02a Understanding Risk Neutral Valuation
Understanding Risk-Neutral
Valuation
Stephen M Schaefer
London Business School
March, 2012
Outline
• The no-arbitrage principle
• Arrow-Debreu (A-D) securities and market completeness • Valuing options with one period to maturity via replication using underlying asset and borrowing / lending replication using A-D securities risk neutral probabilities
• Valuing options with several periods to maturity
Understanding Risk Neutral Valuation
2
No-arbitrage pricing
Understanding Risk Neutral Valuation
3
Arbitrage (Definition)
• An arbitrage opportunity is one which:
a.Requires no invested capital
b.Provides a positive profit with 100% probability
• Or (slightly more generally)
a.Requires no invested capital,
b.Provides a positive profit with a positive probability and has a zero probability of a loss.
• Anyone who prefers more to less would engage in arbitrage because it represents “something for nothing”
• Therefore: in any competitive market there should be no arbitrage Understanding Risk Neutral Valuation
4
“No-Arbitrage Pricing”
• Although absence of arbitrage is simply a necessary requirement for equilibrium it is in some cases sufficient to allow us to price one security in terms of another
• Idea: assets / portfolios with the same cash flows in each state must have the same price
• Pricing via no-arbitrage is relative pricing: we calculate the price on one security in terms of the prices of others.
Understanding Risk Neutral Valuation
5
Example: Valuing a Call option
Understanding Risk Neutral Valuation
6
The next steps …
• We will use three approaches to value the same call option that matures in one period where the price of the underlying stock follows a binomial process
1. Replication with stock and bond
2. Replication with Arrow-Debreu (A-D) securities
3. “Risk-Neutral” valuation
• Later we see how to deal with an option that matures in more than one period in that case we will have to revise the replicating portfolio over time
this
Valuation
Stephen M Schaefer
London Business School
March, 2012
Outline
• The no-arbitrage principle
• Arrow-Debreu (A-D) securities and market completeness • Valuing options with one period to maturity via replication using underlying asset and borrowing / lending replication using A-D securities risk neutral probabilities
• Valuing options with several periods to maturity
Understanding Risk Neutral Valuation
2
No-arbitrage pricing
Understanding Risk Neutral Valuation
3
Arbitrage (Definition)
• An arbitrage opportunity is one which:
a.Requires no invested capital
b.Provides a positive profit with 100% probability
• Or (slightly more generally)
a.Requires no invested capital,
b.Provides a positive profit with a positive probability and has a zero probability of a loss.
• Anyone who prefers more to less would engage in arbitrage because it represents “something for nothing”
• Therefore: in any competitive market there should be no arbitrage Understanding Risk Neutral Valuation
4
“No-Arbitrage Pricing”
• Although absence of arbitrage is simply a necessary requirement for equilibrium it is in some cases sufficient to allow us to price one security in terms of another
• Idea: assets / portfolios with the same cash flows in each state must have the same price
• Pricing via no-arbitrage is relative pricing: we calculate the price on one security in terms of the prices of others.
Understanding Risk Neutral Valuation
5
Example: Valuing a Call option
Understanding Risk Neutral Valuation
6
The next steps …
• We will use three approaches to value the same call option that matures in one period where the price of the underlying stock follows a binomial process
1. Replication with stock and bond
2. Replication with Arrow-Debreu (A-D) securities
3. “Risk-Neutral” valuation
• Later we see how to deal with an option that matures in more than one period in that case we will have to revise the replicating portfolio over time
this