Zero-Volatility Call Pricing and Replicating Hedges
Summary
The document explains how a European call behaves in the Black–Scholes setting when volatility is zero. With no dividends, the stock follows a deterministic path growing at the risk-free rate. The call’s payoff is therefore known at expiry; discounting that payoff at the same rate gives its fair value today. In the stated example, the stock and strike both begin at 100, and the annual rate is 5%, producing a positive but discounted call value.
Because the expiry payoff is certain in this limiting case, the seller can hedge it by holding the required amount in a risk-free bond. The answer contrasts this with nonzero volatility, where the hedge uses stock and a bond and must be adjusted as the option’s delta changes. It cites the Black–Scholes delta hedge and its link to the pricing equation. The discussion assumes the model’s idealized conditions and describes continuous adjustment conceptually; it does not address transaction costs, discrete rebalancing, or market frictions.
Key ideas
- At zero volatility, the stock path and the call payoff are deterministic under the stated assumptions.
- The call value equals its known expiry payoff discounted at the risk-free rate.
- A bond position can replicate the certain payoff in the zero-volatility case.
- With nonzero volatility, the seller uses a stock and bond hedge that changes with the option delta.
- The explanation relies on idealized Black–Scholes assumptions and omits real-world trading frictions.
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# Answer by Kevin (score 6, accepted)
# If the volatility is zero (i.e. σ=0), what is the call worth? After valuing the call, how to hedge the call (assuming you sold it)
> Question: All Black-Scholes assumptions hold. Assume no dividends. The stock price is $100. The riskless interest rate is 5% per annum. Consider a one-year European call option struck at-the-money (i.e. strike equals current spot). $(1)$ If the volatility is zero (i.e. σ=0), what is the call worth? $(2)$ After valuing the call, how to hedge the call (assuming you sold it).
My attempt to $(1)$:
Since volatility is zero, it means that return does not deviate from riskless return, that is, $$$100 \times 1.05 = $105.$$ So the call worth $\$105.$
But I have no idea on how to hedge the call.
Any idea would be appreciated.
## Answer by Kevin (score 6, accepted)
https://quant.stackexchange.com/a/48795
If $\sigma=0$, the stock price is deterministic and grows at rate $r$. In one year, it is thus worth $100\cdot e^{0.05}\approx 105.13$. The strike is $K=100$. Your payoff is thus $5.13$. Discounting at rate $r$, you get as today’s fair option price $5.13\cdot e^{-0.05}\approx4.88$. Note that there is no randomness and the stock price is perfectly predictable.
Hedging such a known payoff can be done by simply investing money into a bond. More interestingly, if $\sigma\neq0$, then there is no static hedge and you need to dynamically hedge the option. Black and Scholes (1973) show that the portfolio $C-\Delta S$ is locally risk-free and hence equals the risk-free bond. Here, $\Delta=\frac{\partial C}{\partial S}$. This gives you a way of hedging the call option by investing in the stock and a (default free zero-coupon) bond (which matures when the option expires). From that relationship, Black and Scholes (1973) also derive their famous PDE which gives a way of finding a closed-form solution for the option price. In a nutshell: when hedging, you replicate the payoff of the derivative. For options, you need to continuously adjust your hedging portfolio (because $\Delta$ keeps changing).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.