A Case Study in Financial Engineering

1
A Case Study in Financial Engineering

• SAMO2002
• Masato Koda
• University of Tsukuba
• koda_at_sk.tsukuba.ac.jp

2
Plan of Talk

• Greeks in European Options
• Greeks in Asian Options
• Exotic Options
• Numerical Examples
• Conclusions
• References

3
Greeks in European Options

• European call option Exercise at final time T.
  Purchaser has right to buy underlying security
  option with X.

4
Greeks in European Options

• Use Smart Monte Carlo
• Variance reduction
• Control variate techniques
• Quasi-random numbers
• Low-discrepancy sequences
• Malliavin Calculus
• etc.

5
Greeks in European Options

• Greeks Normal expressions
• A discounted payoff function,
• Delta, ?
• Vega, ?
• Gamma, ?

6
Greeks in European Options

• Greeks The Malliavin ibp expressions
• Normal expressions Delta
• Then, applying the following formula
• with X Y ST, we may perform the integration
  by parts to give,

7
Greeks in European Options

• And then, in order to evaluate ?0TDvSTdv we apply
  the rules of the
• stochastic derivative.
• thus,
• Then the final expression for ? reads,

8
Greeks in European Options

• Vega, ? (X ST and Y ST(WT sT )
• Gamma, ? (X ST and Y ST2 )

9
Greeks in Asian Options

• Asian Option
• Purchaser has right to buy or sell underlying
  security with X.
• Excise price depends on the average price.

BUT
10
Greeks in Asian Options

• Greek Normal Expression (?)
• Greek Malliavin Expression (?)
• (X Y ?0TStdt )

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Greeks in Asian Options

• Using the relation in Malliavin derivative,
• Now we obtain finally the following three
  formulas for Delta, ?i ( i 1, 2, 3 ).

12
Greeks in Asian Options

• Three formulas for Delta, ?i ( i 1, 2, 3 )
• with Hi (i1,2,3)
• where

13
Greeks in Asian Options

• Why does this happen?

?i ( i 1, 2, 3 )
14
Exotic Options

• Barrier options
• Option dies if it strikes the boundry price
  H.

15
Exotic Options

• Path averaged options

(average rate option) (average strike
option) (lookback option)
Malliavin computation has some limitations!
16
Numerical Examples...
17
Conclusions

• In Malliavin expressions of Greeks, we can remove
  the derivative from payoff
• function.
• In Asian Options, we obtain various different
  Malliavin expressions on Delta.
• Malliavin simulations may be numerically
  advantageous than finite difference simulations.

18
References
E. Benhamou, An application of Malliavin
calculus to continuous time Asian options,
Working Paper, London School of Economics,
2000. E. Fournie, J.M. Lasry, J. Lebuchoux, P.L.
Lions, and N. Touzi, Applications of Malliavin
calculus to Monte Carlo methods in finance,
Finance and Stochastics, Vol. 3, 391-412, 1999.
E. Fournie, J.M. Lasry, J. Lebuchoux, and P.L.
Lions, Applications of Malliavin calculus to
Monte Carlo methods in finance II, Finance
and Stochastics, Vol. 5, 201-236, 2001. M. Koda,
A. Kohatsu-Higa, and M. Montero, An application
of stochastic sensitivity analysis to
financial engineering, Discussion Paper
Series, No. 988, March 2002.