Deriving Durrleman’s Butterfly-Arbitrage Condition

# Proof of Durrleman's condition for the implied volatility surface to eliminate butterfly spread arbitrage


# Proof of Durrleman's condition for the implied volatility surface to eliminate butterfly spread arbitrage












I have seen many papers mention the Durrleman condition for an implied volatility surface as a means to eliminate butterfly spread arbitrage, yet none provide a rigorous proof that fully convinces me. For example:

- Durrleman in the paper "Implied Volatility: Market Models" simply states the condition without any proof.

- Jim Gatheral in the paper "Arbitrage-free SVI volatility surfaces" presents a proof that seems overly simplistic, and I have difficulty following it.

- Michael Roper in his paper "Arbitrage Free Implied Volatility Surfaces" , provides a summary of the condition to some extent.

Is Roper’s formulation (3) equivalent to Jim Gatheral’s equation (2.1)?

- Zhang's "A two-step framework for arbitrage-free prediction of the implied volatility surface" obtains another version of the Durrleman condition. Are these formulations equivalent?

- Kim's No-arbitrage implied volatility functions: Empirical evidence from KOSPI 200 index options

What I really seek is a detailed, equivalent proof that derives the Durrleman condition directly from the original price derivative conditions. So far, I have not encountered a proof that isn’t overly simplified. Also, I cannot fully see the equivalence between these 5 statements.

## Answer by Zach-M (score 8)

https://quant.stackexchange.com/a/85349

The apparent inconsistency among the five sources you cited stems from the fact that implied volatility (IV) can be represented in several distinct but equivalent ways:

- volatility ($\sigma$) versus variance ($\sigma^2$)

- raw ($\sigma$, $\sigma^2$) versus total ($\sigma\sqrt{t}$, $\sigma^2{t}$)

- function of strike price ($K$) versus function of moneyness ($k$)

Different papers adopt different variants of the Durrleman condition, depending on context and notation. Below, I show the equivalence among five commonly used formulations and then explain how the original condition is derived.

But first, let's align terminology.

### Notation

- We work at a fixed point in time. The quantities ${S, r, q, t}$ denote the spot price, risk-free interest rate, dividend yield, and time to expiration, respectively, and are treated as constants.

- We have a volatility function $\sigma(k)$ mapping moneyness ($k$) to volatility, where $k = \ln\frac{K}{F}, \enspace F = S e^{(r-q)t}$. Conversely, the strike corresponding to moneyness $k$ is given by $K(k)=Fe^k$.

- Later, we extend the discussion to $\Sigma(K)$, a volatility function expressed directly in terms of strike $K$ (much less common in literature)

- The function $\sigma(k)$ is assumed to be continuous and twice differentiable

- From $\sigma(k)$, we define total volatility $v(k):=\sigma(k)\sqrt{t}$, and total variance $w(k):=v^2(k)=\sigma^2(k)t$

> For brevity, from this point onward functions of $k$ are written without explicit parentheses; for example, $w, v, \sigma, K$ denote $w(k), v(k), \sigma(k), K(k)$

### Basic Identities

From the definitions of $v$ and $w$:

- $v:=\sigma \sqrt{t}=\sqrt{w}$

- $w:=\sigma^2 t=v^2$

we can obtain the following derivatives and cross-relationships:

- $\sigma':=\dfrac{\mathrm{d} \sigma}{\mathrm{d} k}$

- $\sigma'':=\dfrac{\mathrm{d}^2 \sigma}{\mathrm{d} k^2}$

- $v':=\dfrac{\mathrm{d} v}{\mathrm{d} k}=\sigma'\sqrt{t}=\dfrac{w'}{2\sqrt{w}}$

- $v'':=\dfrac{\mathrm{d}^2 v}{\mathrm{d} k^2}=\sigma''\sqrt{t}=\dfrac{2ww''-w'^2}{4w\sqrt{w}}$

- $w':=\dfrac{\mathrm{d} w}{\mathrm{d} k}=2t\sigma\sigma'=2vv'$

- $w'':=\dfrac{\mathrm{d}^2 w}{\mathrm{d} k^2}=2t(\sigma'^2+\sigma\sigma'')=2(v'^2+vv'')$

(Reminder: all quantities above, except $t$, are functions of $k$)

### Durrleman's Condition for Moneyness-based Volatility

The no-butterfly-arbitrage condition of Durrleman can be written in several equivalent forms:

