Option Portfolio Strategies: Tools and Methods

Abstract

As standard option pricing models rest on restrictive assumptions. additional tools which mitigate some of their deficiencies and facilitate their application have been developed. The main tools are based on the notion of implied volatility and on the sensitivities of the value of an option to the different model parameters. From a portfolio management perspective, it is useful to distinguish static from dynamic strategies. Section 12.1 presents the main static strategies, which consist in building an initial portfolio left unchanged until it is sold out. Section 12.2 examines the important notions of historical volatility, implied volatility, smile (or skew), and volatility surface. Section 12.3 analyzes the sensitivities of options to the different variables that influence their values. Section 12.4 studies dynamic strategies involving revisions and using these sensitivities as indicators for monitoring.

Appendices

Appendix 1

1.1 Computing Partial Derivatives (Greeks)

To simplify notation, in this appendix we always consider that the current time is t = 0.

1.1.1 The Black–Scholes Model

Preliminary Result

Let us start with a result that simplifies calculating Greek parameters . Since the Black–Scholes model writes

$$ C=S\ N\left({d}_1\right)\hbox{--} {Ke}^{\hbox{--} rT}\ N\left({d}_2\right) $$

with

$$ {d}_1=\left(\ln \left(S/K\right)+\left(r+{\sigma}^2/2\right)T\right)/\sigma \sqrt{T}\kern1em \mathrm{and}\kern1em {d}_2={d}_1\hbox{--} \sigma \sqrt{T}, $$

we have

• (i) SN^′(d₁) = Ke^–rTN^′(d₂)

where ${N}^{\prime }(z)\equiv \frac{1}{\sqrt{2\pi }}{e}^{-{z}^2/2}$ is the density of the reduced centered normal distribution.

Proof

From the definition of d₁:

$$ \ln \left(S/K\right)+\left(r+{\sigma}^2/2\right)T={d}_1\sigma \sqrt{T} $$

$$ \Rightarrow \ln S\hbox{--} \ln K+ rT={d}_1\sigma \sqrt{T}-\frac{\sigma^2T}{2}=\frac{1}{2}\left({d}_1^2-{\left({d}_1-\sigma \sqrt{T}\right)}^2\right) $$

$$ \Rightarrow \ln S+\ln \left(\frac{1}{\sqrt{2\pi }}\right)-\frac{d_1^2}{2}=\ln K\hbox{--} rT+\ln \left(\frac{1}{\sqrt{2\pi }}\right)-\frac{d_2^2}{2} $$

which proves (i).

• (a) Delta

$$ {\delta}_C=\frac{\partial C}{\partial S}=\frac{\partial }{\partial S}\Big( SN\left({d}_1\Big)\hbox{--} {Ke}^{\hbox{--} rT}N\left({d}_2\right)\right) $$

$$ =N\left({d}_1\right)+\frac{{S N}^{\hbox{'}}\left({d}_1\right)}{S\sigma \sqrt{T}}-\frac{Ke^{- rT}{N}^{\hbox{'}}\left({d}_2\right)}{S\sigma \sqrt{T}}=N\left({d}_1\right)\kern0.5em \mathrm{from}\kern0.5em (i) $$

Moreover, from call-put parity we have P = C – S + Ke^–rT, from which

$$ {\delta}_P=\frac{\partial P}{\partial S}=\frac{\partial C}{\partial S}\hbox{--} 1=N\left({d}_1\right)\hbox{--} 1. $$

• (b) Gamma

$$ {\Gamma}_C=\frac{\partial {\delta}_C}{\partial S}=\frac{\partial N\left({d}_1\right)}{\partial S}=\frac{N^{\hbox{'}}\left({d}_1\right)}{S\sigma \sqrt{T}} $$

since $N\left({d}_1\right)=N\left(\left(\ln \left(S/K\right)+\left(r+{\sigma}^2/2\right)T\right)/\sigma \sqrt{T}\right)$.

