The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Stochastic Calculus Session 1
Option Pricing in Binomial Model
Peng Liu
Haas School of Business Universities of California, Berkeley
http://faculty.haas.berkeley.edu/peliu/MFE Masters in Financial Engineering, 2006
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Outline
1
The Two-Period binomial Model Dynamic Replication Approach Risk-Neutral Valuation Pricing Exotic Option In Binomial Model What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
2
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Dynamic Replication Approach Risk-Neutral Valuation
Outline
1
The Two-Period binomial Model Dynamic Replication Approach Risk-Neutral Valuation Pricing Exotic Option In Binomial Model What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
2
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Dynamic Replication Approach Risk-Neutral Valuation
Two-Period Dynamic Arbitrage: Example
European Plain-Vanilla Option - Call
Suppose Div = 0,S = 100, u = 1.25, d = 0.65, R = 1.05,T = 2,K = 105.
1
Price European Call using Dynamic Replication (Argue the value of this portfolio must be the arbitrage-free price of the call option.) Price European Call using Risk-neutral Valuation
156.25 s p X s X p XX81.25 100 s X 1-p Xs X X p 65 X 1-p Xs XXX X42.25 1-p Xs 125
2
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Dynamic Replication Approach Risk-Neutral Valuation
Two-Period Model: Dynamic Arbitrage
European Plain-Vanilla Option - Call
Questions: Dynamic Replication Approach:
What if the call expire in one period (Q3)? Why? What if the stock price dynamics as in Q4? Why?
Risk-Neutral Valuation:
What is Risk-Neutral Probability? Who chooses measure? Comparison and Conditions on Risk Neutral Valuation
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Dynamic Replication Approach Risk-Neutral Valuation
Two-Period Model: Dynamic Arbitrage
European Plain-Vanilla Option - Call
Questions: Dynamic Replication Approach:
What if the call expire in one period (Q3)? Why? What if the stock price dynamics as in Q4? Why?
Risk-Neutral Valuation:
What is Risk-Neutral Probability? Who chooses measure? Comparison and Conditions on Risk Neutral Valuation
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Dynamic Replication Approach Risk-Neutral Valuation
Two-Period Model: Dynamic Arbitrage
European Plain-Vanilla Option - Call
Questions: Dynamic Replication Approach:
What if the call expire in one period (Q3)? Why? What if the stock price dynamics as in Q4? Why?
Risk-Neutral Valuation:
What is Risk-Neutral Probability? Who chooses measure? Comparison and Conditions on Risk Neutral Valuation
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Dynamic Replication Approach Risk-Neutral Valuation
Two-Period Model: Dynamic Arbitrage
European Plain-Vanilla Option - Call
Questions: Dynamic Replication Approach:
What if the call expire in one period (Q3)? Why? What if the stock price dynamics as in Q4? Why?
Risk-Neutral Valuation:
What is Risk-Neutral Probability? Who chooses measure? Comparison and Conditions on Risk Neutral Valuation
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Dynamic Replication Approach Risk-Neutral Valuation
Outline
1
The Two-Period binomial Model Dynamic Replication Approach Risk-Neutral Valuation Pricing Exotic Option In Binomial Model What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
2
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Dynamic Replication Approach Risk-Neutral Valuation
Two-Period Dynamic Arbitrage: Example
European Plain-Vanilla Option - Put
Find the arbitrage-free price of a put option using three techniques:
Replication Risk-neutral valuation Put-call parity
Suppose the Put trades at the same price as call at date 0, what strategy would you adopt to lock in arbitrage profits?
Under what conditions the call and put should trade at the same price? How does strike affect option prices?
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Dynamic Replication Approach Risk-Neutral Valuation
Two-Period Dynamic Arbitrage: Example
American Option - Call and Put
Find the arbitrage-free price of an American Call/Put? What is the Early exercise premium you would be willing to pay for an American style option? It is never optimal to exercise an American Call early if the underlying stock pays no dividend. Why?
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Outline
1
The Two-Period binomial Model Dynamic Replication Approach Risk-Neutral Valuation Pricing Exotic Option In Binomial Model What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
2
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Types of Options
Call Option v.s. Put Option In the Money Option, Out the Money Option v.s. At the Money Option European Option v.s. American Option Plain-Vanilla Option v.s. Exotic Option
Lookback option (options on the max) Asian Option (options on the average) Digital Option/Binary Options Barrier Option Quanto (currency option) Shout Chooser (compound option)
Peng Liu MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Types of Options
Call Option v.s. Put Option In the Money Option, Out the Money Option v.s. At the Money Option European Option v.s. American Option Plain-Vanilla Option v.s. Exotic Option
Lookback option (options on the max) Asian Option (options on the average) Digital Option/Binary Options Barrier Option Quanto (currency option) Shout Chooser (compound option)
Peng Liu MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Types of Options
Call Option v.s. Put Option In the Money Option, Out the Money Option v.s. At the Money Option European Option v.s. American Option Plain-Vanilla Option v.s. Exotic Option
Lookback option (options on the max) Asian Option (options on the average) Digital Option/Binary Options Barrier Option Quanto (currency option) Shout Chooser (compound option)
Peng Liu MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Types of Options
Call Option v.s. Put Option In the Money Option, Out the Money Option v.s. At the Money Option European Option v.s. American Option Plain-Vanilla Option v.s. Exotic Option
Lookback option (options on the max) Asian Option (options on the average) Digital Option/Binary Options Barrier Option Quanto (currency option) Shout Chooser (compound option)
Peng Liu MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Types of Options
Call Option v.s. Put Option In the Money Option, Out the Money Option v.s. At the Money Option European Option v.s. American Option Plain-Vanilla Option v.s. Exotic Option
Lookback option (options on the max) Asian Option (options on the average) Digital Option/Binary Options Barrier Option Quanto (currency option) Shout Chooser (compound option)
Peng Liu MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Pricing Exotics: Example
Find the arbitrage free price of an option that pays at the end of period 2 the maximum value attained by the stock price over the life of the option minus the value of the stock at maturity, i.e., it pays, max{0, St − S2 } at date= 2. Find the price of European lookback call/put options written on the max/min value of the stock price with a strike of 105. Find the price of European lookback call/put options written on the average value of the stock price with a strike of 105. What is the relationship of call/Put prices between the European,American,Asian,Lookback options?