- $g_\sigma(k):=(1-\dfrac{k\sigma'}{\sigma})^2-\dfrac{(\sigma\sigma't)^2}{4}+\sigma\sigma''t \ge 0$

- $g_v(k):=(1-\dfrac{kv'}{v})^2-\dfrac{(vv')^2}{4}+vv'' \ge 0$

- $g_w(k):=(1-\dfrac{kw'}{2w})^2-\dfrac{w'^2}{4}(\dfrac{1}{w}+\dfrac{1}{4})+\dfrac{w''}{2} \ge 0$

These are 4 of the 5 formulations mentioned in the question:

- $g_v(k)$ is used by Roper

- $g_w(k)$ is used by Gatheral and Kim (with Kim denoting our $w$ by $v$)

- $g_\sigma(k)$ is used by Zhang.

The equivalence among these expressions follows directly from the differential identities established in the previous section.

### Durrleman's Condition for Strike-based Volatility

When the IV function and its derivatives are expressed in terms of strike $K$ rather than moneyness $k$, additional transformations are required.

Let $\Sigma(K)$ denote the strike-based implied volatility, with derivatives $\Sigma'(K)$ and $\Sigma''(K)$. To apply Durrleman’s condition, we first introduce auxiliary functions that express volatility as a function of $K$, while differentiating with respect to $k$:

- $\sigma_K(K)=\sigma(k(K))=\sigma(\ln{\frac{K}{F}})$

- $\sigma'_K(K)=\dfrac{\mathrm{d} \sigma_K}{\mathrm{d} k}=\sigma'(\ln{\frac{K}{F}})$

- $\sigma''_K(K)=\dfrac{\mathrm{d} \sigma'_K}{\mathrm{d} k}=\sigma''(\ln{\frac{K}{F}})$

> From this point onward in this section, functions of $K$ are written without parentheses. This differs from earlier sections, where omitted arguments referred to $k$.

Now we can express the relationship between the $\Sigma$-functions and the $\sigma_K$-functions:

(for the following note that from $k(K)=\ln{\frac{K}{F}}$, we have $\frac{\mathrm{d} k}{\mathrm{d} K}=\frac{1}{K}$)

- $\Sigma:=\sigma_K$

- $\Sigma':=\dfrac{\mathrm{d} \sigma_K}{\mathrm{d} K}=\dfrac{\mathrm{d} \sigma_K}{\mathrm{d} k}\dfrac{\mathrm{d} k}{\mathrm{d} K}=\dfrac{\sigma'_K}{K}$

- $\Sigma'':=\dfrac{\mathrm{d}^2 \sigma_K}{\mathrm{d} K^2}=\dfrac{\mathrm{d} \Sigma'}{\mathrm{d} K}=\dfrac{\frac{\mathrm{d} \sigma'_K}{\mathrm{d} k}\frac{\mathrm{d} k}{\mathrm{d} K}K-\sigma'_K}{K^2}=\dfrac{\sigma''_K-\sigma'_K}{K^2}$

Equivalently,

- $\sigma_K=\Sigma$

- $\sigma'_K=K\Sigma'$

- $\sigma_K''=K^2\Sigma''+K\Sigma'$

Substituting these into $g_\sigma(k)$ yields the strike-based Durrleman condition:

$g_\Sigma(K):=(1-\dfrac{K\ln{\frac{K}{F}}\Sigma'}{\Sigma})^2-\dfrac{(K\Sigma\Sigma't)^2}{4}+K^2\Sigma\Sigma''t+K\Sigma\Sigma't \ge 0$

This is the formulation originally used by Durrleman (first snippet you attached).

### Two Additional Formulations Using $d_1$ and $d_2$

If one works directly with raw volatility ($\sigma$ or $\Sigma$) and computes the Black–Scholes quantities $d_1$ and $d_2$, the Durrleman condition can also be written as:

- $g_\sigma(k):=1+d_1d_2\sigma'^2t+(d_1+d_2)\sigma'\sqrt{t}+\sigma\sigma''t$

- $g_\Sigma(K):=1+d_1d_2K^2\Sigma'^2t+2d_1K\Sigma'\sqrt{t}+K^2\Sigma\Sigma''t$

As before, $\sigma$-quantities are differentiated with respect to $k$, while $\Sigma$-quantities are differentiated with respect to $K$. The corresponding definitions of $d_1$ and $d_2$ are:

$d_{1,2}=\dfrac{-k}{\sigma\sqrt{t}} \pm \dfrac{\sigma\sqrt{t}}{2}=\dfrac{\ln{\frac{F}{K}}}{\Sigma\sqrt{t}} \pm \dfrac{\Sigma\sqrt{t}}{2}$

### Derivation of the Durrleman Condition

The intuition behind the no-butterfly-arbitrage condition is straightforward. In a butterfly strategy, buying calls at $K-d$ and $K+d$ while selling two calls at $K$, the initial cash flow must be negative (a net debit). Otherwise, the strategy would generate a positive payoff in all future states, constituting an arbitrage.