Γ_P = Γ_C from call-put parity (its second derivative).

• (c) Vega

$$ {\upsilon}_C=\frac{\partial C}{\partial \sigma }={SN}^{\hbox{'}}\left({d}_1\right)\;\frac{\partial {d}_1}{\partial \sigma }-{Ke}^{- rT}{N}^{\hbox{'}}\left({d}_2\right)\;\frac{\partial {d}_2}{\partial \sigma }, $$

from which

$$ {\upsilon}_C={SN}^{\prime}\left({d}_1\right)\left(\sqrt{T}\hbox{--} {d}_1/\sigma \right)\hbox{--} {Ke}^{rT}{N}^{\prime}\left({d}_2\right)\ \left(\hbox{--} \sqrt{T}\hbox{--} {d}_2/\sigma \right) $$

since ${d}_1=\left(\ln \left(S/K\right)+r\ T\right)/\sigma \sqrt{T}+\sigma \sqrt{T}/2$ and

$$ \frac{\partial {d}_1}{\partial \sigma }=\frac{-\left(\ln \left(S/K\right)+ rT\right).\sqrt{T}}{\sigma^2T}+\frac{\sqrt{T}}{2}=-\frac{d_1}{\sigma }+\sqrt{T} $$

and

$$ \frac{\partial {d}_2}{\partial \sigma }=\frac{-\left(\ln \left(S/K\right)+ rT\right).\sqrt{T}}{\sigma^2T}-\frac{\sqrt{T}}{2}=-\frac{d_2}{\sigma }-\sqrt{T} $$

Consequently, and since $\frac{d_2}{\sigma }=\frac{d_1}{\sigma }-\sqrt{T}$, we have

$$ {\upsilon}_c={SN}^{\prime}\left({d}_1\right)\sqrt{T}-\frac{d_1}{\sigma}\left({SN}^{\prime}\left({d}_1\right)\hbox{--} {Ke}^{rT}{N}^{\prime}\left({d}_2\right)\right) $$

$$ =S\sqrt{T}{N}^{\prime}\left({d}_1\right) $$

from (i). Furthermore, ${\upsilon}_P=\frac{\partial P}{\partial \sigma }={\upsilon}_C$, by call-put parity.

• (d) Theta

The direct proof gives us

$$ {\theta}_C=\frac{\partial C}{\partial t}=-\frac{\partial C}{\partial T}=-{SN}^{\hbox{'}}\left({d}_1\right)\left[-\frac{1}{2}\frac{\ln \left(S/K\right)}{\sigma }{T}^{-3/2}+\frac{1}{2}\left(\frac{r}{\sigma }+\frac{\sigma }{2}\right){T}^{-1/2}\right] $$

$$ +\left[-{rKe}^{- rT}N\left({d}_2\right)+{Ke}^{- rT}{N}^{\hbox{'}}\left({d}_2\right)\left[-\frac{1}{2}\frac{\ln \left(S/K\right)}{\sigma }{T}^{-3/2}+\frac{1}{2}\left(\frac{r}{\sigma }-\frac{\sigma }{2}\right){T}^{-1/2}\right]\right] $$

$$ =\hbox{--} {SN}^{\prime}\left({d}_1\right)\frac{\sigma }{2}{T}^{-1/2}\hbox{--} {rKe}^{\hbox{--} rT}N\left({d}_2\right) $$

$$ -\left[{SN}^{\hbox{'}}\left({d}_1\right)-{Ke}^{- rT}{N}^{\hbox{'}}\left({d}_2\right)\right]\left[-\frac{1}{2}\frac{\ln \left(S/K\right)}{\sigma }{T}^{-3/2}+\frac{1}{2}\left(\frac{r}{\sigma }-\frac{\sigma }{2}\right){T}^{-1/2}\right] $$

$$ =-\frac{S\sigma}{2\sqrt{T}}{N}^{\prime}\left({d}_1\right)\hbox{--} {rKe}^{\hbox{--} rT}N\left({d}_2\right)\kern0.5em \mathrm{by}\kern0.5em (i). $$

