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Risk–Neutral Approach: Option Pricing Using Binomial Model | Option Derivative Pricing

Introduction

Binomial Model is a very simple and most frequently used for pricing option derivatives. Simple is because of the unrealistic assumptions taken and most frequently because not too much of mathematics as compared to the Black Scholes Merton Model. However, this model can become complex in case of multi-period option pricing (called Multi-Period Binomial Model).



Risk–Neutral Approach: Call Option Pricing Using Binomial Model


Let us understand how to calculate price of a CALL option using an example-

Consider an European CALL option on an underlying asset (stock price) at a strike price of $500, which is presently trading at $480. In 1-year time, price of a stock can either be $625.00 or $368.64. Risk-free interest rate is 6% compounded continuously.


Under Binomial Model, stock price on expiry is certain and can take any of the two values only (in our case, $625.00 or $368.64). Since the payoff is certain, price of a CALL option is calculated using risk-free interest rate so to get the present value of the payoff.


Price of a CALL option on expiry if stock price happens to be $625.00 = Intrinsic value of CALL option (Cu)

Cu = Stock price – Strike price = $625 - $500 = $125

Price of a CALL option on expiry if stock price happens to be $368.64= Intrinsic value of CALL option (Cd)

Since CALL option lapses, Cd = $0


Also, to calculate price of a CALL option using Risk-Neutral Approach of Binomial Model, risk-neutral probabilities must be identified.


Risk-neutral probability for the up move (P1) = ( R – d ) / ( u – d )

Therefore, P1 = ( 1.0583 - 0.7680 ) / ( 1.3021 - 0.7680 ) = 0.5435


Risk-neutral probability for the down move (P2) = 1 – Risk-neutral probability for the up move.

Therefore, P2 = 1 - 0.5501 = 0.4565


Where,

R = periodic compounding risk-free interest rate = e0.06 x 1 = 1.0583

u = up move return factor = $625 / $480 = 1.3021

d = 1 / u = 1 / 1.3021 = 0.7680


Hence, price of a CALL option using RiskNeutral Approach of Binomial Model is calculated by–


Price of CALL option (C0) = ( P1 x Cu + P2 x Cd ) / R

Therefore, C0 = ( 0.5435 x $125 + 0.4565 x $0 ) / 1.0583 = $64.20



Risk–Neutral Approach: Put Option Pricing Using Binomial Model


Let us understand how to calculate price of a PUT option using an example-

Consider an European PUT option on an underlying asset (stock price) at a strike price of $500, which is presently trading at $480. In 1-year time, price of a stock can either be $625.00 or $368.64. Risk-free interest rate is 6% compounded continuously.


As we know, under Binomial Model, stock price on expiry is certain and can take any of the two values only (in our case, $625.00 or $368.64). Since the payoff is certain, price of a PUT option is calculated using risk-free interest rate so to get the present value.


Price of a PUT option on expiry if stock price happens to be $625 = Intrinsic value of PUT option (Pu)

Since PUT option lapses, Pu = $0

Price of a PUT option on expiry if stock price happens to be $368.64= Intrinsic value of PUT option (Pd)

Pd = Strike price – Stock price = $500 - $368.64= $131.36

Also, to calculate price of a PUT option using Risk-Neutral Approach of Binomial Model, risk-neutral probabilities must be identified.

Risk-neutral probability for the up move (P1) = ( R – d ) / ( u – d )

Therefore, P1 = ( 1.0583 - 0.7680 ) / ( 1.3021 - 0.7680 ) = 0. 5435

Risk-neutral probability for the down move (P2) = 1 – Risk-neutral probability for the up move.

Therefore, P2 = 1 - 0.5501 = 0. 4565


Where,

R = periodic compounding risk-free interest rate = e0.06 x 1 = 1.0583

u = up move return factor = $625 / $480 = 1.3021

d = 1 / u = 1 / 1.3021 = 0.7680


Hence, price of a PUT option using RiskNeutral Approach of Binomial Model is calculated by–


Price of PUT option (P0) = ( P1 x Pu + P2 x Pd ) / R

Therefore, P0 = ( 0. 5435 x $0 + 0. 4565 x $131.36 ) / 1.0583 = $56.67

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