On the Optimal Choice of Strike Conventions in Exchange Option Pricing

Abstract

An important but rarely-addressed option pricing question is how to choose appropriate strikes for implied volatility inputs when pricing more exotic multi-asset derivatives. By means of Malliavin calculus, we construct an asymptotically optimal log-linear strike convention for exchange options under stochastic volatility models. This novel approach allows us to minimize the difference between the corresponding Margrabe computed price and the true option price. We show that this optimal convention does not depend on the specific stochastic volatility model chosen and, furthermore, that parameter estimation can be dramatically simplified by using market observables as inputs. Numerical examples are given that provide strong support for the new methodology.

1. Introduction

Spread options are recognized as important contracts in many financial markets, and have been widely studied both by practitioners and academic researchers. In particular, although also traded in other markets, spread options on commodities are closely linked to the physical markets and the hedging or valuation needs of producers and consumers due to their parallels with physical assets like power plants, refineries, storage facilities or pipelines. Such assets all have an option-like nature with operational decisions and corresponding payoffs depending predominantly on the spread between two commodity spots or forward prices. While a variety of different considerations affect different spread option types (ranging from calendar spreads to locational spreads to input/output spreads like crack or spark), the dominant derivative pricing challenges remain the same. In all cases, a major challenge is capturing the implied volatility structure in order to price spread options consistently with market data for the more liquid derivatives, vanilla calls and puts. By tackling this problem from a novel perspective and via the mathematical tools of Malliavin calculus, our contribution is to derive an asymptotically optimal solution to this challenge that avoids dependence on any particular stochastic volatility model, reducing parameter estimation burden and offering wide-ranging applicability across financial markets.

The commonly-used lognormal assumption (e.g., the Geometric Brownian Motion model) for underlying prices $S_t^X$ and $S_t^Y$ leads to a convenient closed-form pricing formula known as Margrabe’s formula (see [1]) given an ‘exchange option’ payoff ${(S_X^1-S_t^Y)}^{+}$. In the context of stochastic volatility models, we do not have an explicit closed-form expression for the corresponding option price. Some approximations can be found, for example, in [2,3,4,5,6,7,8]. However, prior calibration of model parameters and evaluation of pricing formulas can be time-consuming or challenging in such approaches, with some cases requiring simulation or numerical methods. Computation time can be particularly onerous for physical asset valuation or hedging, whereby strings of hourly or daily spread options over many years or even decades may be required. For such reasons, Margrabe’s formula is frequently employed for useful and fast benchmark approximations to spread option prices.

Despite the prominence of such tools, relatively little attention has been paid to the key question of how to choose an appropriate pair of constant volatility inputs ${\sigma }_X$ and ${\sigma }_Y$ for the Margrabe formula, ideally maintaining consistency both with market data and modeling preferences. A natural starting point is the implied volatility of the two legs of the spread, typically observable from more liquidly traded single-asset vanilla calls or puts. However, a significant implied volatility skew or smile (as well as term structure) exists in most markets, meaning that there are many possible choices for both ${\sigma }_X$ and ${\sigma }_Y$ and no obvious rule for which pair is most appropriate. Indeed, there is also no standard yardstick for measuring which so-called ‘strike convention’ rule is best in this setting. In [9], this important issue is highlighted and discussed, along with some numerical examples that indicate that the common industry solution (described as a ‘volatility look-up heuristic’) can lead to significant pricing differences compared to Monte Carlo values in a simple jump-diffusion model.

In this paper, we aim to answer this crucial question by developing a new theory for an asymptotically optimal short-time strike convention, defined as the choice of implied volatilities such that the resulting estimated option price (obtained from Margrabe’s formula) matches the true option price as closely as possible. This is equivalent to the choice such that the corresponding implied correlation (backed out from Margrabe’s formula) matches the model correlation $\rho$. It is interesting to note that both references [9,10] comment on how the choice of strike convention can impact the implied correlation skew, smile or frown observed across different moneyness spread options. As the underlying assets’ returns correlation is clearly unrelated to contract moneyness, [9] describes this as “purely an artifact of the interaction of skew with the Margrabe formulation”, explaining that “skew risk can manifest itself as a spurious correlation risk simply due to the look-up heuristic”.

In order to investigate such effects and recommend a strike convention for consistent spread option pricing, we rely on tools from Malliavin calculus that allow us to derive the short-time limit of the sensitivity of implied volatilities to moneyness in the context of stochastic volatility models. Our proposed optimal strike convention is, to our knowledge, the first systematic approach to this problem. Moreover, it is model-independent since it depends only on the at-the-money implied volatility levels and skews of the corresponding vanilla options. Thus, it can serve as a very useful and practical ‘financial engineering’ tool to improve option pricing accuracy within the financial industry. Note that, in this paper, we do not delve deep into the modeling of asset prices but instead aim to provide a crucial first step to establishing adequate criteria for the strike choices to motivate further work on the topic. More research is needed to deal with more complex models, such as models with jumps or rough volatility models (see, for example, [11]).

The paper is organized as follows. Section 2 is devoted to introducing the main problem and notations. In Section 3, we make use of Malliavin calculus techniques to derive an equation for our strike convention proposal. In Section 4, we explicitly determine this optimal convention in the class of log-linear strike conventions. Section 5 provides a range of numerical examples and tests to investigate the theory presented in the paper and its implications in practice.

2. The Objective, the Price Model and Notation

Assume, for the sake of simplicity, that the interest rate $r=0.$ Consider a two-asset stochastic volatility model of the form

$$\begin{array}{ccc}& & {\displaystyle \frac{d{S}_t^X}{S_t^X}}={\sigma }_t^{X}d{W}_t^X\hfill \\ & & {\displaystyle \frac{d{S}_t^Y}{S_t^Y}}={\sigma }_t^{Y}d{W}_t^Y,\hfill \end{array}$$

(1)

under a risk-neutral probability P. $W^X \mathrm{and} W^Y$ are Brownian motions, and ${\sigma }_t^X \mathrm{and} {\sigma }_t^Y$ are non-negative, right-continuous and square-integrable processes adapted to the filtration generated by another Brownian motion $Z^X,Z^Y$, respectively. We will use the notation

$$〈W_t^X,Z^X〉={\rho }_X,〈W_t^Y,Z^Y〉={\rho }_Y,〈W_t^X,W_t^Y〉=\rho .$$

$$〈W_t^X,Z^Y〉={\rho }_{XY},〈W_t^Y,Z^X〉={\rho }_{YX}.$$

Itô’s representation theorem gives us that, for any fixed s

$${\sigma }_s^i=E\left({\sigma }_s^i\right)+{\int }_0^s{a}^i(s,u)d{Z}_u^i,i=X,Y.$$

for some square-integrable processes $a^i(s,·)$ adapted to the filtration generated by $Z^i$.

Now, we describe some basic notation that is used in this article. For this, we assume that the reader is familiar with the elementary results of the Malliavin calculus, as given, for instance, in [11].