One can, alternatively, recall that the option price satisfies the Black-Scholes partial differential equation in its form (12.10). Hence we deduce the following equation relating sensitivities

$$ \theta +\delta \kern0.28em rS+\frac{1}{2}\Gamma \kern0.28em {S}^2{\sigma}^2= rC $$

and therefore

$$ {\theta}_C=\frac{\partial C}{\partial t}=r\left(C\hbox{--} dS\right)\hbox{--} 0.5{\Gamma \sigma}^2{S}^2. $$

Replacing the gamma and delta by the expressions given above, we recover the expression for theta.

Furthermore, again from call-put parity,

$$ {\theta}_P=\frac{\partial P}{\partial t}=-\frac{\partial P}{\partial T}=-\frac{\partial }{\partial T}\left(C\hbox{--} S+{Ke}^{\hbox{--} rT}\right) $$

$$ =-\frac{\partial C}{\partial T}+{rKe}^{- rT}=-\frac{S\sigma}{2\sqrt{T}}{N}^{\prime}\left({d}_1\right)+{rKe}^{\hbox{--} rT}\left(1\hbox{--} N\left({d}_2\right)\right). $$

• (e) Rho

$$ {\rho}_C=\frac{\partial C}{\partial r}={SN}^{\hbox{'}}\left({d}_1\right)\frac{\sqrt{T}}{\sigma }-\left[{Ke}^{- rT}{N}^{\hbox{'}}\left({d}_2\right)\frac{\sqrt{T}}{\sigma }-T.{Ke}^{- rT}N\left({d}_2\right)\right] $$

$$ ={TKe}^{\hbox{--} rT}N\left({d}_2\right)\left(>0\right)\ \mathrm{from}\ (i). $$

The rho of a put can then be obtained from call-put parity:

$$ {\rho}_P=\frac{\partial P}{\partial r}=\frac{\partial }{\partial r}\ \left(C\hbox{--} S+{Ke}^{\hbox{--} rT}\right)={TKe}^{\hbox{--} rT}N\left({d}_2\right)\hbox{--} {TKe}^{\hbox{--} rT} $$

$$ =\left(N\left({d}_2\right)\hbox{--} 1\right){TKe}^{\hbox{--} rT}\ \left(<0\right). $$

1.1.2 Other Models

We start from the Black-Scholes formula:

$$ C= BS(S)\equiv S\ N\left({d}_1(S)\right)\hbox{--} {Ke}^{\hbox{--} rT}\ N\left({d}_2(S)\right) $$

with ${d}_1(S)=\left(\ln \left(S/K\right)+\left(r+{\sigma}^2/2\right)T\right)/\sigma \sqrt{T}$ and ${d}_2(S)={d}_1(S)\hbox{--} \sigma \sqrt{T}$; moreover, we know BS′ = N(d₁).

• (a) Black’s model (option on a forward contract quoted F)

The model gives the premium as a function of F: C = BS (Fe^–rT) where the Black-Scholes BS(.) is defined above and ${d}_1(F)=\left(\ln \left(F/K\right)+{\sigma}^2T/2\right)/\sigma \sqrt{T}$. From this, using the Greek parameters of the Black-Scholes model, we obtain almost immediately

${\delta}_C=\frac{\partial C}{\partial F}=\frac{\partial }{\partial F}\mathrm{BS}\left({\mathrm{Fe}}^{\hbox{--} \mathrm{rT}}\right)={\mathrm{e}}^{\hbox{--} \mathrm{rT}}{\mathrm{BS}}^{'}\left({\mathrm{Fe}}^{\hbox{--} \mathrm{rT}}\right)$, from which: δ_C = e^-rTN(d₁).

