Stochastic Modelling in Finance - Personal WWW Pages
Transcript of Stochastic Modelling in Finance - Personal WWW Pages
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Stochastic Modelling in Finance
Xuerong Mao FRSE
Department of Mathematics and StatisticsUniversity of Strathclyde
Glasgow, G1 1XH
April 2010
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
OptionOption value
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
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Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
Problem
Assume that on 10 April 2010, Mr King has $100K to invest for1 year and he has two choices:
(a) invest the money in a bank saving account to receive arisk-free interest.
(b) buy a $100K house and then sell it on 10 April 2011.
Which choice should Mr King take?
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OptionOption value
Problem
Assume that on 10 April 2010, Mr King has $100K to invest for1 year and he has two choices:
(a) invest the money in a bank saving account to receive arisk-free interest.
(b) buy a $100K house and then sell it on 10 April 2011.
Which choice should Mr King take?
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Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
Problem
Assume that on 10 April 2010, Mr King has $100K to invest for1 year and he has two choices:
(a) invest the money in a bank saving account to receive arisk-free interest.
(b) buy a $100K house and then sell it on 10 April 2011.
Which choice should Mr King take?
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Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
Assume the annual interest rate r = 1% and let X denote theprice of the house on 10 April 2011. Consider cases:
(i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5. That is, theprice will increase or decrease by 10% equally likely.In this case, EX = $100K so it is better to deposit themoney in the bank which gives Mr King $101K by 10 April2011.
(ii) P(X = $110K ) = 0.6 and P(X = $90K ) = 0.4.In this case, EX = 0.6× $110K + 0.4× $90K = $102K ,which is $1K better than the return of the saving account.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
Assume the annual interest rate r = 1% and let X denote theprice of the house on 10 April 2011. Consider cases:
(i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5. That is, theprice will increase or decrease by 10% equally likely.In this case, EX = $100K so it is better to deposit themoney in the bank which gives Mr King $101K by 10 April2011.
(ii) P(X = $110K ) = 0.6 and P(X = $90K ) = 0.4.In this case, EX = 0.6× $110K + 0.4× $90K = $102K ,which is $1K better than the return of the saving account.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
Assume the annual interest rate r = 1% and let X denote theprice of the house on 10 April 2011. Consider cases:
(i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5. That is, theprice will increase or decrease by 10% equally likely.In this case, EX = $100K so it is better to deposit themoney in the bank which gives Mr King $101K by 10 April2011.
(ii) P(X = $110K ) = 0.6 and P(X = $90K ) = 0.4.In this case, EX = 0.6× $110K + 0.4× $90K = $102K ,which is $1K better than the return of the saving account.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
Assume the annual interest rate r = 1% and let X denote theprice of the house on 10 April 2011. Consider cases:
(i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5. That is, theprice will increase or decrease by 10% equally likely.In this case, EX = $100K so it is better to deposit themoney in the bank which gives Mr King $101K by 10 April2011.
(ii) P(X = $110K ) = 0.6 and P(X = $90K ) = 0.4.In this case, EX = 0.6× $110K + 0.4× $90K = $102K ,which is $1K better than the return of the saving account.
Xuerong Mao FRSE SM in Finance
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Assume that you trust the housing market will obey Case (ii).Should you have $100K available, you would have invested itinto the house to obtain the expected profit of $2K.
The problem is that you do NOT have the capital of $100K andyou just feel unfair to give the opportunity to rich people like MrKing.
However, Professor Mao would like to help. On 10 April 2010,Professor Mao (the writer) writes a European call option thatgives you (the holder) the right to buy 1 house for $100K on 10April 2011 from Prof Mao if you wish.
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European call option
DefinitionA European call option gives its holder the right (but not theobligation) to purchase from the writer a prescribed asset for aprescribed price at a prescribed time in the future.
The prescribed purchase price is know as the exercise price orstrike price, and the prescribed time in the future is known asthe expiry date.
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On 10 April 2011 you would then take one of two actions:
(a) if the actual value of a house turns out to be $110K youwould exercise your right to buy 1 house from ProfessorMao at the cost $100 and immediately sell it for $110Kgiving you a profit of $10K.
