Estimating Delta and Gamma P&L for a Convertible Bond

Summary

The document asks how to estimate the delta and gamma contributions to the profit and loss of a convertible bond position after an equity-price shock. It uses Bloomberg sensitivities, a stated market value of $2,500,000, delta of 0.74, gamma of 0.00524, and a 15% stock move. The proposed delta estimate applies the bond’s delta to the equity return and then to market value, producing an estimated $277,500 gain.

For gamma, the question treats gamma as the change in delta for each one-percentage-point stock move. It estimates the delta increase over the shock and uses half that change to approximate the convexity contribution, arriving at $14,737. The document provides a worked calculation but no answer or validation. Its interpretation depends on how Bloomberg defines and scales gamma, and the calculation assumes the sensitivities can be applied to the position’s current market value; readers should verify units and conventions before using it.

Key ideas

• Delta P&L is estimated by multiplying the equity return, convertible bond delta, and position market value.
• The example assumes delta is 0.74 and gamma is 0.00524 for a $2,500,000 position.
• The proposed gamma estimate uses half the delta change over the shock as a convexity adjustment.
• Gamma scaling conventions must be checked before applying the calculation.

Tags

# Gamma PNL for Convertible Bond


# Gamma PNL for Convertible Bond












so just trying to compute gamma PNL for some CB positions using Bloomberg data for delta/gamma. for a CB, BBG has delta 0.74 and Gamma 0.00524. if I want to compute the delta PNL and Gamma PNL for a position with market value $2,500,000(not face value)

with equity shock of 15%, i.e. the underlying stock price moves up by 15%. then it should clear that the delta PNL is given: 15% x 0.74 x 2,500,000 = $277,500.

that is to say, the CB price will move up by 15%*0.74 = 11.1%, hence we gain 11.1% from the current MV which is the delta pnl $277,500.

as the underlying stock price moves up, the CB's delta also changes, which is given by Gamma. In this case the Gamma will be the absolute change in the delta percentage given a one percent change in the underlying stock price of the convertible.

Gamma is 0.00524, so given the 15% underlying stock move, the CB's delta will increase by 0.00524*15=0.0786. so after the move the CB's delta is 0.74+0.0786 = 0.8186

now the gamma PNL, in my limited understanding, would be (0.5 * 15 * 0.00524) * 0.15 * $MV = 14,737 so basically (0.5 * 15 * 0.00524) is just half of the delta move.

anyone has any comments or better interpretation of the gamma PNL here? or share any formulas by using this example? Many thanks

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.