Equivalently, the option price at the average strike must not exceed the average price of options at surrounding strikes. This requires the call price function $C(K)$ to be convex in $K$:

- $\dfrac{\partial C}{\partial K} < 0$

- $\dfrac{\partial^2 C}{\partial K^2} > 0$

The Durrleman condition is simply a reformulation of the convexity requirement $\partial^2 C/\partial K^2 > 0$ in terms of implied volatility rather than option prices.

#### Technical Details

The Black–Scholes call price is given by:

$C(K)=Se^{-qt}N(d_1)-Ke^{-rt}N(d_2)$

> While convexity can also be analyzed with respect to moneyness ($\partial^2 C/\partial k^2 > 0$), differentiation with respect to $K$ is algebraically simpler. Moreover, $\partial^2 C/\partial K^2$ has a direct interpretation in terms of the risk-neutral density, as discussed in the appendix.

There are several ways to differentiate $C(K)$. The volatility terms in $C(K)$ are hidden inside $d_1$ and $d_2$, so the choice of how to express them (e.g. by $\sigma$, $\Sigma$, $v$ or $w$), will lead to the corresponding Durrleman condition variant. Choosing the $v$-based representation, we write:

- $d_1=\frac{-k}{v}+\frac{v}{2}=d_2+v$

- $d_2=\frac{-k}{v}-\frac{v}{2}=d_1-v$

Some useful results before differentiating $C(K)$:

- $Se^{-qt}n(d_1)=Ke^{-rt}n(d_2) \enspace$ resulting from: $d_1^2-d_2^2=-2k \enspace\Rightarrow\enspace d_1^2=(d_2^2-2k)$, leading to $n(d_1)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}d_1^2} =\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}(d_2^2-2k)} =\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}d_2^2+k} =n(d_2)e^k$, where $e^k=\frac{K}{F}=\frac{Ke^{-rt}}{Se^{-qt}}$

- $d_1'=\frac{\mathrm{d} d_1}{\mathrm{d} k}=d_2'+v' \enspace\Rightarrow\enspace (d_1'-d_2')=v'$

- $k'=\frac{\mathrm{d} k}{\mathrm{d} K}=\frac{1}{K} \enspace\Rightarrow\enspace Kk'=1$

Also, a reminder on standard normal distribution functions, $N(f)$ and $n(f)$: if $f$ is a function of $k$, then differentiating with regards to $k$ gives: $N'=n(f)f'$ and $n'=-fn(f)f'$.

First derivative:

$ \begin{aligned} \dfrac{\partial C}{\partial K} &=Se^{-qt}\dfrac{\mathrm{d} N(d_1)}{\mathrm{d} K}-e^{-rt}\dfrac{\mathrm{d} KN(d_2)}{\mathrm{d} K}\\ &=Se^{-qt}n(d_1)d_1'k'-e^{-rt}(N(d_2)+n(d_2)d_2'k'K) \\ &=Ke^{-rt}n(d_2)d_1'k'-e^{-rt}N(d_2)-Ke^{-rt}n(d_2)d_2'k' \\ &=(Kk')e^{-rt}n(d_2)(d_1'-d_2')-e^{-rt}N(d_2)\\ &=e^{-rt}(n(d_2)v'-N(d_2)) \end{aligned} $

(the strike-based version of this is: $\frac{\partial C}{\partial K}=e^{-rt}(n(d_2)K\Sigma'\sqrt{t}-N(d_2))$)

Second derivative:

$ \begin{aligned} \frac{\partial^2 C}{\partial K^2} &=e^{-rt}\left[(-d_2n(d_2)d_2'k'v'+v''k'n(d_2))-n(d_2)d_2'k'\right] \\ &=e^{-rt}n(d_2)k'(v''-d_2'd_2v'-d_2') \\ &=e^{-rt}\dfrac{n(d_2)}{K}(v''-d_2'(d_2v'+1)) \end{aligned} $

