Computational Finance

Computational Finance Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html Bank Hapoalim

Bonds and Interest Rates Following T. Bjork, ch. 15 Arbitrage Theory in Continuous Time Bank Hapoalim

Bonds and Interest Rates Zero coupon bond = pure discount bond T-bond, denote its price at time t byp(t,T). principal = face value, coupon bond - equidistant payments as a % of the face value, fixed or floating coupons. CF5

Assumptions • There exists a frictionless market for T-bonds for every T > 0 • p(t, t) =1 for every t • for every t the price p(t, T) is differentiable with respect to T. CF5

Interest Rates Let t < S < T, what is IR for [S, T]? • at time t sell one S-bond, get p(t, S) • buy p(t, S)/p(t,T) units of T-bond • cashflow at t is 0 • cashflow at S is -$1 • cashflow at T is p(t, S)/p(t,T) the forward rate can be calculated ... CF5

The simple forward rate LIBOR - L is the solution of: The continuously compounded forward rate R is the solution of: CF5

Definition 15.2 The simple forward rate for [S,T] contracted at t (LIBOR forward rate) is The simple spot rate for [S,T] LIBOR spot rate is (t=S): CF5

Definition 15.2 The continuously compoundedforward rate for [S,T] contracted at t is The continuously compoundedspot rate for [S,T] is (t=S) CF5

Definition 15.2 The instantaneous forward rate with maturity T contracted at t is The instantaneous short rate at time t is CF5

Definition 15.3 The money market account process is Note that here t means some time moment in the future. This means CF5

Lemma 15.4 For t  s  T we have And in particular CF5

Models of Bond Market • Specify the dynamic of short rate • Specify the dynamic of bond prices • Specify the dynamic of forward rates CF5

Important Relations Short rate dynamics dr(t)= a(t)dt + b(t)dW(t) (15.1) Bond Price dynamics (15.2) dp(t,T)=p(t,T)m(t,T)dt+p(t,T)v(t,T)dW(t) Forward rate dynamics df(t,T)= (t,T)dt + (t,T)dW(t) (15.3) W is vector valued CF5

Proposition 15.5 We do NOT assume that there is no arbitrage! If p(t,T) satisfies (15.2), then for the forward rate dynamics CF5

Proposition 15.5 We do NOT assume that there is no arbitrage! If f(t,T) satisfies (15.3), then the short rate dynamics CF5

Proposition 15.5 If f(t,T) satisfies (15.3), then the bond price dynamics CF5

Proof of Proposition 15.5 Left as an exercise … CF5

Fixed Coupon Bonds CF5

Floating Rate Bonds L(Ti-1,Ti) is known at Ti-1 but the coupon is delivered at time Ti. Assume that K =1 and payment dates are equally spaced. Now it is t<T0. By definition of L we have CF5

Floating Rate Bonds implies CF5

This coupon will be paid at Ti. The value of -1 at time t is -p(t, Ti). The value of the first term is p(t, Ti-1). Thus the present value of each coupon is The present value of the principal is p(t,Tn). CF5

The value of a floater is Or after a simplification CF5

T0 T1 Tn-1 Tn Forward Swap Settled in Arrears K - principal, R - swap rate, rates are set at dates T0, T1, … Tn-1 and paid at dates T1, … Tn. CF5

Forward Swap Settled in Arrears If you swap a fixed rate for a floating rate (LIBOR), then at time Ti, you will receive where ci is a coupon of a floater. And at Ti you will pay the amount Net cashflow CF5

Forward Swap Settled in Arrears At t < T0 the value of this payment is The total value of the swap at time t is then CF5

Proposition 15.7 At time t=0, the swap rate is given by CF5

Zero Coupon Yield The continuously compounded zero coupon yield y(t,T) is given by For a fixed t the function y(t,T) is called the zero coupon yield curve. CF5

The Yield to Maturity The yield to maturity of a fixed coupon bond y is given by CF5

