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Introductory analysis
There is a famous quote: "the reader who finds he does not have the prerequisites for the prerequisites should not lose heart" (or something similar,) basic analysis is the prerequisite for the prequisites. Please note that I recommend 3 books on this topic because I think that it will take less time to read all three than to read the third one alone.
"Yet Another Introduction to Analysis " by Victor Bryant is the book that I wish I had had when I was learning analysis, and if I was to write a book on the topic this is the way I would write it, (except that I won't because Bryant has already done it.) Bryant teaches analysis with lots of motivation and examples. The reader he has in mind knows calculus but cannot see the point of analysis. All mathematics is (or should be!) invented to solve problems and Bryant never forgets this, and explains why as well as how as he introduces each theorem. If you find analysis too dry, this is the book for you.
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"Mathematical Analysis: A Straightforward Approach" by K.G. Binmore. If you find the jump from Bryant to Rudin too big, then Binmore is a nice in-between choice. This is actually the first book I read on analysis -- Bryant wasn't available at the time.
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"Principles of Mathematical Analysis" by Walter Rudin. This a great second book on analysis. It starts from first principles but is drier that Bryant. So first read Bryant to get some idea of what is going on, and then work through Rudin to get all the details and to learn enough to prepare you for measure theory.
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Complex analysis
Complex analysis is not essential to learn probability theory and stochastic processes. However, contour integration and Fourier transforms are indispensable tools for the working quant. It is also one of the most beautiful and useful areas of mathematics.
"Introduction to Complex Analysis" by Hilary Priestley. I learned complex analysis using the first edition of this book. I had never studied complex analysis before and I found the treatment rigorous but pleasurable. I can't think of a better place to start than Priestley. The second edition has more exercises and has divided the book into bite-sized chunks to make it easier for the reader. If you find this book hard then you probably need to spend more time learning basic analysis of the real line so you can follow the mathematical arguments
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Probability Theory and Stochastic Processes
Modern financial mathematics relies heavily on probability theory, if you want to do it well, you really need to learn to think probablistically and to study the theory. To really understand stochastic processes, you need to work through a program of
basic probability theory
basis analysis
discrete-time martingales
continuous-time martingales
stochastic integration
I present a sequence of probability books to help you do this.
"Elementary Probability Theory" by Kai Lai Chung. This is the book I first learnt probability theory from. Chung really knows how to write and his target audience is undergraduates doing a first course in probability. The new edition has a section on mathematical finance but I haven't read that bit.
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"Probability and Random Processes" by Geoffrey Grimmett and David Stirzaker. This is the other book I used when studying probability as an undergraduate. It goes faster and further than Chung but the authors have a real desire to teach as well as present material and is well worth reading.
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"Probability with Martingales" by David Williams. This book is a joy to read. The author takes a subject often regarded as hard and makes it easy, whilst making it come alive with a chatty informal style. All this without sacrificing rigour. Definitely one of my favourite maths books. This is not a first book on probability theory but is a first book on discrete time martingales.
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"Diffusions, Markov Processes and Martingales: by Chris Rogers and David Williams. This is a two volume set. It is a natural sequel to Williams' "probability with martingales," although the authors quickly repeat much material from that book. This is a very good choice for getting the basics of Brownian motions and continuous time martingales in a rigorous fashion. The second volume then goes on to discuss stochastic calculus. Whilst the second volume is good too, I would recommend reading it after Chung and R.Williams (below.)
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"Introduction to Stochastic Integration" by K. L. Chung, R.J. Williams. This is the same Chung but a different Williams! I found this to be the most readable account of stochastic integration theory. It assumes knowledge of continuous time martingales, however, so you must learn those elsewhere first. The authors do all the details, and focus on trying to present the most important case in careful and clear detail rather than trying to work in absurd generality.
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Basic mathematical finance (Other than my books...)
There is a rather large number of introductory textbooks on financial mathematics each with its own bent. I haven't read most of them inevitably. I mention a few I found helpful.
"Arbitrage Theory in Continuous Time" by Tomas Bjork. This books presents a clear but fairly rigorous exposition of the basics of financial mathematics. It bears some similarities to my book Concepts but is stronger on rigour and lighter on practicalities. Unusually in a rigorous book, Bjork never loses sight of the underlying ideas and does a good job of conveying them. The author's background is as a professor in probability theory, and it shows in his approach and choice of topics; the book is strong on risk-neutral evaluation and expectations, but spends less time on PDEs.
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"Financial Calculus: An Introduction to Derivative Pricing" by Martin W. Baxter, Andrew J.O. Rennie. This is a light and accessible introduction to the martingale approach to derivatives pricing from a reasonably pure viewpoint. The authors' objective was to teach the reader the basics of the martingale theory without sacrificing too much rigour, and in this they succeeded very well. Be aware, however, that there is not much discussion of practicalities nor of the PDE approach which is barely mentioned.
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"The Mathematics of Financial Derivatives: A Student Introduction" by Paul Wilmott, Sam Howison and Jeff Dewynne. This is written by experts in applied PDEs for someone with a background in PDEs. It is therefore a good exposition of the PDE approach and well worth reading for getting a grounding in it. If you want to start with the PDE approach and then move on later to the martingale approach it can also be a good book to start with.
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"Stochastic Calculus for Finance" volumers I and II by Steven Shreve. OK I haven't read these but lots of other people like them, and they do seem to be a pair of the best introductory books available. In particular, they seem to have a good blend of rigour and intuition. Volume I does the binomial tree in great detail establishing all the concepts necessary to do the continuous time case in Volume II.
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