QF 101 Revision Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 November 12, 2016 Christopher Ting QF 101 Week 13 November 12, 2016 1/49
Table of Contents 1 Mathematics 2 Finance 3 Pricing of Derivatives with Linear Payoffs 4 Pricing of Derivatives with Non-Linear Payoffs Christopher Ting QF 101 Week 13 November 12, 2016 2/49
Mathematics Mathematics Pre-U Math Geometric series Differentiation Integration Vector and matrix UG Math L Hôpital s rule Taylor s expansion Vector differentiation Christopher Ting QF 101 Week 13 November 12, 2016 3/49
Mathematics Underlying Stochastic Process and Itô s Formula Underlying Stochastic Process X t dx t = φ(x t )µ t dt + ϕ(x t )σ t db t. Usually φ(x) = ϕ(x) = 1. Then µ t dt is the deterministic part and σ t db t is the random part. A very important special case is when φ(x) = ϕ(x) = x. Question: What does this special case correspond to? Answer: Simpler (1942) Version of Itô s Formula Let f(x) be a twice differentiable function: Then df(x t ) = df dx dx t + 1 d 2 f ( ) 2 dxt 2 dx 2 Christopher Ting QF 101 Week 13 November 12, 2016 4/49
Mathematics Example 1 Suppose φ(x) = ϕ(x) = 1, µ t = µ and σ t = σ are constants. f(x) = x µ σ Step 1: f (x) = 1 σ, f (x) = 0. Step 2: Apply the simpler version of Itô s formula. df(x t ) = 1 σ dx t + 0 ( ) 2 dxt 2 Step 3: Substitute in the underlying stochastic process. df(x t ) = 1 ( ) µdt + σdbt σ = µ σ dt + db t Step 4: Integrate from time 0 to time t. f(x t ) f(0) = µ σ t + B t. Christopher Ting QF 101 Week 13 November 12, 2016 5/49
Mathematics Example 2 Suppose φ(x) = ϕ(x) = 1, µ t = µ and σ t = σ are constants. f(x) = a 2 x2 with a being a constant. Step 1: f (x) = ax, f (x) = a. Step 2: Apply the simpler version of Itô s formula. df(x t ) = ax t dx t + a ( ) 2 dxt 2 Step 3: Substitute in the underlying stochastic process and apply Itô s table. for ( ) 2 dx t df(x t ) = ax t dx t + a 2 σ2 dt Step 4: Integrate from time 0 to time t. a 2 X2 t = a t 0 X s dx s + a 2 σ2 t = t 0 X t dx t = 1 2 X2 t 1 2 σ2 t Christopher Ting QF 101 Week 13 November 12, 2016 6/49
Mathematics Example 3 Suppose φ(x) = ϕ(x) = 1, µ t = µ and σ t = σ are constants. f(x) = a 3 x3 with a being a constant. Step 1: f (x) = ax 2, f (x) = 2ax. Step 2: Apply the simpler version of Itô s formula. df(x t ) = ax 2 t dx t + ax t ( dxt ) 2 Step 3: Substitute in the underlying stochastic process and apply Itô s table. for ( dx t ) 2 df(xt) = ax 2 t dx t + ax t σ 2 dt Step 4: Integrate from time 0 to time t. (Suppose X 0 = 0) a 3 X3 t = a t 0 t X 2 sdx s + aσ 2 X s ds = 0 t 0 X 2 sdx s = 1 t 3 X3 t σ 2 X s ds 0 Christopher Ting QF 101 Week 13 November 12, 2016 7/49
Mathematics Example 4 Suppose φ(x) = ϕ(x) = X t, X t 0, µ t = µ and σ t = σ are constants. Hence, the SDE for X t is dx t = µx t dt + σx t db t. Step 1: Consider the function f(x) = log(x), so that f (x) = 1 x and f (x) = 1 x 2 Step 2: Apply the simpler version of Itô s formula. d log(x t ) = 1 dx t 1 ( ) 2 X t 2X 2 dxt = µ dt + σ dbt σ2 t 2 dt = (µ 12 ) σ2 dt + σdb t Step 3 Integrate from time 0 to time t, log(x t ) log(x 0 ) = (µ 12 ) σ2 t + σb t = X t = X 0 e ) (µ 12 σ2 t+σb t Christopher Ting QF 101 Week 13 November 12, 2016 8/49
Finance Financial Instruments and P&L Equity, Fixed Income, FX, Commodity Forward and Futures FRA, IRS, CIRS European and American Options P&L (per unit) = selling price buying price = P s P b (1) Christopher Ting QF 101 Week 13 November 12, 2016 9/49
