Financial Options

1
Financial Options Option Valuation

• Session 4 Binomial Model Black Scholes
• CORP FINC 5880 - Spring 2014 Shanghai

2
What determines option value?

• Stock Price (S)
• Exercise Price (Strike Price) (X)
• Volatility (s)
• Time to expiration (T)
• Interest rates (Rf)
• Dividend Payouts (D)

3
Try to guestimatefor a call option price (5 min)
Stock Price ? Then call premium will?
Exercise Price ? Then..?
Volatility ? Then..?
Time to expiration? Then..?
Interest rate ? Then..?
Dividend payout ? Then..?
4
Answer Try to guestimatefor a call option price
(5 min)
Stock Price ? Then call premium will? Go up
Exercise Price ? Then..? Go down.
Volatility ? Then..? Go up.
Time to expiration? Then..? Go up.
Interest rate ? Then..? Go up.
Dividend payout ? Then..? Go down.
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Binomial Option Pricing

• Assume a stock price can only take two possible
  values at expiration
• Up (u2) or down (d0.5)
• Suppose the stock now sells at 100 so at
  expiration u200 d50
• If we buy a call with strike 125 on this stock
  this call option also has only two possible
  results
• up75 or down 0
• Replication means
• Compare this to buying 1 share and borrow 46.30
  at Rf8
• The pay off of this are

Strategy Today CF Future CF if StgtX (200) Future CF if STltX(50)
Buy Stock -100 200 50
Write 2 Calls 2C - 150 0
Borrow PV(50) 50/1.08 - 50 - 50
TOTAL 2C-53.70(0) 0 (fair game) 0 (fair game)
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Binomial model

• Key to this analysis is the creation of a perfect
  hedge
• The hedge ratio for a two state option like this
  is
• H (Cu-Cd)/(Su-Sd)(75-0)/(200-50)0.5
• Portfolio with 0.5 shares and 1 written option
  (strike 125) will have a pay off of 25 with
  certainty.
• So now solve
• Hedged portfolio valuepresent value certain pay
  off
• 0.5shares-1call (written) 23.15
• With the value of 1 share 100
• 50-1call23.15 so 1 call26.85

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What if the option is overpriced? Say 30 instead
of 26.85

• Then you can make arbitrage profits
• Risk free 6.80no matter what happens to share
  price!

Cash flow At S50 At S200
Write 2 options 60 0 -150
Buy 1 share -100 50 200
Borrow 40 at 8 40 -43.20 -43.20
Pay off 0 6.80 6.80
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Class assignment What if the option is
under-priced? Say 25 instead of 26.85 (5 min)

• Then you can make arbitrage profits
• Risk free no matter what happens to share price!

Cash flow At S50 At S200
.2 options ? ? ?
.. 1 share ? ? ?
Borrow/Lend ? at 8 ? ? ?
Pay off ? ? ?
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Breaking Up in smaller periods

• Lets say a stock can go up/down every half year
  if up 10 if down -5
• If you invest 100 today
• After half year it is u1110 or d195
• After the next half year we can now have
• U1u2121 u1d2104.50 d1u2 104.50 or
  d1d290.25
• We are creating a distribution of possible
  outcomes with 104.50 more probable than 121 or
  90.25.

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Class assignment Binomial model(5 min)

• If up5 and down-3 calculate how many
  outcomes there can be if we invest 3 periods (two
  outcomes only per period) starting with 100.
• Give the probability for each outcome
• Imagine we would do this for 365 (daily)
  outcomeswhat kind of output would you get?
• What kind of statistical distribution evolves?

