Options on infectious diseases
aa r X i v : . [ q -f i n . C P ] M a r Options on infectious diseases
Andrew Lesniewski and Nicholas Lesniewski
Department of MathematicsBaruch College55 Lexington AvenueNew York, NY 10010Draft of March 26, 2020First draft: March 17, 2020
Abstract
We present a parsimonious stochastic model for valuation of options onthe fraction of infected individuals during an epidemic. The underlyingstochastic dynamical system is a stochastic differential version of the SIRmodel of mathematical epidemiology.
In the current environment of economic uncertainty related to the spread of COVID-19, it could be of interest to develop financial instruments that allow market partic-ipants to express their views on the economic impact of an epidemic. The simplestsuch instruments are infection options , which are options on the fraction of infectedindividuals in a population affected by an epidemic . Infection options would allowdynamic hedging of risk exposure to pandemic linked catastophe bonds. Investorspurchasing catastrophe bonds provide insurance to issuers and lose invested princ-pal if certain conditions are met .In this note we propose a mathematical model for pricing and risk manage-ment of such options. One of the challenges is to formulate a model which, atthe very least, exhibits typical stylized facts about the spread of highly infectious According to , as of March 17, the fractions of pop-ulations currently infected by COVID-19 is 0.0018% for the USA, 0.0313% for Switzerland, and0.0431% for Italy. In June 2017 the World Bank issued pandemic linked catastrophe bonds and derivatives worth$425 million. ≤ x ≤ , the fraction of individuals who are susceptible to the disease,(ii) ≤ y ≤ , the fraction of individuals who are infected with the disease,(iii) ≤ z ≤ , the fraction of individuals who have recovered and are immuneto the disease.The model dynamics is given as the following dynamical system: ˙ x ( t ) x ( t ) = − βy ( t ) , ˙ y ( t ) y ( t ) = βx ( t ) − γ, ˙ z ( t ) y ( t ) = γ, (1)with the initial condition: x (0) = x ,y (0) = y ,z (0) = 1 − x − y . (2)The constant parameters β > and γ > are called the infection and recoveryrates, respectively.Notice that this dynamics obeys the conservation law x ( t ) + y ( t ) + z ( t ) = 1 , (3)consistent with the assumption that the variables x, y , and z represent populationfractions. This means that the variable z is, in a way, redundant, as its current valuedoes not affect the dynamics of x and y , and it can be computed in a straightforwardmanner from (3).ptions on infectious diseases 3 Stochastic extensions of the SIR model go back to Bartlett [3], who formulateda stochastic jump process describing the evolution of an epidemic. Various otherstochastic extensions have been proposed since, see e.g. [2]. Here, we will focuson two stochastic extensions of the SIR model, which utilize stochastic differentialequations, see e.g. [5] and references therein. These models are particularly adeptat option modeling.The first of the models, a one factor stochastic SIR model is obtained in thefollowing manner. Let W t denote the standard Brownian motion, and let ˙ W t denotethe (generalized) white noise process. We assume that the infection rate β , ratherthan being constant, is subject to random shocks, namely β t = β + σ ˙ W t , while therecovery rate γ remains constant. Here σ > is a constant volatility parameter.This leads to the following stochastic dynamics: dX t X t = − βY t dt − σY t dW t ,dY t Y t = ( βX t − γ ) dt + σX t dW t ,dZ t Y t = γdt, (4)with the initial condition analogous to (2).Notice that the conservation law X t + Y t + Z t = 1 (5)continues to hold in the stochastic model, and so we can focus attention on thedynamics described by X and Y only.Consider now a time horizon T > , and ξ, η > . The backward Kolmogorovequation for the Green’s function G = G T,ξ,η ( t, x, y ) corresponding to the stochas-tic system (4) reads: ˙ G − βxy ∇ x G + ( βx − γ ) y ∇ y G + 12 σ x y (cid:0) ∇ x G − ∇ xy G + ∇ y G (cid:1) = 0 . (6)We assume that G T,ξ,η ( t, x, y ) satisfies the following terminal condition: G T,ξ,η ( T, x, y ) = δ ( x − ξ ) δ ( y − η ) . (7)Similarly, a two factor stochastic SIR model can be formulated in the followingmanner. In addition to allowing