(b) if the actual value of a house turns out to be $90K youwould not exercise your right to buy the house fromProfessor Mao—the deal is not worthwhile.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
On 10 April 2011 you would then take one of two actions:
(a) if the actual value of a house turns out to be $110K youwould exercise your right to buy 1 house from ProfessorMao at the cost $100 and immediately sell it for $110Kgiving you a profit of $10K.
(b) if the actual value of a house turns out to be $90K youwould not exercise your right to buy the house fromProfessor Mao—the deal is not worthwhile.
Xuerong Mao FRSE SM in Finance
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OptionOption value
Note that because you are not obliged to purchase the house,you do not lose money. Indeed, in case (a) you gain $10K whilein case (b) you neither gain nor lose.
Professor Mao on the other hand will not gain any money on 10April 2011 and may lose an unlimited amount.
To compensate for this imbalance, when the option is agreedon 10 April 2010 you would be expected to pay Professor Maoan amount of money to buy the "right". (The fair amount isknown as the value of the option.)
Question: Should Professor Mao charge you $2K, do you wantto sign the option?
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
Let C denote the payoff of the option on 10/04/2011. Then
C =
{$10K if X = $110K ;$0 if X = $90K .
Recalling the probability distribution of X
P(X = $110K ) = 0.6, P(X = $90K ) = 0.4.
we obtain the expected payoff
EC = 0.6× $10K + 0.4× $0 = $6K .
But $1K saved in a bank for a year will only grow to
(1 + 1%)× $1K = $1.01K .
Therefore, the option produces the expected profit
$6K − $1.01 = $4.99K .
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
It is significant to compare your profit with Mr King’s.
Mr King invests his $100K in the house and expects to make$1K more profit than saving his money in a bank.
You pay only $2K for the option and expect to make $4.99Kmore profit than saving your $2K in a bank.
It is even more significant to observe that you only need $2K,rather than $100K, in order to get into the market.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
However, should Professor Mao charge you $5.99K, doyou want to sign for the option?If you save your $5.99K in a bank, you will have
(1 + 1%)× $5.99K = $6.05K
which is $50 better off than EC = $6K , the expected payoffof the option. You should therefore not sign the option.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
However, should Professor Mao charge you $5.99K, doyou want to sign for the option?If you save your $5.99K in a bank, you will have
(1 + 1%)× $5.99K = $6.05K
which is $50 better off than EC = $6K , the expected payoffof the option. You should therefore not sign the option.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
OptionOption value
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
Key question
How much should the holder pay for the privilege of holding theoption? In other words, how do we compute a fair price for thevalue of the option?
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
In the simple problem discussed above, the fair price of theoption is
EC1 + r
=$6K
1 + 1%= $5940.59
However, the idea can be developed to cope with morecomplicated distribution.
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Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
Example
Assume that the house price will increase by 5% per half a yearwith probability 60% but decrease by 4% per half a year withprobability 40%. Then the house price X on 2011 will have theprobability distribution:
X (in K$) | 92.16 100.80 110.25P | 0.16 0.48 0.36
Hence
EC = 0.48× 0.80K + 0.36× $10.25K = $4.074K
and the option value is
EC1 + r
=$4.074K1 + 1%
= $4.034K
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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OptionOption value
Example
Assume that the house price will increase by 3% per quarterwith probability 60% but decrease by 2% per quarter withprobability 40%. Then the house price X on 10/04/2011 willhave the probability distribution:
X (in K$) |92.237 96.943 101.889 107.087 112.551P |0.0256 0.1536 0.3456 0.3456 0.1296
Hence
EC = 0.3456×1.889+0.3456×7.087+0.1296×12.551 = 4.729
and the option value is
EC1 + r
=4.729
1 + 1%= 4.682
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The model discussed before is the well-knownCox–Ross–Rubinstein (CRR) binomial model. This model canbe simulated easily by R.
