This page is created to help those under-grad or grad students who aspire to learn the theory of derivative pricing (mathematical finance) up to a certain level of their choice. The first question is how to learn and which book to follow. I have observed varying levels of aptitude, prior knowledge, and enthusiasm among the student community. Hence no particular suggestion can satisfactorily answer everybody's query. However, I am here trying to answer the question "how to learn". As the answer should depend on various levels of the inquirer, I would first partition this large variety into a few sectors and then write down the answer for each sector of students. I make 8 sectors and identify each by an element in {A, N}3. A & N represent "Advanced" and "Naive" respectively. For example, (A, A, A) or in short AAA denotes Advanced mathematical aptitude at the level of Master's students, Advanced knowledge in Finance at the level of practitioners, seeking to learn Advanced topics in mathematical finance at the level of a researcher. The meaning of every other code is to be understood by following this order, i.e., (Math aptitude, Finance knowhow, the goal in Math Finance).
Disclaimer: The following answers are purely based on my personal perspective. The book authors might or might not agree. I also agree that I have probably missed many fantastic relevant books unknowingly. The purpose of this page does not include the aim of an exhaustive literature survey. This has the sole purpose of bestowing advanced knowledge of math finance to a reader. A reader with advanced knowledge may easily find out and read other texts with more specialized topics. If you are an author and believe that your book should find a place here or should replace one or many texts suggested here, please feel free to contact me.
References
[BR] Baxter, Martin and Rennie, Andrew. Financial Calculus: An Introduction to Derivative Pricing
[BM] Brigo, Damiano and Mercurio, Fabio. Interest Rate Models - Theory and Practice
[DS] Delbaen, Freddy and Schachermayer Walter. The Mathematics of Arbitrage
[KK] Kallianpur, Gopinath and Karandikar Rajeeva L. Introduction to Option Pricing Theory
[KS] Karatzas, Ioannis and Shreve, Steven. Brownian Motion and Stochastic Calculus
[LU] Luenberger, David. Investment science
[MI] Mikosch, Thomas. Elementary Stochastic Caculus
[OK] Øksendal, Bernt. Stochastic Differential Equations
[PR] Protter, Philip E. Stochastic Integration and Differential Equations
[RO] Royden, Halsey. Real Analysis (Fourth Edition)
[SP] Schilling, René L. and Partzsch, Lothar. Brownian Motion
[SH] Shiryaev A. N.. Essentials of Stochastic Finance: facts, models, theory
[S1] Shreve, Steven. Stochastic Calculus for Finance I: The Binomial Asset Pricing Model
[S2] Shreve, Steven. Stochastic Calculus for Finance II: Continuous-Time Models
[SO] Sondermann, Dieter. Introduction to Stochastic Calculus for Finance: A New Didactic Approach
[TC] Tankov, Peter and Cont, Rama. Financial Modelling with Jump Processes
[WP] Wilmott, Paul. Paul Wilmott on Quantitative Finance
Notice: Students and practitioners, interested in learning advanced mathematics required for finance, find the following page useful too. https://sites.google.com/site/anindyagoswami/teaching/pde. On that page, one gets notes, slides, and lecture recordings of a 12-week-long NPTEL course. Worth noting that I did not teach finance in that course. However, I taught some mathematical background, important for understanding mathematical solutions to many finance problems. Wish you all the best in your learning.