An options-pricing approach to forecasting the French presidential election

Figures & data

Figure 1. Schematic diagram of French political parties on the left-right political spectrum.

Table 1. Outline description of major French political parties.

Left to right	Symbol	Anglicised name	Categorisation
Point 1	LO	Workers struggle	Far left
Point 2	NPA	New Anticapitalist Party	far left
Point 3	PCF	French Communist Party	Left wing to far left
Point 4	LFI	La France Insoumise	Left wing to far left
Point 5	EELV	Europe Ecology – the Greens	Centre left to left wing
Point 6	PS	Socialist Party	Centre Left
Point 7	LREM	La Republic en Marche!	Centre
Point 8	Res	Résistons!	Centre
Point 9	LR	The Republicans	Centre Right
Point 10	DLF	Debout La France	Right wing to far right
Point 11	NR	National Rally	Far right
Point 12	REC	Reconqête	Far right

Table 2. Projected vote-transfer probabilities under three scenarios.

First-round	Scenario 1		Scenario 2		Scenario 3
vote	LREM	NR	LREM	LFI	NR	LFI
LO	0.625	0.375	0.333	0.667	0.231	0.769
NPA	0.643	0.357	0.286	0.714	0.182	0.818
PCF	0.667	0.333	0.2	0.8	0.111	0.889
LFI	0.7	0.3	0	1	0	1
EELV	0.75	0.25	0.333	0.667	0.143	0.857
PS	0.833	0.167	0.667	0.333	0.286	0.714
LREM	1	0	1	0	0.429	0.571
Res	0.75	0.25	0.8	0.2	0.571	0.429
LR	0.5	0.5	0.714	0.286	0.714	0.286
DLF	0.25	0.75	0.667	0.333	0.857	0.143
NR	0	1	0.636	0.364	1	0
REC	0.167	0.833	0.615	0.385	0.889	0.111

Scenario 1 s round is between LREM vs. NR. Scenario 2 s round is between LREM and LFI. Scenario 3 s round is between NR and LFI.

Table 3. Summary statistics of 2022 and 2017 presidential election polls.

Election	First day of polling	Last day of polling	No. of polls	No. of days	No. of polling companies
2017 First round	March 19	April 21	88	34	9
2017s round	April 24	May 5	36	13	8
2022 First round	March 7	April 8	118	32	14
2022s round	April 10	April 22	54	13	3

Table 4. Estimated values of $K_i$ and $s_i$ obtained after the first round of polling.

Party	Estimated vote share ($K_i$)	Standard error ($s_i$)
LO	0.005117346	0.004397742
NPA	0.009503101	0.003780203
PCF	0.02677225	0.002903718
LFI	0.1732441	0.002826978
PS	0.02042434	0.002270473
EELV	0.04615099	0.002225763
LREM	0.2627287	0.001436373
LR	0.08540158	0.002565392
RES	0.02373998	0.004231739
DLF	0.02206708	0.006069075
RN	0.2301469	0.004782486
REC	0.08893775	0.002844208

Figure 2. Estimated probability that President Macron wins the popular vote during the first round of the 2022 French presidential election according to Eq. (4)(4) $$\mathit{\text{Pr}}(i\text{beats}j\text{in the popular vote})=e^{-r(T-t)}\Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}(Q_t)+\left(r-\frac{{\sigma }_Q^2}{2}\right)(T-t)}{{\sigma }_Q\sqrt{T-t}}\right),$$(4) (solid line) and Eq. (8)(8) $$\begin{array}{c}\begin{array}{cc}\mathit{\text{Pr}}(i\text{beats}j)\hfill & =\mathit{\text{Pr}}(Q_T>1)=\mathit{\text{Pr}}(\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}{Q}_T>0)\hfill \end{array}\\ \begin{array}{cc}=\hfill & \Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}{Q}_t+{\mu }_Q(T-t)}{{\sigma }_Q\sqrt{T-t}}\right).\hfill \end{array}\end{array}$$(8) (dashed line).

