8.2 Extracting the SPD using Call-Options

Breeden and Litzenberger (1978) show that one can replicate Arrow-Debreu prices using the concept of butterfly spread on European call options. This spread entails selling two call options at exercise price $ K$, buying one call option at $ K^{-}=K-\Delta K$ and another at $ K^{+}=K+\Delta K$, where $ \Delta K$ is the stepsize between the adjacent call strikes. These four options constitute a butterfly spread centered on $ K$. If the terminal underlying asset value $ S_{T}$ is equal to $ K$ then the payoff $ Z(\cdot)$ of $ \frac{1} {\Delta K}$ of such butterfly spreads is defined as:


$\displaystyle Z(S_{T},K;\Delta K)=P(S_{T-\tau},\tau,K;\Delta K)\vert _{\tau=0}=\frac{u_{1}-u_{2}} {\Delta
K}\bigg\vert _{S_{T}=K,\tau=0}=1$     (8.2)

where

$\displaystyle u_{1}=C(S_{T-\tau},\tau,K+\Delta K)-C(S_{T-\tau},\tau,K),$      
$\displaystyle u_{2}=C(S_{T-\tau},\tau,K)-C(S_{T-\tau},\tau,K-\Delta K).$      

$ C(S,\tau,K)$ denotes the price of a European call with an actual underlying price $ S$, a time to maturity $ \tau$ and a strike price $ K$. Here, $ P(S_{T-\tau},\tau,K;\Delta K)$ is the corresponding price of this security ( $ \frac{1} {\Delta K}*butterfly$ $ spread(K;\Delta K)$) at time $ T-\tau$.

As $ \Delta K$ tends to zero, this security becomes an Arrow-Debreu security paying $ 1$ if $ S_{T}=K$ and zero in other states. As it is assumed that $ S_{T}$ has a continuous distribution function on $ \mathbb{R^{+}}$, the probability of any given level of $ S_{T}$ is zero and thus, in this case, the price of an Arrow-Debreu security is zero. However, dividing one more time by $ \Delta K$, one obtains the price of $ (\frac{1} {(\Delta
K)^{2}}*butterfly$ $ spread(K;\Delta K))$ and as $ \Delta K$ tends to 0 this price tends to $ f^{*}(S_{T})e^{-r_{t,\tau}}$ for $ S_{T}=K$. Indeed,

$\displaystyle \lim_{\Delta K \to 0} \bigg(\frac{ P(S_{t},\tau,K;\Delta K)} {\Delta K}
\bigg)\bigg\vert _{K=S_{T}}= f^{*}(S_{T})e^{-r_{t,\tau}}.$     (8.3)

This can be proved by setting the payoff $ Z_{1}$ of this new security

$\displaystyle Z_{1}\left(S_{T}\right)=\left(\frac{1} {(\Delta
K)^{2}}(\Delta K-\vert S_{T}-K\vert)\boldsymbol{1}(S_{T}\in[K-\Delta K, K+\Delta
K])\right)$      

in (8.1) and letting $ \Delta K$ tend to 0. Indeed, one should remark that:

$\displaystyle \forall (\Delta K): \int_{K-\Delta K}^{K+\Delta K}(\Delta
K-\vert S_{T}-K\vert)dS_{T}=(\Delta K)^{2}.$      

If one can construct these financial instruments on a continuum of states (strike prices) then at infinitely small $ \Delta K$ a complete state pricing function can be defined.

Moreover, as $ \Delta K$ tends to zero, this price will tend to the second derivative of the call pricing function with respect to the strike price evaluated at $ K$:

$\displaystyle \lim_{\Delta K \to 0} \bigg(\frac{ P(S_{t},\tau,K;\Delta K)} {\Delta K}
\bigg)$ $\displaystyle =$ $\displaystyle \lim_{\Delta K \to 0} \frac{u_{1}-u_{2}} {(\Delta
K)^{2}}$  
  $\displaystyle =$ $\displaystyle \frac {\partial^{2} C_{t}(\cdot)} {\partial K^{2}}.$ (8.4)

Equating (8.3) and (8.4) across all states yields:

$\displaystyle \frac {\partial^{2} C_{t}(\cdot)} {\partial K^{2}}\bigg\vert _{K=S_{T}}= e^{-r_{t,\tau}
\tau}f_{t}^{*}(S_{T})$      

where $ r_{t,\tau}$ denotes the risk-free interest rate at time $ t$ with time to maturity $ \tau$ and $ f_{t}^{*}({\cdot})$ denotes the risk-neutral PDF or the SPD in $ t$. Therefore, the SPD is defined as:

$\displaystyle f_{t}^{*}(S_{T})= e^{r_{t,\tau} \tau}\frac {\partial^{2} C_{t}(\cdot)} {\partial K^{2}}\bigg\vert _{K=S_{T}}.$ (8.5)

This method constitutes a no-arbitrage approach to recover the SPD. No assumption on the underlying asset dynamics are required. Preferences are not restricted since the no-arbitrage method only assumes risk-neutrality with respect to the underlying asset. The only requirements for this method are that markets are perfect (i.e. no sales restrictions, transactions costs or taxes and that agents are able to borrow at the risk-free interest rate) and that $ C(\cdot)$ is twice differentiable. The same result can be obtained by differentiating (8.1) twice with respect to $ K$ after setting for $ Z$ the call payoff function $ Z(S_{T})=(S_{T}-K)^{+}$.


