stochastic differential equations in finance and monte carlo...

Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations

Stochastic Differential Equations in Financeand Monte Carlo Simulations

Xuerong Mao

Department of Statistics and Modelling ScienceUniversity of Strathclyde

Glasgow, G1 1XH

China 2009

Xuerong Mao SM and MC Simulations

Outline

1 Stochastic Modelling in Asset Prices

2 The Black–Scholes World

3 Monte Carlo SimulationsEM methodEM method for financial quantities

Outline

One of the important problems in finance is the specification ofthe stochastic process governing the behaviour of an asset. Wehere use the term asset to describe any financial object whosevalue is known at present but is liable to change in the future.Typical examples are

shares in a company,commodities such as gold, oil or electricity,currencies, for example, the value of $100 US in UKpounds.

Now suppose that at time t the asset price is S(t). Let usconsider a small subsequent time interval dt , during which S(t)changes to S(t) + dS(t). (We use the notation d · for the smallchange in any quantity over this time interval when we intend toconsider it as an infinitesimal change.) By definition, the returnof the asset price at time t is dS(t)/S(t). How might we modelthis return?

If the asset is a bank saving account then S(t) is the balance ofthe saving at time t . Suppose that the bank deposit interest rateis r . Thus

dS(t)S(t)

= rdt .

This ordinary differential equation can be solved exactly to giveexponential growth in the value of the saving, i.e.

S(t) = S0ert ,

where S0 is the initial deposit of the saving account at timet = 0.

However asset prices do not move as money invested in arisk-free bank. It is often stated that asset prices must moverandomly because of the efficient market hypothesis. There areseveral different forms of this hypothesis with differentrestrictive assumptions, but they all basically say two things:

The past history is fully reflected in the present price,which does not hold any further information;Markets respond immediately to any new information aboutan asset.

With the two assumptions above, unanticipated changes in theasset price are a Markov process.

However asset prices do not move as money invested in arisk-free bank. It is often stated that asset prices must moverandomly because of the efficient market hypothesis. There areseveral different forms of this hypothesis with differentrestrictive assumptions, but they all basically say two things:

The past history is fully reflected in the present price,which does not hold any further information;Markets respond immediately to any new information aboutan asset.

With the two assumptions above, unanticipated changes in theasset price are a Markov process.

Under the assumptions, we may decompose

dS(t)S(t)

= deterministic return + random change.

The deterministic return is the same as the case of moneyinvested in a risk-free bank so it gives a contribution rdt .The random change represents the response to externaleffects, such as unexpected news. There are many externaleffects so by the well-known central limit theorem this secondcontribution can be represented by a normal distribution withmean zero and and variance v2dt . Hence

dS(t)S(t)

= rdt + N(0, v2dt) = rdt + vN(0,dt).

WriteN(0,dt) = B(t + dt)− B(t) = dB(t),

where B(t) is a standard Brownian motion. Then

dS(t)S(t)

= rdt + vdB(t).

In finance, v is known as the volatility rather than the standarddeviation.

The Black–Scholes germetric Brownian motion

If the volatility v is independent of the underlying assert price,say v = σ = const ., then the asset price follows the well-knownBlack–Scholes geometric Brownian motion

dS(t) = rS(t)dt + σS(t)dB(t).

Theta process

However, in general, the volatility v depends on the underlyingassert price. There are various types of volatility functions usedin financial modelling. One of them assumes that

v = v(S) = σSθ−1,

where σ and θ are both positive numbers. Then the asset pricefollows

dS(t) = rS(t)dt + σSθ(t)dB(t),

which is known as the theta process.

Square root process

To fit various asset prices, one could choose various values forθ. For example, θ = 1.3 or 0.5 have been used to fit certainasset prices. In particular, if θ = 0.5, we have the well-knownsquare root process

dS(t) = rS(t)dt + σ√

S(t)dB(t).

This makes the “variance" of the random change termproportional to S(t). Hence, if the asset price volatility does notincrease “too much" when S(t) increases (greater than 1, ofcourse), this model may be more appropriate.

The asset price follows the geometric Brownian motion

The risk-free interest rate r and the asset volatility σ areknown constants over the life of the option.There are no transaction costs associated with hedging aportfolio.The underlying asset pays no dividends during the life ofthe option.There are no arbitrage possibilities.Trading of the underlying asset can take placecontinuously.Short selling is permitted and the assets are divisible.

European call option

Given the asset price S(t) = S at time t , a European call optionis signed with the exercise price K and the expiry date T . Thevalue of the option is denoted by C(S, t).

The payoff of the option at the expiry date is

C(S,T ) = (S − K )+ := max(S − K , 0).

The Black–Scholes PDF

∂V (S, t)∂t

+ 12σ

2S2∂2V (S, t)∂S2 + rS

∂V (S, t)∂S

− rV (S, t) = 0.

C(S,T ) = (S − K )+ := max(S − K , 0).

∂V (S, t)∂t

+ 12σ

2S2∂2V (S, t)∂S2 + rS

∂V (S, t)∂S

− rV (S, t) = 0.

C(S,T ) = (S − K )+ := max(S − K , 0).

∂V (S, t)∂t

+ 12σ

2S2∂2V (S, t)∂S2 + rS

∂V (S, t)∂S

− rV (S, t) = 0.

Given the initial value S(t) = S at time t , write the SDE as

dS(u) = rS(u)du + σS(u)dB(u), t ≤ u ≤ T .