Figure 3. Estimated probability of qualification for the second round of the 2022 French presidential election according to Eq. (4)(4) $$\mathit{\text{Pr}}(i\text{beats}j\text{in the popular vote})=e^{-r(T-t)}\Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}(Q_t)+\left(r-\frac{{\sigma }_Q^2}{2}\right)(T-t)}{{\sigma }_Q\sqrt{T-t}}\right),$$(4) .

Table 5. Estimated probability of election victory after first round of polling obtained by Monte Carlo simulation (results based on 1,000,000 simulations).

Party	Probability of winning
LO	0
NPA	0
PCF	0
LFI	0.000682
PS	0
EELV	0
LREM	0.961071
LR	0
RES	0
DLF	0
RN	0.038247
REC	0

Figure 4. Real-time model estimated vote share for Macron for the second round of the 2022 French presidential election based on Eq. (4)(4) $$\mathit{\text{Pr}}(i\text{beats}j\text{in the popular vote})=e^{-r(T-t)}\Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}(Q_t)+\left(r-\frac{{\sigma }_Q^2}{2}\right)(T-t)}{{\sigma }_Q\sqrt{T-t}}\right),$$(4) (solid line) and Eq. (7)(7) $$\begin{array}{c}\text{Median}{P}_T=\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left(\mu (T-t)+X_t\right).\hfill \end{array}$$(7) (dashed line).

Figure 5. Real-time estimated probability of a Macron victory in the second round of the 2022 French presidential election based on Eq. (4)(4) $$\mathit{\text{Pr}}(i\text{beats}j\text{in the popular vote})=e^{-r(T-t)}\Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}(Q_t)+\left(r-\frac{{\sigma }_Q^2}{2}\right)(T-t)}{{\sigma }_Q\sqrt{T-t}}\right),$$(4) (solid line) and Eq. (8)(8) $$\begin{array}{c}\begin{array}{cc}\mathit{\text{Pr}}(i\text{beats}j)\hfill & =\mathit{\text{Pr}}(Q_T>1)=\mathit{\text{Pr}}(\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}{Q}_T>0)\hfill \end{array}\\ \begin{array}{cc}=\hfill & \Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}{Q}_t+{\mu }_Q(T-t)}{{\sigma }_Q\sqrt{T-t}}\right).\hfill \end{array}\end{array}$$(8) (dashed line).

Figure 6. Estimated probability that President Macron wins the popular vote during the first round of the 2017 French presidential election according to Eq. (4)(4) $$\mathit{\text{Pr}}(i\text{beats}j\text{in the popular vote})=e^{-r(T-t)}\Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}(Q_t)+\left(r-\frac{{\sigma }_Q^2}{2}\right)(T-t)}{{\sigma }_Q\sqrt{T-t}}\right),$$(4) (solid line) and Eq. (8)(8) $$\begin{array}{c}\begin{array}{cc}\mathit{\text{Pr}}(i\text{beats}j)\hfill & =\mathit{\text{Pr}}(Q_T>1)=\mathit{\text{Pr}}(\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}{Q}_T>0)\hfill \end{array}\\ \begin{array}{cc}=\hfill & \Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}{Q}_t+{\mu }_Q(T-t)}{{\sigma }_Q\sqrt{T-t}}\right).\hfill \end{array}\end{array}$$(8) (dashed line).

Figure 7. Estimated probability of qualification for the second round of the 2017 French presidential election according to Eq. (4)(4) $$\mathit{\text{Pr}}(i\text{beats}j\text{in the popular vote})=e^{-r(T-t)}\Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}(Q_t)+\left(r-\frac{{\sigma }_Q^2}{2}\right)(T-t)}{{\sigma }_Q\sqrt{T-t}}\right),$$(4) .

Figure 8. Real-time model estimated vote share for Macron in the second round of the 2017 French presidential election based on Eq. (4)(4) $$\mathit{\text{Pr}}(i\text{beats}j\text{in the popular vote})=e^{-r(T-t)}\Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}(Q_t)+\left(r-\frac{{\sigma }_Q^2}{2}\right)(T-t)}{{\sigma }_Q\sqrt{T-t}}\right),$$(4) (solid line) and Eq. (7)(7) $$\begin{array}{c}\text{Median}{P}_T=\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left(\mu (T-t)+X_t\right).\hfill \end{array}$$(7) (dashed line).

