Futures and Options Report 02/2024

THE STRONG MARKOV PROPERTY APPLIED TO OPTION PRICINGPART IIMASAHIKO EGAMI
Graduate School of Economics, Kyoto University, Sakyo-ku, Kyoto, 606-8501, Japan.

A BSTRACT. In pricing and hedging derivatives, one uses stochastic processes to represent movement ofunderlying assets and other important state variables (such as volatility). In this report, we collect severaltechniques from stochastic analysis which we believe are useful for financial modeling, derivative pricing,and parameter estimation. A key element is the strong Markov property that helps to streamline seemingly
complex equations and to tremendously reduce variance in numerical simulations.
Following Part I where we discussed the definition of the strong Markov property and related mathematical objects, we show more examples and a case where the strong Markov property is not applicable. We
include proofs of some technical, but useful for applications, results in the Appendix.

4. OTHER E XAMPLESLet us keep our general setting in Part I and consider a set of all the possibilities of future outcomes Ωand denote our stochastic process by X = {Xt ;t ≥ 0} with the initial value x. In this report, we assume
that the process X is right continuous. Given a future scenario ω ∈ Ω and a fixed time t, the stochastic
process X takes a value Xt (ω) in some space (such as Rd ). Given this future path ω, X keeps taking
values as time goes and a history of its behavior is recorded. This is called the filtration generated by
X and denoted by F = {Ft ;t ≥ 0}. Intuitively, Ft is the accumulated information of X up to time t.
Recall that a random time U is called a stopping time of F if we can tell whether or not U has happened
up to time t by looking at the record of X up to time t: {ω : U(ω) ≤ t} ∈ Ft for each t.
If {Xt , Ft ,t ≥ 0} is a strong Markov process, it has the property for a certain set A in RdPx (XU+t ∈ A|FU ) = Px (XU+t ∈ A|XU ),where FU is the record of X up to the stopping time U.
For concrete computations and for practical applications, let X = (Xt ,t ≥ 0) be a diffusion process
taking values on Rd . The shift operator θs : Ω → Ω, s ≥ 0 further facilitates rigorous argument:
Xs+t (ω) = Xt (θs (ω)) = (Xt ◦ θs )(ω) for each s,t ≥ 0.
The infinitesimal generator G of a diffusion X is defined by
Ex [g(Xt )] − g(x)tt↓0Gg(x) := lim

1(4.1)

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for a suitable function g. This is how much the value g(Xt ) would change if an infinitely small amountof time has elapsed from time zero. The infinitesimal generator for a linear diffusion dXt = µ(Xt )dt +σ (Xt )dWt is
1d2dGg(x) = σ 2 (x) 2 g(x) + µ(x) g(x),(4.2)
2dxdx
Now we shall deal with two random times. In Part I, we denote the first time that X reaches b by
Hb := min{t ≥ 0 : Xt ≤ b}, which is a good example of stopping time. A generalization of this is, for aBorel set A,
TA := inf{t > 0 : Xt ∈ A}
where inf 0/ = +∞ by convention. This is called the first hitting time. If the Markov process is rightcontinuous, it is a stopping time (see Theorem 2.4.5 of Chung and Walsh (2004)).
If σ ≤ TA where σ is a stopping time and TA is a first hitting time to a Borel set A, then according toChapter III.7 of Peskir and Shiryaev (2006)
TA = σ + TA ◦ θσ .

(4.3)

For example, with (4.3) the strong Markov property provides
Ex [e−αTA 1l{σ ≤TA } ] = Ex [e−α(σ +TA ◦θσ ) 1l{σ ≤TA } ]= Ex [Ex [e−α(σ +TA ◦θσ ) 1l{σ ≤TA } |Fσ ]]= Ex [e−ασ Ex [e−α(TA ◦θσ ) 1l{σ ≤TA } |Fσ ]]= Ex [e−ασ EXσ [e−αTA 1l{σ ≤TA } ]],so that we may use the information about X at time σ .
The relation (4.3) may remind one of the memoryless property of an exponential random variable. Letν be an exponential random variable with rate λ (i.e., E[ν] = 1/λ ). Then we can write
ν = t + ν ◦ θt

if ν > t.