8.2.1 Black-Scholes SPD

The Black-Scholes call option pricing formula is due to Black and Scholes (1973) and Merton (1973). In this model there are no assumptions regarding preferences, rather it relies on no-arbitrage conditions and assumes that the evolution of the underlying asset price $ S_{t}$ follows a geometric Brownian motion defined through
$\displaystyle \frac{dS_{t}}{S_{t}}=\mu dt+\sigma dW_{t}.$     (8.6)

Here $ \mu$ denotes the drift and $ \sigma$ the volatility assumed to be constant.

The analytical formula for the price in $ t$ of a call option with a terminal date $ T=t+\tau$, a strike price $ K$, an underlying price $ S_{t}$, a risk-free rate $ r_{t,\tau}$, a continuous dividend yield $ \delta_{t,\tau}$, and a volatility $ \sigma$, is:

$\displaystyle C_{BS}(S_{t},K,\tau,r_{t,\tau},\delta_{t,\tau};\sigma)$ $\displaystyle =$ $\displaystyle e^{-r_{t,\tau}}
\int_{0}^{\infty} \max (S_{T}-K,0) f^{*}_{BS,t}(S_{T}) dS_{T}$  
  $\displaystyle =$ $\displaystyle S_{t} e^{-\delta_{t,\tau}\tau} \Phi(d_{1})- K
e^{-r_{t,\tau}
\tau} \Phi(d_{2})$  

where $ \Phi(\cdot)$ is the standard normal cumulative distribution function and
$\displaystyle d_{1}$ $\displaystyle =$ $\displaystyle \frac{\log(S_{t}/K) + (r_{t,\tau} - \delta_{t,\tau}+
\frac{1}{2}\sigma^{2})\tau} {\sigma \sqrt{\tau}},$  
$\displaystyle d_{2}$ $\displaystyle =$ $\displaystyle d_{1} - \sigma \sqrt{\tau}.$  

As a consequence of the assumptions on the underlying asset price process the Black-Scholes SPD is a log-normal density with mean $ (r_{t,\tau} - \delta_{t,\tau}- \frac{1}{2} \sigma^{2})\tau$ and variance $ \sigma^{2} \tau$ for $ \log(S_{T}/S_{t})$:

$\displaystyle f^{*}_{BS,t}(S_{T})$ $\displaystyle =$ $\displaystyle e^{r_{t,\tau} \tau}
\frac{\partial^{2} C_{t}} {\partial
K^{2}} \bigg{\vert}_{K=S_{T}}$  
  $\displaystyle =$ $\displaystyle \frac{1}{S_{T}\sqrt{2 \pi
\sigma^{2}\tau}}\exp\bigg{[}-\frac{[\lo...
...delta_{t,\tau}-
\frac{1}{2} \sigma^{2})\tau]^{2}} {2 \sigma^{2} \tau} \bigg{]}.$  

The risk measures Delta ($ \Delta$) and Gamma ($ \Gamma$) are defined as:
$\displaystyle \Delta_{BS} \stackrel{\mathrm{def}}{=}\frac{\partial C_{BS}} {\partial S_{t}} = \Phi(d_{1})$      
$\displaystyle \Gamma_{BS} \stackrel{\mathrm{def}}{=}\frac{\partial^{2} C_{BS}} {\partial S_{t}^{2}} =
\frac{\Phi(d_{1})}{S_{t}\sigma \sqrt{\tau}}$      

The Black-Scholes SPD can be calculated in XploRe using the following quantlet:


bsspd = 16172 spdbs (K,s,r,div,sigma,tau)
estimates the Black-Scholes SPD

The arguments are the strike prices $ ({\tt K})$, underlying price $ ({\tt s})$, risk-free interest rate $ ({\tt r})$, dividend yields $ ({\tt div})$, implied volatility of the option $ ({\tt sigma})$, and the time to maturity $ ({\tt tau})$. The output consist of the Black-Scholes SPD (bsspd.fbs), $ \Delta$ (bsspd.delta), and the $ \Gamma$ (bsspd.gamma) of the call options. Please note that 16175 spdbs can be applied to put options by using the Put-Call parity.

However, it is widely known that the Black-Scholes call option formula is not valid empirically. For more details, please refer to Chapter 6. Since the Black-Scholes model contains empirical irregularities, its SPD will not be consistent with the data. Consequently, some other techniques for estimating the SPD without any assumptions on the underlying diffusion process have been developed in the last years.