Finance Returns Gross return R := P s P b Simple (rate of) return r := P s P b P b = R 1 Log return l := log ( 1 + r ) = log P s log P b Since asset prices cannot be negative, we have 1 < r <. But < l <. Because log function is concave, it must be that l r. Christopher Ting QF 101 Week 13 November 12, 2016 10/49
Finance Variance as a Difference of Two Returns At daily frequency or higher, the asset return r t is generally very small in magnitude, i.e., r t < 1. Pre-U s Maclaurin series suggests that It follows that r 2 t 2 ( r t l t ). log(1 + r t ) = r t 1 2 r2 t + O(r 3 t ). (2) Since the mean E ( r t ) 0, E ( r 2 t ) V ( rt ), i.e. the variance. Twice the difference between the simple return r t and the log return l t := log(1 + r t ) is the instantaneous variance. σ 2 t := 2 ( r t l t ). Christopher Ting QF 101 Week 13 November 12, 2016 11/49
Finance FX Quoting Convention The bid and ask prices refer to the quoting currency, which is the currency after the "/" in the ISO convention: Base Currency / Quoting Currency Customers buy at the higher ask price from the dealer and sell at the lower bid price to the dealer. Treat the currency as if it is a stock or gold. You can short-sell the base currency as easily as you take a long position. Unit or volume of transaction x mil of base currency y mil of quote currency Christopher Ting QF 101 Week 13 November 12, 2016 12/49
Finance FX Market Practice The market practice has it that USD is always the base currency except Euro: EUR British Pound: GBP Australian Dollar: AUD New Zealand Dollar: NZD Interestingly, dealers trade these currencies by their nicknames: Fiber for EUR, Sterling for GBP, Aussie for AUD, and Kiwi for NZD. The U.S. dollar is nicknamed the Greenback or Buck, Swiss franc the Swissy, Canadian dollar the Loonie, and so on Christopher Ting QF 101 Week 13 November 12, 2016 13/49
Finance Compounding Schemes By default, all interest rates are quoted on the annualized basis. Discrete compounding given the interest rate r d ( FV = PV 1 + r ) d nt. n Money market: When T < 1, the compoudning scheme is 1 + r d T. Two important frequencies Semi-annual: n = 2 Monthly: n = 12 Continuous compounding given the interest rate r c FV = PV e rct Christopher Ting QF 101 Week 13 November 12, 2016 14/49
Finance Effective Annual Rate The annual compounding rate is also known as the simple interest rate. To compare different compounding schemes, a common practice is to entertain the notion of effective annual rate r, which is the interest rate that would be obtained if the forward value were to be calculated under the annual compounding scheme. For example, the rate r of continuous compounding is equivalent to r via the following equation: In other words, r = e r 1. FV = PV e rt = PV ( 1 + r ) T. Christopher Ting QF 101 Week 13 November 12, 2016 15/49
Finance Fixed Income Financial Industry Regulatory Authority s Market Data Quoting convention: Percent of par value expressed in $ One-to-one mapping of yield to maturity and price for a fixed-coupon bond of coupon rate c and T years to maturity: p = c 2 = c 2 2T k=1 2T k=1 1 ( 1 + y ) k + 2 ( 1 1 + z k 2 1 ( 1 + y 2 ) 2T (3) 1 ) k + ( 1 + z ) 2T 2T (4) 2 Christopher Ting QF 101 Week 13 November 12, 2016 16/49
Finance Spot zero Rates and Par Rates The term structure of Treasury s zero rates z k is the yield curve. Price (present value) of a discount bond with face value of 1$, 1 PV k = ( 1 + z ) k k = DF k, (5) 2 is also known as the discount factor. The tenor of this discount bond is k half-years. From spot zero rates, you can compute the par rate c k given p = 1 by 1 = c k k 1 1 ( ) 2 i + ( ) k. i=1 1 + z i 2 1 + z k 2 In this way, you can obtain a term structure of par rates. Christopher Ting QF 101 Week 13 November 12, 2016 17/49