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Pricing Exotics: Example
Find the arbitrage free price of an option that pays at the end of period 2 the maximum value attained by the stock price over the life of the option minus the value of the stock at maturity, i.e., it pays, max{0, St − S2 } at date= 2. Find the price of European lookback call/put options written on the max/min value of the stock price with a strike of 105. Find the price of European lookback call/put options written on the average value of the stock price with a strike of 105. What is the relationship of call/Put prices between the European,American,Asian,Lookback options?
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Pricing Exotics: Example
Find the arbitrage free price of an option that pays at the end of period 2 the maximum value attained by the stock price over the life of the option minus the value of the stock at maturity, i.e., it pays, max{0, St − S2 } at date= 2. Find the price of European lookback call/put options written on the max/min value of the stock price with a strike of 105. Find the price of European lookback call/put options written on the average value of the stock price with a strike of 105. What is the relationship of call/Put prices between the European,American,Asian,Lookback options?
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Pricing Exotics: Example
Find the arbitrage free price of an option that pays at the end of period 2 the maximum value attained by the stock price over the life of the option minus the value of the stock at maturity, i.e., it pays, max{0, St − S2 } at date= 2. Find the price of European lookback call/put options written on the max/min value of the stock price with a strike of 105. Find the price of European lookback call/put options written on the average value of the stock price with a strike of 105. What is the relationship of call/Put prices between the European,American,Asian,Lookback options?
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
Outline
1
The Two-Period binomial Model Dynamic Replication Approach Risk-Neutral Valuation Pricing Exotic Option In Binomial Model What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
2
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
First Fundamental Theorem of Asset Pricing
Absence of Arbitrage Essence of relative pricing Note the difference of Arbitrage v.s.good deal Utility assumption: only need non-satiation assumption! (Does investor’s risk aversion play any role in non-arbitrage pricing?) Law of One Price In a frictionless market, AOA implies that two assets(portfolios/projects/investments,firms)with same set of cash flows must have the same price. First Fundamental Theorem of Asset Pricing The existence of a risk-neutral measure(equivalent martingale measure EMM) implies the absence of arbitrage.
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
First Fundamental Theorem of Asset Pricing
Absence of Arbitrage Essence of relative pricing Note the difference of Arbitrage v.s.good deal Utility assumption: only need non-satiation assumption! (Does investor’s risk aversion play any role in non-arbitrage pricing?) Law of One Price In a frictionless market, AOA implies that two assets(portfolios/projects/investments,firms)with same set of cash flows must have the same price. First Fundamental Theorem of Asset Pricing The existence of a risk-neutral measure(equivalent martingale measure EMM) implies the absence of arbitrage.
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
What are Exotic options? AOA and First Fundamental Theorem of Asset Pricing
First Fundamental Theorem of Asset Pricing
Absence of Arbitrage Essence of relative pricing Note the difference of Arbitrage v.s.good deal Utility assumption: only need non-satiation assumption! (Does investor’s risk aversion play any role in non-arbitrage pricing?) Law of One Price In a frictionless market, AOA implies that two assets(portfolios/projects/investments,firms)with same set of cash flows must have the same price. First Fundamental Theorem of Asset Pricing The existence of a risk-neutral measure(equivalent martingale measure EMM) implies the absence of arbitrage.
Peng Liu
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
Summary
The advantages of Cox, Ross and Rubinstein’s Binomial Option Pricing Model: Easy to Understand Why introduce CRR first. Flexible Can price path-dependent options when B/S cannot give closed form solution. Powerful It can be shown that CRR converges to Black-Scholes option pricing formula. Realistic For more possible stock prices, just make discrete binomial steps finer. Risk-Neutral probability q =
R−d u−d
Risk Neutral Valuation for n-period options V =
Peng Liu
1 q E [C(T )] Rn
MFE Stochastic Calculus
The Two-Period binomial Model Pricing Exotic Option In Binomial Model Summary
OTHER QUESTIONS??? I will be around after section for other questions related to the homework and/or lecture.
Peng Liu
MFE Stochastic Calculus