Delta Hedging a Zero-Volatility Stock and Call Option

Summary

The document considers whether dynamically delta-hedging a call can generate gamma profits when the underlying stock has positive drift but zero volatility. Its answer applies the Black–Scholes setting’s no-arbitrage condition: a riskless stock must grow at the risk-free rate. The stock’s terminal price is therefore deterministic, and its relation to the strike determines whether the call finishes out of the money or in the money.

With a known terminal outcome, the hedge is also fixed: zero shares if the call expires worthless, or a short position of one share if exercise will deliver stock to the call holder. The answer characterizes delta as binary and gamma as zero, since delta does not change as the deterministic stock price evolves. Thus, under these assumptions, the proposed large gamma gains do not arise, and hedge frequency does not create them. The conclusion depends on zero volatility and no-arbitrage-consistent drift; a positive drift different from the risk-free rate would conflict with the riskless-stock premise.

Key ideas

• In the Black–Scholes no-arbitrage setting, a zero-volatility stock must grow at the risk-free rate.
• With a deterministic stock price, comparing its terminal value with the strike determines the call’s outcome.
• The hedge is zero shares when the call expires worthless and one share short when exercise is certain.
• Delta is constant along the deterministic path, so gamma is zero in this setup.
• The conclusion depends on zero volatility and a drift consistent with the risk-free rate.

Tags

# Delta-hedging non-volatile stock


# Delta-hedging non-volatile stock












If a stock has zero vol and some positive drift $\alpha$ (in a BS-setting) and we delta hedge a long call option dynamically over a year with some positive implied volatility.... how would that work out for us?

Would the answer depend on how often we hedge, i.e. every day, month, week?

In usual circumstances, when we delta hedge, we make money everytime we get significant $\Delta S$s, due to the call options $\Gamma$ (gamma).

However in the case of positive $\alpha$, zero $\sigma$, $\Delta S$ is going to be huge, but .... do we then make huge amounts of money off of $\Gamma$?

## Answer by mbison (score 2)

https://quant.stackexchange.com/a/33227

If vol is zero, then the stock S is riskless. If it is riskless its drift should equal the risk free rate r. Therefore, if T is the expiry of your call option then at expiry value of stock should be S_T = S_0 exp(rT).

If your strike K <= S0 exp(rT) then you know for sure your call will expire worthless and your delta hedge = 0 stocks. If K > forward, then for sure you will get stock delivered at expiry. So your delta hedge is 1 stock short.

Note that your delta is binary depending on K. The gamma however is 0 because the delta never changes. (had gamma been different from 0, your vol would not be 0).

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