This technique enables the creation of a synthetic hedged portfolio that provides a risk-free payoff and that risk-free payoff must be discounted at a risk-free rate. Risk-free payoff means, regardless of any fluctuation or any value it takes (asset price goes up or down), the payoff from the hedged portfolio must remain the same.

"As per the no-arbitrage principle, if a portfolio is risk-less (i.e., has a certain payoff),

it should be priced to yield risk-free return."

Please refer to the introductory article to understand the basics of-

Let's take an example to understand the delta-hedging approach-

Imagine a 1-Year European call option on an asset at a strike price of $450. Stock is presently trading at $500. In 1-Year time frame, the asset price can either take $625 on the upside or $400 on the downside.

This is derived on the basis that the estimated volatility is 22.315% in 1-Year time.

Therefore,

Up-move factor (u) = e^SIGMA*SQRT(T) = e^0.22135*SQRT(1) = 1.25

Down-move factor (d) = 1 / u = 1 / 1.25 = 0.8

The risk-free interest rate prevailing in the market is 6% p.a compounded continuously.

### In the case of the Call Option,

Let's suppose, we have already written a call option and to delta hedge the short call exposure from the upside, we purchase the equivalent number of shares (i.e., delta shares).

Sensitivities: Option Delta is the amount by which an option value is expected to change with respect to a $1 change in the underlying asset price, other factors being constant.

Call Delta = Change in Call Option Price / Change in Asset Price

(also known as hedge ratio)

Hedge Ratio (h) = ( $175 - $0 ) / ( $625 - $400 ) = 0.7777

This means, if asset price changes by $1, call option price is expected to change by $0.7777 and therefore, we purchase 0.7777 shares (instead of one share) for every one unit of call option to delta-neutralise the exposure and to construct a hedged portfolio.

To check if the hedged portfolio constructed above is indeed risk-free, we have-

The payoff on maturity-

If the call option lands ITM, the up-move Payoff = 0.7777 Long Stock & 1 Short Call

= 0.7777*$625 – $175 = $311.1111

(gain on stock but lose option payoff)

If the call option lands OTM, the down-move Payoff = 0.7777 Long Stock & 1 Short Call

= 0.7777*$400 – $0 = $311.1111

(lose on stock but option lapses)

Therefore, the hedged portfolio constructed above is indeed risk-free as it has a certain payoff of $311.1111 after 1-Year and hence, it should yield a risk-free return-

Price of the hedged portfolio today = $311.1111 / 1.06184 = $292.9925

Using Delta Hedging Approach,

The price of a delta hedge portfolio today is $292.9925 which consists of 0.7777 underlying stocks and 1 short call option and that should be equivalent to the present value.

(i.e., 0.7777 Long Stock & 1 Short Call = $292.9925)

Hence,

0.7777 * $500 & 1 Short Call = $292.9925

1 Short Call = $292.9925 - 0.7777 * $500 = $95.8970

Call Option Price = $95.8970

### In the case of the Put Option,

Let's suppose, we have already bought put option and to delta hedge the same i.e., hedge the risk of put price falling due to a rise in the price of the underlying asset, we purchase the equivalent number of shares (i.e., delta shares).

Sensitivities: Option Delta is the amount by which an option value is expected to change with respect to a $1 change in the underlying asset price, other factors being constant.

Put Delta = Change in Put Option Price / Change in Asset Price

(also known as hedge ratio)

Hedge Ratio (h) = ( $0 - $50 ) / ( $625 - $400 ) = -0.2222

This means, if asset price changes by $1, put option price is expected to change by $0.2222 and therefore, we purchase 0.2222 shares (instead of one share) for every one unit of put option to delta-neutralise the exposure and to construct a hedged portfolio.

To check if the hedged portfolio constructed above is indeed risk-free, we have-

The payoff on maturity-

If the put option lands OTM, the up-move Payoff = 0.2222 Long Stock & 1 Long Put

= 0.2222*$625 + $0 = $138.8889

(gain on stock but option lapses)

If the Put option lands ITM, the down-move Payoff = 0.2222 Long Stock & 1 Long Put

= 0.2222*$400 + $50 = $138.8889

(lose on stock but option payoff)

Therefore, the hedged portfolio constructed above is indeed risk-free as it has a certain payoff of $138.8889 after 1-Year and hence, it should yield a risk-free return-

Price of the hedged portfolio today = $138.8889 / 1.06184 = $130.8002

Using Delta Hedging Approach,

The price of a delta hedge portfolio today is $130.8002 which consists of 0.2222 underlying stocks and 1 long put option and that should be equivalent to the present value.

(i.e., 0.2222 Long Stock & 1 Long Put = $130.8002)

Hence,

0.2222 * 500 & 1 Long Put = $130.8002

1 Long Put = $130.8002 - 0.2222 * $500 = $19.6890

Put Option Price = $19.6890

As per Put-Call Parity, the expected payoff from protective put should be equivalent to that of fiduciary call (i.e., Put Price + Stock Price = Call Price + PV of Strike Price)

Therefore,

Payoff from Protective Put = $500 + $19.6890 = $519.6890

Payoff from Fiduciary Call = ( $450 / 1.06184 ) + $95.8970 = $519.6890

This means, Delta Hedging Approach of Binomial Option Price Model does a good job!

Understand the fact that, options are a non-linear derivative which means the delta of an option is not constant. As the underlying price moves, the probability that the option will land ITM increases/decreases (due to moneyness), and therefore, the delta changes. Hence, the hedged portfolio constructed in both cases needs to be continuously rebalanced as "h" keeps changing.

In other words,

Option hedging is dynamic, not static!

Every moment rebalancing would require continuous churning the portfolio by dynamic rebalancing or self-financing. This means that in the case of a call option, as the share price rises, the probability that the option will land ITM increases (delta rises) and we have to buy more shares. As share price falls, the probability that the option will land ITM decreases (delta falls) and we have to sell shares. So, we are buying when the asset prices are high and selling when the asset prices are low to replicate the call. The present value of the losses in doing the same is essentially the call price today.

And therefore, the market has to be open at all times. Churning would also require a frictionless market- no transaction cost, no taxes, and no regulatory constraints, the asset is marketable, unlimited borrowing/investing funds at a risk-free rate.