The set ${\mathbb{D}}_{Z^i}^{1,2}$, $i=1,2$ will denote the domain of the derivative operator $D^{Z^i}$ with respect to the Brownian Motion $Z^i$. It is well-known that ${\mathbb{D}}_{Z^i}^{1,2}$ is a dense subset of $L^2\left(\mathsf{\Omega }\right)$ and that $D^{Z^i}$ is a closed and unbounded operator from $L^2\left(\mathsf{\Omega }\right)$ into $L^2([0,T]\times \mathsf{\Omega }).$ We will also consider the iterated derivatives $D^n,$ for $n>1,$ whose domains will be denoted by ${\mathbb{D}}_{Z^i}^{n,2}.$ We will also make use of the notation ${\mathbb{L}}_i^{n,2}:=L^2([0,T];{\mathbb{D}}_{Z^i}^{n,2}).$

We notice that, if ${\left({\sigma }^i\right)}^2\in$${\mathbb{L}}^{1,2}$, the Clark–Ocone formula gives us that

$$a^i(s,u)=E_u\left(D_u^{Z^i}{\left({\sigma }_s^i\right)}^2\right),i=X,Y.$$

Then, under suitable integrability conditions, the change rule for the Malliavin derivative operator (see, for example, [12]) gives us that

$$a^i(s,u)=2E_u\left({\sigma }_s^i{D}_u^{Z^i}{\sigma }_s^i\right),i=X,Y.$$

We will also make use of the following notation:

• $BS\left(t,x,k,\sigma \right)$ denotes the classical Black–Scholes call price with time to maturity $T-t,$ log stock price x, log strike price k and volatility $\sigma$.
• ${\mathcal{L}}_{BS}={\partial }_t+{\displaystyle \frac{1}{2}}{\sigma }^2{\partial }_x^2-x{\partial }_x$ denotes the classical Black–Scholes operator. Notice that $\left({\mathcal{L}}_{BS}BS\right)(t,x,k,\sigma )=0$.
• $X_t:=log{S}_t^X,Y_t:=log{S}_t^Y$.
• $V_t=E_t{(S_T^X-S_T^Y)}^{+}$ is the exchange option price under the model (1).
• For every $0<t<T$ and $x,k>0,I_X(t,x,z)$ is the implied volatility of an option with payoff ${(S_T^X-exp\left(z\right))}^{+}$ with $X_t=x.$ That is,

  $$BS\left(t,x,k,I_X(t,x,z)\right)=E_t{(S_T^X-exp\left(z\right))}^{+}.$$
  Analogously, $I_Y(t,y,z)$ is the implied volatility of an option with payoff ${\left(S_T^Y-exp\left(z\right)\right)}^{+}$ with $Y_t=y.$
• ${\tilde{\sigma }}_t:=\sqrt{{\left({\sigma }_t^X\right)}^2+{\left({\sigma }_t^Y\right)}^2-2\rho {\sigma }^X{\sigma }^Y}$
• ${\tilde{v}}_t:=\sqrt{{\displaystyle \frac{1}{T-t}}\left({\int }_t^T{\tilde{\sigma }}_s^{2}ds\right)}$
• $M_t^i:=E_t{\int }_0^T{\left({\sigma }_s^i\right)}^{2}ds,i=X,Y.$
• ${\tilde{M}}_t:=E_t{\int }_0^T{\left({\tilde{\sigma }}_s\right)}^{2}ds$

For the sake of simplicity, we will take $t=0$ and we will denote $I_X(x,z)=I_X(0,x,z)$ and $I_Y(y,z)=I_Y(0,y,z)$. Moreover, we denote $x=X_0$ and $y=Y_0$.

It is well-known that, under the Black–Scholes model, ${\sigma }_t^X={\sigma }_X$ and ${\sigma }_t^Y={\sigma }_Y$, for all $t\in [0,T]$ and for some positive constants ${\sigma }_X$ and ${\sigma }_Y$. In this case, the option price $V_0$ can be computed analytically by means of Margrabe’s formula. More precisely, in this case, the price is given by

$$BS\left(0,x,y,\sqrt{{\sigma }_X^2+{\sigma }_Y^2-2\rho {\sigma }_X{\sigma }_Y}\right)$$

(2)

In the general stochastic volatility case, there is no analytical formula for this option price. One common strategy is to substitute ${\sigma }_X$ and ${\sigma }_Y$ by the vanilla-implied volatilities $I_X(x,k_X)$ and $I_Y(y,k_Y)$, for some log strikes $k_X,k_Y$. However, notice that, as these implied volatilities are not constant as a function of the strike, the corresponding price estimation

$$BS\left(0,x,y,\sqrt{I_X^2(x,k_X)+I_Y^2(y,k_Y)-2\rho I_X(x,k_X)I_Y(y,k_Y)}\right)$$

(3)

will depend strongly on the choice of the log strikes $k_X$ and $k_Y$. Despite the relevance of this problem, there is currently no standard rule for choosing $k_X$ and $k_Y$ (see, for example, [9]). Our aim in this paper is to develop a standard rule that will allow us to choose these strikes in such a way that the approximation (3) will be as close as possible to the true option price $V_0$ for a range of moneyness cases. More precisely, we want to find the pair $k_X:=k_X(x,y)$ and $k_Y:=k_Y(x,y)$ that minimizes the difference

$$|V_0-BS\left(0,x,y,\gamma (x,y)\right)|,$$

(4)

for short-time and near-the-money ($x\approx y$) options, where

$$\gamma (x,y):=\sqrt{I_X^2(x,k_X)+I_Y^2(y,k_Y)-2\rho I_X(x,k_X)I_Y(y,k_Y)}.$$

(5)

Notice that, if we define $\widehat{\gamma }\left(x,y\right)$ as the quantity such that

$$V_0=BS(0,x,y,\widehat{\gamma }\left(x,y\right)),$$

(6)

to minimize (4), it is sufficient to minimize

$$|\widehat{\gamma }(x,y)-\gamma (x,y)|.$$

Remark 1. 

Notice that straitghtforward computations give us that $\widehat{\gamma }(x,y)=\gamma (x,y)$ is equivalent to the equality $\rho =\widehat{\rho }$, where $\widehat{\rho }$ denotes the implied correlation, defined by the equality

$$\widehat{\gamma }(x,y):=\sqrt{I_X^2(x,k_X)+I_Y^2(y,k_Y)-2\widehat{\rho }{I}_X(x,k_X)I_Y(y,k_Y)}.$$

In the following section, we will develop a methodology to choose the pair $(k_X,k_Y)$. As we have no explicit expressions for $\gamma$ and $\widehat{\gamma }$, the main idea is to approximate these two quantities and to find the pair $(k_X,k_Y)$ that makes these approximations equal. Towards this end, we will consider for any fixed x the short-time limit of the Taylor expansion of the function $\gamma (x,·)-\widehat{\gamma }(x,·)$. This motivates the following definition of strike conventions of any order.

Definition 1. 

Assume the model (1). We will say that a pair $(k_1,k_2)\in L^2({\mathbb{R}}^2;{\mathbb{R}}^2)$ is a short-time optimal strike convention of order n (a n-STOSC) if

$$\underset{T\to 0}{lim}{\displaystyle \frac{{\partial }^i\gamma }{{\partial }^{i}y}}(x,x)=\underset{T\to 0}{lim}{\displaystyle \frac{\partial {\widehat{\gamma }}^i}{{\partial }^{i}y}}(x,x),$$

(7)

for any $i=0,\dots ,n$, and where γ and $\widehat{\gamma }$ are defined as in (5) and (6), respectively.

Remark 2. 

Notice that, as n increases, $\widehat{\gamma }$ is expected to be closer to γ (and $\widehat{\rho }$ closer to ρ) for short-term and near-the-money options.

3. The Construction of Optimal Strike Conventions

This section contains the mathematical derivation of an optimal strike convention, as motivated above. Although it is quite technical, and readers may need to refer to the cited references for a full understanding, the eventual result is a straightforward and easily applicable formula for choosing $k_X$ and $k_Y$, which we then analyze further in the following sections. We will make use of the following hypotheses.

Hypothesis 1 (H1). 

For any $x\in \mathbb{R}$, $k_X(x,x)=k_Y(x,x)=x$.

Hypothesis 2 (H2). 

$\sigma \in {\mathbb{L}}_i^{2,4},i=X,Y$.

Hypothesis 3 (H3). 

There exist two positive constants a and b such that, for any $t\in [0,T]$, $a<{\sigma }_t<b$.

Hypothesis 4 (H4). 

Hypothesis (H2) holds and there exists a positive constant $C>0$ such that, for any $0<r<s<T$,

$$E_r\left[D_r^{Z^i}{\left({\sigma }_s^i\right)}^2\right]\le C,i=X,Y.$$

Hypothesis 5 (H5). 

