Tutorial on Option Pricing Using Risk-Neutral Valuation

Introduction

• The tutorial describes the methodology to price an option using the risk-neutral valuation approach.

Stock Price Dynamics

• Current stock price: ( S_0 )
• Over time period ( T ):
  • Stock price could go up to ( SU ) with probability ( Q )
  • Stock price could go down to ( SD ) with probability ( 1 - Q )

Option Contract Details

• Strike price: ( K )
• Time to maturity: ( T )
• Discount rate: ( r ) percent (continuously compounded)
• Methodology applies to both continuously and discretely compounded rates

Payoff Scenarios

• When stock price goes up to ( SU ):
  • Option payoff: ( P_U )
• When stock price goes down to ( SD ):
  • Option payoff: ( P_D )

Expected Value Calculation

• At time ( T ), two possible states:
  • Up State: Stock price goes up to ( SU )
  • Down State: Stock price goes down to ( SD )
• Expected value of terminal stock price ( S_T ):
  • ( E[S_T] = SU \times Q + SD \times (1 - Q) )

Risk-Neutral World

• No risk premium; investors indifferent to risk
• Expected return on any security: risk-free rate
• Expected value of stock price at time ( T ) equals current stock price compounded at risk-free rate over time period ( T )

Risk-Neutral Probability ( Q )

• Formula to solve for ( Q ):
  • ( Q = \frac{S_0 \times e^{rT} - SD}{SU - SD} )

Expected Future Payoff of Option

• ( E[Payoff] = P_U \times Q + P_D \times (1 - Q) )

Option Pricing in Risk-Neutral World

• Discount rate is risk-free rate
• Value of option = Expected payoff discounted at risk-free rate
• Formula for option price:
  • ( Price = \frac{(P_U \times Q + P_D \times (1 - Q))}{e^{rT}} )

Conclusion

• Learned to use risk-neutral valuation approach to price an option in a discrete-time framework
• Open for questions or comments