MMath Mathematics and Statistics

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The course starts with discrete-time market models. Fundamental theorems of Asset Pricing are introduced, and then the pricing and hedging problems for derivatives are discussed with a focus on incomplete markets. For continuous-time market models, such as the Black-Scholes model and local/stochastic volatility models, the course delves into discussions on hedging and pricing for various derivatives, including European options, American options, and exotic options. Finally, interest rate models are presented.  
 

Pre-Requisites: MATH27720 and one of MATH37002 or MATH47201

Students are not permitted to take, for credit, MATH49102 in an undergraduate programme and then MATH69012 in a postgraduate programme at the University of Manchester, as the courses are identical.

The unit aims to:
provide precise mathematical formulations for some problems in financial markets such as pricing and hedging derivatives, interest rate modelling. Tools from stochastic calculus and martingale theory are used in a rigorous manner within the framework of the no-arbitrage pricing theory.
 

On successful completion the students will be able to: 

• apply modern probability theory, including martingale theory, stochastic calculus, and the no-arbitrage theory, to stochastic models in finance.
   
• compute the hedging strategies and fair prices of various options in discrete time financial market models.
   
• compute and analyse the fair prices and the hedging strategies for options in simple continuous time market models.
   
• apply interest rate models, analyse various affine term structure models, short rate models and compute fair prices of some interest rate derivatives. 
   

Syllabus
1. Discrete-time market models: self-financing portfolios; The Cox-Ross-Rubinstein model; trinomial model; contingent claims; arbitrage opportunity; risk-neutral measure; fundamental theorem of asset pricing; market completeness; pricing and hedging. [4 hours ~ 2 weeks]
2. Continuous-time market models: the Black-Scholes models; self-financing portfolios; contingent claims; arbitrage opportunity; risk-neutral measure; fundamental theorem of asset pricing; market completeness; delta hedging, pricing and hedging; Greeks; implied volatility; local/stochastic volatility models. [8 hours ~ 4 weeks] 
3. American (put and call) options: Optimal stopping and free boundary problems; some exotic options, e.g. Knock-out barrier option; lookback option; Asian option; chooser option; digital option; forward-Start option; basket option. [4 hours ~ 2 weeks]
4. Interest rate models: Short rate models; the Vasicek model; the Ho-Lee model; Cox-Ingersoll-Ross model; Heath-Jarrow-Morton framework; Hull-White models; Interest rate derivatives. [6 hours ~ 3 weeks]

Lectures (22 hours) and tutorials (11 hours) for 11 weeks. The last week (Week 12 is for revision.
 

General feedback provided after the exam.

Recommended reading
• Shreve, S. E. (2004). Stochastic calculus for finance II: Continuous-time models (Vol. 11). New York: Springer.
• Shiryaev, A. N. (1999). Essentials of stochastic finance: facts, models, theory (Vol. 3). World scientific.
• Björk, T. (2009). Arbitrage theory in continuous time. Oxford university press.
• Lamberton, D., & Lapeyre, B. (2011). Introduction to stochastic calculus applied to finance. CRC press.
• Musiela, M., & Rutkowski, M. (2006). Martingale methods in financial modelling (Vol. 36). Springer Science & Business Media.

Further reading
• Brigo, D., & Mercurio, F. (2006). Interest rate models-theory and practice: with smile, inflation and credit (Vol. 2). Berlin: Springer.
• Jeanblanc, M., Yor, M., Chesney, M. (2009). Mathematical methods for financial markets. Springer Science & Business Media.