文档介绍：Valuation and Risk Models‐(30%)
Option Valuation
Pricing options using binomial trees
The Black‐Scholes‐Merton Model
The Greeks
Fixed Income Valuation
Discount factors, spot rates, forward rates, and yield to maturity
Arbitrage and the law of one price
One‐Factor measures of price sensitivity
Key‐rate exposures and multi‐Factor measures of price sensitivity
Hedging and immunization
Value at Risk
Applied to stocks, currencies, and commodities
Applied to linear and non‐linear derivatives, and securities with embedded options
Structured Monte Carlo, stress testing, and scenario analysis
Limitations as a risk measure
Coherent risk measures
Volatility models
Operational Risk
External and internal credit ratings
Expected and Unexpected Loss
Country and sovereign risk models and management
Section 1 Option Valuation
Pricing options using binomial trees
The Black‐Scholes‐Merton Model
The Greeks
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Binomial Trees
AIMS: Calculate the value of an American and a European call or put option using a one‐ step and two‐step binomial model.
Example
The stock price of ABC company is $20 presently. The stock will go up to 22 or down to 18 three months later. What is the European call price of this stock three months from now? Suppose the strike price is X=$21; continuously compounded risk-free rate is 12%.
●
●
Stock price=$22
Option price=$1
p
Stock price=$20 Option price f
●
Stock price=$18 Option price=$0
1-p
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Binomial Trees
●
S0u
u
0
f =max(0, S u-X)
A one‐step binomial model
p
S0
f
●
S0d
1-p
●
d
0
f =max(0, S d-X)
long  stocks and short 1 option to hedge the risk:
risk neutral probability
f  e rt  pf  (1 p) f 
u d
d
rT
p  e
u  d
4-177
Binomial Trees
Risk‐Neutral Valuation
In a Risk‐Neutral World all individuals are indifferent to risk.
In such a world, investors require no compensation for risk, and the expected return on all securities is the risk free int