A Convex-Combination View of Black–Scholes Call Prices

# Black-Scholes formula is a (probabilistic) convex combination


# Black-Scholes formula is a (probabilistic) convex combination












A call price is bounded when $\sigma\sqrt{T}$ goes to $0$ and $\infty $ by:

$$C_{inf} = e^{-rT}[F-K] \leq C \leq C_{sup}=S $$

Now a simple rearrangement of Black-Scholes formula gives: $$ C = N_1S - e^{-rT}N_2K = e^{-rT}N_2[F-K] + [N_1-N_2]S$$ $$ = \frac{N_2}{N_1}N_1e^{-rT}[F-K] + [1-\frac{N_2}{N_1}]N_1S$$ $$ = \frac{N_2}{N_1}C_\inf + [1-\frac{N_2}{N_1}]N_1C_\sup$$ $$ = \frac{N_2}{N_1}\widehat{C_\inf} + [1-\frac{N_2}{N_1}]\widehat{C_\sup}$$ $$ = \alpha\widehat{C_\inf} + [1-\alpha]\widehat{C_\sup}$$

The last formula reads as a convex combination of two functions which are the extreme values of a call contract. It is actually a probabilistic convex combination as the coefficient $\alpha = \frac{N_2}{N_1} $ is not constant but depends on the state $(S,\sigma,T-t)$:

$$\alpha = \alpha(s,\sigma,ttm)$$

- $\widehat{C_\inf} = N_1e^{-rT}[F-K] $ is the "forward intrinsic value" or your expected payoff.

- $\widehat{C_\sup} = N_1S = \Delta S$ is simply your delta hedging portfolio value.

Being a convex combination between these two bounds ensures the no-arbitrage of the price.

- The occurence of the ratio $\frac{N_2}{N_1}$ is interesting. Still looking for a good interpretation.

For now: $$\alpha = \frac{N_2}{N_1} = \mathbb{P}(C_T > 0|S_t,\sigma,ttm) = \frac{\mathbb{P}(S_T > K)}{N_1}$$ As the $ttm \rightarrow 0$ , ITM $N_1 \rightarrow 1$ then $$\mathbb{P}(C_T > 0) \rightarrow \mathbb{P}(S_T > K) = N_2 $$

So the interpolation bounds (dark blue and green curves in the picture) and the interpolation coefficient $\alpha$ change dynamically with the contract $\Delta = N_1$ which is the degree of the contract's linearity (w.r.t to the underlying) that depends on the state $(s, \sigma,ttm) $.

Example:

-For the forward contract (lower bound) $C^{inf} = e^{-rT}[F-K]$, $\Delta = N_1 = 1 $: $$\mathbb{P}(C^{inf}_T > 0) = \mathbb{P}(S_T - K > 0) = \mathbb{P}(S_T > K) = N_2 = \frac{N_2}{N_1} $$

Any corrections or additional interpretations are much appreciated ?

EDIT

I found an interpretation to the ratio $\frac{N_2}{N_1}$ in terms of the elasticity $e$. The elsaticity refers to the ratio of the option return covariance to the stock return covariance: $$e = \frac{\frac{\Delta C}{C}} { \frac{\Delta S}{S} } $$

Delta-hedging in BS gives for small change: $$\Delta C = \frac{\partial C}{\partial S}\Delta S = N_1\Delta S$$ Hence, $$ e = \frac{N_1 S}{C} = \frac{N_1 S}{N_1S - Ke^{-rT}N_2} $$ $$= 1 + \frac{Ke^{-rT}N_2}{N_1S - Ke^{-rT}N_2} $$ $$= 1 + \frac{\frac{N_2}{N_1}}{\frac{S}{Ke^{-rT}} - \frac{N_2}{N_1}}$$

Finally, $$\frac{N_2}{N_1} = \frac{F}{K} \frac{e -1}{e} $$

Note that the option's elasticity $e \geq e_{stock}=1$.

Also, from CAPM perspective: $$\beta_{option} = e \cdot \beta_{stock}$$