${\Gamma}_C=\frac{\partial^2 BS\left({Fe}^{- rT}\right)}{\partial {F}^2}={e}^{-2 rT}{BS}^{\prime \prime}\left({Fe}^{- rT}\right)={e}^{-2 rT}\frac{N^{\hbox{'}}\left({d}_1\right)}{F{e}^{- rT}\sigma \sqrt{T}}$, from which ${\Gamma}_C={e}^{- rT}\frac{N^{\hbox{'}}\left({d}_1\right)}{F\sigma \sqrt{T}}$.

$$ {\upsilon}_C=\frac{\partial C}{\partial \sigma }={Fe}^{- rT}\sqrt{T}{N}^{\hbox{'}}\left({d}_1\right), $$

and

$$ {\theta}_C=-\frac{\partial C}{\partial T}=-\frac{dBS\left(T,{Fe}^{- rT}\right)}{dT}=-\frac{\partial BS}{\partial T}+{Fre}^{- rT}{BS}^{\hbox{'}},\kern1em \mathrm{whence}: $$

$$ {\theta}_C=\hbox{--} \frac{Fe^{- rT}\sigma }{2\sqrt{T}}{N}^{\hbox{'}}\left({d}_1\right)\hbox{--} {rKe}^{- rT}N\left({d}_2\right)+{Fre}^{- rT}N\left({d}_1\right). $$

The Greek parameters related to a put can be obtained from those for the analogous call using the parity relationship written as P = C – e^–rT (F – K), whence:

$$ {\delta}_P={\delta}_C\hbox{--} {e}^{\hbox{--} rT};\kern1em {\Gamma}_P={\Gamma}_C;\kern1em {\upsilon}_P={\upsilon}_C;\kern1em {\theta}_P={\theta}_C\hbox{--} {Fre}^{\hbox{--} rT}. $$

• (b) Merton’s Model (When the Underlying Distributes a Continuous Dividend c)

This model writes: $C= BS\left({Se}^{\hbox{--} cT}\right);\kern0.5em {d}_1=\left(\ln \left(S/K\right)+\left(r\hbox{--} c+{\sigma}^2/2\right)T\right)/\sigma \sqrt{T}$ and ${d}_2={d}_1\hbox{--} \sigma \sqrt{T}$.

From this, we have:

$$ {\delta}_C={e}^{- cT}{BS}^{\prime}\left({S e}^{- cT}\right)={e}^{- cT}N\left({d}_1\right);\kern0.5em {\Gamma}_C={e}^{-2 cT}{BS}^{\prime \prime}\left({S e}^{- cT}\right)={e}^{-2 cT}\frac{N^{\hbox{'}}\left({d}_1\right)}{{S e}^{- cT}\sigma \sqrt{T}}={e}^{- cT}\frac{N^{\hbox{'}}\left({d}_1\right)}{S\sigma \sqrt{T}}; $$

$$ {\upsilon}_C=\frac{\partial BS\left({Se}^{- cT}\right)}{\partial \sigma }={Se}^{- cT}\sqrt{T}{N}^{\hbox{'}}\left({d}_1\right);\kern0.5em {\theta}_C=\hbox{--} \frac{S{e}^{- cT}\sigma }{2\sqrt{T}}{N}^{\hbox{'}}\left({d}_1\right)\hbox{--} r\ K\ {e}^{- rT}N\left({d}_2\right)+{cSe}^{- cT}N\left({d}_1\right). $$

The Greek parameters for a put can be obtained from those for a call using the parity relation written as P = C – (Se^–cT – Ke^–rT).

Appendix 2

1.1 Option Prices and the Underlying Price Probability Distribution

The following proposition highlights the relationship between the prices of standard options and the probability distribution for the future value S_T of the underlying.

