:: Leland model (modelling transaction costs) ::

• Leland model (Hayne E. Leland: Option Pricing and Replication with Transactions Costs, 1985)
• Bid and ask price of the stock: S_bid, S_ask - we define
  • S=(S_bid + S_ask)/2
  • c=(S_ask - S_bid)/S
• Transaction costs for one transaction: (c/2)S

• What is the value of c in practice? Compute its value from the following data:
  
  
  
• PDE for the derivatives prices:
  • Portfolio in the Black-Scholes model: 1 option, stocks (determined by delt hedging), continuous trading
  • In the presence of transaction costs: 1 option, stocks (determined by delt hedging), we adjust the portfolio in the intervals of length , number of transactions is
    
  • Change of the portfolio value:
    
  • We have obtained the PDE:
    
• For call and put options:
  • We define Leland number
    
  • Leland number has to be from the interval (0,1), then we obtain the price of a call/put option by using the Black-Scholes formula with adjusted volatility:
    

:: Exercises (1) ::

1. Write a function which computed the value of Leland number (imput parameters are: stock volatility, constant ,c characterizing transaction costs, interval between two adjustments of the portfolio). Check, for selected value of the parameters, whether they are feasible, i.e., whether the Leland number is from the interval (0,1).
   
   
2. Compute, for feasible values of the parameters, prices of options. The parameters are: stock price, strike price, volatility, time to expiration, interest rate, constant c, interval between two adjustments of the portfolio. Write a function which compute the option value for these input parameters.
   
   
3. Choose parameters of the option and the underlying stock, and the interest rate - all the input parameters except for the interval between two adjustments of the portfolio. Plot a graph of Leland function as a function of this interval between two adjustments of the portfolio. What times are feasible? Compute option prices for some of them and compare them with the Black-Scholes price. What it the effect of the interval between two adjustments of the portfolio on the option price?

:: Modelling bid-ask spreads in the Leland model ::

• Price from the previous section can be seen as the bid price.
• If we want to derive the ask price, we condider the portfolio with minus one option. In the same way we obtain:
  
• Call/put option price can be again computed using the Black-Scholes formula, the adjusted volatility is:
  
• this means that adjusted volatilities are:
  
  and the difference between ask and bid price is:
  
  
  

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• There results enable us to estimate model parameters from the data:
  1. We compute implied bid a ask volatilities and from bid aks ask prices of the option:
     
  2. We solve the system of eqution to obtain the parameters (implied volatility), Le:
     
  3. From the bid and ask prices of the stock we compute the constant c.

  4. Finally, we compute the implied time between two adjustments of the portfolio when hedging the option:
     

:: Exercises (2) ::

1. Use the algorithm above to compute the implied parameters using the following data for ACN options (choose some of these options). The data come from the time before opening the stock exchange on 28 November 2014.
   

:: Practice problems ::

1. Implied parameters in Leland model for put option. The algorithm is the same, we only need to compute implied Black-Scholes volatility for a put option.
   
   Do this computation for a selected ACN option with expiration in March:
   
   
   
2. Consider transaction costs according to the Leland model. Suppose that the difference between ask and bid price of a stock equals to half percent of their average value. Volatility of the stock is 0.5 and its price today is 140 USD. Interest rate is 0.5 percent. Consider a call option with exercise price 150 USD and expiration in 1/4 years. Find all feasible time intervals between two changes of the hedging portfolio. Then, select one of them and compute bid and ask prices of the given option.
   
   
3. Consider the difference between bid and ask price of an option as a function of the stock price S. (The remaining parameters - stock volatility, parameter c from the transaction costs, interest rate, strike price, expiration time - are constant).
   • Graphically display this differece for selected sets of parameters.
   • When is this difference maximal? Analytically derive the stock price for which this maximum is attained (for general values of parameters).

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