Hypothesis (H2) holds and, for any $t\in [0,T]$ and $i=X,Y$, there exists a constant $D^{+,i}{\sigma }_0^i$ such that as $T\to 0$,

$$\underset{r,s\in [0,T]}{sup}{E}_t|D_s^{Z^i}{\sigma }_r^i-D^{+,i}{\sigma }_0^i|\to 0.$$

While certain assumptions on the stochastic volatility model are needed for our derivations, we note that the choices above encompass a broad range of commonly used models in the financial industry, such as the Heston model, the CEV model, the 3/2 model and the SABR model, as well as GARCH-type models from econometrics. We choose (H3) and (H4) for the sake of simplicity, but these hypotheses can be substituted by adequate integrability conditions. On the other hand, (H2) and (H5) are satisfied by the classical stochastic volatility models, where the volatility is assumed to be a diffusion process (see, for example, [13] for the Heston case). In the case of fractional volatility models with $H<{\displaystyle \frac{1}{2}}$ (see, for example, [14,15,16]), (H5) is not satisfied. Adapting our results to these models is left for future research. Finally, we note that (H1) ensures that an at-the-money (ATM) exchange option should be consistent with ATM call options for both assets.

Our first result establishes that all the strike conventions satisfying hypotheses (H1)–(H5) are 0-STOSCs.

Proposition 1. 

Consider model (1) and assume that $(k_X,k_Y)$ is a strike convention such that Hypotheses (H1)–(H5) hold. Then, $(k_X,k_Y)$ is a 0-STOSC.

Proof. 

It suffices to say that ${lim}_{T\to 0}\gamma (x,x)={lim}_{T\to 0}\widehat{\gamma }(x,x)$. This proof will be decomposed into two steps.

Step 1: Let us prove that

$$\underset{T\to 0}{lim}\gamma (x,x)={\tilde{\sigma }}_0$$

(8)

It is well-known (see for example [17]) that the vanilla at-the-money implied volatilities $I_X,I_Y$ tend to the corresponding spot volatility. That is,

$$\underset{T\to 0}{lim}\left(I_i(x,x)-{\sigma }_0^i\right)=0,i=X,Y.$$

Now, taking into account (H1) it follows that

$$\underset{T\to 0}{lim}{I}_i(x,k_i)={\sigma }_0^i,i=X,Y,$$

where $k_i=k_i(x,x)$. Now, as

$$\gamma (x,y):=\sqrt{I_X^2(x,k_X)+I_Y^2(y,k_Y)-2\rho I_X(x,k_X)I_Y(y,k_Y)},$$

(8) follows.

Step 2: Let us see that

$$\underset{T\to 0}{lim}\widehat{\gamma }(x,x)={\tilde{\sigma }}_0.$$

(9)

By its definition, we have that

$$\widehat{\gamma }(x,x)=B{S}^{-1}(0,x,y,V_0),$$

where $B{S}^{-1}$ is the inverse of the Black–Scholes function in the sense that

$$V_0=BS(0,x,y,B{S}^{-1}(0,x,y,V_0)).$$

Then, Theorem 5 in [2] gives us that

$$V_0=E\left(BS(0,x,x,{\tilde{v}}_0)\right)+o\left(1\right),$$

which implies that

$$\widehat{\gamma }(x,x)=B{S}^{-1}\left(E\left(BS(0,x,x,{\tilde{v}}_0)\right)+o\left(1\right)\right).$$

(10)

Moreover, the martingale representation theorem allows us to obtain

$$E\left(BS(0,x,x,{\tilde{v}}_0)\right)=BS(0,x,x,{\tilde{v}}_0)+{\int }_0^T(A^X(T,s)d{Z}_s^X+A^Y(T,s)d{Z}_s^Y),$$

for some adapted and square-integrable processes $A^i(T,·),i=X,Y$. This, jointly with (10), gives us that

$$\begin{array}{ccc}& & \underset{T\to 0}{lim}\widehat{\gamma }(x,x)\hfill \\ & =& \underset{T\to 0}{lim}B{S}^{-1}\left(0,x,x,\left(BS(0,x,x,{\tilde{v}}_0)+\sum _{i=X,Y}{\int }_0^T{A}^i(T,s)d{Z}_s^i\right)+o\left(1\right)\right)\hfill \\ & =& \underset{T\to 0}{lim}B{S}^{-1}\left(0,x,x,\left(BS(0,x,x,{\tilde{v}}_0\right)\right)\hfill \\ & =& \underset{T\to 0}{lim}{\tilde{v}}_0\hfill \\ & =& {\tilde{\sigma }}_0,\hfill \end{array}$$

(11)

and this allows us to complete the proof. □

In order to identify the strike conventions that are 1-STOCs, we will need the following result, which is an application of the results in [11,14].

Theorem 1. 

Consider that model (1) holds with ${\rho }_{1,2}\in [-1,1]$ and suppose that Hypotheses (H1)–(H5) hold. Then,

$$\underset{T\to t}{lim}{\displaystyle \frac{\partial I_{1,2}}{\partial z}}={\displaystyle \frac{\rho }{2{\sigma }_0^i}}\underset{T\to 0}{lim}{\displaystyle \frac{1}{{(T-t)}^2}}{E}_t{\int }_t^T\left(D_s^W{\int }_s^T{\sigma }_r^{2}dr\right)ds.$$

(12)

Theorem 2. 

Consider model (1) and assume that Hypotheses (H1)–(H5) hold. Then, for $i=X,Y$,

$$\underset{T\to 0}{lim}{\displaystyle \frac{\partial I_i}{\partial z}}={\displaystyle \frac{{\rho }_i{D}^{+,i}{\sigma }_0^i}{2{\sigma }_0^i}}={\displaystyle \frac{1}{2{\left({\sigma }_0^i\right)}^3}}\underset{T\to 0}{lim}{\displaystyle \frac{{\langle log{S}^i,M^i\rangle }_T}{T}}$$

Proof. 

Theorem 1 gives us that

$${\displaystyle \frac{\partial I_Y}{\partial y}}=-{\displaystyle \frac{{\rho }_i{D}^{+,i}{\sigma }_0^i}{2{\sigma }_0^i}},{\displaystyle \frac{\partial I_X}{\partial x}}=-{\displaystyle \frac{{\rho }_i{D}^{+,i}{\sigma }_0^i}{2{\sigma }_0^i}}.$$

Now, as ${\displaystyle \frac{\partial I_Y}{\partial z}}=-{\displaystyle \frac{\partial I_Y}{\partial y}}$ and ${\displaystyle \frac{\partial I_X}{\partial z}}=-{\displaystyle \frac{\partial I_X}{\partial x}}$, the first equality follows. For the second one, notice that the Clark–Ocone formula (see, for example, [12]) gives us that

$${\left({\sigma }_t^i\right)}^2=E\left({\left({\sigma }_t^i\right)}^2\right)+{\int }_0^t{E}_r\left(D_r^{Z^i}{\left({\sigma }_t^i\right)}^2\right)d{Z}_r^i,i=X,Y,$$

(13)

from which we can easily deduce that

$$d{M}_t^i=\left({\int }_t^T{E}_t\left(D_t^{Z^i}{\left({\sigma }_u^i\right)}^2\right)du\right)d{Z}_t^i=2\left({\int }_t^T{E}_t\left({\sigma }_u^i{D}_t^{Z^i}{\sigma }_u^i\right)du\right)d{Z}_t^i,i=X,Y,$$

from which the second equality holds. □

Remark 3. 

The above result gives us that the derivatives ${\displaystyle \frac{\partial I_i}{\partial z}},i=X,Y$ depend only on the quadratic covariation between M and $log{S}^i$ and on the volatility ${\sigma }^i$.

Now, we will study the short-time limit of the derivative of $\widehat{\gamma }$. To this end, we define ${\displaystyle \frac{d\widehat{P}}{dP}}=e^{Y_T-Y_0}$. The set ${\mathbb{D}}_{{\widehat{Z}}^i}^{1,2}$, $i=1,2$ will denote the domain of the derivative operator ${\widehat{D}}^i$ under $\widehat{P}$, with respect to ${\widehat{Z}}^i$. We will write ${\mathbb{L}}_{{\widehat{Z}}^i}^{1,2}=L^2([0,T],{\mathbb{D}}_{{\widehat{Z}}^i}^{1,2})$. Notice that as $T\to 0$, ${sup}_{r,s\in [0,T]}{\widehat{E}}_t|{\widehat{D}}_s^{Z^i}{\sigma }^i-D^{+,i}{\sigma }_0^i|\to 0$, for $i=X,Y$, where $D^{+,i}{\sigma }_0^i$ are defined as in (H5), and ${\widehat{E}}_t$ is the conditional expectation with respect to $\widehat{P}$.

