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I'm reading a book called Quant Job Interview Questions and Answers and came across the following question and its answer, but cannot make sense of it, so I really appreciate your advice:

Question 2.4:

Suppose two assets in Black Scholes world have the same volatility but different drifts. Suppose one of the assets undergoes downward jumps at random times. How will this affect option prices?

Answer:

We construct a portfolio with initial cost $C_{BS}(0,S_0)$ which sometimes finishes with the same value as the option (if no jumps occur) and sometimes finishes with a lower value (if a jump occurs). Thus by no arbitrage considerations, the value of the option on the stock with jumps must be greater than $C_{BS}(0,S_0)$.

So my doubt is: how come "by arbitrage considerations" can lead to the above conclusion?

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No arbitrage means that you can't have a portfolio with a positive expectation without risk. let's suppose that the value of option with jumps is lower than $C_{BS}(0,S_0)$ Please consider the following portfolio at time 0:

  1. Sell BS hedge on option with jumps with price $C_{BS}(0,S_0)$.
  2. Buy option with jumps with price $P_J(0,S_0)$.
  3. Keep the rest of money $C_{BS}(0,S_0)-P_J(0,S_0)$ in the portfolio.

At time t:

  1. Keep selling BS hedge on option with jumps with price $C_{BS}(t,S_t)$.
  2. keep the option with jumps with price $P_J(t,S_t)$.
  3. Keep the rest of money $C_{BS}(0,S_0)-P_J(0,S_0)$ in the portfolio.

At t=0, the value of the portfolio is: $$-C_{BS}(0,S_0)+P_J(0,S_0)+C_{BS}(0,S_0)-P_J(0,S_0)=0$$ At maturity, if no jump occurs, the hedge is perfect so : $C_{BS}(T,S_T)=P_J(T,S_T)$ and the value of the portfolio is: $$-C_{BS}(T,S_T)+P_J(T,S_T)+C_{BS}(0,S_0)-P_J(0,S_0)>0$$ if downward jumps occurs, you have $C_{BS}(T,S_T)<P_J(T,S_T)$ as the hedge finish at a lower value at maturity. It means that the portfolio value at maturity is: $$-C_{BS}(T,S_T)+P_J(T,S_T)+C_{BS}(0,S_0)-P_J(0,S_0)>0$$ ==> Arbitrage!

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  • $\begingroup$ Thanks a lot! @user1987, What is the meaning of "BS hedge on option" (from the 4th line)? $\endgroup$ – M00000001 Feb 2 at 0:10
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    $\begingroup$ sell bs hedge means to sell black & scholes delta stock and invest the rest of the premium on a bond $\endgroup$ – Valometrics.com Feb 2 at 8:57
  • $\begingroup$ Thank you very much! $\endgroup$ – M00000001 Feb 2 at 23:32

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