Finance Class Exercise: Sample Question 1 Suppose the term structure of (fictitious) zero rates is given below: k half-years 1 2 3 4 z k 0.7% 1.2% 1.8% 2.0% Compute the term structure of par rates c 1, c 2, c 3 and c 4. Christopher Ting QF 101 Week 13 November 12, 2016 18/49
Finance Forward Interest Rates The spot rate is essentially the geometric average of the forward-forward rates. ( 1 + z k 2 ) ( k = 1 + f ) ( (0,1) 1 + f ) ( (1,2) 1 + f ) (k 1,k) 2 2 2 (6) The implicit relationship between the spot and forward interest rates is ( 1 + z ) k k 2 1 + f (k 1,k) 2 = ( 1 + z k 1 2 ) k 1 = DF k 1 DF k. Christopher Ting QF 101 Week 13 November 12, 2016 19/49
Finance Class Exercise: Sample Question 2 Given the spot rates in Question 1 (Slide 18), construct the correponding discount factors. Based on the discount factors, construct the term structure of forward rates. Christopher Ting QF 101 Week 13 November 12, 2016 20/49
Finance Principles of Quantitative Finance According to the first principle, the Treasury zero rates should be all equal, i.e., the yield curve should be flat, if there is absolutely no risk. Based on the second principle, the long-term and short-term risks render the term structure into a curve with level, slope, and curvature. A parsimonious model of 1- to 10-year yield curve is ( ) Y T = r + β (l) T β (s) 1 e T/τ e T/τ. The third principle provides the mechanism by which the first and second principles are observed in the market. T τ Christopher Ting QF 101 Week 13 November 12, 2016 21/49
Finance Interest Rate Risk and Return The change in interest rate δr is, in percentage terms, r := δr 1 + r In terms of r, the return corresponding to the interest rate risk is (7) R s = D r + 1 2 C( r)2, (8) where the duration D and convexity C are, respectively, D := (1 + r)d m, C := (1 + r) 2 C m. (9) The modified duration D m and modified convexity C m are, respectively, D m := 1 P P r, C m := 1 P 2 P r 2. Christopher Ting QF 101 Week 13 November 12, 2016 22/49
Pricing of Derivatives with Linear Payoffs Linear Payoff Forward Price F 0 Time to maturity T, asset s spot price S 0, and risk-free rate r 0 Forward price F 0 = S 0 (1 + r 0 T ) (10) Payoff at maturity for the buyer: S T F 0 Interest Rate Parity and Forward FX f 0 Risk-free rate for quote currency r q = r 0, risk-free rate for base currency r b, f 0 = S ( 0 1 + r0 T ). (11) 1 + r b T Payoff at maturity for the buyer: S T f 0 Christopher Ting QF 101 Week 13 November 12, 2016 23/49
Pricing of Derivatives with Linear Payoffs Forward Interest Rate y a : risk-free yield of tenor t 1 t 0 y b : risk-free yield of tenor t 2 t 0 g 0 : (implied) forward interest rate Strategy A: y a g 0 t 0 t 1 t 2 Strategy B: t 0 y b t 2 Two Strategies that Give Rise to the Same Forward Value Christopher Ting QF 101 Week 13 November 12, 2016 24/49
Pricing of Derivatives with Linear Payoffs Forward Interest Rate (Cont d) By the first and third principles of QF, Solving for f 0, we obtain (1 + y a ) t 1 t 0 (1 + f 0 ) t 2 t 1 = (1 + y b ) t 2 t 0 (12) ( (1 + yb ) T 2 f 0 = (1 + y a ) T 1 ) 1 T 2 T 1 1. For notational convenience, we have let T 1 := t 1 t 0 and T 2 := t 2 t 0. Christopher Ting QF 101 Week 13 November 12, 2016 25/49
