Pricing Convertible Bonds Using PDEsPricing Convertible Bonds Using PDEs

Dragan BezanovicDragan Bezanovic

Leversys –TeslaLeversys –TeslaSolutions and services for financial markets

Split, June 2011Split, June 2011

Solutions and ServicesSolutions and ServicesLeversys and Tesla Risk Analytics provide financial market clients with innovative and leading edge solutions and services that enable them to leverage their skills and resources.

SystemSolutions

Data Services

ValuationModels

Custom Software

Development

Operation Services

Solutions and ServicesSolutions and Services

System SolutionsLeversys Drina, cross-asset class system for

instrument analytics and portfolio risk management

Valuation ModelsPricing libraries for convertible bonds, fixed income,

equity and credit derivatives

Data Services Setup and maintenance of convertible bonds terms and

conditions

Solutions and ServicesSolutions and Services

Custom Software DevelopmentCustom development of projects that are not related to

Leversys or Tesla solutions.

Operation Services Trade capture, confirmation and settlement, positions

and cash reconciliation, valuation and P&L reporting, custom reporting, collateral management, documentation management, security master and static data maintenance.

Instruments – Coverage and development planInstruments – Coverage and development planDevelopment pipelineExisting coverage

bonds •zero coupon, fixed, floating,

step up/down, accreting, structured

convertibles•callable, putable, accreting,

resettable, variable conversion, exchangeable

PERCS like mandatoriesPEPS like mandatoriesconvertible asset swapsequity optionsequity default swapscredit default swaps

bonds (PIK, sinking schedule)

Convertibles (contingent conversion, dividend protection, takeover protection)

options•futures•FX•credit default swap

interest rate swapsvariance swapsdividend swaps

CBs-definition

Convertible Bonds (CBs) are bonds that can be converted into a fixed number of shares of the issuer.

Convertibles are hybrid securities: - They have the benefits of debt instruments that pay fixed

coupons and will be redeemed at maturity at a pre-specified price.

- On the other hand, the embedded conversion option provides the investor with a participation in the upside potential of the underlying equity.

So, at maturity, the CBs are worth the higher of (a) redemption value (the price at which the issuer had agreed

to buy the bonds back) or (b) the value of the underlying shares

CBs-definition (cont)

CBs-definition (cont)

Types of CBs

• Vanilla CBs

• Exchangeables

• Mandatory CBs

• Contingent convertables

• Cross FX CBs

• Resettable CBs

and many more…

CBs - pricing

• Underlying equity follows Geometric Brownian Motion with jumps:

S(t-) and S(t+) are left and right limits of stock price

Wt is a standard Brownian motion, Nt is a Poisson (counting) process

When jump occurs we have

CBs - pricing

• Following standard Ito’s calculus, we end up with Black Scholes PDE

B(t) is the value recovered in the case of default (B(t)=RR*Notional(t))

Numerical method

Crank Nicholson method for time integration

Finite difference for spatial discretisation

Example (CB Vedanta 2017, price VS spot)

Screenshots - Tesla (excel)

Screenshots - Drina (CB Analysis)

Screenshots - Drina (Portfolio Monitor)

References

TeslaLib – Model document, Leversys

Leversys-Tesla Risk Analytics, Presentation, Leversys, 2011

Marcus Overhaus et al.Equity Hybrid Derivatives, John Wiley and Sons, Inc.,2007