The BlackScholes Model

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Derivative Securities

The Black-Scholes Model
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The Black-Scholes Model

• Stocks trade continuously during the day how to
  price them at tltT using the Binomial Model and
  when/how to adjust the hedge (frequency? T?)
• The Black-Scholes Model for continuous options
  pricing (Fischer Black and Myron Scholes The
  1997 Nobel Prize in Economics)
• This model is like a huge number of the Binomial
  models over infinitesimally small periods.
• When these periods are small enough in limit
  (n??) they can approximate continuous trading
  (this is like discrete mp3 files!).

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The Black-Scholes Model

• European call
• This can be related to the continuous Binomial
  Model

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The Black-Scholes Model

• Suppose S100, K100, u1.02, d0.98, n100,
  r1.001, T1 (year), s?
• Pt(T1)price of a zero-coupon bond that pays 1
  in 1 year. From the continuously compounded r

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The Black-Scholes Model

• Now, lets price this option at t0 d0.61525
• C(100,100,1,0)100?(d)-0.90529x100?(d-0.198)
• 100?(0.61525)-90.529?(0.403525)13.18
• NB ?(0.61525)Prob(xlt 0.61525)0.50.2263

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The Black-Scholes Model

• To find the volatility (s) can be problematic
• The one we used is the volatility that market
  expects assuming that the B-S formula is a
  correct option pricing model implied volatility
• Alternatively, we could have found the variance
  of the past log price changes and take the square
  root.
• We can hedge one call by selling h units of the
  stock. For the B-S formula
• ht?(d)0.7263

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The Black-Scholes Model

• To summarize the determinants of ct are St,
  Pt(T), K, T-t, s.
• The Greeks of the B-S formula
• 1. Delta (d) the exposure of ct with respect to
  St.
• Ignoring the approximation error
• This is a good hedge for small stock changes.

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The Black-Scholes Model

• 2. Vega (not a Greek letter sometimes called
  kappa - k) the exposure of ct with respect to
  s. Ignoring the approximation error
• 3. Rho (r) the exposure of ct with respect to
  r.

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The Black-Scholes Model

• 4. Theta (?) the exposure of ct with respect to
  (T-t). Ignoring the approximation error
• 5. (No letter) the exposure of ct with respect
  to K

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The Black-Scholes Model

• European put
• From the put-call parity formula we have

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Hedging with Options

• Suppose S100, K100, s19.8, Pt(T1)0.90529
  p(100,100,1,0)3.70238
• The Greeks for the put
• Put dCall d - 1
• Put vegaCall vega
• Put rCall r (T-t)K x Pt(T)
• Put qCall q r x K x Pt(T)
• ?p/?K ?c/?K Pt(T)

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Hedging with Options

• Extentions
• Dividends
• Time-varying volatility - ss(t)
• ss(S)
• rr(t)
• American options
• Non-normal distribution of St
• etc.

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Hedging with Options

• Empirical evidence on the B-S formula
• Whaley(1982) overpricing on high volatility
  stocks, underpricing on low volatility stocks
• Bakshi (1996) finds biases (for short
  maturity/deep out-of-money options)
• Gencay (2000) and Qi and Maddala (1996) improve
  over the B-S model using a more sophisticated
  non-linear model called an Artificial Neural
  Network model

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Empirical evidence on the B-S formula (Bakshi)
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Empirical evidence on the B-S formula (Gencay)
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Hedging with Options

• Currency options (European)

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Hedging with Options

• Example rFX10, r5, s20, K0.6/foreign
  currency, S0.5, T-t0.5 (six months)
• Pt(T1)e-0.05x0.50.97531
• PtFX(T1)e-0.10x0.50.951229
• calculate d -1.395
• calculate c(0.5 0.6 0.5 0)0.0024