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Convertible bond on a binomial stock tree: bond floor, conversion option, premium and delta

ToolConvertible Bond
SourceIllustrative case computed by the tool with the method of Tsiveriotis, K., and Fernandes, C. (1998), "Valuing Convertible Bonds with Credit Risk", Journal of Fixed Income, Vol. 8, No. 2, pp. 95–102, on the tree of Cox, J. C., Ross, S. A., and Rubinstein, M. (1979), Journal of Financial Economics, Vol. 7, No. 3, pp. 229–263.
LevelGraduate

Problem

A 5-year bond of face value 1,000 pays a 2 % coupon in two half-yearly payments and can be converted into 20 shares at any time. The stock is at 40 with a volatility of 30 % a year and no dividend; the risk-free rate is 4 % and the issuer's credit spread 3 %, both compounded twice a year. Find the conversion value, the bond floor, the value of the convertible on a 200-step binomial stock tree with cash payments discounted at the issuer's rate and shares at the risk-free rate, the value of the conversion option, the conversion premium and the delta.

Inputs to Enter

  • Face value 1,000, coupon 2 % paid half-yearly, maturity 5 years, conversion ratio 20 (conversion price 50)
  • Stock price 40, volatility 30 %, risk-free rate 4 %, credit spread 3 %; 200 tree steps
  • Advanced: issuer call at 1,000 from year 2 with a 130 % soft-call trigger; holder put at 1,000 in year 3; market price 1,100
Open the Convertible Bond tool (loaded with this example)

Answers

  • Conversion value 20 · 40 = 800.00; parity 80.00 % of face value
  • Bond floor at 7 %: 10 · 8.31661 + 1,000 · 0.708919 = 792.08
  • Tree: step 0.025 years, u = exp(0.30 · √0.025) = 1.04858, d = 0.95367, up-move probability 0.49858; one-step discount factors 0.999010 (risk-free) and 0.998281 (issuer)
  • Today's node: equity part 0.999010 · (0.49858 · 539.69 + 0.50142 · 449.63) = 494.05; cash part 0.998281 · (0.49858 · 522.71 + 0.50142 · 560.09) = 540.52; convertible value 1,034.57
  • Conversion option 1,034.57 − 792.08 = 242.48; conversion premium (1,034.57 − 800) / 800 = 29.32 %
  • Delta (1,062.403 − 1,009.724) / (41.943 − 38.147) = 13.877 per unit of stock price, 69.4 % of the conversion ratio
  • Advanced: with the soft call the value is 1,014.67 (1,056.01 with the put alone); a market price of 1,100 corresponds to an implied volatility of 40.46 %, and 1,000 to an implied credit spread of 4.485 points

Notes

The tool's default inputs are this case; press Submit (Advanced mode for the last line). The bond is illustrative. The numbers are the tool's own calculation, checked against an independent numpy / scipy implementation and closed forms (test_convertible_bond.php): with a zero spread the tree returns the straight bond plus 20 Black-Scholes calls struck at 50 (1,112.96) within 0.02 %. Not a worked example printed in the cited papers. A calculator for education and analysis, not investment advice.