Equivalent Replication and Hedging in One-Period Binomial Option Pricing
Summary
The note explains why alternative introductions to one-period binomial option pricing, such as a covered-call argument and a replicating-portfolio construction, lead to the same valuation. Both set up positions in the underlying stock and the option so their payoffs match across possible outcomes; no choice of presentation changes the theoretical price when the assumptions are the same.
It describes delta hedging as offsetting stock-price exposure and distinguishes it from vega hedging, which targets volatility exposure. The risk-free asset is part of the replication framework because it supplies the financing component needed alongside the risky asset to match the option’s payoff and express discounted value. The answer gestures toward the martingale rationale and a volatility-hedging extension in a Heston setting, but provides little derivation. It is an introductory explanation rather than a full proof or treatment of practical hedging costs and model limits.
Key ideas
- A covered-call derivation and a replicating-portfolio derivation are alternative presentations of the same binomial pricing logic.
- Delta hedging offsets exposure to movements in the underlying asset price.
- Vega hedging targets sensitivity to volatility rather than stock price.
- The risky asset and risk-free account together provide the components needed to replicate an option payoff.
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# Do different hedging strategies affect the theoretical pricing of options in one period binomial model? # Do different hedging strategies affect the theoretical pricing of options in one period binomial model? I just started my financial maths master and was introduced to binomial option pricing for European options. I am slightly confused by the derivation as I saw a different version. Some straightly get into "we want to replicate an option with a riskless and a risky asset" while others use a covered call to introduce the derivation. - I am wondering if different hedging strategies (I am not entirely sure it is hedging) affect the valuation of the option. - When replicating a portfolio for an option, why do we need a risky and a riskless asset at the same time? can we just use either one? ## Answer by THATS MY QUANT MY QUANTITATIVE (score 1) https://quant.stackexchange.com/a/77002 They are both doing the same thing. 1 derivation is constructing a portfolio containing shorted calls and longed stocks (or the reverse), the other is setting up a replicating portfolio. They both achieve the same result. There are different types of hedging strategies. The one you are currently learning is to be "delta-neutral", which is hedging the price of the stock. So if S goes from 20 to 10, you would have not lost money because you profited from the call options you sold. There are other types of hedging like Vega hedging, which is hedging the volatility of a stock. To your second question, we have a risk-free account because the discounted call option is a martingale. More intuitively, the call option requires money and you are looking at the difference of your bankroll going up and down versus how much the call option is. If you look at the derivation of the Heston PDE, it's similar to what you would have done in class with the black-scholes, but we have a new account to hedge the volatility, so the portfolio has the value: $$\Pi = V + \Delta S + \phi U$$ https://www.frouah.com/finance%20notes/The%20Heston%20model%20short%20version.pdf If you want a more technical explanation of why we require a risk-free account, try to prove that the call option option (without discounting) is a martingale (it won't be possible).
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