OSP-7 Conditional Monte Carlo

Basic idea:

\Bbb{E}[Y]=\Bbb{E}[\Bbb{E}[Y|X]]\text{ and }Var(Y)\ge Var[\Bbb{E}(Y|X)]

Hence,

\hat{\theta}=\frac{1}{n}\sum_{i=1}^n\mu(X_i)\text{ where }\mu(x)=\Bbb{E}[Y|X=x]

has less variance than the standard Monte Carlo estimator

n^{-1}\sum_{i=1}^n Y_i

In other words, part of the expectation calculation can help reduce the variance of the simulation.

Remarks:

For any choice of the random variable $X$ , the conditional Monte Carlo method yields a variance reduction. But a good choice of $X$ yields a high correlation between $\Bbb{E}[Y|X]$ and $Y$

We can find $\Bbb{E}[Y|X]$ precisely in an analytical form only when we assume a “nice” model for the underlying stochastic process

Under several examples of SDE, such conditional expectations are readily computable

Example (1): Rare event probability

Suppose we want to estimate

\theta=\Bbb{P}(S_T>u)

for some

u

. This arises in the case of a digital option. For example,

S_T=\exp(\sum_{i=1}^T Y_i)

for some random variables

Y_1,...,Y_T

. Hence

\theta=\Bbb{P}(\sum_{i=1}^T Y_i>\log(u))

Denote

L_T=\sum_{i=1}^T Y_i

and

M_T = \max_{1\le i\le T}Y_i

, we can then introduce the conditioning on the maximum,

\bold{1}\{L_T>\log(u)\}=\sum_{j=1}^T\bold{1}\{L_T>\log(u),M_T=Y_j\}

Therefore

\begin{aligned} \theta &=\Bbb{E}\Big[\bold{1}\{L_T>\log(u)\}\Big]\\ &=\Bbb{P}\Big(L_T>\log(u)\Big)\\ &=\sum_{j=1}^T\Bbb{P}\Big(L_T>\log(u),M_T=Y_j\Big)\\ &=T·\Bbb{P}\Big(L_{T-1}+Y_T>\log (u),\max_{1\le j\le T-1}X_j<Y_T\Big)\\ &=T·\Bbb{P}\Big(Y_T>\log (u)-L_{T-1},\max_{1\le j\le T-1}X_j<Y_T\Big)\\ &=T·\Bbb{P}\Big(Y_T>\max\{\log (u)-L_{T-1},\max_{1\le j\le T-1}X_j\}\Big)\\ &=T·\Bbb{E}\Big[\Bbb{P}\Big(\max\{\log (u)-L_{T-1},\max_{1\le j\le T-1}X_j\}|X_1,...,X_{T-1}\Big)\Big] \end{aligned}

Denote

\bar{F}(t)=P(Y>t)

, which is the complementary of the CDF

F(t)=P(Y\le t)

, then

\theta=T·\Bbb{E}\Big[\bar{F}\Big(\max\{\log (u)-L_{T-1},\max_{1\le j\le T-1}X_j\}\Big)\Big]

Example (2): Barrier Options

Suppose we want to find the price of a European option that has a payoff at expiration given by

h(S_T)=(S_T-K_1)_+\bold{1}\{S_{T/2}\le L\}+(S_T-K_2)_+\bold{1}\{S_{T/2}>L\}

The price of this option is given by

\theta=e^{-rT}\Bbb{E}\Big[(S_T-K_1)_+\bold{1}\{S_{T/2}\le L\}+(S_T-K_2)_+\bold{1}\{S_{T/2}>L\}\Big]

If we know the transition density of

S_T

given

S_{T/2}=x

, then we can use the conditional Monte Carlo method:

\begin{aligned} \theta =&e^{-rT}\Bbb{E}\Big[\bold{1}\{S_{T/2}\le L\}\Bbb{E}[(S_T-K_1)_+|S_{T/2}]\Big]\\ &+e^{-rT}\Bbb{E}\Big[\bold{1}\{S_{T/2}>L\}\Bbb{E}[(S_T-K_2)_+|S_{T/2}]\}\Big] \end{aligned}

where

e^{-rT/2}\Bbb{E}[(S_T-K_1)_+|S_{T/2}]

and

e^{-rT/2}\Bbb{E}[(S_T-K_2)_+|S_{T/2}]

can be calculated with BS European call formula (with

S_0=S_{T/2}

, maturity time

T/2

, interest rate

r

, and strike price

K_1

K_2

Example (3): Knock-in Option

Consider the digital knock-in option with a payoff

h({S}_T)=\bold{1}\{S_T>K\}\bold{1}\{\min_{1\le k\le m} S(t_k)< H\}

With

S(t_n)=S(0)\exp(L_n),L_n=\sum_{i=1}^n X_i

, we get

h({S}_T)=\bold{1}\{L_m>\log(K/S(0))\}\bold{1}\{\tau < m\}

where

\tau

is the first

k

such that

L_k<\log(H/S(0))

To find

\Bbb{E}[h(\bold{S}_T)]

, we can use the importance sampling method, which requires choices of two changes of measure. Alternatively, we have

\theta=\Bbb{E}[\Bbb{E}[h(\bold{S}_T)|(\tau, S_\tau)]]