Macaulay Duration Definition of duration, assuming t=0. CF5

Macaulay Duration What is the duration of a zero coupon bond? A weighted sum of times to maturities of each coupon. CF5

$ r Meaning of Duration CF5

Proposition 15.12 TS of IR With a term structure of IR (note yi), the duration can be expressed as: CF5

$ r Convexity CF5

FRA Forward Rate Agreement A contract entered at t=0, where the parties (a lender and a borrower) agree to let a certain interest rate R*, act on a prespecified principal, K, over some future time period [S,T]. Assuming continuous compounding we have at time S: -K at time T: KeR*(T-S) Calculate the FRA rate R* which makes PV=0 hint: it is equal to forward rate CF5

Exercise 15.7 Consider a consol bond, i.e. a bond which will forever pay one unit of cash at t=1,2,… Suppose that the market yield is y - flat. Calculate the price of consol. Find its duration. Find an analytical formula for duration. Compute the convexity of the consol. CF5

Change of Numeraire Following T. Bjork, ch. 19 Arbitrage Theory in Continuous Time Bank Hapoalim

Change of Numeraire P - the objective probability measure, Q - the risk-neutral martingale measure, We will introduce a new class of measures such that Q is a member of this class. CF5

Intuitive explanation Assuming that X and r are independent under Q, we get In all realistic cases that X and r are not independent under Q. However there exists a measure T (forward neutral) such that CF5

Risk Neutral Measure Is such a measure Q that for every choice of price process (t) of a traded asset the following quotient is a Q-martingale. Note that we have divided the asset price (t) by a numeraire B(t). CF5

Conjecture 19.1.1 For a given financial market and any asset price process S0(t) there exists a probability measure Q0 such that for any other asset (t)/S0(t) is a Q0-martingale. For example one can take p(t,T) (fixed T) as S0(t) then there exists a probability measure QT such that for any other asset (t)/p(t,T) is a QT-martingale. CF5

Using p(T,T)=1 we get Using a derivative asset as (t,X) we get CF5

Assumption 19.2.1 Denote an observable k+1 dimensional process X=(X1, …, Xk, Xk+1) where Xk+1(t)=r(t) (short term IR) Denote by Q a fixed martingale measure under which the dynamics is: dXi(t)=i(t,X(t))dt + i(t,X(t))dW(t), i=1,…,k+1 A risk free asset (money market account): dB(t)=r(t)B(t)dt CF5

Proposition 19.1 The price process for a given simple claim Y=(X(T)) is given by (t,Y)=F(t,X(t)), where F is defined by CF5

Practical Numeraire Approach Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html Bank Hapoalim

Options with uncertain strike Stock option with strike fixed in foreign currency. How it can be priced? Margarbe 78 or Numeraire approach 1. Price it using this currency as a numeraire. foreign interest rate foreign current price foreign volatility! 2. Translate the resulting price into SHEKELS using the current exchange rate. CF5

Options with uncertain strike Endowment warrants strike is increasing with short term IR. strike is decreasing when a dividend is paid What is an appropriate numeraire? A closed Money Market account. Result – price by standard BS but with 0 dividends and 0 IR. CF5

Options with uncertain strike An option to choose by some date between dollar and CPI indexing (may be with some interest). Margrabe can be used or one can price a simple CPI option in terms of an American investor and then translate it to SHEKELS. CF5

Convertible Bonds A convertible bond typically includes an option to convert it into some amount of ordinary shares. This can be seen as a package of a regular bond and an option to exchange this regular bond to shares of the company. If the company does not have traded debt there is a problem of pricing this option. CF5

Convertible Bonds This is an option to exchange one asset to another and can be priced with Margrabe approach. However in order to use this approach one need to know the correlation between the two assets (stock and regular bond). When there is no market for regular bonds this might be a problem. CF5

Convertible Bonds An alternative approach is with a numeraire. Denote by St stock price at time t, Bt price at time t of a regular bond (may be not observable). CBt price of a convertible bond. C - value of the conversion option, so that CB = C(B) + B at any time CF5

Computational Finance