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Black-Scholes Option Valuation

• Assuming that the risk free rate stays the same
  over the life of the option
• Assuming that the volatility of the underlying
  asset stays the same over the life of the option
  s
• Assuming Option held to maturity(European style
  option)

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Without doing the math

• Black-Scholes value call
• Current stock priceprobability present value
  of strike priceprobability
• Note that if dividend0 that
• CoSo-Xe-rtN(d2)The adjusted intrinsic value
  So-PV(X)

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Class assignment Black Scholes

• Assume the BS option model
• Call Se-dt(N(d1))-Xe-rt(N(d2))
• d1(ln(S/X)(r-ds2/2)t)/ (svt)
• d2d1- svt
• If you use EXCEL for N(d1) and N(d2) use
  NORMSDIST function!
• stock price (S) 100
• Strike price (X) 95
• Rf ( r)10
• Dividend yield (d)0
• Time to expiration (t) 1 quarter of a year
• Standard deviation 0.50
• A)Calculate the theoretical value of a call
  option with strike price 95 maturity 0.25 year
• B) if the volatility increases to 0.60 what
  happens to the value of the call? (calculate it)

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In Excel
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Homework assignment 9 Black Scholes

• Calculate the theoretical value of a call option
  for your company using BS
• Now compare the market value of that option
• How big is the difference?
• How can that difference be explained?

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Implied Volatility

• If we assume the market value is correct we set
  the BS calculation equal to the market price
  leaving open the volatility
• The volatility included in todays market price
  for the option is the so called implied
  volatility
• Excel can help us to find the volatility (sigma)

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Implied Volatility

• Consider one option series of your company in
  which there is enough volume trading
• Use the BS model to calculate the implied
  volatility (leave sigma open and calculate back)
• Set the price of the option at the current market
  level

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Implied Volatility Index - VIX
Investor fear gauge
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Class assignmentBlack Scholes put option
valuation (10 min)

• P Xe-rt(1-N(d2))-Se-dt(1-N(d1))
• Say strike price95
• Stock price 100
• Rf10
• T one quarter
• Dividend yield0
• A) Calculate the put value with BS? (use the
  normal distribution in your book pp 516-517)
• B) Show that if you use the call-put parity
• PCPV(X)-S where PV(X) Xe-rt and C 13.70 and
  that the value of the put is the same!

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The put-call parity

• Relates prices of put and call options according
  to
• PC-So PV(X) PV(dividends)
• X strike price of both call and put option
• PV(X) present value of the claim to X dollars
  to be paid at expiration of the options
• Buy a call and write a put with same strike
  pricethen set the Present Value of the pay off
  equal to C-P

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The put-call parity

• Assume
• S Selling Price
• P Price of Put Option
• C Price of Call Option
• X strike price
• R risk less rate
• T Time then Xe-rt NPV of realizable risk less
  share price (P and C converge)
• SP-C Xe-rt
• So P C (Xe-rt - S) is the relationship
  between the price of the Put and the price of the
  Call

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Class AssignmentTesting Put-Call Parity

• Consider the following data for a stock
• Stock price 110
• Call price (t0.5 X105) 14
• Put price (t0.5 X105) 5
• Risk free rate 5 (continuously compounded rate)
• 1) Are these prices for the options violating the
  parity rule? Calculate!
• 2) If violated how could you create an arbitrage
  opportunity out of this?

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Black Scholes

• The Black-Scholes model is used to calculate a
  theoretical call price (ignoring dividends paid
  during the life of the option) using the five key
  determinants of an option's price stock price,
  strike price, volatility, time to expiration, and
  short-term (risk free) interest rate.

Myron Scholes and Fischer Black
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Some spreadsheets will show you the option Greeks

• Delta (d) Measures how much the premium changes
  if the underlying share price rises with 1.-
  (positive for Call options and negative for Put
  options)
• Gamma (?) Measures how sensitive delta is for
  changes in the underlying asset price (important
  for risk managers)
• Vega (?) Measures how much the premium changes
  if the volatility rises with 1 higher
  volatility usually means higher option premia
• Theta (?) Measrures how much the premium falls
  when the option draws one day closer to expiry
• Rho (?) Measrures how much the premium changes
  if the riskless rate rises with 1 (positive for
  call options and negative for put options)

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Example

• Results Calc type Value
• Price P 0.25517 Price of the call
  option
• Delta D 0.28144 Premium changes
  with 0.28144 if share price is up 1
• Gamma G 0.21606 Sensitivity of delta
  for changes in price of share
• Vega V 0.01757 Premium will go up
  with 0.01757 if volatility is up 1
• Theta T -0.00419 1 day closer to
  expiry the premium will fall 0.00419
• Rho R 0.00597 If the risk less rate
  is up 1 the premium will increase 0.00597