(4.4)

Suppose that the next bus arrives after an exponentially distributed random time of rate λ and there is aperson who has already waited for a t amount of time. But at time t, this person’s average waiting time
for the bus is still 1/λ : the fact that the passenger has waited for the t amount of time is forgotten (infuture scenarios). If we write (4.4) as ν − t = ν ◦ θt , additional waiting time over t (the left-hand side) isequal to ν reckoned from time t (the right-hand side).
The equations (4.3) and (4.4) are useful, for example, in studying options with multiple exercisingrights (see Carmona and Dayanik (2008)) and financial instruments under regime switching models (seeHamilton (2008)). Regime switching is widely used since it allows model parameters to alter according
to economic conditions. A switch from one regime to another is often assumed to be governed by a
Poisson process, so that the waiting time before switching follows exponential distribution. In case there2

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are two regimes (e.g. bulls and bears), X takes the values of 1 or 2, so that we consider two-state Markov
chain with the infinitesimal generator!
−λ1 λ1(4.5)λ2 −λ2where λ1 is the rate of waiting time of a jump from state 1 to state 2 and λ2 is the rate of waiting timeof a jump from state 2 to state 1. While the appearance of this generator is very different from (4.2)(linear diffusion case), it can be shown this is so. See Appendix B for a derivation of this matrix fromthe definition (4.1).

5. B EYOND THE STRONG M ARKOV PROPERTY
The key element of the strong Markov property and its applications is that the random time at which
the process is evaluated has to be a stopping time. Recall that a random time U is called a stopping timeof F if we can tell whether or not U has happened up to time t by looking at the record of X up to time
t: that is, {ω : U(ω) ≤ t} ∈ Ft for each t. The first hitting time is a good example of stopping times.
On the other hand, the last exit time is a typical example of a random time that is not a stopping time.
Suppose that a Markov process X = {Xt : t ≥ 0} is transient. This means heuristically that the lifetime,say ζ , of X is finite, so that X disappears at some point of time.1
For a Borel set B, the last exit time from B is defined by
LB (ω) := sup{t ≥ 0 : Xt (ω) ∈ B}
where sup 0/ = 0 by convention. Since X is assumed to be transient, it disappears at some time and the set{t ≥ 0 : Xt ∈ B} is finite. Suppose that X exits from B at time s. But we cannot tell at that time s whether
this is the last time for the process to exit from B. Hence LB is not a stopping time and we cannot discuss
the strong Markov property at LB . Despite of this difficulty, the last exit time is increasingly applied in
the finance literature as indicated in Nikeghbali and Platen (2013).
Consider an investment in the Nikkei 225. Suppose that the investor sets a stop-loss level at c andan alarming price level at α(> c) since he needs to prepare well in advance for a possible loss. Theinvestor models the Nikkei 225 by a geometric Brownian motion X and wishes to estimate how longit would take from the last exit time from level α to reaching the stop-loss threshold c. He needs thisinformation perhaps for some liquidity concerns when loss seems inevitable. In this case the set B in theabove definition is a singleton set {α}. In sum, the investor wishes to compute
Ex [Hc − Lα ].
1In financial engineering this concept of transience is used, for example, in barrier options of up-and-out and down-and-out types. A down-and-out option becomes worthless once the underlying asset hits a threshold level b(< x). X is considered
to disappear at the first time when X hits b.

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Fig. 3. THE DENSITY OF THE DISTRIBUTION P[Hc − Lα ∈ dt : Hc < ∞].

Since Hc and Lα are closely related random times, it is not easy to efficiently compute the distribution
even by simulations. To overcome this difficulty, Egami and Kevkhishvili (2020) employs the reversed
process: the original process X reversed from the time Hc . Let it be called Z which starts at c. The
infinitesimal generator of the reversed process Z is identified (see Corollary 3.8 of the paper). Then thequantity (Hc − Lα ) with respect to X is the same as the first hitting time to α of Z. Hence the problem
transforms into computing the distribution of the first hitting time of Z. Note that the result of the said
paper is expressed in terms of a Brownian motion with non-zero drift and unit variance, so that we need
1 2to transform a geometric Brownian motion Xt = xe(µ− 2 σ )t+σWt to 1
Xtlog
=σx

