Fin 500Q – Quantitative Risk Management Homework #6
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Fin 500Q – Quantitative Risk Management
Homework #6 Solutions
1. (Delta VaR) Consider a put option on a stock in the Black-Scholes model. The exercise price of the option is K = $100 and the maturity is T = 1 year. The stock’s current price is S = $100, the volatility is 19.8%, and the dividend yield is 5%. The risk-free rate is 2%, and the expected (price) return of the stock is 5%. The volatility, the dividend yield, the risk-free rate, and the expected return are all annual values.
(a) What is the current value of the put option?
Answer: First, compute
d1 = 
= _0.05, d2 = d1 _ σ ^T = _0.25.
The value of the put option is
P = Ke −RF T Φ(_d2 ) _ Se −qT Φ(_d1 ) = 9.151.
(As a reminder, Φ(x) can be calculated as NUaM .DⅠST(x, 0, 1, 1) in Excel.)
(b) What is the 5% daily Delta VaR of the long put option position in dollars?
Answer: The delta of the put is ∆P = _e −qT Φ(_d1 ) = _0.495. Hence, we compute the 5% daily Delta VaR of the long put option position as
匝[∆P] = ∆P．S．(µ + q)．∆t + (P _ ∆P．S)．RF．∆t = _0.015 var[∆P] = ∆P(2)．var[∆S] = 0.382
VaR5% = _(匝[∆P] _ 1.645 ^var[∆P]) = 1.032.
(c) Now, assume that we make the simplifying assumption that the expected change in the option value is zero. What is the 5% daily Delta VaR in this case? Explain why the number is bigger or smaller than the result in (b).
Answer: With this simplifying assumption, the VaR is
VaR5% = 1.645 ^var[∆P] = 1.017.
Since a put option provides insurance against low payoffs of the stock, and the stock has a positive risk premium, the expected change in the put value is negative. Hence, the VaR is larger if we appropriately take into account this expected loss in option value. However, since expected returns are small on a daily basis, the difference is modest.
2. (Model Risk) Suppose an investor assumes that the Black-Scholes assumptions hold and he plans to use the BS formula to compute his hedging strategy to replicate the payoff of a put option. The parameter values are as follows:
S = 20, K = 15, σ = 14%, RF = 5%, q = 3%, T = 5.
(a) Find the dollar position in stocks and bonds he must hold in his replicating portfolio using a delta hedging strategy.
Answer: First, compute
d1 = 
= 1.3949, d2 = d1 _ σ ^T = 1.0819.
The value of the put option is
P = Ke −RF T Φ(_d2 ) _ Se −qT Φ(_d1 ) = 0.2282.
The delta of the put is ∆P = _e −qT Φ(_d1 ) = _0.0702. The replicating portfolio has ∆P．S =
_1.40 dollars of stock (short in stocks) and P _ ∆P．S = 1.63 dollars of riskless bonds.
Now suppose the model in fact does not hold and the stock price in the next instant either jumps up 30% or jumps down 30%. Then:
(b) Find the value of his replicating position in each case (price jumps up or jumps down). Answer: If the stock price jumps up 30%, the value of the stock position falls to _1.82, while the bond position remains the same. Therefore, the replicating portfolio is now valued at _0.19. If the stock price jumps down 30%, the stock position rises to _0.98 and the replicating portfolio is worth 0.65.
(c) Find the value of the put option at the new stock price in each case after the jump, assuming that after the jump the assumptions of BS hold.
Answer: Using similar calculations as in part (a), we find that after the price jumps up (S =
1.3．20), the put option is valued at 0.0346. If the price jumps down (S = 0.7．20), the option is
(d) Are the replicating portfolio values the same as the option values? Why?
Answer: The put values are different from the values of the replicating portfolio, that is, the option payoffs have not been replicated. The replicating portfolio does a good job of replicating the payoffs of the option if the delta remains approximately constant following the price change. Because the option value is a convex function of the stock price, in fact the delta does not remain constant. It increases following a price increase and falls following a price decrease. Thus the replicating portfolio makes a delta-hedging error, which is large because the price jumped by a large amount.
(e) Find the new replicating portfolio in each case, assuming that after the jump the BS model assumptions hold.
Answer: After the stock price increases, the delta rises to ∆P = _0.01, and ∆P．S is now
Therefore, the replicator must now sell more stock. This is the paradox with replicating options.
2023-05-03