Proposition

The probability density under Q of the final value S_T of the underlying, ${\phi}_T(k)=\frac{Q\left({S}_T\in \left[k,k+ dk\right]\right)}{dk}$, is equal to the capitalized value of the second derivative of the option (a call or a put) with respect to the strike denoted by k:

$$ {\phi}_T(k)={e}^{rT}\frac{\partial^2C}{\partial {k}^2}(k)={e}^{rT}\frac{\partial^2P}{\partial {k}^2}(k). $$

(12.13)

Proof

This result follows directly from the definition of the premium for calls or puts as the risk-neutral expectation of the discounted final payoff. As an example, take a call with strike k whose value at a distance T from its expiration date is

$$ C(k)={E}^Q\left[\operatorname{Max}\left({S}_T-k,\kern0.5em 0\right){e}^{- rT}\right]={\int}_k^{\infty}\left(x-k\right){e}^{- rT}{\phi}_T(x) dx. $$

Taking the derivative of the right-hand side with respect to k, we have

$$ \frac{\partial C}{\partial k}(k)={\int}_k^{\infty }{e}^{- rT}{\phi}_T(x) dx. $$

Taking the derivative a second time, we obtain

$$ \frac{\partial^2C}{\partial {k}^2}(k)={e}^{- rT}{\phi}_T(k). $$

An analogous proof can be used for the put.

This proposition shows that knowing the prices of vanilla options in theory allows us to deduce the distribution for the underlying’s future value implicitly used by the market for valuing options. This would be perfect knowledge in the purely theoretical case of a continuum of prices. In practice, of course, price discontinuity makes it imperfect and not so easy to use. In any event, it is theoretically possible to compare the distribution for S_T which is implicit in option market prices with the theoretical distributions used in valuation models. In the special case of BS, the underlying’s distribution is assumed to be log-normal. In practice, the distribution deduced from vanilla option prices using Eq. (12.13) (T-density) seems to differ from the log-normal model. Empirical studies clearly show that, for stocks as underlyings, the left tail of the distribution for small values of S_T is noticeably thicker than what follows from the Gaussian model. And this form of distribution for S_T can explain why the smile is downward sloping (a skew , as in Fig. 12.12a), which is precisely the usual case for stock options. This connection between the shape of the distribution for S_T and the smile or skew can be seen from Fig. 12.12a, b. Suppose the smile is decreasing like the solid line in Fig. 12.12a; also in Fig. 12.12a the horizontal smile assumed in the BS model is shown as a dotted line which serves as a reference. Assume further that the distribution of ln(S_T) is asymmetrical, with a thick left tail, as drawn with a solid line in A-b ; the normal distribution (as hypothesized in BS) is also shown in the same figure, drawn dotted as a reference. The solid (dotted) lines in A-a thus correspond to the solid (dotted) lines in A-b.

Fig. 12.12

Relation between the shape of the smile and that of the distribution of S_T

On the abscissa axis K in Fig. 12.12a we find the underlying’s value S₀ on date 0 (today). For an option with a low strike, such that K₁ < S₀, options are expensive in comparison to a standard BS price; the opposite is true for an option with a high strike, such that K₂ > S₀, which is relatively cheap.

Now let us consider a put deep out-of-the-money with strike K₁; it is logical that it should be relatively costly (see A-a) since, as A-b shows, the probability of exercise (S_T < K₁) and the expected gain upon exercise are larger than for a log-normal distribution. It also is reasonable that a call deep out-of-the-money K₂ should be cheap since the right tail of the distribution for S_T is thinner than in the log-normal case. Finally, let us remark that, from the call-put parity, a relatively expensive put implies a relatively expensive call, and conversely, since C – P = S – Ke^–rT, independently of any assumption regarding the actual distribution of S_T.

The relative dearness of stock options with low strikes and the relative cheapness of options with high strikes (tantamount to a decreasing implied volatility curve) are therefore compatible with empirical results on the distributions of stock yields.

Similarly, a U-shaped smile curve (s_i (K) decreasing then increasing again) can be explained by an empirical distribution for S_T with thick tails on both left and right.

This result shows that some assumptions of the Black–Scholes (BS) model are not realistic and, because of this, the model yields biased results. When applying BS formula, taking the smile into account corrects for this bias only imperfectly. The fundamental assumption that should be questioned is the constancy of the volatility parameter (which implies the log-normality of the underlying’s distribution).