the infection rate being stochastic, we assume A.Lesniewski and N.Lesniewskithat the recovery rate is subject to random shocks delivered by a second Brownianmotion B t , γ t = γ + ζ ˙ B t . Again, ζ > is a volatility parameter. The twoBrownian motions are not required to be independent, and we let ρ denote thecorrelation coefficient between them, dW t dB t = ρdt. (8)This leads to the following stochastic dynamics: dX t X t = − βY t dt − σY t dW t ,dY t Y t = ( βX t − γ ) dt + σX t dW t − ζdB t ,dZ t Y t = γdt + ζdB t , (9)with the initial condition analogous to (2). The two factor stochastic model con-tinues to obey the conservation law (5). Notice that the two factor model has twoadditional parameters compare to the one factor model, namely the volatility of therecovery rate ζ and the correlation coefficient ρ in (8).The backward Kolmogorov equation for the stochastic system (9) reads as fol-lows. Set η ( x ) = p σ x − ρσζx + ζ . (10)Then ˙ G − βxy ∇ x G + ( βx − γ ) y ∇ y G + 12 y (cid:16) σ x ∇ x G − σx ( σx − ρζ ) ∇ xy G + η ( x ) ∇ y G (cid:17) = 0 , (11)where the Green’s function G = G T,ξ,η ( t, x, y ) satisfies terminal condition (7). The payoff of the call option on the infection fraction struck at K is given by ( Y T − K ) + multiplied by the notional amount on the contract. Here, as usual, x + = max( x, . Likewise, the payoff of the put option on the infection fractionstruck at K is ( K − Y T ) + multiplied by the notional amount on the contractInfection options are written on an underlying which is not a financial asset.For this reason, they are much closer in spirit to weather options (or real options),than to traditional equity or FX options. Consequently, there is no natural riskneutral approach to their valuation.ptions on infectious diseases 5For this reason, we adopt an approach that is based on the process (4) (or, al-ternatively, the more complicated process (9)), which is written under the physicalmeasure. Today’s value of an infection call option is thus given by Call(
T, K ) = e − rT E (( Y T − K ) + ) × notional amount , (12)where r is the discount rate, and E denotes expected value with respect to theprobability measure generated by the process (4) (or (9)). In terms of the Green’sfunction G , this can be expressed as a two dimensional integral over the terminalprobability distribution: E (( Y T − K ) + ) = Z Z [0 , G T,ξ,η (0 , X , Y )( η − K ) + dξ dη. (13)Note that while the current value of the option depends on the current reading ofthe susceptible fraction, its value at expiration is not set. Analogous formulas holdfor put options.Closed form solutions to equations (6) and (11) are not available and the eval-uation of the expected values requires numerical computations. Two methods areof practical importance.Substituting t → T − t we convert the (two-dimensional) backward Kol-mogorov equation to a diffusion equation, which can be solved through a numberof standard methods, such as the ADI method, see e.g. [7].Alternatively, one can calculate the expected values using Monte Carlo simu-lations [4]. Specifically, a convenient discretization to (4) is formulated as follows.In order to guarantee the positivity of X t and Y t throughout the simulation, we set X t = exp( − l t ) and Y t = exp( − m t ) . Denoting by δ the magnitude of the timestep, we are lead to the following Euler scheme: l n +1 = l n + ( βδ + σ √ δ z n ) e − m n + 12 σ δe − m n ,m n +1 = m n + γδ − ( βδ + σ √ δ z n ) e − l n + 12 σ δe − l n , (14)for n = 0 , , . . . . Here, z n are independent variates drawn from the standard normaldistribution. At each iteration step one has to floor l n +1 and m n +1 at , l n +1 =max( l n +1 , , m n +1 = max( m n +1 , , so that X t , Y t ≤ .Similarly, the two factor system (9) can be discretized as follows: l n +1 = l n + ( βδ + σ √ δ z n ) e − m n + 12 σ δe − m n ,m n +1 = m n + γδ − ( βδ + σ √ δ z n ) e − l n + ζ √ δ b n + 12 η ( e − l n ) δ, (15) A.Lesniewski and N.Lesniewskiwith η ( x ) defined in (10), for n = 0 , , . . . . The two dimensional random nor-mal vectors ( z n , b n ) T are independently drawn from the Gaussian distribution withcovariance C = (cid:18) ρρ (cid:19) . (16) References [1] Brauer, F., Castillo-Chavez, C., and Feng, Z.:
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