Portfolio strategies

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Portfolio strategies

• Hans Dewachter

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Overview

• Ito processes and the Ito lemma
• Numerical calculation of the Greeks
• Portfolio strategies

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Ito processes and theIto lemma

• Assumptions
• Itos lemma in one dimension
• The Greeks

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Ito processes and theIto lemma

• Assumptions
• All underlying stochastic processes can be
  represented by an Ito-process
• We assume interest rates to be uncorrelated with
  underlying asset
• Restriction to the class of derivatives with
  continuous and twice differentiable functionals
  in the underlying asset.

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Ito processes and theIto lemma

• Itos lemma constant interest rate
• Let the underlying asset, X, follow an Ito
  process, dX(t) A(X, t).dt B(X,t).dW(t), and
  denote the derivative price by C(X,t)
• Then the derivative price dynamics are
• dC(X,t) Ct dt CX dX ½CXX B( ) dt
• With Ci the partial derivative of C w.r.t. i

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Ito processes and theIto lemma

• The partial derivatives are called the Greeks
• CX the Delta
• Ct the Theta
• CXX the Gamma
• Other  Greeks 
• Cvariance the Vega
• Cr the Rho

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Numerical approximation of Greeks

• How to compute numerically a partial derivative?
• Definition of partial derivative
• Limh 0 (C(Xh,t) C(X,t) ) / h
• Second partial derivative is then the derivative
  of the derivative
• Easily computed!

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Computing derivatives in practice

• Calculate the delta of a derivative C(X,t) given
  a current price of 100.
• Forward method
• Compute C at price 100 and again at 100 h.
• Take the difference and divide by h.
• So (C(100h,t)-C(100,t))/h
• How large should h be?

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Computing derivatives in practice

• Calculate the delta of a derivative C(X,t) given
  a current price of 100.
• Backward method
• Compute C at price 100 and again at 100 - h.
• Take the difference and divide by h.
• So (C(100,t) - C(100-h,t))/h
• How large should h be?

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Computing derivatives in practice

• Calculate the delta of a derivative C(X,t) given
  a current price of 100.
• Two-sided average
• Compute derivatives using forward and backward
  method and take the average.
• Take the difference and divide by 2h.
• So (C(100h,t) - C(100-h,t))/(2 h)
• How large should h be?

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Example in excel

• montecarlooptionfinal.xls

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Portfolio strategies

• Example 1 Insuring the writer of a call against
  stock price movements.
• Portfolio V(X,t) - C(X,t) gX h Bond
• Dynamics of portfolio?
• dV -Ct dt - CX dX 1/2 CXX B()2 dt g dX h.r
  dt
• Immunization dV0 implies that
• g CX
• h - ( Ct dt 1/2 CXX B()2 ) / r
• Note that the Greeks are time- and X- dependent
  and hence that portfolio composition needs to be
  adjusted

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Portfolio strategies

• Example 2 Insuring the writer of a call against
  stock price movements.
• Portfolio V(X,t) - C(X,t) gX h Bond
• Dynamics of portfolio?
• dV -Ct dt - CX dX 1/2 CXX B()2 dt g dX h.r
  dt
• Making portfolio riskless dV independent of dW
  implies that
• g CX
• Note that the Greeks are time- and X- dependent
  and hence that portfolio composition needs to be
  adjusted

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Portfolio strategies
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Portfolio strategies

• Interpretation of Gamma ?
• d(Delta)/dX Gamma
• So d (Delta) Gamma. dX
• So high gamma implies a large delta portfolio
  adjustment next period, small gamma means small
  adjustment!
• Relevant for replication with transaction costs!!!

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Portfolio strategies

• Producing  click funds 
• Investment period five years. For every 1
  invested there is the guarrantee that
• When stock markets go down you get the money back
• Else you get as profit half of the increase of
  the stock market (without dividends).
• Construct this porfolio (using bonds and call
  options). Does the financial institution gain on
  this contract or not?