Pricing of Derivatives with Linear Payoffs FRAs of Short-Term Maturities The fair value K is given by the following relationship: (1 + τ 1 r 1 )(1 + τ k K) = 1 + (τ 1 + τ k )r 2, (13) where r 1 is the spot rate with a shorter maturity τ 1. τ k is the FRA maturity r 2 is the spot rate with maturity τ 1 + τ k. It follows from (13) that the FRA rate is given by K = 1 ( ) 1 + (τ1 + τ k )r 2 1. (14) τ k 1 + τ 1 r 1 Christopher Ting QF 101 Week 13 November 12, 2016 26/49
Pricing of Derivatives with Linear Payoffs Payoff of FRA is Linear At time τ 1 when the FRA expires, the LIBOR rate R of tenor τ k is observed. The cash flow to the buyer is then given by ( ) 1 Notional Amount (R K)τ k. 1 + Rτ k The cash flow generated by the interest rate differential is 1 discounted by the discount factor. 1 + Rτ k This is because instead of entering into the physical or actual borrowing over the tenor of τ k starting from τ 1, the anticipated cash flow at τ 1 + τ k, namely, notional Amount (R K)τ k, is settled at τ 1 by discounting it back from τ 1 + τ k to τ 1. Christopher Ting QF 101 Week 13 November 12, 2016 27/49
Pricing of Derivatives with Linear Payoffs Pricing of IRS Swap Rate K The net present value of the IRS at time 0 is n NPV 0 = DF j Floating CF j + DF n 1 j=1 ( n ) DF i Fixed CF i + DF n 1. i=1 In this form, IRS is effectively a long-short strategy on two bonds. The IRS buyer is effectively betting on a position that is long in the floating rate security and short in the fixed rate bond. Christopher Ting QF 101 Week 13 November 12, 2016 28/49
Pricing of Derivatives with Linear Payoffs Pricing of IRS Swap Rate K (Cont d) At time 0, since both bonds are issued at par, by the third law of QF, we must have NPV 0 = 0. Accordingly, we set the floating bond to its par value to obtain 0 = 1 n DF i Fixed CF i DF n 1. i=1 Result: Pricing the IRS swap rate K per period (e.g. semi-annual) K = 1 DF n. (15) n DF i i=1 Christopher Ting QF 101 Week 13 November 12, 2016 29/49
Pricing of Derivatives with Linear Payoffs Overnight Index Swaps (OIS) Overnight indexed swaps are interest rate swaps in which a fixed rate of interest (OIS rate) is exchanged for a floating rate that is the geometric mean of a daily overnight rate. The overnight rates include Federal Funds rate (USD) EONIA (EUR) SONIA (GBP) CHOIS (CHF) TONAR (JPY) There has recently been a shift away from LIBOR-based swaps to OIS indexed swaps due to the scandal. Discounting with OIS is now the standard practice for pricing collateralized deals and is being mandated by clearing houses. Christopher Ting QF 101 Week 13 November 12, 2016 30/49
Pricing of Derivatives with Linear Payoffs NPV Pricing of CIRS Swap Rate K Given the spot FX rate S 0, which is the units of quote currency needed to exchange for one unit of base current, the net present value for the CIRS buyer is n NPV 0 =S 0 DF j Floating CF j + DF n 1 j=1 ( n ) DF i Fixed CF i + DF n 1. i=1 The buyer receives the base currency in exchange for the quote currency at the spot rate S 0. Christopher Ting QF 101 Week 13 November 12, 2016 31/49
Pricing of Derivatives with Linear Payoffs NPV Pricing of CIRS Swap Rate K (Cont d) Again, this is a long-short strategy. The CIRS buyer is long a floating bond denominated in the base currency and short in a fixed rate bond in the quote currency. What is the value of NPV 0 at time 0? Answer: Floating leg s bond is valued at par. ( n ) S 0 1 = S 0 DF i Fixed CF i + DF n 1. i=1 Solving for K, we find that the fixed rate is still given by the same formula: (15)! Christopher Ting QF 101 Week 13 November 12, 2016 32/49