Theorem 3. 

Consider model (1) and assume that (H1)–(H5) hold. Then,

$$\begin{array}{ccc}\hfill & & \underset{T\to 0}{lim}{\displaystyle \frac{\partial \widehat{\gamma }}{\partial y}}(x,x)\hfill \\ & & ={\displaystyle \frac{1}{2{\tilde{\sigma }}^3}}\left[D^{+,i}{\sigma }_0^X({\sigma }_0^X-\rho {\sigma }_0^Y)({\rho }_X{\sigma }_0^X-{\rho }_{YX}{\sigma }_0^Y)\right.\hfill \\ & & +\left.D^{+,i}{\sigma }_0^Y({\sigma }_0^Y-\rho {\sigma }_0^X)({\rho }_{XY}{\sigma }_0^X-{\rho }_Y{\sigma }_0^Y)\right].\hfill \end{array}$$

(14)

Proof. 

We have that

$$V_0=BS(0,x,y,\widehat{\gamma }).$$

(15)

On the one hand, a direct computation gives us that

$$BS(0,x,y,\widehat{\gamma })=e^{y}BS(0,x-y,0,\widehat{\gamma })$$

(16)

On the other hand,

$$\begin{array}{ccc}\hfill & & V_0=E{\left(e^{X_T}-e^{Y_T}\right)}^{+}\hfill \\ & & =e^{Y_0}\widehat{E}{\left(e^{X_T-Y_T}-1\right)}^{+}\hfill \\ & & =e^y\widehat{E}{\left(e^{X_T-Y_T}-1\right)}^{+}\hfill \end{array}$$

(17)

where $\widehat{E}$ denotes the expectation with respect to the probability measure $\widehat{P}$. Notice that, under $\widehat{P}$, the process $U_t:=e^{X_t-Y_t}$ satisfies

$$d{U}_t=U_t({\sigma }_t^{X}d{\widehat{W}}_t^X-{\sigma }_t^{Y}d{\widehat{W}}_t^Y),$$

where ${\widehat{W}}^X,{\widehat{W}}^Y$ are $\widehat{P}$-Brownian motions. Then, (16) and (17) gives us that (15) is equivalent to

$$\widehat{E}{(U_T-1)}^{+}=BS(0,x-y,0,\widehat{\gamma }).$$

Notice that $\widehat{\gamma }$ is the implied volatility of a vanilla option with strike 1 on an underlying $U_t$, with volatility $\tilde{\sigma }$. Then, Theorem 2 gives us that

$$\underset{T\to 0}{lim}{\displaystyle \frac{\partial \widehat{\gamma }}{\partial z}}(x,x)={\displaystyle \frac{1}{2{\tilde{\sigma }}_0^3}}\underset{T\to 0}{lim}{\displaystyle \frac{{\langle lnU,\tilde{M}\rangle }_T}{T}}.$$

Now, the classical Itô’s formula, jointly with (13), gives us that

$$\begin{array}{ccc}\hfill d{\tilde{M}}_t& =& \left({\int }_t^T{\widehat{E}}_t({\widehat{D}}_t^{Z^X}{\left({\sigma }_r^X\right)}^2\right)d{Z}_t^X\hfill \\ & +& \left({\int }_t^T{\widehat{E}}_t({\widehat{D}}_t^{Z^Y}{\left({\sigma }_r^Y\right)}^2\right)d{Z}_t^Y\hfill \\ & +& {\sigma }_t^X\left({\int }_t^T{\widehat{E}}_t\left({\widehat{D}}_t^{Z^Y}\left({\sigma }_r^Y\right)\right)dr\right)d{Z}_t^Y\hfill \\ & +& {\sigma }_t^Y\left({\int }_t^T{\widehat{E}}_t\left({\widehat{D}}_t^{Z^X}\left({\sigma }_r^Y\right)\right)dr\right)d{Z}_t^X\hfill \end{array}$$

(18)

Then,

$$\begin{array}{ccc}\hfill {\langle lnU,\tilde{M}\rangle }_T& =& {\rho }_X{\int }_0^T\left({\int }_t^T{\widehat{E}}_t\left({\widehat{D}}_t^{Z^X}{\left({\sigma }_r^X\right)}^2\right)dr\right){\sigma }_t^{X}dt\hfill \\ & -& {\rho }_{YX}{\int }_0^T\left({\int }_t^T{\widehat{E}}_t\left({\widehat{D}}_t^{Z^X}{\left({\sigma }_r^X\right)}^2\right)dr\right){\sigma }_t^{Y}dt\hfill \\ & +& {\rho }_{XY}{\int }_0^T\left({\int }_t^T{\widehat{E}}_t\left({\widehat{D}}_t^{Z^Y}{\left({\sigma }_r^Y\right)}^2\right)dr\right){\sigma }_t^{X}dt\hfill \\ & -& {\rho }_Y{\int }_0^T\left({\int }_t^T{\widehat{E}}_t\left({\widehat{D}}_t^{Z^Y}{\left({\sigma }_r^Y\right)}^2\right)dr\right){\sigma }_t^{Y}dt\hfill \\ & +& 2\rho {\rho }_{XY}{\int }_0^T{\left({\sigma }_t^X\right)}^2\left({\int }_t^T{\widehat{E}}_t\left({\widehat{D}}_t^{Z^Y}\left({\sigma }_r^Y\right)\right)dr\right)dt\hfill \\ & -& 2\rho {\rho }_Y{\int }_0^T{\sigma }_t^X{\sigma }_t^Y\left({\int }_t^T{\widehat{E}}_t\left({\widehat{D}}_t^{Z^Y}\left({\sigma }_r^Y\right)\right)dr\right)dt\hfill \\ & +& 2\rho {\rho }_X{\int }_0^T{\sigma }_t^X{\sigma }_t^Y\left({\int }_t^T{\widehat{E}}_t\left({\widehat{D}}_t^{Z^X}\left({\sigma }_r^X\right)\right)dr\right)dt\hfill \\ & -& 2\rho {\rho }_{YX}{\int }_0^T{\left({\sigma }_t^Y\right)}^2\left({\int }_t^T{\widehat{E}}_t\left({\widehat{D}}_t^{Z^X}\left({\sigma }_r^X\right)\right)dr\right)dt\hfill \end{array}$$

(19)

Now, a direct computation, jointly with (H5), gives us that

$$\begin{array}{ccc}\hfill \underset{T\to 0}{lim}{\displaystyle \frac{{\langle lnU,\tilde{M}\rangle }_T}{T}}& =& {\rho }_X{\left({\sigma }_t^X\right)}^2\left({\mathsf{\Phi }}_r{\sigma }_0^X\right)\hfill \\ & -& {\rho }_{YX}{\sigma }_0^X{\sigma }_0^Y\left(D^{+,i}{\sigma }_0^X\right)\hfill \\ & +& {\rho }_{XY}{\sigma }_0^X{\sigma }_0^Y\left(D^{+,i}{\sigma }_0^Y\right)\hfill \\ & -& {\rho }_Y{\left({\sigma }_0^Y\right)}^2\left(D^{+,i}{\sigma }_0^Y\right)\hfill \\ & +& \rho {\rho }_{XY}{\left({\sigma }_0^X\right)}^2\left(D^{+,i}{\sigma }_0^Y\right)\hfill \\ & -& \rho {\rho }_Y{\sigma }_0^X{\sigma }_0^Y\left(D^{+,i}{\sigma }_0^Y\right)\hfill \\ & +& \rho {\rho }_X{\sigma }_0^X{\sigma }_0^Y\left(D^{+,i}{\sigma }_0^X\right)\hfill \\ & -& \rho {\rho }_{YX}{\left({\sigma }_0^Y\right)}^2\left(D^{+,i}{\sigma }_0^X\right),\hfill \end{array}$$

(20)

and this allows us to complete the proof. □

In the next theorem, we establish a condition for a strike convention to be a 1-STOSC. This is the main result of this paper.