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> n=4 # number of periods > up=0.03 # increase percentage > dw=0.02 # decrease percentage > upno <- 0:n > p0=100 # initial house price > pT <-p0*(1+up)^upno*(1-dw)^(n-upno) #prices at expiry date > pT [1] 92.23682 96.94278 101.88884 107.08725 112.55088 > upprob <- 0.6 # prob of increase > prob <- dbinom(0:n,n,upprob) \#binomial distribution > prob [1] 0.0256 0.1536 0.3456 0.3456 0.1296 > K=100 #strike price > payoff <-numeric() > {for (i in 1:(n+1)) + if (pT[i]>K) payoff[i]=pT[i]-K else payoff[i]=0} > payoff [1] 0.000000 0.000000 1.888836 7.087246 12.550881 > meanpayoff <- sum(payoff*prob) > meanpayoff [1] 4.728728 > r =0.01 # riskfree interest rate > optionvalue <- meanpayoff/(1+r) > optionvalue [1] 4.681909
x
x
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OptionOption value
R-simulation for the12-month CRR binomial model > n=12 > up=0.01 > dw=0.009 > upno <-0:n > p0=100 > pT <- p0*(1+up)^upno*(1-dw)^(n-upno) > upprob= 0.6 > prob<-dbinom(0:n,n,upprob) > K=100 > payoff <- numeric() > {for (i in 1:(n+1)) + if (pT[i]>K) payoff[i]=pT[i]-K else payoff[i]=0} > meanpayoff <-sum(payoff*prob) > r=0.01 > optionvalue <- meanpayoff/(1+r) > optionvalue [1] 3.238385
x
x
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R-simulation for 365-day CRR model:
CRR(365, 0.00028, 0.00028, 100, 0.6, 100, 0.01)
produces the option value $2.044K
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However, the housing price, or more generally, an asset price ismuch more complicated than the binomial distributionsassumed above.
How might we model an asset price?
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Now suppose that at time t the underlying asset price is x(t).Let us consider a small subsequent time interval dt , duringwhich x(t) changes to x(t) + dx(t). (We use the notation d · forthe small change in any quantity over this time interval when weintend to consider it as an infinitesimal change.) By definition,the intrinsic growth rate at t is dx(t)/x(t). How might we modelthis rate?
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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Linear modellingNonlinear modelling
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
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If, given x(t) at time t , the rate of change is deterministic, say r ,then
dx(t)x(t)
= rdt .
This gives the ordinary differential equation (ODE)
dx(t)dt
= rx(t).
Thenx(t) = x(0)ert
For example, if you invest x(0) into a bond with the risk-freeinterest rate r , then your return (capital plus interest) by time tis x(t).
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However the rate of change is in general not deterministic as itis often subjective to many factors and uncertainties e.g.system uncertainty, environmental disturbances. To model theuncertainty, we may decompose
dx(t)x(t)
= deterministic change + random change.
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The deterministic change may be modeled by
µdt
where µ is the average rate of change. So
dx(t)x(t)
= µdt + random change.
How may we model the random change?
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In general, the random change is affected by many factorsindependently. By the well-known central limit theorem thischange can be represented by a normal distribution with meanzero and and variance σ2dt , namely
random change = N(0, σ2dt) = σ N(0,dt),
where σ is the standard deviation of the rate of change , andN(0,dt) is a normal distribution with mean zero and andvariance dt . Hence
dx(t)x(t)
= µdt + σN(0,dt).
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A convenient way to model N(0,dt) as a process is to use theBrownian motion B(t) (t ≥ 0) which has the followingproperties:
B(0) = 0,dB(t) = B(t + dt)− B(t) is independent of B(t),dB(t) follows N(0,dt).
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The stochastic model can therefore be written as
dx(t)x(t)
= µdt + σdB(t),
or
dx(t) = µx(t)dt + σx(t)dB(t)
which is a linear stochastic differential equation (SDE)—theNobel prize winning Black–Scholes model.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Linear modellingNonlinear modelling
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
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If the rate of change and the standard deviation depend on x(t)at time t , the the model become nonlinear. In this case, thedeterministic change may be modeled by
R̄dt = R̄(x(t), t)dt
where R̄ = r̄(x(t), t) is the average rate of change given x(t) attime t , while
random change = N(0,V 2dt) = V N(0,dt) = VdB(t),
where V = V (x(t), t) is the standard deviation of the rate ofchange given x(t) at time t .
