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Rafal Wojakowski

Reader in Finance

Academic and research departments

Surrey Business School.


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

Previous roles

2013 - 2015
Research seminar convenor

Affiliations and memberships

Member of Advisory Board
Centre for Quantitative Finance, University of Kent

In the media

CNN Money: Continuous Workout Mortgages 
Coauthor of research talked about and discussed
Media Contacts

Contact the press team


Phone: +44 (0)1483 684380 / 688914 / 684378
Out-of-hours: +44 (0)7773 479911
Senate House, University of Surrey
Guildford, Surrey GU2 7XH


Research interests

My teaching

My publications


Wojakowski RM (2013) How should firms selectively hedge in incomplete markets? Explaining reluctance towards long-term hedging.
Wojakowski R, Shahid Ebrahim M, Shackleton M (2016) Reducing the Impact of Real Estate Foreclosures with Amortizing Participation Mortgages Journal of Banking and Finance 71 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.
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.
Ebrahim S, Shackleton M, Wojakowski RM (2008) Participation Mortgages and the Efficiency of Financial Intermediation
Wojakowski RM, Ebrahim S, Shackleton M (2008) Valuing participation mortgage loans using profit caps and floors
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).
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
Shiller RJ, Wojakowski RM, Ebrahim S, Shackleton M (2011) On Continuous Workout Mortgages
Wojakowski RM, Shackleton M (2003) Real probability of exercising and expected values of option payoff Futures Market (Rynek Terminowy) 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, 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.
Wojakowski RM, Ebrahim S, Shackleton M (2009) On Pricing Continuous Workout Mortgages
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.
Ebrahim S, Shackleton M, Wojakowski RM (2009) Participating Mortgages and the Efficiency of Financial Intermediation
Wojakowski RM, Shackleton M (2001) On Option Expected Returns pp. 265-374
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.
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.
Gang X, Taylor S, Wojakowski RM (2009) Information Flow, Volatility Measurement and Jump Prediction
Chesney M, Marois B, Wojakowski RM (1997) Foreign Exchange Options: Pricing (Options de change: evaluation) pp. 1398-1422
Wojakowski RM (2011) Continuous Workout Mortgages
Shackleton M, Wojakowski RM (2001) Reversible Real Options pp. 339-344
Gang X, Taylor S, Wojakowski RM (2009) Information Flow, Volatility Measurement and Jump Prediction
Ebrahim S, Shackleton M, Wojakowski RM (2008) Valuing Participation Mortgage Loans Using Profit Caps and Floors
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.
Wojakowski RM (2010) On Continuous workout mortgages
Wojakowski RM, Ebrahim S, Shackleton M (2009) Participating Mortgages and the Efficiency of Financial Intermediation
Chesney M, Marois B, Wojakowski RM (1995) Foreign Exchange Options: Pricing and Using (Les options de change: Evaluation et utilisation) pp. 1-129
Quittard-Pinon F, Wojakowski RM (1994) On Term Structure & Options (Sur la structure par terme & des options)
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.
Ebrahim S, Shackleton, Wojakowski RM (2008) Participation Mortgages and the Efficiency of Financial Intermediation
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) pp. 1-179
Gang X, Taylor S, Wojakowski RM (2009) Information Flow, Volatility Measurement and Jump Prediction
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 S, Shackleton M, Wojakowski RM (2010) Participating Mortgages and the Efficiency of Financial Intermediation
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, Ebrahim S, Shackleton M (2012) Optimal amortization schedule of repayment participation mortgages
Tian L (2017) Banking market competition and corporate innovation.
By using the U.S. data, this study shows that state-level banking market competition, measured by Panzar-Rosse (1984) H-statistic, increases the number of patents and citations generated by firms and improves corporate innovation efficiency, supporting market power hypothesis. Greater banking competition is also found to enable innovative firms to adopt more ambitious innovation policies and have more flexibility to experiment with new technologies. The investigation of the heterogeneity of banking competition effects at the state-level shows that regional innovation in the states with lower R&D intensity benefits more from improved competition in local banking markets where additional innovation makes a greater marginal economic contribution. At the firm-level, such favourable effects vary across firm characteristics and innovative firms, with greater dependence on external finance and being financially constrained, enjoy a greater benefit from increased banking competition. The research also examines the role of information specialisation and finds that the banking competition effects are stronger for firms operating in informationally opaque industries and having more specialised information. It implies that banks benefit from the economies of scale in more specialised information acquisition when allocating credit supply. Finally, the research reveals novel evidence on the substitution effects of competition in a wider region and neighbour-state to local banking market in financing corporate innovations. The finding shows ?how local is local banking market? depends on the operating scope and information transparency of borrowing firms and local banks have an information advantage over distant banks in financing local businesses and informationally opaque corporate innovation activities.
Shiller R, Wojakowski R, Ebrahim M, Shackleton M (2017) Continuous Workout Mortgages: Efficient Pricing
and Systemic Implications
Journal of Economic Behavior and Organization
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).

Additional publications