Theorem 4. 

Consider model (1) and assume that Hypotheses (H1)–(H5) hold. Then, a strike convention $(k_X,k_Y)$ is a 1-STOSC if and only if

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{1}{2{\tilde{\sigma }}^3}}\left[D^{+,i}{\sigma }_0^X({\sigma }_0^X-\rho {\sigma }_0^Y)({\rho }_X{\sigma }_0^X-{\rho }_{YX}{\sigma }_0^Y)\right.\hfill \\ & & +\left.D^{+,i}{\sigma }_0^Y({\sigma }_0^Y-\rho {\sigma }_0^X)({\rho }_{XY}{\sigma }_0^X-{\rho }_Y{\sigma }_0^Y)\right]\hfill \\ & & =\underset{T\to 0}{lim}\left\{{\displaystyle \frac{1}{\gamma }}\left[(I_X-\rho I_Y){\displaystyle \frac{\partial I_X}{\partial z}}{\displaystyle \frac{\partial k_X}{\partial y}}\right.\right.+\left.\left.(I_Y-\rho I_X)\left({\displaystyle \frac{\partial I_Y}{\partial z}}{\displaystyle \frac{\partial k_Y}{\partial y}}+{\displaystyle \frac{\partial I_Y}{\partial y}}\right)\right](x,x)\right\}\hfill \end{array}$$

(21)

Proof. 

We have to prove that

$$\underset{T\to 0}{lim}\left({\displaystyle \frac{\partial \gamma }{\partial y}}-{\displaystyle \frac{\partial \widehat{\gamma }}{\partial y}}\right)(x,x)=0.$$

Theorems 2 and 3 directly give us the desired result.  □

Remark 4. 

If ${\rho }_X\ne 0$ and ${\rho }_Y\ne 0$, then $D^{+,i}{\sigma }_0^i={\displaystyle \frac{{\sigma }_0^i}{{\rho }_i}}{lim}_{T\to 0}{\displaystyle \frac{\partial I_i}{\partial z}}$ for $i=X,Y$. Moreover, if ${\rho }_{XY}={\rho }_X$ and ${\rho }_{YX}={\rho }_Y$, the left-hand side in (21) can be written as

$$\underset{T\to 0}{lim}{\displaystyle \frac{{\rho }_X{I}_X-{\rho }_Y{I}_Y}{{\gamma }^3}}\left[{\displaystyle \frac{\partial I_X}{\partial z}}{\displaystyle \frac{I_X}{{\rho }_X}}(I_X-\rho I_Y)+{\displaystyle \frac{\partial I_Y}{\partial z}}{\displaystyle \frac{I_Y}{{\rho }_Y}}(I_Y-\rho I_X)\right](x,x).$$

(22)

This gives us a model-free condition for a 1-STOSC in the sense that a specific model for the volatility processes is not needed.

While various different cases of the general rule above may be considered, a convenient particular case of a strike convention is obtained if ${\sigma }_t^X={\lambda }_X{\sigma }_t$ and ${\sigma }_t^Y={\lambda }_Y{\sigma }_t$, where ${\lambda }_X$ and ${\lambda }_Y$ are positive constants and ${\sigma }_t$ is a non-negative, right-continuous and square-integrable process adapted to the filtration generated by $Z_t$. We note that this case of a single volatility process shifted by a constant for each of the two assets is a generalization of the model introduced for correlation options in [18] and also for spread options in [8]). For convenience we shall refer to this model as the one-volatility two-level (1V2L) model. The following corollary demonstrates that for the 1V2L model, a strike convention can be derived either in terms of model parameters or market observables, namely the short-time limits of the corresponding vanilla-implied volatility levels and skews.

Corollary 1. 

Assume the 1V2L model. Then, a strike convention $(k_X,k_Y)$ is a 1-STOSC if and only if

$$\begin{array}{ccc}\hfill & & \underset{T\to 0}{lim}\left[\left(1-\rho {\displaystyle \frac{I_Y}{I_X}}\right){\displaystyle \frac{\partial I_X}{\partial z}}{\displaystyle \frac{\partial k_X}{\partial y}}+\left({\displaystyle \frac{I_Y}{I_X}}-\rho \right)\left({\displaystyle \frac{\partial I_Y}{\partial z}}{\displaystyle \frac{\partial k_Y}{\partial y}}+{\displaystyle \frac{\partial I_Y}{\partial y}}\right)\right.\hfill \\ & & -\left.{\displaystyle \frac{\partial I_X}{\partial z}}+{\displaystyle \frac{I_Y}{I_X}}{\displaystyle \frac{\partial I_Y}{\partial z}}\right](x,x)=0.\hfill \end{array}$$

(23)

or equivalently (in terms of model parameters):

$$\left(1-\rho {\displaystyle \frac{{\lambda }_Y}{{\lambda }_X}}\right){\displaystyle \frac{{\rho }_X}{{\rho }_Y}}{\displaystyle \frac{\partial k_X}{\partial y}}+\left({\displaystyle \frac{{\lambda }_Y}{{\lambda }_X}}-\rho \right)\left({\displaystyle \frac{\partial k_Y}{\partial y}}-1\right)={\displaystyle \frac{{\rho }_X}{{\rho }_Y}}-{\displaystyle \frac{{\lambda }_Y}{{\lambda }_X}}$$

(24)

Proof. 

In the 1V2L model (with ${\sigma }_t^X={\lambda }_X{\sigma }_t$, ${\sigma }_t^Y={\lambda }_Y{\sigma }_t$), Theorem 2 implies that

$${\displaystyle \frac{\partial I_Y}{\partial z}}={\displaystyle \frac{{\rho }_Y}{{\rho }_X}}{\displaystyle \frac{\partial I_X}{\partial z}}.$$

Expanding the expression in (22) and substituting for ${\displaystyle \frac{\partial I_Y}{\partial z}}$, we obtain

$$\begin{array}{ccc}& =& \underset{T\to 0}{lim}\left[{\displaystyle \frac{1}{{\gamma }^3}}\left({\rho }_X{I}_X-{\rho }_Y{I}_Y\right){\displaystyle \frac{\partial I_X}{\partial z}}{\displaystyle \frac{1}{{\rho }_X}}\left(I_X(I_X-\rho I_Y)+I_Y(I_Y-\rho I_X)\right)\right](x,x)\hfill \\ & =& \underset{T\to 0}{lim}\left[{\displaystyle \frac{1}{{\gamma }^3}}\left(I_X{\displaystyle \frac{\partial I_X}{\partial z}}-I_Y{\displaystyle \frac{{\rho }_Y}{{\rho }_X}}{\displaystyle \frac{\partial I_X}{\partial z}}\right)\left(I_X(I_X-\rho I_Y)+I_Y(I_Y-\rho I_X)\right)\right](x,x)\hfill \\ & =& \underset{T\to 0}{lim}\left[{\displaystyle \frac{1}{{\gamma }^3}}\left(I_X{\displaystyle \frac{\partial I_X}{\partial z}}-I_Y{\displaystyle \frac{\partial I_Y}{\partial z}}\right){\gamma }^2\right](x,x)\hfill \\ & =& \underset{T\to 0}{lim}\left[{\displaystyle \frac{1}{\gamma }}\left(I_X{\displaystyle \frac{\partial I_X}{\partial z}}-I_Y{\displaystyle \frac{\partial I_Y}{\partial z}}\right)\right](x,x)\hfill \end{array}$$

Equating this with the right-hand side of (21) and rearranging gives the desired result. The equivalent result in terms of model parameters ${\rho }_X,{\rho }_Y,{\lambda }_X,{\lambda }_Y$ can be found by Theorem 2 and the fact that the at-the-money (ATM) implied volatility tends to the corresponding spot volatility at time zero. □

