My research is currently focused on pricing new types of mortgages, including the Continuous Workout Mortgage --- a new type of product designed to insure homeowners against decrease in house prices --- in collaboration with Robert Shiller (Yale University, USA).
My contributions to Finance Theory are published in leading journals including the Journal of Corporate Finance, Journal of Banking and Finance, Journal of Economic Dynamics and Control, Journal of Economic Behavior & Organization, European Financial Management, Journal of Business Finance and Accounting and Journal of Applied Probability.
I have a PhD in Finance from HEC Paris, an M.Res in Economic Policy from Paris School of Economics and an M.Eng in Mathematical Physics from Wroclaw University of Technology.
Areas of specialism
University roles and responsibilities
- PGR Director, 3-year PhD in Management, Surrey Business School
- PGR Director, PhD in Finance and Accounting, Department of Finance and Accounting
- PGR Director, PhD in Healthcare Management, Surrey Business School
Affiliations and memberships
In the media
Incomplete Markets. Derivatives and Risk Management. Mortgage contracts and Real Estate Finance. Executive compensation. Real Options. Corporate Finance. Household finance. Behavioral finance. Mathematical finance. Stochastic processes and continuous-time finance. Auction theory.
- Doctoral Finance (PhD)
- Derivative Securities (MSc) MANM137
- Lancaster University Management School (LUMS): Doctoral Finance (PhD), Derivatives pricing (MSc), Advanced Topics in Finance (MSc), Quantitative Methods for Finance (MSc), Financial Markets (MSc), Advanced Principles of Finance, Bond Markets, Fixed income securities (MSc);
- HEC School of Management, Paris, France: Finance Theory, Microeconomics, Corporate Finance, Econometrics, Mathematical Finance (MSc Finance);
- Sciences Po, Paris, France: Emerging Markets;
- Warsaw University of Technology, Poland: Financial Markets (MBA);
- Warsaw School of Economics, Poland: Investments (MBA).
The continuous-time formula for expected payoff to holding an option, which nests several major pricing tools, is derived. It is shown also that under current market conditions the true exercise probability, N(d4), lies halfway between the two more familiar terms: N(d1) and N(d2).
Models of interest rate caps and floors are typically based on discrete rates over finite horizons while existing real option models describe perpetual claims on the maximum of two continuous flows. In this paper, we produce formulae for finite maturity caps and floors that are contingent on continuous flows. We present hedge ratios and discuss applications where a lognormally distributed flow variable is suitable. For other situations where practitioners use proprietary models, the formula presented is useful as a quick, tractable and universal means for mapping quoted implied to prices and vice versa.
This paper studies symmetries between fixed and floating-strike Asian options and exploits this symmetry to derive an upper bound for the price of a floating-strike Asian. This bound only involves fixed-strike Asians and vanillas, and can be computed simply given one of the many efficient methods for pricing fixed-strike Asian options. The bound is exact until after the averaging has begun and again at maturity. The bound is compared to benchmark prices obtained via Monte Carlo simulation in numerical examples.
There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating- and fixed-strike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation, and in this case the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options.
We analyse the entry decisions of competing firms in a two-player stochastic real option game, when rivals earn different but correlated uncertain profitabilities from operating. In the presence of entry costs, decision thresholds exhibit hysteresis, the range of which is decreasing in the correlation between competing firms. A measure of the expected time of each firm being active in the market and the probability of both rivals entering within a finite time are explicitly calculated. The former (latter) is found to decrease (increase) with the volatility of relative firm profitabilities implying that market leadership is shorter-lived the more uncertain the industry environment. In an application of the model to the aircraft industry, we find that Boeing's optimal response to Airbus' launch of the A380 super carrier is to accommodate entry and supplement its current product line, as opposed to the riskier alternative of committing to the development of a corresponding super jumbo. © 2002 Elsevier B.V. All rights reserved.
We analyse the rate of return and expected exercise time of Merton-style options (1973) employed in many real option situations where the possibility of exercise is both perpetual and American in nature. Using risk-neutral and risk-adjusted pricing techniques, Merton-style options are shown to have an expected return that is a "constant percentage" of the option value and independent of the proximity to the critical exercise boundary. Merton options thus remain at the same point on the Security Market Line, unlike European options whose position and rate of return change dynamically. We also present formulae for the expected time and discounted times to exercise and analyse the dependency of these variables on volatility.
This study focuses on structuring tangible asset backed loans to inhibit their endemic option to default. We adapt the pragmatic approach of a margin loan in the configuring of collateralized debt to yield a quasi-default-free facility. We link our practical method to the current Basel III (2017) regulatory framework. Our new concept of the Loan Valuation Adjustment (LVA) and novel method to minimize the LVA converts the risky loan into a quasi risk-free loan and achieves value maximization for the lending financial institution. As a result, entrepreneurial activities are promoted and economic growth invigorated. Information asymmetry, costly bailouts and resulting financial fragility are reduced while depositors are endowed with a safety net equivalent to deposit insurance but without the associated moral hazard between risk-averse lenders and borrowers.
This paper studies the Continuous Workout Mortgage (CWM), a two in one product: a fixed rate home loan coupled with negative equity insurance, to advocate its viability in mitigating financial fragility. In order to tackle the many issues that CWMs embrace, we perform a range of tasks. We optimally price CWMs and take a systemic market-based approach, stipulating that mortgage values and payments should be linked to housing prices and adjusted downward to prevent negative equity. We illustrate that amortizing CWMs can be the efficient home financing choice for many households. We price CWMs as American option style, defaulting debt in conjunction with prepayment within a continuous time, analytic framework. We introduce random prepayments via the intensity approach of Jarrow and Turnbull (1995). We also model the optimal embedded option to default whose exercise is motivated by decreasing random house prices. We adapt the Barone-Adesi and Whaley (1987) (BAW) approach to work within amortizing mortgage context. We derive new closed-form and new analytical approximation methodologies which apply both for pricing CWMs, as well as for pricing the standard US 30-year Fixed Rate Mortgage (FRM).
