Dr Rafal Wojakowski

Reader in Finance

Qualifications: PhD (Finance) HEC Paris; MRes (Economics) Paris School of Economics; MEng (Theoretical Physics) Wroclaw University of Technology

Phone: Work: 01483 68 3477
Room no: 08 MS 02

Office hours

Office Hours (term time): Weds 16:30-18:30 & Thu 12:00-13:00 or any mutually convenient time. Please call or email for an appointment. Always happy to see you!

Further information


Rafal has published his research in leading journals including: 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. Recently, Rafal developed leadership in applying mathematical methods to pricing innovative new types of mortgages. His current research focus on Continuous Workout Mortgages --- a new type of product designed to insure homeowners against drops in house prices --- and involves collaboration with Robert Shiller (Yale University, USA), Shahid Ebrahim (Bangor Business School, UK) and Mark Shackleton (Lancaster University, UK).

Research Interests

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.

Research Collaborations

Robert Shiller (Yale University, USA)

Shahid Ebrahim (Bangor Business School, UK) 

Mark Shackleton (Lancaster University, UK)

Vicky Henderson (University of Oxford, UK)

David Hobson (University of Warwick, UK)

William Shaw (University College London, UK)

Greg Pawlina (Lancaster University, UK)

Jinsha Zhao (Kingston University London, UK)

Radu Tunaru (University of Kent, UK)

Dogan Tirtiroglu (University of Adelaide, Australia)

Bernard Dumas (INSEAD, France)

Peter Carr (Morgan Stanley & New York University, USA)

Marc Chesney (University of Zurich, Switzerland)


Journal articles

  • Wojakowski RM, Shahid Ebrahim M, Shackleton MB. (2016) 'Reducing the Impact of Real Estate Foreclosures with Amortizing Participation Mortgages'. Journal of Banking and Finance, 71 (October 2016), pp. 62-74.


    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.

  • Shiller RJ, Wojakowski RM, Ebrahim MS, Shackleton MB. (2013) 'Mitigating financial fragility with Continuous Workout Mortgages'. Journal of Economic Behavior and Organization, 85 (1), pp. 269-285.


    This paper models Continuous Workout Mortgages (CWMs) in an economic environment with refinancings and prepayments. CWMs are home loans whose balance and payments are indexed using a market-observable house price index of the pertaining locality. Our main results include: (a) explicit modelling of repayment and interest-only CWMs; (b) closed form formula for mortgage payment and mortgage balance of a repayment CWM; (c) a closed form formula for the actuarially fair mortgage rate of an interest-only CWM. For repayment CWMs we extend our analysis to include two negotiable parameters: adjustable " workout proportion" and adjustable " workout threshold." These results are of importance as they not only help in the understanding of the mechanics of CWMs and estimating key contract parameters, but they also provide insight on how to enhance the resilience of the financial architecture and mitigate systemic risk. © 2012 Elsevier B.V.

  • Tsekrekos AE, Shackleton MB, Wojakowski R. (2012) 'Evaluating Natural Resource Investments under Different Model Dynamics: Managerial Insights'. European Financial Management, 18 (4), pp. 543-575.


    We focus on factors that drive the dynamics of commodity prices. We highlight the capital budgeting implications of three highly-cited, nested, multi-factor models for commodity prices that have been successful in empirical investigations. Competing assumptions regarding commodity prices and their convenience yields can account for differences close to 40% on average, and in excess of 60% in cases, in the valuation of typical natural resource investments. These value differences are found to increase with the maturity and the intrinsic value of the investment, and also with the level and the volatility of the resource's convenience yield. Resources such as oil or copper, that are used for production purposes, usually exhibit high and volatile convenience yields; thus our findings should be more relevant for decision-makers in such sectors. © 2010 Blackwell Publishing Ltd.

  • Wojakowski RM. (2012) 'How should firms selectively hedge? Resolving the selective hedging puzzle'. Journal of Corporate Finance, 18 (3), pp. 560-569.