4. Optimal Linear Log-Strike Conventions

Several strike conventions have been proposed in the literature. Some classical examples (see, for example [9,10]) are of the form

$$\left\{\begin{array}{c}{k}_X(x,y)=(1-a)x+ay\\ k_Y(x,y)=ax+(1-a)y\end{array}\right.,$$

(25)

for some real number a. For example, in [9], the author suggests taking $k_X=ln{S}_t^Y$ and $k_Y=ln{S}_t^X.$ This choice corresponds to (25) in the case $a=1.$ On the other hand, in [10], the authors mostly study the strike convention $k_X=ln{S}_t^X$ and $k_Y=ln{S}_t^Y$, which is the case $a=0$. In this section, we will find an optimal linear log-strike option of the form (25). Given two strikes $k_X,k_Y$ of the form (25), we have

$${\displaystyle \frac{\partial k_X}{\partial y}}=a,{\displaystyle \frac{\partial k_Y}{\partial y}}=1-a,$$

and thus, Equation (24) reduces to

$$a\left[{\displaystyle \frac{{\rho }_X}{{\rho }_Y}}\left(1-{\displaystyle \frac{\rho {\lambda }_Y}{{\lambda }_X}}\right)-\left({\displaystyle \frac{{\lambda }_Y}{{\lambda }_X}}-\rho \right)\right]={\displaystyle \frac{{\rho }_X}{{\rho }_Y}}-{\displaystyle \frac{{\lambda }_Y}{{\lambda }_X}}.$$

(26)

or, alternatively,

$$a\left[{\rho }_X({\lambda }_X-\rho {\lambda }_Y)-{\rho }_Y({\lambda }_Y-\rho {\lambda }_X)\right]={\lambda }_X{\rho }_X-{\rho }_Y{\lambda }_Y.$$

Then, if

$$[{\rho }_X({\lambda }_X-\rho {\lambda }_Y)-{\rho }_Y({\lambda }_Y-\rho {\lambda }_X)\ne 0$$

there exists a unique 1-STOSC, given by $a=a^{\star }$, where

$$a^{\star }={\displaystyle \frac{{\rho }_X{\lambda }_X-{\rho }_Y{\lambda }_Y}{{\rho }_X({\lambda }_X-\rho {\lambda }_Y)-{\rho }_Y({\lambda }_Y-\rho {\lambda }_X)}}$$

(27)

Remark 5. 

We note several interesting special cases related to this result:

1. 

The underlying prices $S^X$ and $S^Y$ are uncorrelated ($\rho =0$):

$$a^{\star }={\displaystyle \frac{{\rho }_X{\lambda }_X-{\rho }_Y{\lambda }_Y}{{\rho }_X{\lambda }_X-{\rho }_Y{\lambda }_Y}}=1$$

Intuitively, thinking of an exchange option as a regular option with a floating strike, if the strike is uncorrelated, then it is optimal to use the implied volatility corresponding to that floating strike (to the opposite leg of the spread), the ‘volatility look-up heuristic’ of [9].

2. 

The two volatilities have the same level (${\lambda }_X={\lambda }_Y$):

$$a^{\star }={\displaystyle \frac{{\rho }_X-{\rho }_Y}{{\rho }_X(1-\rho )-{\rho }_Y(1-\rho )}}={\displaystyle \frac{1}{1-\rho }}$$

Notice that, in this case, $a^{\star }$ is no longer dependent on ${\rho }_X,{\rho }_Y,{\lambda }_X,{\lambda }_Y$.

3. 

The two asset-to-volatility correlations are equal (${\rho }_X={\rho }_Y$):

$$a^{\star }={\displaystyle \frac{{\lambda }_X-{\lambda }_Y}{{\lambda }_X-\rho {\lambda }_Y-{\lambda }_Y+\rho {\lambda }_X}}={\displaystyle \frac{1}{1+\rho }}$$

Again, here, $a^{\star }$ no longer depends on correlations ${\rho }_X,{\rho }_Y$ or levels ${\lambda }_X,{\lambda }_Y$.

4. 

The asset-to-volatility correlation is zero (${\rho }_Y=0$):

$$a^{\star }={\displaystyle \frac{{\lambda }_X{\rho }_X}{{\rho }_X({\lambda }_X-\rho {\lambda }_Y)}}={\displaystyle \frac{{\lambda }_X}{{\lambda }_X-\rho {\lambda }_Y}}$$

Similarly, if ${\rho }_X=0$, then $a^{\star }={\displaystyle \frac{{\lambda }_Y}{{\lambda }_Y-\rho {\lambda }_X}}$. In these cases, we can also conclude (since ${\lambda }_X,{\lambda }_Y>0$) that $\rho >0$ corresponds to $a^{\star }>1$ (and $\rho <0$ to $a^{\star }<1$), which is intuitive for a floating strike option in which the strike tends to move away as S moves towards it.

Remark 6. 

Similarly to the equivalence expressions within Corollary 1, we note that since ${\displaystyle \frac{{\lambda }_Y}{{\lambda }_X}}={lim}_{T\to 0}{\displaystyle \frac{I_Y}{I_X}}$ and ${\displaystyle \frac{{\rho }_Y}{{\rho }_X}}={lim}_{T\to 0}{\displaystyle \frac{\partial I_Y/\partial z}{\partial I_X/\partial z}}$, our results can be transformed from model parameters to market observables. Thus, the optimal strike convention can be computed from Equation (26), needing to only know ρ (often estimated from price histories) and the short-time limits of the corresponding vanilla implied volatility levels and skews.

5. Numerical Examples

In order to investigate the performance of the optimal log-linear strike convention given by (27), we consider an extensive set of numerical examples of spread option pricing, corresponding to a wide range of correlation assumptions and implied volatility shapes. In each case, we compare our optimal choice of $a^{\star }$ to results from using the only other known strike conventions of ‘at-the-money’ (ATM)-implied volatilities for each asset (i.e., $a=0$) or the volatility look-up heuristic (i.e., $a=1$). Although a key strength of our approach is model independence and the opportunity to use market observables, as discussed in Remark 6, in this section, we choose the Heston Model as a simple and commonly-used benchmark to generate realistic option prices and to conduct tests under a large variety of different parameter sets. Volatility dynamics are given by:

$$d{\sigma }_t^2=\kappa \left(\theta -{\sigma }_t^2\right)dt+\nu \sqrt{{\sigma }_t^2}d{Z}_t^{\left(3\right)},$$

within the 1V2L model, a version of model (1) introduced before Corollary 1.

5.1. Test Cases

For now, we consider two test cases with parameters, as described below, varying only ${\rho }_Y$ between cases:

• Option maturity: $T=0.05$ (a few weeks)
• Volatility process (${\sigma }_t$) parameters: $\kappa =1.5,\theta =0.15,\nu =0.5,{\sigma }_0=0.15$
• Volatility scaling factors: ${\lambda }_X=1.5,{\lambda }_Y=1$
• Correlation parameters: $\rho =0.5,{\rho }_X=-0.5$, and ${\rho }_Y=-0.6$ or ${\rho }_Y=0.4$

Note that Test Case 1 with ${\rho }_Y=-0.6$ corresponds to two downward-sloping implied volatility skews for the two assets, while Test Case 2 with ${\rho }_Y=0.4$ produces an upward skew for the second asset. The top row of Figure 1 shows these implied volatility plots, generated by pricing single-asset options under the Heston model (via the commonly-used characteristic function approach). We then use these saved implied volatilities to price an exchange option with payoff ${(S_T^X-S_T^Y)}^{+}$ across a range of moneyness, with $S_0^X=100$ fixed and $S_0^Y\in [80,120]$. Margrabe’s formula, with the three different strike conventions (choices of a), is compared against an ‘exact solution’ using 1,000,000 simulated paths (with the constant volatility solution as a control variate).