We rearrange the term $(v''-d_2'(d_2v'+1))$ and use it to define $g_v(k)$:

$ d_2'=\dfrac{\mathrm{d} d_2}{\mathrm{d} k}=\dfrac{\partial d_2}{\partial k}+\dfrac{\partial d_2}{\partial v}\dfrac{\mathrm{d} v}{\mathrm{d} k} =-\dfrac{1}{v} + (\dfrac{k}{v^2}-\dfrac{1}{2})\cdot v'=\dfrac{-2v+2kv'-v^2v'}{2v^2} \\ d_2v'=\dfrac{-2kv'-v^2v'}{2v} \\ \begin{aligned} v''-d_2'(d_2v'+1)&= \dfrac{1}{v} \left[ vv''-\dfrac{-2v+2kv'-v^2v'}{2v}(\dfrac{-2kv'-v^2v'}{2v}+1) \right]\\ :&=\dfrac{1}{v} \cdot g_v(k) \end{aligned} $

Organizing $g_v(k)$ to get the Durrleman result:

$ \begin{aligned} g_v(k) &=vv''-\frac{(-2v+2kv'-v^2v')(-2kv'-v^2v')+2v(-2v+2kv'-v^2v')}{4v^2} \\ &=vv''-\frac{8kvv'-4k^2v'v'+v^4v'v'-4v^2}{4v^2} \\ &=vv''-\frac{2kv'}{v}+(\frac{kv'}{v})^2-\frac{(vv')^2}{4}+1 \\ &=(1-\frac{kv'}{v})^2-\frac{(vv')^2}{4}+vv'' \end{aligned} $

Collecting all terms we get:

$ \dfrac{\partial^2 C}{\partial K^2}= e^{-rt}\dfrac{n(d_2)}{Kv}g_v(k)= e^{-rt}\dfrac{n(d_2)}{Kv}\left[ (1-\dfrac{kv'}{v})^2-\dfrac{(vv')^2}{4}+vv'' \right] $

(reminding again, all quantities here: $d_2$, $K$, $v$, $v'$, and $v''$, are functions of $k$)

Since all prefactors are strictly positive, the sign of $\partial^2 C/\partial K^2$ is entirely determined by the term in square brackets. Hence, the convexity condition $\partial^2 C/\partial K^2 > 0$ is equivalent to requiring $g_v(k) > 0$, which is precisely the Durrleman condition.

> In the Black–Scholes setting without a volatility smile (constant $v$, so $v' = v'' = 0$), we have $g_v(k) = 1$ for all strikes, and the condition is automatically satisfied.

### Appendix: Risk-Neutral Density (RND)

Devising $\partial^2 C/\partial K^2$ has another useful application in addition to no-butterfly-arbitrage enforcement.

Under risk-neutral assumptions (which of course are unrealistic) the price of a Call option is:

$ C(K)=e^{-rt}\,\mathbb{E}\left[\max{(S_T-K,0)}\right]=e^{-rt}\int_{K}^{\infty}{(s-K)p(s)ds} $

Where $p(s)$ is the density function of the market expectation for the price of the underlying asset at expiration, $S_T$. A famous result from Breeden and Litzenberger (1978) shows that by differentiating $C(K)$ twice with respect to $K$, we isolate $p(s)$ completely:

$ p(K)=e^{rt}\dfrac{\partial^2 C}{\partial K^2} \qquad p(k)=e^{rt}\dfrac{\partial^2 C}{\partial K^2} \Bigr|_{K=Fe^k} $

Which leads to direct relationship between the $g(k)$ functions we have and the RND of the underlying asset, $p(k)$:

$ p(K)=\dfrac{n(d_2)}{K\Sigma\sqrt{t}}g_\Sigma(K) \qquad p(k)=\dfrac{n(d_2)}{Kv}g_{\sigma/v/w}(k) $

This can be expanded to more elaborate forms, such as:

$ p(K)=n(d_2)\left[ \dfrac{1}{K\Sigma\sqrt{t}}+ \dfrac{d_1d_2K\sqrt{t}}{\Sigma}\Sigma'^2+ \dfrac{2d_1}{\Sigma}\Sigma'+ K\sqrt{t}\Sigma'' \right] \\ p(k)=\dfrac{n(\frac{-k}{v}-\frac{v}{2})}{Fe^k v}\left[(1-\dfrac{kv'}{v})^2-\dfrac{(vv')^2}{4}+vv''\right] $