課程名稱︰數理金融導論 課程性質︰數學系選修 課程教師︰韓傳祥 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰104/1/12 考試時限（分鐘）：9:10 ~ 12:10 試題 : (30%) Multiple Choices 1. Which event is mainly caused by algo trading? (A) 2008 finanacial crisis (B) 2010 flash crash (C) both 2. What kind of sample data is suitable for estimators derived from stochastic differential equations? (A) low frequency (ex, year and season) (B) medium frequency (ex, month and week) (C) high frequency (ex, intraday) 3. The self-financing condition assumes the change of a portfolio is because of (A) market price (B) chain rule (C) optimal selection 4. The perfect replication of a stock option value consists of (A) stock (B) bond (C) both 5. When volatility increases, its value of European call or put will (A) decrease (B) increase (C) both are possible 6. Which probability measure is undertaken by the option evaluation as a conditional expectation of discounted payoff? (A) historical (B) risk-neutral (C) both 7. A futures option can be viewed as an option and its underlying asset pays dividend with (A) growth rate (B) risk-free interest rate (C) volatility 8. The binomial tree model is a discretized version of (A) Brownian motion (B) geometric Brownian motion (C) mean-reverting process 9. What can be used for arbitrage trading (A) box spread (B) put-call parity (C) both 10.Which result is not based on the model-free assumption (A) put-call parity (B) Black-Scholes formula (C) VIX 11.Historical volatility contains market information of the (A) backward (B) instant (C) forward 12.Implied volatility contains market information of the (A) backward (B) instant (C) forward 13.What can be viewed as a market filter (A) trading volume (B) VIX (C) technical analysis 14.A variance swap such as VIX can be replicated by (A) static hedging (B) dynamic hedging (C) both 15.VIX and S&P 500 index often illustrate a correlation of (A) negativeness (B) positiveness (C) none (30%) Fill in blanks 1.According to the definition of historical volatility, annualized variance = ___(1)___ Daily variance. 2.List 6 unrealistic assumptions of the Black-Scholes option pricing theory _________________(2-7)___________________ 3.Given the same market and contract condition, compare options of European, barrier, and American, then order their prices from small to large. ____(8)____ < ____(9)____ < ____(10)____ 4.Given the necessary information flow so that each conditional expectation can be used to define an option price -r(T-t) European option P(t,(11)) = E*{e h(S )|(12)} T -r(T-t) + Barrier option P(t,(13)) = E*{e I( max S < B)(S > K) |(14)} 0≦t≦T t T -r(T-t) + Lookback option P(t,(15)) = E*{e ( max S － S )|(16)} 0≦t≦T t T -r(T-t) + Compound option P(t,(17)) = E*{e (C (T,S ;T ,K )-K) |(18)} BS T 1 1 -r(T-t) 1 T Asian option P(t,(19)) = E*{e h(——∫ S dt)|(20)} T 0 t (40%) Calculation and proof 1.[Implied Density Function] Let C(t,x;T,K) denote a European call option price by -r(T-t) + ∞ -r(T-t) + C(t,x;T,K) = E*[e (S - K) | S = x] = ∫ e (y-K) p(T,y;t,x)dy, T t 0 where p(T,y;t,x) is the transition density function from the initial state (t,x) to the terminal state (T,y). (a) Prove a mathematical result 2 -r(T-t) ∂C(t,x;T,K) p(T,K;t,x) = e ————————. 2 ∂K (b) Give a financial interpretation. 2.[Vege and implied volatility] Under the Black-Scholes model, prove that one ∂P Greek letter Vega: ——— ∂σ -rT e 2 (a) (general payoff function Φ(x)) V = ———E*[Φ(S )(W* - σTW* - T)] σT T T T ∂P (b) (call payoff function) V(Vega) = ——— = x√(T-t) N'(d ) ∂σ 1 (c) Prove that the implied volatility is unique. 3.[Basket Option Pricing] Consider a two-dimensional Black-Scholes system, consisted of two stocks dS = μ S dt + σS dW 1t 1 1t 1 1t 1t ~ dS = μ S dt + σS dW 2t 2 2t 2 2t 2t and one bond dB = rB dt, B = 1. Brownian motions W and W are independent, and t t 0 1t 2t ~ 2 W = ρW + √(1-ρ ) W with correlation coefficient ρ between 0 and 1. 2t 1t 2t (a) Derive the European option pricing PDE by the no arbitrage method. (b) Define the European option pricing function under the risk-neutral probability measure. 4.[Futures Option] Given the stock dynamics under the risk-neutral probability measure P* dS = (r-q)S dt + σS dW*, t t t the futures price F expired at T (≧t≧0) can be defined as t 2 F = E*[S | S ]. t T t 2 (a) Write down the futures pricing PDE of F(t,x) = E*[S | S = x] and verify T t 2 (r-q)(T -t) 2 that its solution is F(t,x) = e x. (b) Now we consider the evaluation problem of a futures option. Assume that the maturity of the option is T (≦T ), then the futures option price can be 1 2 defined by -r(T -t) 1 P (t,x) = E*[e h(F ) | S = x]. S T t 1 Derive the futures option pricing PDE of P (t,x) S (r-q)(T -t) 2 (c) From the closed-form of futures price F = e S , prove that by t t Ito's lemma dF = σF dW*. t t t (d) One can use futures to defined the futures option price, instead of using stock, by -r(T -t) 1 P (t,x') = E*[e h(F ) | F = x']. F T t 1 Derive a closed-form solution for futures option price. (e) Derive the futures option pricing PDE of P (t,x'). F (f) Use a proper change of variable and show that these two pricing PDEs are identical, i.e., P (t,x) = P (t,x'). S F 2 5.(Breeden and Litzenberger) Prove that for a C real-valued function f(y),y∈R, 2 2 df(x) ∞ d f(k) + x d f(k) + f(y) = f(x) + ———(y-x) + ∫ ————(y-k) dk +∫ ————(k-y) dk 2 2 dx x dk 0 dk --

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