Rows 2–4 of Figure 1 provide three alternative methods of visualizing the performance of our optimal strike convention (the darkest line) across moneyness compared with the other approaches:

(i) By converting spread option prices back to implied correlations $\widehat{\rho }$ in order to compare with the model correlation of $\rho =0.5$, recalling Remark 1; (ii) By plotting the ratio of Margrabe price to exact price; (iii) By plotting the difference between Margrabe price and exact price. Note that the ATM values ($S_0^Y=100$) are equal across strike conventions since they all coincide at this point, and $\rho \approx \widehat{\rho }$. However, moving away from the ATM point ($S_0^Y=100$), we can clearly see that the optimal $a^{\star }$ performs significantly better than the other contenders.

In Test Case 1 (left column), we see that $a=0$ significantly overprices the spread option ($\widehat{\rho }<\rho$) when $S_0^Y<S_0^X$ (the ‘in the money’, or ITM, case) and underprices ($\widehat{\rho }>\rho$) when $S_0^Y>S_0^X$ (the ‘out of the money’, or OTM, case), while $a=1$ does the opposite. It might, therefore, appear that a rather arbitrary midpoint convention of $a=1/2$ could work as a compromise between the other rules, but this is not surprising considering that $a^{\star }=0.429$ is optimal in this case. In contrast, in Test Case 2 (right column), $a^{\star }=1.917$ is optimal, and thus, both the other strike conventions overprice ITM and underprice OTM, sometimes by a large amount. Our approach keeps absolute errors in Case 1 below 0.01 and in Case 2 below 0.03 across different $S_0^Y$ values. This consistent pricing of options at different moneyness levels is a major advantage. In practice an indicative quote on a different spread option in the market could, therefore, more accurately be used to price another contract.

Although dominated by skew, the implied correlation plots in Figure 1 reveal a slight ‘frown’ in the first test case, as sometimes witnessed in the market. Ideally, we would like to observe a flat line at $\widehat{\rho }=0.5$, as the theory dictates we should hold with short enough T and near the money, but our results are nonetheless encouraging. Note that when looking at relative pricing errors in the third row of plots, errors unsurprisingly dominate for OTM options, which always have zero intrinsic value and much lower prices than ITM. It is more interesting to note the patterns in the case of absolute errors just below, in particular, that deep ITM and OTM options show less pricing error than moderate ITM and OTM. This effect can be explained by the fact that there is less (model-dependent) extrinsic value to accurate pricing.

5.2. Extensive Numerical Investigations

Instead of considering individual cases of parameter sets as above, we now test the approach across a wide range of different parameter values, and in particular, correlation structures. Note that parameters here have been chosen to limit the occurrence of extreme cases where the denominator of $a^{\star }$ in (27) is very close to 0, leading to unrealistically large values of $|a^{\star }|$. However, some such cases still exist in this set, and we shall investigate their impact shortly. We use the following ranges for our parameters:

• $T\in [0.05,0.1,0.25,0.5,1]$
• $S_0^X=100$, $S_0^Y\in [80,84,\dots ,116,120]$
• ${\lambda }_X=1$, ${\lambda }_Y=1.24$
• Heston parameters (as before): $\kappa =1.5,\theta =0.15,\nu =0.5,{\sigma }_0=0.15$
• $\rho \in [-0.9,-0.7,-0.5,-0.3,-0.1,0.1,0.3,0.5,0.7,0.9]$
• ${\rho }_X\in [-0.72,-0.42,-0.12,0.18,0.48]$
• ${\rho }_Y\in [-0.61,-0.31,-0.01,0.29,0.59]$

The parameter set above represents over 10,000 individual cases of spread option pricing, across different moneyness levels, maturities and implied volatility shapes. In this way, instead of relying on a sample dataset, we are effectively mimicking a huge range of market conditions that can occur in different financial or commodity markets and at different times. The resulting $a^{\star }$ values cover a wide range, with about two-thirds of cases lying within $[0.5,1.5]$. Although we do not vary the volatility levels ${\lambda }_X=1$ and ${\lambda }_Y$ in these results, we note that tests for different $\lambda$s perform similarly. We shall compare results using a variety of commonly-used pricing errors such as Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), Maximum Absolute Error (MaxAE, i.e., worst case), as well as considering the Mean Standard deviation (MStd) of errors across moneyness ($S_0^Y$ grid). The first two correspond to the price ratio and price difference plots in Figure 1 while the last of these is a way to assess the methodology’s aim of pricing consistently across moneyness, or in other words, flattening the implied correlation skew or frown we would otherwise observe.

Table 1. Comparison of strike conventions by Mean Absolute Error (MAE) averaged across ${\rho }_X$, ${\rho }_Y$ and $S_0^Y$ grids, varying $\rho$ and T as labeled on the left.

	$\mathit{\rho }$	$\mathit{a}=0$	$\mathit{a}=1$	${\mathit{a}}^{\mathbf{\star }}$	Bounded ${\mathit{a}}^{\mathbf{\star }}$	ATM Error
$T=0.05$	−0.7	0.0646	0.053	0.0069	0.0069	0.0036
	−0.3	0.0591	0.0241	0.0066	0.0066	0.0048
	0.1	0.0567	0.0064	0.0107	0.01	0.005
	0.5	0.0408	0.0189	0.0193	0.0117	0.0039
	0.9	0.0147	0.0121	0.005	0.0052	0.0018
$T=0.1$	−0.7	0.1108	0.0838	0.0117	0.0117	0.01
	−0.3	0.1064	0.0395	0.0132	0.0132	0.012
	0.1	0.1093	0.0159	0.0217	0.0203	0.0121
	0.5	0.091	0.0439	0.0426	0.0277	0.0102
	0.9	0.0369	0.0304	0.0126	0.0138	0.004
$T=0.25$	−0.7	0.1943	0.1413	0.039	0.039	0.0426
	−0.3	0.1932	0.0787	0.0465	0.0465	0.0487
	0.1	0.2074	0.0497	0.0561	0.0536	0.0441
	0.5	0.1933	0.1004	0.095	0.0681	0.0345
	0.9	0.1145	0.0948	0.0434	0.0465	0.0181
$T=1$	−0.7	0.3634	0.3097	0.2243	0.2243	0.2252
	−0.3	0.3976	0.2855	0.2632	0.2632	0.2724
	0.1	0.4134	0.2436	0.2519	0.2488	0.2398
	0.5	0.3898	0.2575	0.2464	0.2195	0.1856
	0.9	0.3246	0.2735	0.1471	0.156	0.102

shows the MAE (between simulated prices and Margrabe prices), averaging over the $S_0^Y$, ${\rho }_X$ and ${\rho }_Y$ grids. Each number in the table is thus an average of $11\times 5\times 5=275$ cases (grid points) for the different choices of $\rho$, T and, of course, a. We only show half of our $\rho$ values here as a reasonable sample. When calculating average errors, we first exclude parameter sets that lead to a non-valid (non-positive definite) correlation matrix. This is 19.6% of the cases overall, and around half of the cases for the most extreme values of $\rho =\pm 0.9$. We also exclude a very small number of OTM cases where Monte Carlo prices are less than 1 cent. While columns 1 to 3 of the table compare the alternative strike conventions of $a=0$ and $a=1$ with our optimal $a^{\star }$, the final column shows the ‘at-the-money (ATM) error’, meaning the error averaged over only the cases where $S_0^Y=S_0^X=100$. Recall from Figure 1 that ATM prices agree across all strike conventions (for any a) since they all collapse onto the same choice of $k_X,k_Y$. As discussed in Section 2, ATM error is zero as $T\to 0$ but is non-zero here since $T\ge 0.05$. In some sense, ATM error is thus the best we could hope for our strike convention to reach when averaging across all moneyness values.