µ − 21 σ 2σ!
t +Wt ,

where W = {Wt : t ≥ 0} is a Brownian motion.
Let us take a hypothetical situation where c = 30, 000 and α = 35, 000 for the Nikkei 225. Theparameters are set to be σ = 0.235 and µ − 21 σ 2 = −0.125. Note that an estimation procedure of theparameters is discussed in Egami (2022). In this case, X reaches the level c in finite time with probability
one. Figure 3 shows the probability density P[Hc − Lα ∈ dt : Hc < ∞]. The parameters are standard whenthe market is slightly bearish. The density function has the sharp peak around t = 0.08, indicating thatthe time interval between the last exit from level α and the stop-loss level c is rather short. This kindof information could be useful from the risk management point of view. Moreover, when one needs toevaluate options of a look-back type in relation to a last exit time, the concept of reversed process can behelpful.
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A PPENDIX A. D ERIVATION OF (3.3) IN PART I
Only the second term of (3.3) needs a proof. The derivation of
Z ∞
−αt
Exef (Xt )dt = Ex [e−αT U α f (XT )].
Tis done by making substitution t = T + u as below.
Z ∞ Z ∞
−αt−αtExef (Xt )dt = Ex Exef (Xt )dt FTTT Z ∞−α(T +u)= Ex Exef (XT +u )du FT0Z ∞
−αT−αu= Ex eExef (XT +u )du FT.
0R ∞ −αu

Let us consider 0 ef (Xu )du as a random variable, say Y . With this in mind, we can continue

Z ∞

Z ∞

−αT−αu−αT−αuEx e
Exef (XT +u )du FT= Ex eExef (Xu ◦ θT )du FT00

Z ∞


−αT−αu= Ex e
Exef (Xu )du ◦ θT FT0
= Ex e−αT Ex {Y ◦ θT |FT }
= Ex e−αT EXT {Y }
Z ∞


−αT−αu= Ex eEXTef (Xu )du = Ex e−αT U α f (XT )0by using (3.2) in Part I for the fourth line and the definition of U α f (·) in the fifth line.
A PPENDIX B. T HE GENERATOR OF A M ARKOV CHAIN
Let us recall an indicator function for a set A1,1lA (x) =0,if x ∈ A,otherwise.

Our state variable X takes the value of 1 or 2. First we show why the (1,2) entry of the matrix (4.5) is λ1 .
The (1,2) entry corresponds to the situation where X moves from state 1 to state 2 within a small amountof time. Now X is initially in state 1 and we write the generator according to the definition (4.1):E[1l{not in state 1} (Xt )] − 1l{not in state 1} (x)P(Xt 6= 1) − 0= lim
ttt↓0t↓0lim

P(T1 < t)
1 − e−λ1t
λ1 t= lim'= λ1tttt↓0t↓0= lim

where we apply g(x) = 1lA (x) in (4.1) and A = {not in state 1}. T1 denotes the waiting time until a jump
to state 2 from state 1 and is exponentially distributed with rate λ1 .
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Next the (1,1) entry corresponds to the situation where X starts in state 1 and remains the same withina small amount of time. Similar to the case of the (1,2) entry,
E[1l{stay in state 1} (Xt )] − 1l{stay in state 1} (x)P(Xt = 1) − 1= limttt↓0t↓0lim

P(T1 > t) − 1e−λ1t − 1 −λ1t= lim'= −λ1 ,tttt↓0t↓0= lim
which proves that the (1,1) entry of (4.5) is −λ1 .

R EFERENCES
Carmona, R., Dayanik, S., 2008. Optimal multiple stopping of linear diffusions. Mathematics of Operations Research 33 (2), 446–460.
Chung, K. L., Walsh, J. B., 2004. Markov Processes, Brownian Motion, and Time Symmetry, 2nd Edition. Springer.
Egami, M., 2022. A financial engineering view of drawdowns in the stock market part I. Futures andOptions Report No. 1, Osaka Exchange.
Egami, M., Kevkhishvili, R., 2020. Time reversal and last passage time of diffusions with applications
to credit risk management. Finance and Stochastics 24 (3), 795–825.
Hamilton, J. D., 2008. Regime switching models. In: Durlauf, S. N., Blume, L. E. (Eds.), The New
Palgrave Dictionary of Economics, 2nd Edition. Palgrave Macmillan.
Nikeghbali, A., Platen, E., 2013. A reading guide for last passage times with financial applications inview. Finance and Stochastics 17 (3), 615–640.
Peskir, G., Shiryaev, A. N., 2006. Optimal stopping and Free-Boundary Problems (Lectures in Mathematics, ETH Zürich). Birkhauser, Basel.

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