    We provide a model of intertemporal hedging consistent with selective hedging, a widespread practice corroborated by recent empirical studies. We argue that the optimal hedge is a value hedge involving total current value of future earnings. More importantly, the hedging decision is independent of risk preferences of the firm or agent. Our closed-form solutions imply several implications for the risk management policy in a firm. In order to lock in profits a hedge increase is recommended in favorable states of nature, while in bad states the firm should decrease the hedge and wait. Our main new empirical implication is that selective hedging should be more prevalent in industries where managers are exposed to convex cash flow structures and are more likely to "value hedge" their exposures. © 2012 Elsevier B.V.

  • Ebrahim MS, Shackleton MB, Wojakowski RM. (2011) 'Participating mortgages and the efficiency of financial intermediation'. Journal of Banking and Finance, 35 (11), pp. 3042-3054.


    This paper establishes a basic framework to study three different variants of Participating Mortgages (PMs). We obtain results for Shared Appreciation Mortgages (SAMs), Shared Income Mortgages (SIMs) and Shared Equity Mortgages (SEMs) in closed-form. We illustrate our findings with examples that show PMs are also attractive in an environment where prepayment can occur. Finally we conclude with the public policy implications of employing PMs as workout loans, especially post sub-prime crisis. We argue that by facilitating better risk sharing, PMs offer a means to enhance the efficiency and resiliency of the financial system. © 2011 Elsevier B.V.

  • Shiller RJ, Wojakowski RM, Ebrahim S, Shackleton M. (2011) 'Continuous Workout Mortgages'. National Bureau of Economic Research Working Paper,
  • Shackleton M, Wojakowski RM. (2007) 'Finite Maturity Caps and Floors on Continuous Flows'. Journal of Economic Dynamics and Control, 31 (12), pp. 3843-3859.


    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.

  • Henderson V, Hobson D, Shaw W, Wojakowski RM. (2007) 'Bounds for In-progress Floating-strike Asian Options Using Symmetry'. Annals of Operations Research, 151 (1), pp. 81-98.


    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.

  • Shackleton MB, Tsekrekos AE, Wojakowski R. (2004) 'Strategic entry and market leadership in a two-player real options game'. Journal of Banking and Finance, 28 (1), pp. 179-201.


    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.

  • Chung SL, Shackleton M, Wojakowski RM. (2003) 'Efficient Quadratic Approximation of Floating Strike Asian Option Values'. Finance, 24 (1), pp. 49-62.


    We derive a new formula for Asian options with floating strike, which proves more accurate for both low and higher volatility values. Average Strike Options are less often considered in the literature because their valuation is more complex. Compared to a benchmark our analytical formula is very efficient in the sense of accuracy vs speed, whereas numerical methods: Monte-Carlo, numerical integration of the partial differential equation or numerical inversion of the Laplace transform all require considerable calculating time.

  • Wojakowski RM, Shackleton M. (2003) 'Real probability of exercising and expected values of option payoff'. Futures Market (Rynek Terminowy), Poland: 20 (2), pp. 125-127.


    We derive the continuous-time formula for expected payoff to holding an option, which nests several major pricing tools. We also show that under current market conditions the true exercise probability, N(d4), lies halfway between the two more familiar terms: N(d1) and N(d2).

  • Henderson V, Wojakowski R. (2002) 'On the equivalence of floating-and fixed-strike Asian options'. Journal of Applied Probability, 39 (2), pp. 391-394.


    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.

  • Shackleton M, Wojakowski R. (2002) 'The Expected Return and Exercise Time of Merton-style Real Options'. Journal of Business Finance & Accounting, 29 (3&4), pp. 541-555.


    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.

  • Shackleton M, Wojakowski R. (2001) 'On the expected payoff and true probability of exercise of European options'. Applied Economics Letters, 8 (4), pp. 269-271.


    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).

