Butterfly Arbitrage and the Implied Density in Call Prices

# Strike Arbitrage


# Strike Arbitrage












In Stochastic Volatility Modelling, Chapter 2, the author derived the Dupire equation $$\mathbb{E}[\sigma_T^2|S_T = K] = 2\frac{\frac{dC}{dT} + qC +(r-q)K\frac{dC}{dK}}{K^2 \frac{d^2C}{dK^2}}.$$

The author discussed its denominator: he linked the denominator to a butterfly strategy. Then I cannot understand the following parts:

- Options’ markets are arbitraged well enough that butterfly spreads do not have negative prices:3 the denominator in the Dupire formula is positive.

- In a model, $\frac{d^2C(K,T)}{dK^2} = e^{-rT}\mathbb{E}[\delta(S_T - K)]$, where $\delta(\cdot)$ denotes the Dirac delta function. The condition $\frac{d^2C(K,T)}{dK^2} > 0$ is equivalent to requiring that the market implied density (what's implied density?) be positive.

Any comments and advice would be greatly appreciated! Thank you for your time and help!

## Answer by ir7 (score 1, accepted)

https://quant.stackexchange.com/a/56011

For the second question:

The implied density is the density function we integrate call payoffs against to match market call prices, denoted $f$ here.

So, ignoring discount factors, the answer comes from Dirac delta function's properties:

$$ \mathbf{E}[\delta(S-K)] = \int_{-\infty}^\infty \delta(S-K)f(S) dS = f(K) $$

Alternatively:

$$ C = C(K) = \int_K^\infty (S-K)f(S) dS $$

$$\frac{\partial C}{\partial K} = \frac{\partial }{\partial K} \left(\int_K^\infty (S-K)f(S) dS \right) $$

$$ = \int_K^\infty \frac{\partial }{\partial K}\left((S-K)f(S)\right)dS - (K-K)f(K) $$

$$ = - \int_K^\infty f(S)dS $$

Then:

$$ \frac{\partial^2 C}{\partial K^2} = \frac{\partial C}{\partial K} \left( - \int_K^\infty f(S)dS\right) = \left[-f(S)\right]\bigl\vert_{S=K}^{S=\infty} = f(K)$$

## Answer by StackG (score 2)

https://quant.stackexchange.com/a/55999

If ${\frac {\partial^2 C} {\partial K ^2}}$ was zero, then the price-strike curve would just be a straight sloping-downwards line, and it would cost the same to buy either two call options at strike $K$ (portfolio A), or one option each at strike $K-1$ and strike $K+1$ (portfolio B).

If you think about the payoffs at expiry where spot=$S_t$ of these two portfolios, you'll see that they are the same for $S_t < K-1$ (ie. both payoffs are 0) and the same for $S_t > K+1$ (ie. both payoffs are $2(S_t - K)$). BUT, between $K-1 < S_t < K+1$, portfolio B always pays more, because the option with strike $K-1$ is in the money earliest.

I've shown some graphs of this below. What this means is that portfolio B must cost more than portfolio A in a fairly priced market, and if you think about the shape of the price vs. strike curve, it means it must be concave (ie. ${\frac {\partial^2 C} {\partial K ^2}} > 0$).