Xuerong Mao FRSE SM in Finance
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The stochastic model can therefore be written as
dx(t)x(t)
= R̄(x(t), t)dt + V (x(t), t)dB(t),
or
dx(t) = R̄(x(t), t)x(t)dt + V (x(t), t)x(t)dB(t)
which is a nonlinear stochastic differential equation (SDE).
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
The Black–Scholes worldNon-linear SDE models
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
The Black–Scholes worldNon-linear SDE models
The Nobel prize winning Black–Scholes model
dx(t) = µx(t)dt + σx(t)dB(t)
is also known as the geometric Brownian motion.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
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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.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
The Black–Scholes worldNon-linear SDE models
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.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
The Black–Scholes worldNon-linear SDE models
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.
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Regardless whatever the growth rate µ the individual holdermay think, the fair option value should be priced based on thefollowing SDE
dx(u) = rx(u)du + σx(u)dB(u), t ≤ u ≤ T , x(t) = S,
where r is the risk-free interest rate, rather than the individualSDE
dy(u) = µy(u)du + σy(u)dB(u), t ≤ u ≤ T , y(t) = S
that the holder may think. Hence the expected payoff at theexpiry date T is
E(x(T )− K )+
Discounting this expected value in future gives
C(S, t) = e−r(T−t)E[max(x(T )− K ,0)].
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The solution
x(T ) = S exp[(r − 1
2σ2)(T − t) + σ(B(T )− B(t))
]gives
log(x(T )) = log(S)+(
r−12σ2)
(T−t)+σ(B(T )−B(t)) ∼ N(µ̂, σ̂2),
where
µ̂ = log(S) +(
r − 12σ2)
(T − t), σ̂ = σ√
T − t .
HenceZ :=
log(x(T ))− µ̂σ̂
∼ N(0,1)
which givesx(T ) = eµ̂+σ̂Z .
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TheoremThe explicit BS formula for the value of the European call optionis
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
∫ x
−∞e−
12 z2
dz,
while
d1 =log(S/K ) + (r + 1
2σ2)(T − t)
σ√
T − tand
d2 =log(S/K ) + (r − 1
2σ2)(T − t)
σ√
T − t.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
The Black–Scholes worldNon-linear SDE models
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
The Black–Scholes worldNon-linear SDE models
Square root process
IfR̄(x(t), t) = µ, V (x(t), t) =
σ√x(t)
,
then the SDE becomes the well-known square root process
dx(t) = µx(t)dt + σ√
x(t)dB(t).
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Mean-reverting square root process
IfR̄(x(t), t) =
α(µ− x(t))
x(t), V (x(t), t) =
σ√x(t)
,
then the SDE becomes
dx(t) = α(µ− x(t))dt + σ√
x(t)dB(t).
This is the mean-reverting square root process.
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Theta process
IfR̄(x(t), t) = µ, V (x(t), t) = σ(x(t))θ−1,
then the SDE becomes
dx(t) = µx(t)dt + σ(x(t))θdB(t),
which is known as the theta process.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Most of SDEs used in practice do not have explicit solutions.Monte Carlo simulations have been widely used to simulate thesolutions of nonlinear SDEs. There are two main motivationsfor such simulations:
using a Monte Carlo approach to compute the expectedvalue of a function of the underlying underlying quantity, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.
Question: Can we trust the Monte Carlo simulations?
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Most of SDEs used in practice do not have explicit solutions.Monte Carlo simulations have been widely used to simulate thesolutions of nonlinear SDEs. There are two main motivationsfor such simulations:
using a Monte Carlo approach to compute the expectedvalue of a function of the underlying underlying quantity, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.
Question: Can we trust the Monte Carlo simulations?
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Most of SDEs used in practice do not have explicit solutions.Monte Carlo simulations have been widely used to simulate thesolutions of nonlinear SDEs. There are two main motivationsfor such simulations:
using a Monte Carlo approach to compute the expectedvalue of a function of the underlying underlying quantity, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.
Question: Can we trust the Monte Carlo simulations?
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Solution of a linear SDEThe linear SDE
dX (t) = 2X (t)dt + X (t)dB(t), X (0) = 1
has the explicit solution
x(t) = exp(1.5t + B(t)).
