Summary: Market Risk Measurement & Management | Harper

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• 1 Hull Chapter 19: Volatility smiles

• 1.1 Define volatility smile and volatility skew

• In a plot of the volatility smile, what is each axis and how is the function of the line plotted

  The y-axis is implied volatility (not realized historical volatility!). The x-axis is strike price (K), K/S, K/Forward, or option delta. 

• Identify the axes in a plot of the (i) volatility term structure and (ii) volatility surface.

  (i) Implied volatility versus option term. (ii) Three dimensions: implied volatility (Y axis) versus strike or K/S (X axis) versus term (Z axis)

   

• Which is consistent with the lognormal price distribution (GBM) that underlies classic Black-Scholes Merton: volatility smile or volatility skew?  

  Neither. A horizontal line (Y-axis = implied volatility and X-axis = strike price) is consistent with the lognormal price distribution. A smile (i.e., high implied volatility for OTM puts and calls) implies heavier tails, while a skew (i.e., high implied volatility for ITM call/OTM put and low implied volatility for OTM call/ITM put) implies heavy left tail and light right tail.

• True or false: the implied option volatility smile should be the same for all market participants (traders).  

  False. Implied volatility, by definition, is model-dependent. It varies with the model employed. Only if all traders use the same OPM will they all perceive the same volatility smile.

   

• 1.2 Explain how put-call parity indicates that the implied volatility used to price call options is the same used to price put options

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• Value of a European call according to BSM

  C=S*N(d1)-e^(-rT)*X*N(d2)

   

• Six assumptions of the Black Scholes Merton model  

  1) The stock pays no dividends during the option's life

  2) European exercise terms are used

  3) Markets are efficient

  4) No commissions are charged

  5) Interest rates remain constant and known

  6) Returns are lognormally distributed

   

• Interpretation and formula of d1 (in BSM)

   , however, does not lend itself to a simple probability interpretation.  is correctly interpreted as the present value, using the risk-free interest rate, of the expected asset price at expiration, given that the asset price at expiration is above the exercise price

• 1.3 Compare the shape of the volatility smile (or skew) to the shape of the implied distribution of the underlying asset price and to the pricing of options on the underlying asset

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• What price distribution causes a flat volatility smile?

  lognormal asset price distribution

• What price distribution causes a volatility smile?

  heavy tails (leptokurtosis; kurtosis > 3 or excess kurtosis >0)

• 1.4 Explain why foreign exchange rates are not necessarily lognormally distributed

• When does an asset price have a lognormal distribution? (2 conditions)

  - Constant volatility

  - Smooth price changes without jumps

   Neither is satisfied for exchange rates