Pricing of Derivatives with Non-Linear Payoffs Options Contract specification: call or put, strike price K, maturity T, exercise style Underlying asset S t, risk-free interest rate r t Option pricing depends on volatility of the underlying, a lot! Money-ness Intrinsic value, time value, early exercise premium (for American option) Christopher Ting QF 101 Week 13 November 12, 2016 33/49
Pricing of Derivatives with Non-Linear Payoffs Put-Call Parity The net cash flow at time T is zero, regardless of the outcomes (either S T < K or S T > K or S T = K). By the first principle of QF, the cash flow at time 0 must also be zero because there is no uncertainty and hence no risk. Why no uncertainty? All the prices and the interest rate are known at time 0! Hence Ke rt + c 0 S 0 p 0 = 0. and this put-call parity is more commonly written as At time t, it is written as c 0 p 0 = S 0 Ke r 0T. c t p t = S t Ke rt(t t) (16) Christopher Ting QF 101 Week 13 November 12, 2016 34/49
Pricing of Derivatives with Non-Linear Payoffs Option Price Curves as Functions of Strike K $ c 0 (K) p 0 (K) K K Christopher Ting QF 101 Week 13 November 12, 2016 35/49
Pricing of Derivatives with Non-Linear Payoffs Monotonicity, Gradient Boundedness, and Convexity K 1 < K 2 < K 3 Monotonicity in the option price level c 0 (K 2 ) c 0 (K 1 ); p 0 (K 1 ) p 0 (K 2 ). (17) Boundedness in the gradient 1 c 0(K 2 ) c 0 (K 1 ) K 2 K 1 0; 0 p 0(K 2 ) p 0 (K 1 ) K 2 K 1 1. Convexity (18) c 0 (K 2 ) c 0 (K 1 ) K 2 K 1 c 0(K 3 ) c 0 (K 2 ) K 3 K 2 ; p 0 (K 2 ) p 0 (K 1 ) K 2 K 1 p 0(K 3 ) p 0 (K 2 ) K 3 K 2. (19) Christopher Ting QF 101 Week 13 November 12, 2016 36/49
Pricing of Derivatives with Non-Linear Payoffs f(s)= f(λ) + f (λ)(s λ) + λ 0 Static Replication f (K)(K S) + dk + λ f (K)(S K) + dk (20) The payoff f(s) contingent on the outcome S at maturity T can be replicated by f(λ): number of risk-free discount bonds, each paying $1 at T f (λ): number of forward contracts with delivery price λ (K S) + : European put option s payoff at T of strike K (S K) + : European call option s payoff at T of strike K f (λ)dk is the number of put options of all strikes K < λ, and call options of all strikes K > λ The payoff replication is static, and model-free of Type 1. Christopher Ting QF 101 Week 13 November 12, 2016 37/49
Pricing of Derivatives with Non-Linear Payoffs Model-Free Approach to VIX ( σmf 2 T = 2er 0T c 0 F 0 K 2 dk + F0 No requirement for an option pricing model = No model risk! 0 ) p 0 K 2 dk. (21) No worry about parameters The only exogenous inputs are risk-free interest rate and dividend yields No bias σ MF reflects volatility across all out-of-the-money strike prices and thus reflects the option skew Uses both put and call options = σ MF is less sensitive to individual option prices. The formula is beautiful! Christopher Ting QF 101 Week 13 November 12, 2016 38/49
Pricing of Derivatives with Non-Linear Payoffs Binomial Tree Model for Option Pricing In addition to the stock price S 0, the most important quantity needed for option is volatility σ. A model for up and down factors is u = e σ t, and d = e σ t. For each t of the binomial tree, the risk-neutral valuation of a pair of future payoffs is c t = e r 0 ( pc + t+1 + (1 p)c t+1) = e r 0 E ( c t+1 ), (22) where the risk-neutral probability of up movement is p = er 0 d u d. (23) Christopher Ting QF 101 Week 13 November 12, 2016 39/49
Pricing of Derivatives with Non-Linear Payoffs A Numerical Example of Binomial Option Pricing Asset prices for all nodes S 0 = Put option s days to maturity = 15 days σ = 73% u = 1.0892 d = 0.9181 $32.68 $35.59 $38.76 $32.68 Since N = 3, each period is 15/3 = 5 days 5 days is t = 5/365 = 1/73 years $30.00 $27.54 $30.00 $25.29 $27.54 risk-free rate r 0 = 0.25% $23.22 Christopher Ting QF 101 Week 13 November 12, 2016 40/49
