sims by wangnuanzg

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```									Options, Dynamic Strategies
and Simulations

FIN285a: Lecture 5.4-5.5
Fall 2010
Jorion 6.1-6.2 (6.3 skim),6.4, 6.5
Monte-carlo Simulations
Examples
 Harder problems
 Simulations necessary
Applications
 Simple  option portfolio
 Exotic options (Asian)
   Asian options
   Barrier options
Code
 putdow.m
 asianopt.m
 barrier.m
 pairstrat.m
Simple Option Example
 100 shares of Dow
 Current price = 1
 Cover 75 shares with put options
   European
   Expiration in 1 day
 Strike   price = 1 (at the money)
Two Big Assumptions
 Option   value today
   Assume Black/Scholes formula
 Value   option tomorrow as
   Price increase : Zero
   Price decrease: 1-P(tomorrow)
   Note: this is sensible because of expiration
Matlab Example
 putdow.m
 Note:
   Very nonnormal distribution
   Importance of VaR when options are
present
Applications
 Simple  option portfolio
 Exotic options (Asian)
   Asian options
   Barrier options
Exotic Option (Asian Option)
 Standard       call option
   Payoff = max(price-strike,0)
 Standard       put option
   Payoff = max(strike-price,0)
 Asian   call
   Payoff = max( mean(price)-strike,0)
 Asian   put
   Payoff = max( strike - mean(price),0)
Example (matlab)
 Asianopt.m
 European option expiring in 20 days
 Option valuation
   Risk neutral pricing (pp 153-154)
   Monte-carlo
   Use this to get today’s value
   Future value = max(strike-mean(price))
 VaR : usual calculations
 Compare : plain and Asian option
Risk Neutral Note
Pt 1  Pt
r                       Pt 1 / Pt  1
Pt
R  log(Pt 1 )  log(Pt )  log(Pt 1 / Pt )
R ~ N( ,  2 )
   2 /2
E(e )  e
R

E(r)  E(e R )  1
   2 /2
E(r)  e                1
2                 2
E(r)  1                  1  
2           2
Desired (Risk neutral) mean E(r)  rf (risk free)
2
  rf 
2
Why Asian Option?
 Firm with continuous cash flows over
the period
 Multinational:
   Ford get BP inflows over month
   Want to cushion average exchange rate,
not end of period exchange rate
Barrier Option
 Option depends on price path again
 Expires or comes to life depending on max or
min of recent path
 Down and out
   Option expires when price hits lower barrier from
above
 Down and in
 Up and out
 Up and in
Barrier Option
 Down  and out put option
 barrier.m
 Similar to Asian example
Why Use Barrier?
 Exporter  to Japan
 Hedge \$/Yen fx rate risk with a put
option
 If \$/Yen rate drops below a certain level,
then firm closes out export operation -
don’t need option
 Use a down and out put for this
Applications
 Simple  option portfolio
 Exotic options (Asian)
   Asian options
   Barrier options
 Market neutral strategy
 Implementation
   Two stocks that ought to move together
   Short high priced one
   Long low priced one
   Wait
   When (and if) they converge clear out position and
take profits
Example
 Simulated   price series
 Two stocks with common factor
 Prices driven by common factor
(industry) and own factors
 Own factors eventually go away, and
prices should never get too far apart
 pairsgen.m, pairsplot.m
Implementing the Strategy
   p1 and p2 = prices
   p1 starts at 100, p2 at 100+spread
 Initially long \$100 of p1, and sell \$100 of p2
short
 (Ignore margins, dividends, and transaction
costs)
 Otherwise clear position after maximum days
VaR versus Liquidity VaR
 Simple discounted end of period value is VaR
 What if there are intermediate cash flows?
   Margin calls on shorts
 For simplicity: model like a futures
 Mark to market each day
   (See picture: next slide)
   Futures would require you to have available
this much cash
Marking to Market

price
110
2
105

100

95
1
90

time
Example
 pairstrat.m
 Note:
   Great strategy in terms of VaR
   However, may require large amounts of
cash before it becomes profitable
 VaR  is not the whole story here
 Try changing parameters

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