Convertible Bond

The computation result is as below,

From the results above, we can see several points.

1. The Bond Price increases when the asset price rises. Intuitively, it is reasonable because the conversion price is fixed, the higher the asset price, the more likely the holder is going to convert and the more profit the holder can get. I draw a chart of bond price with asset price from 10 to 90. We can see the result more clearly.

2. The Delta of the bond increases when the asset price rises. Intuitively, it is reasonable because the Delta measures the change in the convertible’s price with respect to the change in the underlying common stock price. It is the slope of the line that is tangent to the convertible’s fair value line at a particular stock price. From the convexity of the Price curve, we can see the slope does increase when the underlying asset price increases. I draw a chart of Delta with asset price from 3 to 110 (I get rid of some noise points).

From the chart, we can see when the bond is deep out of money, the Delta is approaching 0 since it is more like a straight bond. When it is deep in the money, Delta is approaching a fixed number since it is more like equity. Another point I think should mention is that Delta is supposed to have most change, in another word, the Gamma should have the maximum number when the bond is at the money. We cannot see that from this chart and model, maybe it suggests that this model still has some points to improve.

3. The Vega is higher when the asset price is 47 than it is when asset price is 15 or 85. It makes sense because Vega measures the price sensitivity to volatility change and the Bond Price should be more sensitive to volatility when it is at the money. When asset price is 15 or 85, it is obviously deep out or in the money,