London Graduate School in Mathematical Finance and MRes in Finance and Stochastics
Doctoral Course MF17
Nonlinear valuation under credit gap risk, initial and variation margins and funding costs
Course Syllabus:
Click here.
Lecture notes (PDF): click here
IMPORTANT: THE FIRST LECTURE HAS BEEN MOVED FROM MONDAY 27 TO TUESDAY 28, SAME ROOM, SAME TIME.
IMPORTANT: THE THIRD LECTURE HAS BEEN MOVED FROM MONDAY APRIL 10 TO TUESDAY APRIL 11, SAME ROOM, SAME TIME.
IMPORTANT: Due to room capacity, it is required that all students planning to follow this course register.
London Graduate School of Mathematical Finance PhD students: Please register following instructions
here.
MRes students: Please follow contact details
here.
Dates: March 28 and April 3, 11, 19 and 24, 2017.
Times: 2pm - 5pm (all five dates). 15 hours in total.
Location: Imperial College London, South Kensington campus, Department of Mathematics, Huxley Building, Lecture room 130.
The department is located at 180 Queen's Gate, SW7 2AZ. Directions to the lecture theatre 130 are given
here
Nonlinear valuation under credit gap risk, initial and variation margins and funding costs
Syllabus
The market for financial products and derivatives reached an outstanding notional size of 708
USD Trillions in 2011, amounting to ten times the planet gross domestic product.
Even discounting double counting, derivatives appear to be an important part of the world economy
and have played a key role in the onset of the financial crisis in 2007. After briefly reviewing
the Nobel-awarded option pricing paradigm by Black Scholes and Merton, hinting at precursors
such as Bachelier and de Finetti, we explain how the self-financing condition and Ito's formula
lead to the Black Scholes Partial Differential Equation (PDE) for basic option payoffs.
We hint at the Feynman Kac theorem that allows to interpret the Black Scholes PDE solution as
the expected value under a risk neutral probability of the discounted future cash flows,
and explain how no arbitrage theory followed. Following this quick introduction, we describe
the changes triggered by post 2007 events. We re-discuss the valuation theory assumptions and
introduce valuation under counterparty credit risk, collateral posting, initial and variation margins,
and funding costs. We explain model dependence induced by credit effect, hybrid features,
contagion, payout uncertainty, and nonlinear effects due to replacement closeout at default
and possibly asymmetric borrowing and lending rates in the margin interest and in the funding strategy
for the hedge of the relevant portfolio. Nonlinearity manifests itself in the valuation equations
taking the form of semi-linear PDEs or Backward SDEs. We discuss existence and uniqueness of
solutions for these equations. We also present a high level analysis of the consequences of
nonlinearities, both from the point of view of methodology and from an operational angle.
We discuss the Modigliani Miller theorem and whether this really implies that funding costs
should be zero at overall institution level.
Finally, we connect these developments to interest rate theory under multiple discount curves,
thus building a consistent valuation framework encompassing most post-2007 effects.
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