Exotic interest-rate options
Marco Marchioro www.marchioro.org
November 10th, 2012
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 1 Lecture Summary
• Exotic caps, floors, and swaptions • Swaps with exotic floating-rate legs • No-arbitrage methods (commodity example) • Numeraires and pricing formulas • stochastic differential equations (SDEs) • Partial differential equations (PDEs) • Feynman-Kac formula • Numerical methods: analytical approximations
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 2
Common exotic interest-rate options
There are all sorts of exotic interest-rate options.
The least exotic, i.e. commonly traded, types are
• Caps and floors with a digital payoff
• Caps and floors with barriers
• Bermuda and American Swaptions
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 3
Caps and floors with a digital payoff
A vanilla Cap has caplets with payoff
Payoff CapletV = NT (6m, 9m) max [(3m-Libor − K), 0] (1)
A digital Cap has caplets with payoff NT (6m, 9m) R for 3m-Libor ≥ K Payoff CapletD = (2) 0 for 3m-Libor < K
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 4
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 5
Caps and floors with barriers
A barrier Cap has caplets with payoff
Payoff CapletV = NT (6m, 9m) max [(3m-Libor − K), 0] (3)
• Knock-in: paid if 3m-Libor touches a barrier rknock-in
• Knock-out: paid if 3m-Libor does not reach rknock-out
Discrete barriers are checked at certain given barrier dates Continuous barriers are checked on each daily Libor fixing Digital payoffs are also available. A rebate is paid upon knock out.
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 6
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 7
Bermuda and American Swaptions
Very similar to standard Swaptions (known as European swaptions) can be exercised at other dates
• Bermuda swaptions: exercised dates are discrete (typically with the same tenor as one of the legs)
• American Swaption can be exercised at any time
Both are found in two different types of payoffs: co-terminal swap and constant-maturity swap
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 8
Questions?
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 9
More exotic interest-rate options
• Caps and floors on Constant-Maturity-Swaps rates (CMS)
• Look-back options
• TARN (Target Accumulator Redemption Note) swap legs
• Spread options
Usually these options are gift-wrapped within a swaps paying a fixed or a Libor rate
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 10
Exotic swap legs
• Constant-maturity swaps (CMS)
• Spread options
• Ratchet swap
• Range accruals
• Look-back options
• TARN (Target Accumulator Redemption Note) swap legs
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 11
Constant-maturity swaps (CMS)
A swap with a standard fixed-rate leg and a floating-rate leg that pays at times Ti, with i = 1, . . . , n cash flows given by
Ci = N τi−1 CMSM (Ti−1) (4) where
• τi is the year fraction between dates Ti and Ti+1
• CMSM (Ti) is the fair rate, observed at time Ti, of a vanilla swap (fixed leg against Libor leg) with maturity date Ti + M
Constant-maturity swaps usually have Caps or Floors on the swap rate Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 12
Swaps with spread options
A swap with a standard fixed-rate leg and a floating-rate leg that pays a cash flow given by the difference of two swap rates.
For example the floating leg fixing at time Ti−1, and paying at time Ti, payments are given by Ci = N τi−1 max rfloor, CMS10Y (Ti−1) − CMS2Y (Ti−1)
• We are betting on the swap curve to steepen
• Usually a multi-factor model is needed to price this instrument
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 13
Swap with ratchet options
Also known as cliquet option. A swap with a standard fixed-rate leg and a floating-rate leg that pays at times T2,...,Tn Caplet-like coupons, Ci = N τi−1 max 0,Li−1 − Ki−1 (5)
where Li is the Libor rate observed at time Ti, and Ki is the strike satisfying Ki−1 = max Ki−2,Li−2 for i = 3,... (6)
Note: coupon rates are not decreasing. Usually K1 = F wd12 Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 14
Swaps with range accruals
A swap with a floating-rate leg paying coupons
Ci = N τi−1 hri (7) where the average rate hri is given by 1 hri = X f[L(t )] (8) K k k∈[Ti−1,Ti) K business dates between Ti−1 and T1, f being the range function of Libor rate r for L ≤ L ≤ Lmax in min f(L) = (9) rout otherwise Lmin minimum Libor rate and Lmax maximum Libor rate Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 15
Look-back options
A swap with a floating-rate leg paying coupons on the maximum Libor fixing over a certain past period
Ci = N τi−1 rmax(Ti−1,Ti) (10) where
rmax = max [L(t )] (11) k∈[Ti−2,Ti−1) k
• We always receive the best Libor fixing in a range
• Sometimes look-back periods extend further back than one coupon
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 16
TARN (Target Accumulator Redemption Note) swap legs
A swap with a floating-rate leg paying coupons up to a maximum accumulated rate rmax
Ci = N τi−1 ci (12) where
c1 = min [rmax,L0] c2 = min [rmax − c1,L1] c3 = min [rmax − c1 − c2,L2] ... Coupon rates are subtracted to the maximum rate until a zero rate is reached Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 17
Questions?
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 18
Crash course on commodity contracts
(Example of no-arbitrage methods)
• Introduction to commodities
• The forward-spot and the forward-forward relationship
• Convenience yields
• Example of commodity derivatives: The forward contract and the futures spread
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 19
Exposure to commodity prices
If you are managing an hedge fund and want some exposure on the price of live cattle what can you do?
