingRisk management
Full text
# How to hedge a put under the Black-Scholes model?


# How to hedge a put under the Black-Scholes model?












To hedge a call, one would invest the option price proceeds into $\Delta_t*S_t + B_t = c_t$. (ok)

However, a put has negative delta, so I would short $\Delta_t*S_t$ and invest $p_t+\Delta_t*S_t>p_t$ into a risk-free bond?

It just seems a bit odd to me that I would invest more than the actual put price into a bond.

## Answer by Gordon (score 2, accepted)

https://quant.stackexchange.com/a/18368

Assuming zero interest, the put option has the price \begin{align*} KN(-d_2)-S_0N(-d_1), \end{align*} and delta $-N(-d_1)$. When $N(-d_1)$ units of stocks are shorted and invested in bonds, the total value in bonds is $KN(-d_2)$, which is indeed greater than the option price. However, as you have shorted $N(-d_1)$ units of stocks, your portfolio value is \begin{align*} KN(-d_2) -N(-d_1) S_t \end{align*} at time $t$. That is, the portfolio value is not necessarily greater than the option price, and at the deal inception, is the same as the option price.

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.