No Arbitrage Condition

• Payoff: the gross outcome of an investment or trade. The amount you earned from that trade, regardless of commissions, extraneous costs, etc.
• Profits: the gross outcome of an investment or trade, including commissions, extraneous cost
  • → the distinction happens in Options (Finance). Payoffs don’t consider the option premium, while the profits do.
• Return: short for “rate of return”. The percentage of profit (not payoff!) per original investment. Quoted in percentage (%) or log-returns. def. No arbitrage condition (=Law of One Price means without any risk, there cannot be excess return (=return above risk-free rate).

1. let security $A$ have payoff $p_A$, cost $c_A$ and $B$ payoff $p_B$, cost $c_B$.
2. If $p_A=p_B$ then $c_A=c_B$ Alternative formulation:
3. If security $A$ is risk-free and has payoff $p_A$…
4. …then the current price of $A$ should simply be the DCF of $p_A$ discounted at the risk-free rate

Risk-Neutral Assumption §

Motivation. People in general are risk-averse. Observe in the following example¹: Consider:

1. A bond with principal \1$thatmaydefault,andincaseofdefaultonlypays$60%$ of the principal
2. A put option that acts as insurance on this bond, that pays nothing if solvent, and $40\%$ if default.
3. Your portfolio contains both. Assume for simplicity risk-free rate $r=0$. Our objective is to price the put option so that the law of one price applies. The naive approach: $p=0.1$ is the historical default rate from data.
4. The expected payoff of bond calculated from $p=0.1$ is $0.96$
5. The expected value of put calculated from $p=0.1$ is $0.12$
6. Total payoff of portfolio is always \1$ regardless of solvency
7. And both sum up to \1$ which is same as total payoff, thus LoP applies
8. ! …but this is NOT the observed market price of bond: \0.88$! The investors are demanding a risk-premium (=acting irrational) because they’re risk-averse So we try a different method:
9. From the market price \0.88$wecalculatethe\ast \ast impliedrateofreturn\ast \ast :$0.88=(1-q)\cdot1+q\cdot 0.6$,andwefind$q=0.3$
10. Calculate the put option from this $q$ to get price \0.12$
11. And both sum up to $1 which is same as total payoff thus LoP applies
12. $\dots ANDitisusingthemarketpriceofthebond,$0.88.

def. Risk-Neutral Assumption. A method of pricing derivatives despite the fact that investors are acting irrationally, that makes arbitrage impossible (=makes law of one price hold.)

Footnotes §

1. Quantitative Risk Management ↩