Closed-form interpolation-based formulas for European call options written on defaultable assets

Orosi, Greg

doi:10.1057/jam.2015.1

Cited by 5 publications

(8 citation statements)

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“…Assuming zero recovery when the expected recovery is positive can lead to mispricing of options and misestimation of the probability of default from equity options. There are several ways of estimating the probability of default from options, namely, Carr and Wu (), Orosi (), and Taylor et al (). In this section, we focus on the method proposed in Orosi ().…”

Section: Extracting Probability Of Default From Option Prices When Eqmentioning

confidence: 99%

“…There are several ways of estimating the probability of default from options, namely, Carr and Wu (), Orosi (), and Taylor et al (). In this section, we focus on the method proposed in Orosi (). As the formula in Orosi () does not allow for positive recovery, we first derive an extension of the formula in Orosi () for the case of positive recovery.…”

Section: Extracting Probability Of Default From Option Prices When Eqmentioning

confidence: 99%

“…In this section, we focus on the method proposed in Orosi (). As the formula in Orosi () does not allow for positive recovery, we first derive an extension of the formula in Orosi () for the case of positive recovery. Using and the results of Orosi (), an analytic expression for European call options can be derived as:

C (K, T) = \{\begin{matrix} S_{0} e^{- italicdT} - {Ke}^{- italicrT} & if K \leq R \\ S_{0} e^{- italicdT} - {Ke}^{- italicrT} + e^{- italicrT} (K - R) \cdot italicPD & if R \leq K \leq italicdb \\ (S_{0} e^{- dT} - e^{- rT} \cdot R \cdot PD) \cdot c & if italicdb < K \end{matrix},

where

\begin{matrix} right c & center = & left \frac{- B_{2} + \sqrt{{(B_{2})}^{2} - 4 A_{2} C_{2}}}{2 A_{2}} \\ right A_{2} & center = & left 1 - G (1 - PD) \\ right B_{2} & center = & left x (1 - PD) - 1 + 2 G \cdot DB (1 - PD), \\ right C_{2} & center = & left - G \cdot {italicDB}^{2} (1 - PD), \\ right DB & center = & left S_{0} e^{- dT} - e^{...} \end{matrix}

…”

Section: Extracting Probability Of Default From Option Prices When Eqmentioning

confidence: 99%

“…To derive , we extend the methodology in Orosi () by allowing non‐zero equity recovery at default. We start by observing that Equation gives the following equations for call option prices when

K \leq italicdb :

C (K, T) = \{\begin{matrix} S_{0} e^{- italicdT} - {Ke}^{- italicrT} & if K \leq R \\ S_{0} e^{- italicdT} - {Ke}^{- italicrT} + e^{- italicrT} (K - R) \cdot italicPD & if R \leq K \leq italicdb \end{matrix} .

To obtain analytic call option prices for

K > italicdb,

apply the following transformations to the strikes and call option prices:

\begin{matrix} right c & center = & left \frac{C ((), K, T)}{S_{0} e^{- dT} - e^{- rT} \cdot R \cdot PD} \\ right x & center = & left \frac{{italicKe}^{- rT}}{S_{0} e^{- dT} - e^{- rT} \cdot R \cdot PD} . \end{matrix}

Note that if

R \leq K \leq italicdb,

the relation between c and x is given by

c = 1 - false(1 - italicPD false) \cdot x,

or equivalently

x = \frac{1 - c}{1 - italicPD} .

To ensure that the transformed call option prices, c , are decreasing and convex functions of x when

…”

Section: Extracting Probability Of Default From Option Prices When Eqmentioning

confidence: 99%

“…In this section, we focus on the method proposed in Orosi (2015a). As the formula in Orosi (2015a) does not allow for positive recovery, we first derive an extension of the formula in Orosi (2015a) for the case of positive recovery. Using (11) and the results of Orosi (2015a), an analytic expression for European call options can be derived as:…”

Section: Extracting Probability Of Default From Option Prices When Eqmentioning

confidence: 99%

See 4 more Smart Citations

Equity Option Implied Probability of Default and Equity Recovery Rate

Chang

Orosi

2016

Journal of Futures Markets

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There is a close link between prices of equity options and the default probability of a firm. We show that in the presence of positive expected equity recovery, standard methods that assume zero equity recovery at default misestimate the option‐implied default probability. We introduce a simple method to detect stocks with positive expected equity recovery by examining option prices and propose a method to extract the default probability from option prices that allows for positive equity recovery. We demonstrate possible applications of our methodology with examples that include financial institutions in the United States during the 2007–09 subprime crisis. © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:599–613, 2017

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Section: Extracting Probability Of Default From Option Prices When Eqmentioning

confidence: 99%