Conference papers

  • Wojakowski RM. (2013) 'How should firms selectively hedge in incomplete markets? Explaining reluctance towards long-term hedging.'. Chicago, USA: Midwest Finance Association
  • Wojakowski RM, Ebrahim S, Shackleton M. (2012) 'Optimal amortization schedule of repayment participation mortgages'. Sydney, Australia: Bachelier Finance Society, 7th World Congress
  • Shiller RJ, Wojakowski RM, Ebrahim S, Shackleton M. (2011) 'On Continuous Workout Mortgages'. Bangor, UK: Conference on Financial Sector Performance & Risk (Post-Crisis)
  • Wojakowski RM. (2011) 'Continuous Workout Mortgages'. Denver, USA: ASSA Meetings, American Real Estate and Urban Economics Association
  • Wojakowski RM. (2010) 'On Continuous workout mortgages'. Toronto, Canada: Bachelier Finance Society, 6th World Congress
  • Ebrahim S, Shackleton M, Wojakowski RM. (2010) 'Participating Mortgages and the Efficiency of Financial Intermediation'. Palais Brongniart, Paris, France: Households, Risk and Insurance
  • Gang X, Taylor S, Wojakowski RM. (2009) 'Information Flow, Volatility Measurement and Jump Prediction'. University of Edinburgh: Royal Statistical Society Annual Conference
  • Wojakowski RM, Ebrahim S, Shackleton M. (2009) 'Participating Mortgages and the Efficiency of Financial Intermediation'. University of Glasgow: ISA RC43 International Housing Conference
  • Gang X, Taylor S, Wojakowski RM. (2009) 'Information Flow, Volatility Measurement and Jump Prediction'. National University of Singapore: Third Annual Risk Management Conference
  • Gang X, Taylor S, Wojakowski RM. (2009) 'Information Flow, Volatility Measurement and Jump Prediction'. University of Venice, Ca'Foscari: International Risk Management Conference
  • Wojakowski RM, Ebrahim S, Shackleton M. (2009) 'On Pricing Continuous Workout Mortgages'. Washington D.C.: American Real Estate and Urban Economics Association (AREUEA) 37th annual Mid-Year Meeting
  • Ebrahim S, Shackleton M, Wojakowski RM. (2009) 'Participating Mortgages and the Efficiency of Financial Intermediation'. San Francisco: ASSA Meetings, American Real Estate and Urban Economics Association (AREUEA) annual conference
  • Ebrahim S, Shackleton , Wojakowski RM. (2008) 'Participation Mortgages and the Efficiency of Financial Intermediation'. Imperial College, London: Bachelier Finance Society, 5th World Congress
  • Ebrahim S, Shackleton M, Wojakowski RM. (2008) 'Participation Mortgages and the Efficiency of Financial Intermediation'. Coimbra, Portugal: Portuguese Finance Network, 5th Finance Conference
  • Wojakowski RM, Ebrahim S, Shackleton M. (2008) 'Valuing participation mortgage loans using profit caps and floors'. Warsaw University, Poland: Warsaw Economics Meetings (WIEM)
  • Ebrahim S, Shackleton M, Wojakowski RM. (2008) 'Valuing Participation Mortgage Loans Using Profit Caps and Floors'. Istanbul, Turkey: American Real Estate and Urban Economics Association (AREUEA) international conference


  • Chesney M, Hernandez-Trillo F, Marois B, Wojakowski RM. (2001) Management of Foreign Exchange Risk: Currency Options (El manejo del riesgo cambiario: Las opciones sobre divisas). Mexico : Limusa-CIDE , pp. 1-179.
  • Chesney M, Marois B, Wojakowski RM. (1995) Foreign Exchange Options: Pricing and Using (Les options de change: Evaluation et utilisation). Paris : Economica , pp. 1-129.

