1. Pricing Convertible Bonds Using PDEs Dragan Bezanovic
Leversys –Tesla
Solutions and services for financial markets
Split, June 2011 1

2. Solutions and Services
Leversys 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.
System
Solutions
Data
Services
Operation
Services
Custom
Software
Development
Valuation
Models 2

3. Solutions and Services
System Solutions
Leversys Drina, cross-asset class system for instrument analytics and portfolio risk management
Valuation Models
Pricing libraries for convertible bonds, fixed income, equity and credit derivatives
Data Services
Setup and maintenance of convertible bonds terms and conditions 3

4. Solutions and Services
Custom Software Development
Custom 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. 4

5. Instruments – Coverage and development plan
Existing coverage
Development pipeline
bonds
zero coupon, fixed, floating, step up/down, accreting, structured
convertibles
callable, putable, accreting, resettable, variable conversion, exchangeable
PERCS like mandatories
PEPS like mandatories
convertible asset swaps
equity options
equity default swaps
credit default swaps
bonds (PIK, sinking schedule)
Convertibles (contingent conversion, dividend protection, takeover protection)
options
futures FX
credit default swap
interest rate swaps
variance swaps
dividend swaps 5

6. 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 6

7. CBs-definition (cont)7

8. CBs-definition (cont)8

9. Types of CBs Vanilla CBs Exchangeables Mandatory CBs
Contingent convertables
Cross FX CBs
Resettable CBs
and many more… 9

10. 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 10

11. 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)) 11

12. Numerical method Crank Nicholson method for time integration
Finite difference for spatial discretisation
Example (CB Vedanta 2017, price VS spot) 12

13. Screenshots - Tesla (excel)13

14. Screenshots - Drina (CB Analysis)14

15. Screenshots - Drina (Portfolio Monitor)15

16. 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 16