Why an Equally Spaced Call Butterfly Has Nonnegative Payoff
Summary
The document examines the terminal payoff of a long call butterfly with three equally spaced strikes. It establishes that the middle strike is the midpoint of the lower and upper strikes, then compares the combined payoffs of calls at the outer strikes with twice the payoff of the middle-strike call. This pointwise inequality shows that the standard butterfly payoff, long one call at each outer strike and short two at the middle strike, cannot be negative at maturity.
Discounting and taking a risk-neutral conditional expectation preserves nonnegativity, so the butterfly’s value is nonnegative before expiry under the stated setup. The argument is algebraic and applies across terminal stock prices; it does not depend on estimating a probability or using the geometric Brownian motion assumption to prove the payoff inequality. The question’s phrasing says the value is positive, but the derivation guarantees only nonnegativity; strict positivity requires conditions under which the payoff is positive with positive probability.
Key ideas
- With equally spaced strikes, the middle strike is the average of the two outer strikes.
- The sum of the outer call payoffs is at least twice the middle call payoff at every terminal price.
- The standard long call butterfly therefore has a nonnegative terminal payoff.
- A risk-neutral expectation of a nonnegative payoff remains nonnegative before expiry.
- Strict positivity requires additional conditions beyond the pointwise payoff inequality.
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# Reference request: Quantitative Trading Strategies # Reference request: Quantitative Trading Strategies I intend to thoroughly prepare for an internship that I will start in a couple of months, and therefore wanted to clarify what topics I need to study and some recommended references for them. The description for the internship is as follows: the intern will be a part of research for quant trading strategies, will be given a live project and will learn how to develop and backtest trading strategies for equities, forex, commodities, etc. As for my existing background, I once did a project in mathematical finance on option pricing in Markov modulated markets (which I think may not be related a whole lot to the internship) and I also learned a bit of C and Python programming languages earlier. I have absolutely no prior experience in the area of quant trading. I'd be grateful for any advice regarding what to study, and for suggestions on what book(s) or papers (e.g. which among the quant papers on sites like arxiv.org) to read, or if there are any websites/software that simulate the kind of work I will need to undertake. Thanks in advance! ## Answer by Uditg_ucla (score 3, accepted) https://quant.stackexchange.com/a/22812 I did a similar internship at a quant equity shop and based on my experience, I think there are a few common aspects to such work, which you can try and work on, to have a more productive internship experience: 1) handle on programming language - check with the firm what programming language they'd want you to work in. And in case, you have never worked in it, spend some time getting used to it. Some firms can be flexible on this but some want you to work in specific language only (e.g in my case I learnt SAS, since the firm mostly used SAS) 2) broader understanding of 'quant trading' and what you are after - incase you have had a coursework on this, revise that. Or pick a book to get a general idea about the discipline. e.g. Inside the black box by Rishi Narang, could be a useful read 3) experience with data - its good if you have had some prior experience with the dataset you'd be using over the internship, but in your case, since you don't know what you'd be working on - equity, forex, commodities or something else, don't worry about it 4) experience with backtesting - you could practice by running a backtest of a simple well documented strategy (such as value/ momentum/ Fama- French factors etc). This will give you a better hold on some of the things that you should look out for, such as avoiding 'look ahead' bias, selecting relevant securities etc. Also, you should check with your guide at the company, if there are any relevant research papers/ background reading material, that they would like you to read in the meantime. All the best.
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