Book chapters

  • Wojakowski RM, Shackleton M. (2001) 'On Option Expected Returns'. in Kohlmann M, Tang S (eds.) Mathematical Finance Basel, Switzerland : Birkhauser Verlag , pp. 265-374.
  • Shackleton M, Wojakowski RM. (2001) 'Reversible Real Options'. in Kohlmann M, Tang S (eds.) Mathematical Finance Basel, Switzerland : Birkhauser Verlag , pp. 339-344.
  • Chesney M, Marois B, Wojakowski RM. (1997) 'Foreign Exchange Options: Pricing (Options de change: evaluation)'. in (ed.) Encyclopaedia of financial markets (Encyclopedie des marches financiers) Paris : Economica Article number 69 , pp. 1398-1422.

Other publications

  • Quittard-Pinon F, Wojakowski RM. (1994) On Term Structure & Options (Sur la structure par terme & des options). Ecole Normale Supérieure


Current: Derivative Securities (MSc) MANM137


Departmental Duties



Research activities

  • Session Chair & Discussant "Real options and Hedging Theory" Midwest Finance Association conference, Chicago, USA, March 2013.
  • Member of Scientific Committee, conference on Banking, Finance, Money and Institutions: The Post Crisis Era, Centre for Money, Banking & Institutions (University of Surrey) and the Center for Research in Contemporary Finance (Fordham University, USA), Guildford, UK November 2013.
  • Member of Scientific Committee, EFMA Conference, Reading, UK 2013.
  • Member of Scientific Committee, French Finance Association conference, Lyon, May 2013.
  • Member of Scientific Committee, French Finance Association Conference, Montpellier, May 2011.
  • Member of Scientific Committee, French Finance Association Conference, St Malo, May 2010.
  • Discussant, Workshop Households, Risk and Insurance, Universite Paris-Dauphine & Institut Louis Bachelier, Paris, January 2010.
  • Discussant, ASSA Meetings, Atlanta, January 2010
  • Discussant, ISA RC43 International Housing Conference, University of Glasgow, 1-4 September 2009.
  • Participant, Risk Management and Financial Crisis, Paris, March 2009.
  • Session chair, Warsaw Economics Meetings (WIEM), Warsaw University, Poland, July 2008.
  • Participant, Dynamic Portfolio Choice, Asset Pricing and Mathematical Finance, London Business School 2004.
  • Discussant, European Finance Association, London, Summer 2000.
  • Participant, AFFI International Conference, La Baule, France, June 1993.
  • Participant, Topological Methods in Quantum Field Theories, International Centre for Theoretical Physics,
    Trieste, Italy, June 1990.

Invited presentations at research seminars

  • University of Kent, England, September 2012.
  • Universite de Cergy-Pontoise, France, September 2012.
  • University of Adelaide, Australia, June 2012.
  • Universite de Cergy-Pontoise, France, February 2012.
  • Edinburgh University, Scotland, June 2011.
  • Glasgow University, Scotland, June 2011.
  • Morgan Stanley, New York City, January 2011.
  • Bangor University, UK, December 2010.
  • ICMA Centre, University of Reading, UK, November 2009.
  • City University, London, UK, October 2009.
  • Maastricht University, Netherlands, March 2009.
  • Manchester University, UK, October 2008.
  • Nottingham University, UK, November 2007.
  • Central European University (CEU), Budapest, Hungary, March 2006.
  • Ecole Superieure de Commerce de Paris (ESCP), France, January 2003.
  • Warwick University, UK, February 2003.
  • ESSEC, Paris, France, April 2002.
  • Pole Universitaire Leonard de Vinci, Paris, France, March 2002.
  • ENSAE Mathematical Economics Seminar, Institut Henri Poincare, Paris, France, January 2002.
  • Financial Options Research Centre (FORC), Warwick University, UK, 2001.
  • Strathclyde University, Glasgow, UK, 2000.
  • Hugo Steinhaus Centre for Stochastic Methods in Science and Technology, Wroclaw, Poland, December 1998.
  • Concordia University, Montreal, Canada 1998.
  • HEC Montreal, Canada 1998.
  • Rotterdam University, Netherlands 1998.
  • Lancaster University, UK 1998.
  • HEC Lausanne, Switzerland 1996.

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