An approach that is theoretically more rigorous and more general is to look for a model of which BS would be a special case, and which would be compatible with the whole volatility surface observed at a given instant (and not only with the smile curve relative to a specific expiration date T). Mathematically one has to find a model for the evolution of the underlying price which provides, for any future value of the underlying, a probability distribution equal to the distribution implied by the surface.

Several answers appear satisfactory and are used in practice. The first is to work with a model for the volatility s(t,S) that depends entirely on the time remaining until the option’s expiration date and the current price of the underlying. Such models can also explain the smile. It is, for example, in line with the empirical evidence and the theory that a stock’s volatility is a decreasing function s(S) of its price S. At the theoretical level, a drop in the stock price leads to an increase in the ratio (debt/market capitalization) (financial leverage); this increase in leverage implies an increase in the risk affecting the equity, and so an increased stock volatility. The fact that the stock volatility is higher for small than for large values of S explains the thick left tail of the distribution for S_T (see Fig. 12.12b) and the volatility skew (see Fig. 12.12a).

A complementary explanation for this dual phenomenon of volatilities asymmetric to the right and returns asymmetric to the left is the feedback effect. Following a large positive jump in the price, the stock volatility increases which exerts pressure to reduce its return. Indeed, since the expected (ex ante) return must increase to offset the increased risk, the price must fall and with it the realized (ex post) return. This feedback phenomenon thus mitigates the amplitude of positive returns. In the same way, a large negative jump in the price causes an increase in volatility which forces the return down for the same reason as above. Thus the feedback effect accentuates the amplitude of negative returns. Consequently, this dissymmetric feedback effect contributes to the left asymmetry in the return distribution and the right asymmetry in the volatility distribution.

The interest of these models which express the volatility as a function s(t,S) lies in their relative simplicity; in particular, they can be implemented using Cox-Ross-Rubinstein recombining trees. An alternative solution consists of models with stochastic volatility, in which the variable s_t is itself the solution to a stochastic differential equation. Such models present the advantage to take into account the specific risk of a change in the volatility.²⁶

Appendix 3

1.1 Replication of an Arbitrary Payoff with a Static Option Portfolio

Let us consider European calls and puts with the same underlying and the same expiration date T. Let us denote by F_0,T the forward price²⁷ of the underlying on date 0 for an expiration date T. An option with strike k is said to be at-the-money-forward when F_0,T = k. Subsequently, C(k) and P(k) respectively denote the premium at 0 of a call and a put with expiration date T and strike k. The following proposition describes the replication of an arbitrary payoff Ψ(S_T) by a portfolio of calls and puts out-of-the-money-forward .

Proposition

Let us consider a security Ψ which generates a single payoff on date T defined by a sufficiently regular function Ψ(S_T).

• This security is worth on date 0:

$$ {V}_0=\Psi \left({\Phi}_{0,T}\right){e}^{- rT}+{\int}_0^{\Phi_{0,T}}\frac{\partial^2\Psi}{\partial {k}^2}(k)P(k) dk+{\int}_{\varPhi_{0,T}}^{\infty}\frac{\partial^2\Psi}{\partial {k}^2}(k)C(k) dk. $$

(12.14)

• The static duplication of the security Y is guaranteed by a position with $\frac{\partial^2\Psi}{\partial {k}^2}(k) dk$ options (calls or puts) out-of-the-money and with strike k.

One says the vanilla calls and puts span the set of regular payoffs, for a given expiration date.