The Monte Carlo simulation can be carried out based on theEuler-Maruyama (EM) method
x(0) = 1, x(i + 1) = x(i)[1 + 2∆ + ∆Bi ], i ≥ 0,
where ∆Bi = B((i + 1)∆)− B(i∆) ∼ N(0,∆).
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
0.0 0.5 1.0 1.5 2.0
24
68
1012
t
X(t)
or x
(t)
true solnEM soln
0.0 0.5 1.0 1.5 2.0
05
1015
2025
t
X(t)
or x
(t)
true solnEM soln
0.0 0.5 1.0 1.5 2.0
020
6010
014
0
t
X(t)
or x
(t)
true solnEM soln
0.0 0.5 1.0 1.5 2.0
24
68
10
t
X(t)
or x
(t)
true solnEM soln
x
x
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Example - the Black-Scholes model
Consider a BS model
dS(t) = 0.05S(t)dt + 0.03S(t)dB(t), S(0) = 10
and a European call option with the exercise price K = 10.05 atexpiry time T = 1, where 0.05 is the risk-free interest rate and0.03 is the volatility. By the well-known Black-Scholes formulaon the option, we can compute the value of a European calloption at time zero is
C = 0.4487318.
On the other hand, we can let ∆ = 0.001, simulate 1000 pathsof the SDE, compute the mean payoff at T = 1, discounting itby e−0.05, we get the estimated option value
C∆ = 0.4454196
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
To be more reliable, we can carry out such simulation, say 10times, to get 10 estimated values:
0.4454196, 0.4611569, 0.4512847, 0.4490462, 0.4294038,
0.4618921, 0.4556195, 0.4559547, 0.4399189, 0.4489594.
Their mean valueC̄∆ = 0.4498656
gives a better estimation for C.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
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 .
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
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).
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Continuous-time EM approximation
s(t) := s0 + r∫ t
0s̄(u))du + σ
∫ t
0
√|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.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
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
E(
sup0≤t≤T
|s(t)−S(t)|2)
= lim∆→0
E(
sup0≤t≤T
|s̄(t)−S(t)|2)
= 0.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
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
E(
sup0≤t≤T
|s(t)−S(t)|2)
= lim∆→0
E(
sup0≤t≤T
|s̄(t)−S(t)|2)
= 0.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Outline1 Option and Probability
OptionOption value
2 Stochastic ModellingLinear modellingNonlinear modelling
3 Well-known ModelsThe Black–Scholes worldNon-linear SDE models
4 Monte Carlo SimulationsWhy Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Bond
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
|sn|
), where N = T/∆.
Theorem
lim∆→0|β − β∆| = 0.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
Bond
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
|sn|
), where N = T/∆.
Theorem
lim∆→0|β − β∆| = 0.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
European call option
A European call option with the exercise price K at expiry timeT pays S(T )− K if S(T ) > K otherwise 0.
TheoremLet r be the risk-free interest rate and define
C = e−rT E[(S(T )− K )+
],
C∆ = e−rT E[(s̄(T )− K )+
].
Thenlim
∆→0|C − C∆| = 0.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
European call option
A European call option with the exercise price K at expiry timeT pays S(T )− K if S(T ) > K otherwise 0.
TheoremLet r be the risk-free interest rate and define
C = e−rT E[(S(T )− K )+
],
C∆ = e−rT E[(s̄(T )− K )+
].
Thenlim
∆→0|C − C∆| = 0.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
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.
TheoremDefine
V = E[(S(T )− K )+I{0≤S(t)≤c, 0≤t≤T}
],
V∆ = E[(s̄(T )− K )+I{0≤s̄(t)≤c, 0≤t≤T}
].
Thenlim
∆→0|V − V∆| = 0.
Xuerong Mao FRSE SM in Finance
Option and ProbabilityStochastic ModellingWell-known Models
Monte Carlo Simulations
Why Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities
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.
TheoremDefine
V = E[(S(T )− K )+I{0≤S(t)≤c, 0≤t≤T}
],
V∆ = E[(s̄(T )− K )+I{0≤s̄(t)≤c, 0≤t≤T}
].
Thenlim
∆→0|V − V∆| = 0.
Xuerong Mao FRSE SM in Finance