Delta of a Down-and-Out Call When the Barrier Equals Debt
Summary
The document derives the sensitivity to the initial asset value of a down-and-out call whose barrier is set equal to its strike, interpreted here as a debt level. It rewrites the option value as a vanilla Black–Scholes call minus a scaled call on a transformed asset value. Differentiating this expression applies the product rule to the scale factor and the chain rule to the transformed call; the resulting delta combines the vanilla call delta, a term proportional to the transformed call price, and a barrier-related normal-distribution term.
The derivation supplies symbolic expressions rather than numerical examples or empirical validation. It assumes the stated Black–Scholes pricing setup and the barrier-option formula used in the question, so the result depends on those model conventions and parameter definitions. The document focuses on the equality case for barrier and strike and does not compare it with other barrier placements or discuss hedging performance.
Key ideas
- The option value is expressed as a vanilla call price less a scaled call price on a transformed asset value.
- Differentiating the scaled term requires both the product rule and the chain rule.
- The delta contains the vanilla call delta and additional terms tied to the barrier adjustment.
- The derivation is analytical and does not provide numerical or empirical validation.
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Full text
# Derivative: Delta of a Down and Out Call Option with Barrier=Debt(K)
# Derivative: Delta of a Down and Out Call Option with Barrier=Debt(K)
I am trying to compute the derivative of this function with respect to V0:
This is the price of a down and out call option, assuming the barrier equal to the level of debt K. In other terms, I need to compute the Delta of this DOC Option, in the case of Barrier=K (neither Barrier higher than K nor Barrier lower than K) and I cannot find this case anywhere in the literature. Furthermore, the derivative of the first two terms of the equation equals N(d1), the delta of a plain vanilla call option. Therefore, I just need the derivative of what is in the parenthesis [...] with respect to V_0.
Can someone of you help me?
Anything will be really appreciated!
## Answer by Gordon (score 2)
https://quant.stackexchange.com/a/25629
Let \begin{align*} C(S, K, t) = SN(d_1) - e^{-rt}KN(d_2) \end{align*} denote the Black-Scholes call option price with initial asset value $S$, strike $K$, and maturity $t$. Note that \begin{align*} \frac{\partial C}{\partial S} = N(d_1). \end{align*}
For the above barrier option, note that \begin{align*} E_0 &= V_0 N(d_1)-e^{-rt}KN(d_2) -\bigg[V_0 \Big(\frac{K}{V_0}\Big)^{2\lambda}N(d_1^B) -e^{-rt}K \Big(\frac{K}{V_0}\Big)^{2\lambda-2} N(d_2^B) \Big) \bigg]\\ &=C(V_0, K, t) - \Big(\frac{K}{V_0}\Big)^{2\lambda-2}\bigg[\frac{K^2}{V_0}N(d_1^B) -e^{-rt}K N(d_2^B) \Big) \bigg]\\ &=C(V_0, K, t) - \Big(\frac{K}{V_0}\Big)^{2\lambda-2} C\Big(\frac{K^2}{V_0}, K, t \Big). \end{align*} Therefore, \begin{align*} \frac{\partial E_0}{\partial V_0} &=N(d_1) + (2\lambda-2)\frac{K^{2\lambda-2}}{V_0^{2\lambda-1}}C\Big(\frac{K^2}{V_0}, K, t \Big)-\Big(\frac{K}{V_0}\Big)^{2\lambda-2}\frac{\partial C\Big(\frac{K^2}{V_0}, K, t \Big)}{\partial V_0}\\ &=N(d_1) + (2\lambda-2)\frac{K^{2\lambda-2}}{V_0^{2\lambda-1}}C\Big(\frac{K^2}{V_0}, K, t \Big)+\Big(\frac{K}{V_0}\Big)^{2\lambda}N(d_1^B), \end{align*} since \begin{align*} \frac{\partial C\Big(\frac{K^2}{V_0}, K, t \Big)}{\partial V_0} &= \frac{\partial C\Big(\frac{K^2}{V_0}, K, t \Big)}{\partial \frac{K^2}{V_0}}\frac{\partial \frac{K^2}{V_0}}{\partial V_0}\\ &=-\Big(\frac{K}{V_0}\Big)^{2}N(d_1^B). \end{align*}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.