Figure 9. Real-time estimated probability of a Macron victory in the second round of the 2017 French presidential election based on Eq. (4)(4) $$\mathit{\text{Pr}}(i\text{beats}j\text{in the popular vote})=e^{-r(T-t)}\Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}(Q_t)+\left(r-\frac{{\sigma }_Q^2}{2}\right)(T-t)}{{\sigma }_Q\sqrt{T-t}}\right),$$(4) (solid line) and Eq. (8)(8) $$\begin{array}{c}\begin{array}{cc}\mathit{\text{Pr}}(i\text{beats}j)\hfill & =\mathit{\text{Pr}}(Q_T>1)=\mathit{\text{Pr}}(\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}{Q}_T>0)\hfill \end{array}\\ \begin{array}{cc}=\hfill & \Phi \left(\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}{Q}_t+{\mu }_Q(T-t)}{{\sigma }_Q\sqrt{T-t}}\right).\hfill \end{array}\end{array}$$(8) (dashed line).

Table 6. Comparison with the options-pricing approach and raw vote-intention polling numbers for the 2022 French presidential election after the first round of polling.

Party	Options-pricing model	Raw polling numbers	Actual
LO	0.005117346	0.005714286	0.0056
NPA	0.009503101	0.010000000	0.0077
PCF	0.02677225	0.02714286	0.0228
LFI	0.1732441	0.1735714	0.2195
PS	0.02042434	0.02071429	0.0175
EELV	0.04615099	0.04642857	0.0463
LREM	0.2627287	0.2628571	0.2785
LR	0.08540158	0.08571429	0.0478
RES	0.02373998	0.02428571	0.0313
DLF	0.02206708	0.02285714	0.0206
RN	0.2301469	0.2307143	0.2315
REC	0.08893775	0.08928571	0.0707
RMSE	0.01876124	0.01876718	$·$
MAE	0.01146484	0.01151905	$·$

Table 7. Comparison with the options-pricing approach and raw vote-intention polling numbers for the second round of the 2022 French presidential election.

Date	Options-pricing model	Raw polling numbers
Apr 13	0.5361323	0.5370000
Apr 14	0.5337876	0.5341667
Apr 15	0.5388382	0.5390000
Apr 16	0.5547051	0.5550000
Apr 18	0.5456631	0.5460000
Apr 19	0.5547377	0.5550000
Apr 20	0.5573433	0.5575000
Apr 21	0.5515747	0.5516667
Apr 22	0.5599435	0.5600000
RMSE	0.0385388	0.03822347
MAE	0.03741939	0.03712962

Table 8. Comparison with the options-pricing approach and raw vote-intention polling numbers for the 2017 French presidential election after the first round of polling.

Party	Options-pricing model	Raw polling numbers	Actual
LO	0.003026144	0.003333333	0.0064
NPA	0.01269327	0.01333333	0.0109
LFI	0.1897006	0.19	0.1958
PS	0.07488916	0.075	0.0636
EM	0.2399117	0.24	0.2401
Res	0.007769815	0.008333333	0.0121
LR	0.1914305	0.1916667	0.2001
DLF	0.04128121	0.04166667	0.0470
UPR	0.007300721	0.008333333	0.0092
FN	0.2282986	0.2283333	0.2130
S&P	0.003089621	0.003333333	0.0018
RMSE	0.00704024	0.006947207	$·$
MAE	0.005449996	0.005278781	$·$

Table 9. Comparison with the options-pricing approach and raw vote-intention polling numbers for the second round of the 2017 French presidential election.

Date	Options-pricing model	Raw polling numbers
Apr 27	0.6010233	0.6012500
Apr 28	0.5950509	0.5950000
Apr 29	0.6000272	0.6000000
Apr 30	0.6001014	0.6000000
May 1	0.5934568	0.5933333
May 2	0.5963662	0.5962500
May 3	0.6067211	0.6066667
May 4	0.6170280	0.6170000
May 5	0.6266846	0.6266667
RMSE	0.05789811	0.05793319
MAE	0.05694896	0.05698148