Comparing a Jump-Diffusion Call with a Black-Scholes Call
Summary
The document asks how downward jumps affect a vanilla call’s price relative to a Black-Scholes call when the assets are described as having the same volatility but different drifts. The supplied answer sketches a comparison portfolio: short the Black-Scholes value, buy the jump-model option, and invest any initial price difference. It argues that if the jump-model option were cheaper, the portfolio would finish positive whether a jump occurs or not, which it presents as an arbitrage contradiction.
The argument is offered as a brief explanation of the no-arbitrage intuition, not a full pricing derivation. Its portfolio calculations leave financing and the mechanics of maintaining or rebalancing the hedge unspecified. The document therefore does not establish the result under general jump-diffusion assumptions; the conclusion depends on how the hedge, cash account, and model assumptions are defined.
Key ideas
- The response compares a jump-model call with a Black-Scholes call using a proposed portfolio.
- It argues that a cheaper jump-model option would create a positive terminal portfolio value in both stated cases.
- The argument relies on a Black-Scholes hedge matching the option payoff when no jump occurs.
- Financing, hedge maintenance, and model assumptions are not fully specified in the explanation.
Tags
Full text
# Vanilla Call Option Priced Using Jump Diffusion Model
# Vanilla Call Option Priced Using Jump Diffusion Model
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?
## Answer by Valometrics.com (score 2, accepted)
https://quant.stackexchange.com/a/50956
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:
- Sell BS hedge on option with jumps with price $C_{BS}(0,S_0)$.
- Buy option with jumps with price $P_J(0,S_0)$.
- Keep the rest of money $C_{BS}(0,S_0)-P_J(0,S_0)$ in the portfolio.
At time t:
- Keep selling BS hedge on option with jumps with price $C_{BS}(t,S_t)$.
- keep the option with jumps with price $P_J(t,S_t)$.
- 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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.