Arbitrage Opportunity
Valuation Methods
- Discount factor (denominator)
- Take into account the risky nature of a cash flow
- Certainty equivalent (numerator)
- Transform the expected risky cash flow into certainty
- Risk neutral evaluation
- Correct the probability
- Free of the structure assumptions
Hedging
No arbitrage price for option. Assume the risk-free rate is 4%. There is a stock with payoff
Assume no arbitrage, to price a European call option written on the stock with exercise price $21 in 3 months:
- Intuition
- Set up a portfolio to eliminate uncertainty
- Its return must equals risk-free rate
- Strategy: Construct a riskless portfolio
- Long shares of the stock
- Short one call option
To eliminate the uncertainty:
With NA (no arbitrage), riskless portfolios must earn the risk-free rate
Generalization: Delta Hedging
Delta of a derivative: the ratio of the change in the price of a derivative with the price of the underlying asset
Delta hedging: The price of a portfolio is insensitive to small changes in the price of the underlying asset. Delta-neutral means a portfolio with a delta of zero.
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RN Probability and AD Price
RN probability
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The first term is the EN expected value of the acquired asset if the option is exercised at T
The second term is the RN expected cost of acquiriin git at T
The difference is RN expected value of the option at T
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Asian Option is cheaper because its volatility is lower. It is aimed for consumers or producers applied where there are multiple cash flows during a period of time.
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Analyze NA:
- construct portfolio, test cash flows
- risk neutral pricing, test whether have solutions
- test whether EMM exists (corresponding probability satisfies 0<p<1)
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