Proof

The proposition can be deduced from the following result: For any x > 0,

$$ \Psi (x)=\Psi \left({\Phi}_{0,T}\right)+\frac{\mathrm{\partial \Psi }}{\partial k}\left({\Phi}_{0,T}\right)\left(x-{\Phi}_{0,T}\right)+{\int}_{\varPhi_{0,T}}^{\infty}\frac{\partial^2\Psi}{\partial {k}^2}(k){\left(x-k\right)}^{+} dk+{\int}_0^{\Phi_{0,T}}\frac{\partial^2\Psi}{\partial {k}^2}(k){\left(k-x\right)}^{+} dk. $$

This equation can be justified with a bit of functional analysis. By definition of the first derivative, we have

$$ \Psi (x)-\Psi \left({\Phi}_{0,T}\right)={\int}_{\varPhi_{0,T}}^x\frac{\mathrm{\partial \Psi }}{\partial k}(k) dk. $$

Integrating by parts (consider u = x – k and $v=\frac{\mathrm{\partial \Psi }}{\partial k}(k)$), we find

$$ \Psi (x)-\Psi \left({\Phi}_{0,T}\right)=-{\left[\left(x-k\right)\frac{\mathrm{\partial \Psi }}{\partial k}(k)\right]}_{\Phi_{0,T}}^x+{\int}_{\varPhi_{0,T}}^x\left(x-k\right)\frac{\partial^2\Psi}{\partial {k}^2}(k) dk. $$

But x − k = (x − k)⁺ − (k − x)⁺. Hence

$$ \Psi (x)-\Psi \left({\Phi}_{0,T}\right)=\left(x-{\Phi}_{0,T}\right)\frac{\mathrm{\partial \Psi }}{\partial k}\left({\Phi}_{0,T}\right)+{\int}_{\varPhi_{0,T}}^x{\left(x-k\right)}^{+}\frac{\partial^2\Psi}{\partial {k}^2}(k) dk-{\int}_{\varPhi_{0,T}}^x{\left(k-x\right)}^{+}\frac{\partial^2\Psi}{\partial {k}^2}(k) dk. $$

Remark that for any x > 0, one may replace the integration limit x in the first integral by infinity (since, if k > x, (x – k)⁺ vanishes), and the limit x in the second integral by 0 (since, if k < x, (k – x)⁺ vanishes); as a result we get

$$ \Psi (x)-\Psi \left({\Phi}_{0,T}\right)=\left(x-{\Phi}_{0,T}\right)\frac{\mathrm{\partial \Psi }}{\partial k}\left({\Phi}_{0,T}\right)+{\int}_{\varPhi_{0,T}}^{\infty }{\left(x-k\right)}^{+}\frac{\partial^2\Psi}{\partial {k}^2}(k) dk-{\int}_{\varPhi_{0,T}}^0{\left(k-x\right)}^{+}\frac{\partial^2\Psi}{\partial {k}^2}(k) dk. $$

Rewriting this, we have

$$ \Psi (x)=\Psi \left({\Phi}_{0,T}\right)+\left(x-{\Phi}_{0,T}\right)\frac{\mathrm{\partial \Psi }}{\partial k}\left({\Phi}_{0,T}\right)+{\int}_{\varPhi_{0,T}}^{\infty }{\left(x-k\right)}^{+}\frac{\partial^2\Psi}{\partial {k}^2}(k) dk+{\int}_0^{\Phi_{0,T}}{\left(k-x\right)}^{+}\frac{\partial^2\Psi}{\partial {k}^2}(k) dk $$

This last equation shows that the replication of the payoff Y(S_T) is ensured by a static position containing $\frac{\partial^2\Psi}{\partial {k}^2}(k) dk$ options (call or put) out-of-the-money with strike k.²⁸

Expressing the discounted risk-neutral expectation of each member of the last equation for x = S_T and noting, on the one hand, that E^Q[(S_T − Φ_{0, T})e^−rT] = 0, and, on the other, that E^Q[(S_T − k)⁺e^−rT] = C(k) and E^Q[(k − S_T)⁺e^−rT] = P(k), we end up with Eq. (12.14).

The proposition proves that any payoff can be replicated by a static strategy based on vanilla call and put options. In a very intuitive sense, it is the gamma (second derivative of the payoff) on the expiration date that determines the quantity of standard options to hold in the replicating portfolio.