Hence the expected payoff at the expiry date T is

E(S(T )− K )+

Discounting this expected value in future gives

C(S, t) = e−r(T−t)E[max(S(T )− K ,0)],

which is known as the Cox formula.

The SDE has the explicit solution

S(T ) = S exp[(r − 1

2σ2)(T − t) + σ(B(T )− B(t))

]= exp

[log(S) + (r − 1

2σ2)(T − t) + σ(B(T )− B(t))

]= eµ̂+σ̂Z ,

where Z ∼ N(0,1),

µ̂ = log(S) +(

r − 12σ2)

(T − t), σ̂ = σ√

T − t .

Noting that S(T )− K ≥ 0 iff

Z ≥ log(K )− µ̂σ̂

compute

E(S(T )− K )+ =

log(K )−µ̂σ̂

(eµ̂+σ̂z − K

) 1√2π

e−12 z2

=1√2π

∫ ∞−d2

eµ̂+σ̂z−12 z2

dz − K√2π

∫ ∞−d2

e−12 z2

d2 := − log(K )− µ̂σ̂

=log(S/K ) +

(r − 1

(T − t)

T − t.

1√2π

∫ ∞−d2

e−12 z2

dz =1√2π

∫ d2

−∞e−

dz := N(d2),

and1√2π

∫ ∞−d2

eµ̂+σ̂z−12 z2

=eµ̂+

12 σ̂

√2π

N(d1),

where d1 = d2 + σ̂. Hence

E(S(T )− K + =eµ̂+

12 σ̂

√2π

N(d1)− KN(d2).

BS formula for call option

TheoremThe explicit formula for the value of the European call option is

C(S, t) = SN(d1)− Ke−r(T−t)N(d2),

where N(x) is the c.p.d. of the standard normal distribution,namely

N(x) =1√2π

−∞e−

while d1 = d2 + σ̂ and

d2 =log(S/K ) + (r − 1

2σ2)(T − t)

T − t.

EM methodEM method for financial quantities

The Black–Scholes formula benefits from the explicit solution ofthe geometric Brownian motion. However, most of SDEs usedin finance do not have explicit solutions. Hence, numericalmethods and Monte Carlo simulations have become more andmore popular in option valuation. There are two mainmotivations for such simulations:

using a Monte Carlo approach to compute the expectedvalue of a function of the underlying asset price, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.

Typically, let us consider the square root process

dS(t) = rS(t)dt + σ√

S(t)dB(t), 0 ≤ t ≤ T .

A numerical method, e.g. the Euler–Maruyama (EM) methodapplied to it may break down due to negative values beingsupplied to the square root function. A natural fix is to replacethe SDE by the equivalent, but computationally safer, problem

dS(t) = rS(t)dt + σ√|S(t)|dB(t), 0 ≤ t ≤ T .

Outline

Discrete EM approximation

Given a stepsize ∆ > 0, the EM method applied to the SDEsets s0 = S(0) and computes approximations sn ≈ S(tn), wheretn = n∆, according to

sn+1 = sn(1 + r∆) + σ√|sn|∆Bn,

where ∆Bn = B(tn+1)− B(tn).

Continuous-time EM approximation

s(t) := s0 + r∫ t

0s̄(u))du + σ

√|s̄(u)|dB(u),

where the “step function” s̄(t) is defined by

s̄(t) := sn, for t ∈ [tn, tn+1).

Note that s(t) and s̄(t) coincide with the discrete solution at thegridpoints; s̄(tn) = s(tn) = sn.

The ability of the EM method to approximate the true solution isguaranteed by the ability of either s(t) or s̄(t) to approximateS(t) which is described by:

Theorem

lim∆→0

sup0≤t≤T

|s(t)−S(t)|2)

= lim∆→0

sup0≤t≤T

|s̄(t)−S(t)|2)

The ability of the EM method to approximate the true solution isguaranteed by the ability of either s(t) or s̄(t) to approximateS(t) which is described by:

Theorem

lim∆→0

sup0≤t≤T

|s(t)−S(t)|2)

= lim∆→0

sup0≤t≤T

|s̄(t)−S(t)|2)

Outline

If S(t) models short-term interest rate dynamics, it is pertinentto consider the expected payoff

β := E exp

(−∫ T

0S(t)dt

from a bond. A natural approximation based on the EM methodis

β∆ := E exp

(−∆

N−1∑n=0

), where N = T/∆.

Theorem

lim∆→0|β − β∆| = 0.

If S(t) models short-term interest rate dynamics, it is pertinentto consider the expected payoff

β := E exp

(−∫ T

0S(t)dt

from a bond. A natural approximation based on the EM methodis

β∆ := E exp

(−∆

N−1∑n=0

), where N = T/∆.

Theorem

lim∆→0|β − β∆| = 0.

Up-and-out call option

An up-and-out call option at expiry time T pays the Europeanvalue with the exercise price K if S(t) never exceeded the fixedbarrier, c, and pays zero otherwise.

Using the discrete numerical solution to approximate the truepath gives rise to two distinct sources of error:

a discretization error due to the fact that the path is notfollowed exactly—the numerical solution may cross thebarrier at time tn when the true solution stays below, or viceversa,a discretization error due to the fact that the path is onlyapproximated at discrete time points—for example, thetrue path may cross the barrier and then return within theinterval (tn, tn+1).

TheoremDefine

V = E[(S(T )− K )+I{0≤S(t)≤c, 0≤t≤T}

V∆ = E[(s̄(T )− K )+I{0≤s̄(t)≤B, 0≤t≤T}

Thenlim

∆→0|V − V∆| = 0.

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