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 – d / u – d = (1.06184 – 0.8) / (1.25 – 0.8) = 0.58187

Risk-neutral Probability (P2) = 1 – P1 = 1 – 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 Delta-Hedging Approach, 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.

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 – d / u – 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.