A standard DCF technique is a valuation method used to estimate the value of any financial instrument by taking the present value of expected cash flows discounted using a certain discount rate.

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

If a financial product is risk-free, we use a risk-free interest rate to discount the expected cash flows but if a financial product involves risk then we have to use a risk-adjusted rate which is subject to a never-ending dispute whether to use the Capital Asset Pricing Model or Multi-Factor Pricing Model.

In the case of equity, the expected cash flows are discounted at a risk-adjusted rate (i.e., risk-free rate plus risk premium). In the case of bonds, the expected cash flows are discounted at the YTM rate on similar bonds available in the market.

**Let's take an example to understand the risk-neutralization 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^0.22315*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.

The risk Neutralisation Approach states that the expected payoff from the option is discounted at a risk-free interest rate provided that the expected payoff calculated is by using the risk-neutral probabilities. Thus, the risk-neutral probabilities are not subjective/judgemental instead these probabilities are derived in such a manner that the expected return from all the risky assets is indeed risk-free.

P1 * S * u + ( 1 - P1) * S * d = S * e^r*t

P1 * 625 + ( 1 - P1) * 400 = 500 * 1.06184 --(solve for P1)

Therefore,

Risk-neutral Probability (P1) = R â€“ d / u â€“ d = (1.06184 â€“ 0.8) / (1.25 â€“ 0.8) = 0.58187

Risk-neutral Probability (P2) = 1 â€“ P1 = 1 â€“ 0.58187 = 0.41813

Where,

Risk-free interest rate factor (R) = e^r*t = e^0.06*1 = 1.06184

Using Risk-Neutralisation Approach,

Option Price = Expected Payoff / Risk-free Interest Rate Factor

Where,

Expected Payoff is calculated using Risk-Neutral Probabilities

(i.e., up-move payoff * up-move probability + down-move payoff * down-move probability)

### In the case of Call Option, the payoff on maturity-

If the call option lands ITM, the up-move Payoff = S1 - Strike Price = $625 - $450 = $175

(option exercised)

If the call option lands OTM, the down-move Payoff = 0

(option lapses)

Therefore,

Expected Payoff = $175 * P1 + $0 * P2 = $175 * 0.58187 + $0 * 0.41813 = $101.82725

Call Option Price = Expected Payoff / R

= $101.82725 / 1.06184

= $95.8970

### In the case of Put Option, the payoff on maturity-

If the put option lands OTM, the up-move Payoff = 0

(option lapses)

If the put option lands ITM, the down-move Payoff = Strike Price - S2 = 450 - 400 = 50

(option exercised)

Therefore,

Expected Payoff = $0 * P1 + $50 * P2 = $0 * 0.58187 + $50 * 0.41813 = $20.9065

Put Option Price = PV of the Expected Payoff = Expected Payoff / R

= $20.9065 / 1.06184

= $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, Risk-Neutralisation Approach of Binomial Option Price Model does a good job!

Unlike ** Replicating Portfolio Approach** and

**, this approach is very simple and easy to price an option. Although the risk-neutral probabilities are not the true probabilities of up-move and down-move of the underlying asset (as these probabilities are calculated based on the assumption that- risky instrument will generate risk-free cash flow) but somehow it manages to produce the correct option price.**

__Delta-Hedging Approach__This is because, to price any risky asset, we got to take the risk-adjusted rate as the discount rate to calculate the present value of the expected future cash flows. However, the risk-adjusted rate is not available (as it is subject to never-ending dispute) and we are forced to take a risk-free rate. In that case, the expected payoff calculated for option pricing should also be the expected payoff in the risk-neutral world and to ensure that, we take risk-neutral probabilities being not subjective/judgemental instead calculated in such a manner that expected return from all risky assets managed to generate a risk-free return.

On the other hand, if the risk-adjusted rate is available by any means and we were to calculate the option price using the risk-adjusted rate then the expected payoff will also be calculated considering the risk-averse world having risk-averse probabilities. In that case, the probabilities will accordingly change as P1 is the positive function of R and therefore, [ P1 = R â€“ d / u â€“ d ] would also change (higher R implies higher P1). The ultimate results of call and put option price in both- the risk-neutral world and the risk-averse world will be the same.

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