![The price of an Asian option by MCS
An Asian option (call) has discounted payoff:
Y = e-rT
[S - K]+
S =
1
m
S(ti )
i=1
m
å For fixed dates 0<t1<…<tm<T
Since for Monte Carlo simulation we describe the risk-neutral dynamics of the stock price, we
need to use the stochastic differential equation for modeling the price movement of the
underlying asset
S(T) = S(0)exp([r -
1
2
s 2
]T +sW(T))
Where W(T) is the random variable,
normally distributed with mean 0 and
variance T.](https://image.slidesharecdn.com/assignment15borisova-150306084810-conversion-gate01/85/The-Delta-Of-An-Arithmetic-Asian-Option-Via-The-Pathwise-Method-4-320.jpg)
![S(T) = S(0)exp([r -
1
2
s 2
]T +sW(T))
Monte Carlo simulation of the
lognormal asset price movement
Model:
S = 100, r = 0.045, σ=15%,
trials = 100000
For fixed dates 0<t1<…<tm<T](https://image.slidesharecdn.com/assignment15borisova-150306084810-conversion-gate01/85/The-Delta-Of-An-Arithmetic-Asian-Option-Via-The-Pathwise-Method-5-320.jpg)
![Option value estimation and its
confidence level
The sample standard deviation
sC =
1
n-1
(Yi - ˆYn )2
i=1
n
å
1-δ quantile of the standard normal distribution zdConfidence interval:
ˆYn ± zd/2
sC
n
ˆYn =
1
n
Yi
1
n
å
E[ ˆYn ]= Y
ˆYn -Y
sC / n
Þ N(0,1)
The estimation of the option value is unbiased
As number of replications increases, the standardized estimator converges in
distribution to the standard normal](https://image.slidesharecdn.com/assignment15borisova-150306084810-conversion-gate01/85/The-Delta-Of-An-Arithmetic-Asian-Option-Via-The-Pathwise-Method-10-320.jpg)
Uploaded byAnna Borisova
The Delta Of An Arithmetic Asian Option Via The Pathwise Method
The document discusses the computation of the delta of an arithmetic Asian option using Monte Carlo simulation and the pathwise method. It includes a detailed mathematical model, MATLAB code for simulations, and the estimation of option values and confidence intervals. The pathwise estimator is highlighted for its unbiased nature and efficiency in reducing variance and computational time compared to finite-difference methods.
The Delta Of An Arithmetic Asian Option Via The Pathwise Method
- 1. THE DELTA OFAN ARITHMETIC ASIAN OPTION VIA THE PATHWISE METHOD Anna Borisova University of Bocconi 1/12/2014
- 2. Assignment 1. Compute theMonte Carlo simulation for the price of an Asian option on a lognormal asset (with descrete monitoring at dates t1, t2, … , tM); 2. Provide the pathwise estimate of the Delta of these options.
- 3. Structure ❏ Mathematical model; ❏The code features; ❏ Output.
- 4. The price ofan Asian option by MCS An Asian option (call) has discounted payoff: Y = e-rT [S - K]+ S = 1 m S(ti ) i=1 m å For fixed dates 0<t1<…<tm<T Since for Monte Carlo simulation we describe the risk-neutral dynamics of the stock price, we need to use the stochastic differential equation for modeling the price movement of the underlying asset S(T) = S(0)exp([r - 1 2 s 2 ]T +sW(T)) Where W(T) is the random variable, normally distributed with mean 0 and variance T.
- 5. S(T) = S(0)exp([r- 1 2 s 2 ]T +sW(T)) Monte Carlo simulation of the lognormal asset price movement Model: S = 100, r = 0.045, σ=15%, trials = 100000 For fixed dates 0<t1<…<tm<T
- 6. Constructing the Brownianmotion The properties of Brownian motion: • W(0) = 0; • The increments {W(t1)-W(t0), W(t2)-W(t1), … , W(tk)-W(tk-1)} are independent; • W(t) – W(s) is normally distributed N(0, t-s) for any 0 ≤ s < t ≤ T. S(t) = S(0)exp([r -0.5s 2 )t + ts Zi i=0 t å
- 7. 0.5697 -1.0916 -1.3929 -0.1540 -1.7215 -0.1277 -0.9524 -1.9553 -0.5648 Numberofsimulations Number ofaveraging periods + + + + + + -0.5367 -4.7684 -2.0854 0.4157 -2.8131 -1.5206 0.5697 -1.0916 -1.3929 S(t) = S(0)exp([r -0.5s 2 )t + ts Zi i=0 t å
- 8.
- 9. Model: S = 100,r = 0.045, σ=15% 12 averaging points 100simulations5000simulations 1000simulations10000simulations
- 10. Option value estimationand its confidence level The sample standard deviation sC = 1 n-1 (Yi - ˆYn )2 i=1 n å 1-δ quantile of the standard normal distribution zdConfidence interval: ˆYn ± zd/2 sC n ˆYn = 1 n Yi 1 n å E[ ˆYn ]= Y ˆYn -Y sC / n Þ N(0,1) The estimation of the option value is unbiased As number of replications increases, the standardized estimator converges in distribution to the standard normal
- 11.
- 12. TRADE OFF: Thevalue of the option, confidence interval and computational time Number of simulations 100 1000 3000 5000 7000 10000 Value of the option 0.9306 1.3199 1.125 6 1.3533 1.3347 1.2666 1.1797 1.1634 1.2084 1.2256 1.2575 1.1560 Comput. time 0.15 0.11 0.13 0.17 0.42 0.47 0.76 0.85 1.13 1.16 1.67 2.14 95% c.l. lower bound 0.4123 0.929 8 1.2043 1.0898 1.1318 1.1917 95% c.l. upper bound 1.4490 1.321 4 1.4652 1.2696 1.2850 1.3233 99% c.l. lower bound 0.3496 1.1336 1.1454 1.0755 1.1472 1.0932 99% c.l. upper bound 2.2901 1.5730 1.3878 1.2512 1.3040 1.2188
- 13. The pathwise estimatorof the option delta This estimator has great practical value • This estimator is unbiased; • mean(S) is simulated in estimating the price of the option also, so finding the delta requires just a little additional effort; • This method reduces variance and computing time compared to finite-difference. dY dS(0) = dY dS dS dS(0) = e-rT 1{S > K} S S(0) dS dS(0) = 1 m dS(ti ) dS(0) = 1 m S(ti ) S(0) = i=1 m å i=1 m å S S(0) dY dS(0) = e-rT 1{S > K} S S(0)
- 14.
- 15. Model: Asian call option S(0)= from 0 to 200 with step 1; r = 4,5%; sigma = 1; K=50; T = 1 with 24 averaging points (two times a month); Trials = 10000.
- 16.
- 17. References: • Glasserman “MonteCarlo Methods in Financial Engineering” • Mark Broadie, Paul Glasserman “Estimating Security Price Derivatives Using Simulations” • John C. Hull “Options, futures and other derivatives” • Huu Tue Huynh, Van Son Lai, Issouf Sourmare “Stochastic Simulation and Applications in Finance with MATLAB® Programs”