We employ Amortizing Participation Mortgage (APM) to offer a novel ex post renegotiation method of a foreclosure. APM belongs to the family of home loan credit facilities advocated in the Dodd-Frank Wall Street Reform and Consumer Protection Act 2010. In our framework, APMs reduce the endemic agency costs of debt by improving affordability. These benefits increase the demand for real estate in bust times and reduce fragility of the financial system thereby preventing foreclosures. We evaluate APMs in a stochastic control framework and provide solutions for an optimal amortization schedule. We generalize our approach to partially amortizing and commercial mortgages which encompass balloon payments. Finally, we provide concrete numerical examples of home loan modifications. We also offer detailed sensitivity analysis to market parameters such as house price volatility and interest rates.
Papers in refereed journals
 Robert J. Shiller, Rafal M. Wojakowski, M. Shahid Ebrahim, and Mark B. Shackleton. Continuous Workout Mortgages: Efficient pricing and systemic implications. Journal of Economic Behavior & Organization, forthcoming, 2017.
 Rafał M. Wojakowski, M. Shahid Ebrahim, and Mark B. Shackleton. Reducing the impact of real estate foreclosures with amortizing participation mortgages. Journal of Banking and Finance, 71:62 – 74, October 2016.
 Robert J. Shiller, Rafal M. Wojakowski, M. Shahid Ebrahim, and Mark B. Shackleton. Mitigat- ing financial fragility with Continuous Workout Mortgages. Journal of Economic Behaviour and Organisation, 85:269–285, 2013.
 Rafal M. Wojakowski. How should firms selectively hedge? resolving the selective hedging puzzle. Journal of Corporate Finance, 18(3):560 – 569, 2012.
 M. Shahid Ebrahim, Mark B. Shackleton, and Rafał M. Wojakowski. Participating mortgages and the efficiency of financial intermediation. Journal of Banking and Finance, 35(11):3042–3054, November 2011.
 Andrianos Tsekrekos, Mark B. Shackleton, and Rafal M. Wojakowski. Evaluating natural re- source investments under different model dynamics: Managerial insights. European Financial Management, 18(2):543–577, 2012.
 Mark Shackleton and Rafal M.W̃ ojakowski Wojakowski. Finite maturity caps and floors on con- tinuous flows. Journal of Economic Dynamics and Control, 31(12):3843–3859, October 2007.
 Vicky Henderson, David Hobson, William Shaw, and Rafał Wojakowski. Bounds for in-progress floating-strike Asian options using symmetry. Annals of Operations Research, 151(1):81–98, April 2007.
 Mark Shackleton, Andrianos Tsekrekos, and Rafał Wojakowski. Strategic entry and market leadership in a two-player real options game. Journal of Banking & Finance, 28(1):179–201, January 2004.
 San-Lin Chung, Mark Shackleton, and Rafał Wojakowski. Efficient quadratic approximation of floating strike Asian option values. Finance, 24(1):49–62, June 2003.
 Rafał Wojakowski and Mark Shackleton. Real probability of exercising and expected values of option payoff. Rynek Terminowy (Futures Markets), 20(2):125–127, 2003. Refereed paper.
 Vicky Henderson and Rafał Wojakowski. On the equivalence of floating- and fixed-strike Asian options. Journal of Applied Probability, 39(2):391–394, June 2002.
 Mark Shackleton and Rafał Wojakowski. The expected return and exercise time of Merton style Real Options. Journal of Business Finance and Accounting, 29(3-4):541–555, April/May 2002.
 Mark Shackleton and Rafał Wojakowski. On the expected payoff and true probability of exercise of European options. Applied Economics Letters, 8(4):269–271, April 2001.
 Marc Chesney, Fausto Hernandez Trillo, Bernard Marois, and Rafał Wojakowski. El Manejo Del Riesgo Cambiario: Las Opciones Sobre Divisas (Foreign Exchange Risk Management: Currency Options). Editorial Limusa-CIDE, Mexico, 2001. ISBN: 968-18-6052-7.
 Marc Chesney, Bernard Marois, and Rafał Wojakowski. Les options de change: évaluation et utilisation (Foreign Exchange Options: Pricing and Using). Collection: Gestion, Série: Politique générale, Finance et Marketing. Economica, Paris, 1995. In French, ISBN 2–7178–2967–9, 129 pages.
Papers in edited collections
 Rafał Wojakowski and Mark Shackleton. On option expected returns. In Michael Kohlmann and Shanjian Tang, editors, Mathematical Finance, Trends in Mathematics, pages 265–374. Birkhäuser Verlag, Basel Switzerland, 2001. ISBN 3–7643–6553–6.
 Mark Shackleton and Rafał Wojakowski. Reversible real options. In Michael Kohlmann and Shanjian Tang, editors, Mathematical Finance, Trends in Mathematics, pages 339–344, Basel Switzerland, 2001. Birkhäuser Verlag. ISBN 3–7643–6553–6.
 Marc Chesney, Bernard Marois, and Rafał Wojakowski. Options de change: évaluation (For- eign exchange options: Pricing). In Yves Simon, editor, Encyclopédie des marchés financiers, chapter 69, pages 1398–1422. Economica, Paris, 1997. Commissioned chapter, in French, ISBN 2–7178–3043–X.