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Delta Hedging a Black–Scholes Call with Stock and Bonds

Article Quant Q&A · Author: frank
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Summary

This exchange explains how to form the initial replicating portfolio for a European call in the Black–Scholes model. The call’s stock delta gives the number of shares to hold, while the bond position supplies the remaining value needed to match the option price. Because that bond position is negative in the setup described, it represents borrowing money or taking a short position in the bond rather than an impossible holding.

The evidence is a brief conceptual answer: negative holdings are implemented through borrowing or short selling. The exchange does not derive the Black–Scholes formula, discuss rebalancing as the option delta changes, or address trading frictions, collateral, and short-sale constraints. Its scope is the interpretation of the initial bond position in a frictionless model.

Key ideas

  • The stock position in the initial replicating portfolio equals the call option’s delta.
  • The bond position makes the combined portfolio value equal the call price.
  • A negative bond holding represents borrowing or shorting the bond.
  • The explanation addresses initial portfolio construction, not ongoing hedge rebalancing.

Tags

OptionsDerivatives pricingFixed income
Full text
# Initial holdings of bonds with delta hedging (Black Scholes model)


# Initial holdings of bonds with delta hedging (Black Scholes model)












Consider the Black Scholes model so $$dS_t = \mu S_t dt + \sigma S_t dW_t, \;\;\; dB_t = rB_t dt$$ I want to delta hedge an European call option with strike price $K$ and strike time $T$. It is known that the price of the option at $t=0$ is $C_0 = S_0\Phi(d_1) - \exp(-rT)K\Phi(d_2)$ where $d_1, d_2$ are well known but not relevant for my question.

Now the delta of the stock is $\Phi(d_1)$, so my initial portfolio consists of $\Phi(d_1)$ shares in $S$ the portfolio value is then $S_0\Phi(d_1)$. But as I understand we want the initial portfolio value ($V_0$) equal to $C_0$. But that would mean we need to put another $- \exp(-rT)K\Phi(d_2)/B_0$ bonds in the portfolio so that $V_0 = C_0$. This confuses me. How can we buy a negative amound of bonds? What am I doing wrong?

## Answer by alec (score 3)

https://quant.stackexchange.com/a/21036

It just simply means you have to borrow money

## Answer by mbison (score 0)

https://quant.stackexchange.com/a/20984

Holding a negative amount is also known as short sale.

https://en.wikipedia.org/wiki/Short_(finance)

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