This technique enables the creation of a replicating portfolio that provides a payoff equivalent to a payoff from an option at expiration.

"As per the no-arbitrage principle, If two portfolios have an identical payoff, the price of both the portfolio should be same."

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

This replicating portfolio can be constructed using any form of asset. In the case of equity, we purchase/sell an equivalent number of shares (i.e., delta shares) by borrowing/investing the funds at a risk-free interest rate until expiration.

At expiration, regardless of any fluctuation or any value, it takes (i.e., asset price goes up or down), the payoff from the replicating portfolio must remain the same. If the cash flows of both- a call option and the replicating portfolio are the same, the price of both the portfolios should be equal.

**Let's take an example to understand the replicating portfolio 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, to replicate a long call option, we purchase the equivalent number of shares (i.e., delta shares) by borrowing the funds at a risk-free interest rate until expiration.

**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, the 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 a call option by borrowing the funds at a risk-free rate to delta-neutralize the exposure and to construct a replicating portfolio to call option.

The amount which is required to be borrowed-

If the share price takes to $625 on maturity,

The payoff from Long Stock - Repayment of Borrowings with Interest = Payoff from Call

0.7777 * $625 – $Funds * 1.06184 = $175.00 --(call option exercised)

Therefore,

$Funds * 1.06184 = $175.00 - 0.7777 * $625

$Funds = -$292.9925 (minus sign represents borrowing)

If the share price takes to $400 on maturity,

Payoff from Long Stock - Repayment of Borrowings with Interest = Payoff from CallOption

0.7777 * $400 – $Funds * 1.06184 = $0 --(call option lapses)

Therefore,

$Funds * 1.06184 = $0 - 0.7777 * $400

$Funds = -$292.9925 (minus sign represents borrowing)

Using Replicating Portfolio Approach,

The price of a call option today should be equivalent to the price of the replicating portfolio-

0.7777 * $500 – $292.9925 = $Call Option

Hence,

Call Option Price = 0.7777 * $500 – $292.9925 = 95.8970

### In the case of Put Option,

Let's suppose, to replicate a long put option, we short sell the equivalent number of shares (i.e., delta shares) and invest the proceeds at a risk-free interest rate until expiration.

**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, the put option price is expected to change by $0.2222, and therefore, we short sell 0.2222 shares (instead of one share) for every one unit of a put option and invest the proceeds at a risk-free rate to delta-neutralize the exposure and to construct a replicating portfolio to put option.

The amount which is required to be invested-

If the share price takes to $625 on maturity,

Investment Proceeds with Interest - Buyback of Stock = Payoff from PutOption

$Funds * 1.06184 - 0.2222 * $625 = $0 --(put option lapses)

Therefore,

$Funds * 1.06184 = $0 + 0.2222 * $625

$Funds = $130.8002 (plus sign represents investing)

If the share price takes to $400 on maturity,

Investment Proceeds with Interest - Buyback of Stock = Payoff from PutOption

$Funds * 1.06184 - 0.2222 * $400 = $50 --(put option exercised)

Therefore,

$Funds * 1.06184 = $50 + 0.2222 * $400

$Funds = $130.8002 (plus sign represents investing)

Using Replicating Portfolio Approach,

The price of a put option today should be equivalent to the price of the replicating portfolio-

$130.8002 - 0.2222 * $500 = $Put Option

Hence,

Put Option Price = $130.8002 - 0.2222 * $500 = 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, Replicating Portfolio 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 replicating 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 of 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 the 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.

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