Pricing of Derivatives with Non-Linear Payoffs Put Option Prices $0.00 $0.00 Strike price = $28 $0.12 $0.00 Upward probability p = 47.89% $0.86 $1.53 $0.24 $0.46 $2.71 $4.78 Christopher Ting QF 101 Week 13 November 12, 2016 41/49
Pricing of Derivatives with Non-Linear Payoffs Two-Period Binomial Tree Algorithm Two-step binomial tree given by the parameters: S 0 = 4 u = 2 d = 1/2 r = 22.31% (artificially made very large to get nice numbers) t = 1 8 16 4 4 2 1 Christopher Ting QF 101 Week 13 November 12, 2016 42/49
Pricing of Derivatives with Non-Linear Payoffs Two-Period Binomial Tree for European Put Compute the risk-neutral probability of upward movement p, and set q := 1 p. To value a European put option struck at K = 5, we evaluate V n = e r t E Q n (V n+1 ) = e r t ( pv + n+1 + qv n+1). The result is V 0 = p 0 = 24 25 24 25 2 5 2 0 1 4 Christopher Ting QF 101 Week 13 November 12, 2016 43/49
Pricing of Derivatives with Non-Linear Payoffs Binomial Tree for American Put At each time step prior to the expiry nodes, the early exercise provision in the American option gives you the choice of either to exercise immediately and receive the intrinsic value of the option, or to hold on to the option to the next step. ( V n = max e r t [ pv n+1 + + qv n+1], (K Sn ) +). Continuing from the earlier example in Slide 42, 34 25 2 5 3 0 1 4 Christopher Ting QF 101 Week 13 November 12, 2016 44/49
Pricing of Derivatives with Non-Linear Payoffs Model-Free Properties of American Options Put-call Inequality S 0 K C 0 P 0 S 0 Ke rt. (24) Irrational to early exercise American calls on stocks that don t pay dividends C t c t > S t K Irrational to early exercise American puts on stocks that don t pay dividends and when C t K ( 1 e r(t t)) Irrational to early exercise a margined put or call option on futures Christopher Ting QF 101 Week 13 November 12, 2016 45/49
Pricing of Derivatives with Non-Linear Payoffs Binomial to Continuous Binomial random walk becomes a Brownian motion as t 0. Einstein and Bachelier s theories lead to a proportional relationship between variance and time. Binomial tree pricing model becomes the Black-Scholes pricing formula as the number of periods becomes very large. In the original Blakc and Scholes (1973), Itô s calculus is needed to arrive at the Black-Scholes equation. The Black-Scholes model works for European options only Christopher Ting QF 101 Week 13 November 12, 2016 46/49
Pricing of Derivatives with Non-Linear Payoffs Black-Scholes Option Pricing Formulas d 1 and d 2 d1 = log ( ) ( S t K + r + 1 2 σ2) τ σ τ, d 2 = log ( ) ( S t K + r 1 2 σ2) τ σ τ Standard normal cumulative distribution function: Φ(x) := 1 2π x e v2 2 dv =: P ( X x ) The Black-Scholes pricing formulas for European calls and puts c(t, S t ) = S t Φ(d 1 ) Ke r(t t) Φ(d 2 ) (25) p(t, S t ) = Ke r(t t) Φ( d 2 ) S t Φ( d 1 ) (26), Christopher Ting QF 101 Week 13 November 12, 2016 47/49
Pricing of Derivatives with Non-Linear Payoffs Real World: Implied Volatilities Christopher Ting QF 101 Week 13 November 12, 2016 48/49
Pricing of Derivatives with Non-Linear Payoffs Final Words So, depending on how you look at it, Quantitative Finance can be either practically incorrect, or incorrectly practical. That, in a nutshell, is the deadly ugliness and beauty of Quantitative Finance intertwined in All is Vanity (Ecclesiastes 1:2). Christopher Ting QF 101 Week 13 November 12, 2016 49/49