As we see in Table 1, the optimal $a^{\star }$ outperforms the other strike conventions in the vast majority of cases, often cuts MAE by more than 50% versus $a=0$ or $a=1$, and comes much closer to the ATM error. Interestingly, $a=1$ is much more competitive than $a=0$ and seems to slightly outperform $a^{\star }$ as a convention when $\rho$ is near zero. However, this is not so surprising considering that $a^{\star }$ is often near 1 anyway in such cases, which is in line with the first special case in Remark 5 earlier. Furthermore, the weakest cases of performance can often be attributed to unusually large (or very negative) values of $a^{\star }$, since they imply picking implied volatilities from deep ITM or OTM vanilla options, especially when $\left|S_0^X-S_0^Y\right|$ is not small. This is of course also impractical in the real world. As a possible improvement, in the final column of the table, we show the average pricing errors for bounding $a^{\star }$ in the range $[-1,2]$. The extreme $a^{\star }$ situation is more common for cases of positive and fairly high $\rho$. For example, for $\rho =0.5$ here, $a^{\star }$ happens to reach as high as 7.6 and as low as −3.7 at some grid points. Therefore, while the bounding of $a^{\star }$ in $[-1,2]$ does not affect all rows, for $\rho =0.5$, it narrows the gap between $a^{\star }$ and the ATM error by about 50%. Tests on data would be required to better assess the impact of this point, but we leave this for further studies.

In addition to our earlier parameter set with very short maturity $T=0.05$, we are also interested in investigating the performance of the approach for larger T. Moving down Table 1, results for longer maturity reveal that even without bounding (or excluding) trickier cases of high $|a^{\star }|$, our approach continues to perform well, always substantially outperforming $a=0$ and often significantly outperforming $a=1$ especially for higher $\left|\rho \right|$. Interestingly, although the theory for $a^{\star }$ was derived for a short time to maturity, we see that the approach maintains a competitive advantage for large T, even $T=1$. Overall, MAE levels are higher in all cases when T increases, but the increase stems from option prices being higher and from the ATM error increasing, while the gap between $a^{\star }$ and ATM error narrows to near zero. Since larger T clearly implies larger option prices, it is insightful here to also consider MAPE. Figure 2 reveals the average MAPE across all cases (including the 10 values of $\rho$) split by T this time. Seen in percentage terms, ATM error grows steadily with T, but error from all strike conventions actually falls. Our strike convention $a^{\star }$ maintains a 0.5–1.0% advantage over $a=1$ across maturities, and the bounded version improves this slightly. Moreover, if we exclude the more challenging grid points with $a^{\star }\notin [-1,2]$, the plot shows that MAPE falls significantly to be very close to the ATM error, especially for larger T.

We focused more on MAE above primarily due to the observation in Figure 1 that relative errors show a clear asymmetry between ITM and OTM, which could distort strike convention comparisons in different cases. However,

Table 2. Comparison of all results for five different error measures with all points included (left) and excluding $a^{\star }<-1,a^{\star }>2$ cases (right). Results shown for all $T=0.05$ scenarios (top) and all $T=0.25$ scenarios (bottom).

		No Exclusions (Normal Case)					${\mathit{a}}^{\mathbf{\star }}\notin [-1,2]$ Excluded
	$\mathsf{\rho }$	$\mathit{a}=\mathbf{0}$	$\mathit{a}=\mathbf{1}$	${\mathit{a}}^{\mathbf{\star }}$	Bounded	ATM	$\mathit{a}=\mathbf{0}$	$\mathit{a}=\mathbf{1}$	${\mathit{a}}^{\mathbf{\star }}$	ATM
$T=0.1$	MAE	0.0982	0.0502	0.022	0.0177	0.01	0.0949	0.0453	0.013	0.01
	MAPE	4.23%	2.76%	1.78%	1.48%	0.14%	3.18%	1.89%	0.62%	0.15%
	RMSE	0.1288	0.0649	0.038	0.0248	0.0121	0.1235	0.0588	0.016	0.0122
	MaxAE	0.3616	0.1768	0.2338	0.1041	0.0245	0.3445	0.1563	0.0408	0.0244
	MStd	0.1137	0.0567	0.0169	0.0149	n/a	0.1098	0.051	0.0098	n/a
$T=0.25$	MAE	0.1908	0.1029	0.0586	0.0512	0.0381	0.1806	0.0908	0.04	0.0381
	MAPE	3.99%	2.77%	1.93%	1.73%	0.36%	2.53%	1.51%	0.63%	0.37%
	RMSE	0.2499	0.1307	0.0897	0.0688	0.0459	0.2356	0.1154	0.0481	0.0455
	MaxAE	0.7359	0.3678	0.4695	0.274	0.0914	0.6899	0.3145	0.119	0.0881
	MStd	0.2188	0.1067	0.0365	0.0336	n/a	0.2078	0.0928	0.0205	n/a

illustrates how our 1-STOSC approach compares to the other conventions across all our different error measures when averaging over all the scenarios for $T=0.1$ and $T=0.25$. The left half of the table includes all cases of $a^{\star }$ (as in Table 1), while the right half simply excludes cases where $a^{\star }<-1$ or $a^{\star }>2$, as mentioned above in Figure 2. The fourth column also shows the middle ground of a ‘bounded’ $a^{\star }$ within this range instead of excluding these grid points. Note that the $a=0$, $a=1$ and ATM columns also change slightly (often improve a little) when excluding these more extreme cases from the average error. Throughout the table, the optimal strike convention performs very well again, and depending on the error measure used, bounding $a^{\star }$ can cut the gap to ATM error in half, while exclusions may bring us almost all the way. However, what is especially crucial is the clear benefit $a^{\star }$ already provides relative to a commonly-used choice, such as ATM implied vols ($a=0$), often reducing the error by a factor of about three or four.

Finally, before concluding, we return to the question of consistency across moneyness—a key strength of the approach, which is captured well by the impressive final row of the table called ‘MStd’ (maximum standard deviation) but is also visually striking in Figure 3. Here, we plot average errors across moneyness (against $S_0^Y$ again) average over all the $T=0.1$ grids. Backing up the theory derived in earlier sections, the stability of errors across moneyness is very prominent, especially in comparison with the commonly-used alternatives of $a=0$ or $a=1$, which could be described more as ‘rules of thumb’ not supported by any particular theory or derivation. Indeed, to our knowledge, there is no other approach that adapts a strike convention rule to different scenarios in order to achieve clear-cut error reduction, as demonstrated in this section.

In addition to reducing pricing errors across different asset types, $a^{\star }$ adapts dynamically to changes in market correlations, allowing it to perform more consistently over time than a static rule like the ‘look-up heuristic’. On the other hand, as noted earlier, unlike the static rules, it does come with the downside of occasionally leading to large values of $|a^{\star }|$, for which no available implied volatility may exist in the market, and for which a bounded $a^{\star }$ may perform better as we saw. However, it is important to remember that $a^{\star }$ can also be estimated directly from the short-time limits of vanilla-implied volatility levels and skews, as highlighted in Remark 6. In financial and commodity markets, quantitative analysts and traders are often faced with a time-consuming daily process of re-calibrating their models to new market data each morning, sometimes observing their model parameters move significantly overnight. The opportunity to reduce model risk and bypass this parameter estimation and recalibration step via market observables is a major advantage of our novel approach, which can provide both accuracy and efficiency benefits across many derivative pricing and asset valuation challenges.

6. Conclusions

We have presented a new and systematic methodology to construct an optimal strike convention for spread option pricing in the context of stochastic volatility models. Although its derivation is rather technical, this approach is simple to use and is based on the computation of the corresponding vanilla-implied volatility levels and skews. Thus, market observables can be taken as inputs in a model-independent setting, strengthening the appeal of the technique. The obtained numerical results in Section 5 confirm its strong performance, especially compared to the limited simplistic alternatives commonly used in the industry. There is certainly more interesting work to be undertaken in this direction, and we hope to pave the way for this research with our original first contribution on the topic. Expanding into other model classes or derivative structures would be valuable, for example, extending from exchange options to any spread options (see [19]) or to three-asset spreads, or adapting to other stochastic volatility processes such as jump processes or fractional models (see [20,21]). Data analysis and further numerical investigations would also be useful, especially in these new settings. Therefore, we see this paper as the starting point for a broadly applicable and valuable new pricing tool designed to nicely complement existing practices in the financial markets.