A. Become a cowboy overnight and buy some live cattle
B. Enter in a derivative contract that gives you an exposure to the live-cattle price (e.g. a futures contract)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 20
Commodity futures contracts
• Commodity futures are among the oldest financial instrument traded on any trading floor
• Futures on the cotton price, for example, have traded on the market for longer than a century
• Refer to specialized literature for more details
• In this talk we describe how to generate simulated spot prices and convenience-yield curves (to be used in the computation of risk figures)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 21
Copper, Corn, and WTI Oil
We consider three samples commodities
• Copper (a.k.a Dr. Copper): non-ferrous industrial metal quoted, in $ per ton, on the London Metal Exchange (LME)
• Corn: an agriculture commodity quoted, in $-cents per bushel (27,216 kg), on the Chicago Board of Trade (CBoT)
• West-Texas-Intermediate Oil: an energy commodity quoted, in $ per barrel (158.987 liters), on the New York Mercantile Exchange (NYMEX)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 22
Commodity futures contracts
A contract that allows the delivery of a commodity at a certain future date (however settled daily according to the close price).
E.g. on 2011-11-01 we observed the following market quotes
Commodity Maturity Date Price copper 2012G 2012-02-24 7993.00 copper 2012H 2012-03-27 7995.00 copper 2012J 2012-04-25 7995.50
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 23
Commodity forward contracts
A commodity forward contract is struck between two parties that agree to buy/sell a given commodity at a future date at a pre- determined price (the strike price).
Settlement could be physical or cash. In case of cash settlement the payoff P at the maturity date T (for the long side) is given by the difference between the value of the underlying commodity spot price S(T ) and a strike price K.
For W lots we have, h i P = W · S(T ) − K (13)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 24
Arbitrage-free strategies
An arbitrage-free strategy is ...
A series of physical or financial transactions that starting with a portfolio with a zero value end up with a risk-less portfolio.
The no-arbitrage assumption states that the final portfolio value is zero with 100% probability (otherwise we could buy the cheaper part and sell the dearer one making a risk-free profit with some probability).
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 25
Spot-forward relationship (1/2)
For a storable commodity. Consider a forward contract on a cer- tain storable commodity (no costs no benefits) asset and set up the following strategy:
Today borrow an amount S of currency, exactly enough to buy the spot asset (since T is a short maturity the borrowing can be made at reasonable, i.e. risk-free like, interest rates) and enter into a short forward contract to sell the asset at a future date T at a price f; then store the asset until the date T ; when the date T comes, enforce the contract and sell the asset for a price fT , finally, payback the loan with interests.
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 26
Spot-forward relationship (2/2)
The strategy can be summarized in the following table Date Description Cash flows Asset exch. borrow an amount S of cash +S t = 0 purchase the asset at spot price −S + asset enter into a short forward at T 0 0 < t < T store the asset use the asset to service the forward f - asset t = T T repay debt with interests −S[1 + R(T ) T ]
Because of no arbitrage we have h i fT = S 1 + R(T ) T (14)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 27
Adding costs and benefits
In the case of a commodity, for a dividend-paying stock, for a bond, and for some other asset types, in general there are costs and/or benefits associated in holding the asset,
fT = S[1 + R(T ) T ] + Cos(T ) − Ben(T ) (15)
Assuming costs and benefits to be proportional to the asset price, using simple compounding, h i h i fT = S 1 + RT + S YCos T − YBen T (16)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 28
Convenience yield
We can define the convenience yield as the difference between benefits and costs,
y = YBen − YCos (17) so that h i e−y T f = S 1 + RT − YT = S (18) T e−r T where we used continuous compounding. Using the discount factor we have, S f = e−y T (19) T D(T )
Note: the convenience yield y usually depends on the maturity T . Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 29
Forward-forward relationship
Strategy: at time t = 0 enter into a long forward contract to buy the asset at T1 for f1 units of currency, at the same time enter into a short contract to sell the asset at date T2 for a price f2; at date t = T1 use the long forward contract to buy the asset for a price of f1 financing the purchase by borrowing the money from the market; at a later date T2 sell the commodity for f2
From arbitrage-free assumption we have
D(T1) −y (T −T ) f2 = f1 e 12 2 1 (20) D(T2) where y12 is the forward convenience yield between T1 and T2.
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 30
General spot-forward relationship
The forward-forward relationship between T2 and T3 is
D(T2) −y (T −T ) f3 = f2 e 23 3 2 (21) D(T3)
Chain linking the spot-forward formula and multiple forward-forward relationships we can write
S −y T fj = e j j (22) D(Tj) where
yj Tj = y1 T1 + y12 (T2 − T1) + y23 (T3 − T2) + ... + yij (Tj − Ti)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 31
Forward-futures relationship
The main difference between a futures contract and a forward con- tract is that the former must be settled daily while the latter is settled at the contract maturity. It can be shown that we have,
T T F = f + D(T ) σS σr ρr−S (23)
• σs is the asset-price volatility
• σr is the interest-rate volatility
• ρS−r is the correlation between the asset price and the money- market account.
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 32
Convenience yields from futures quotes
We assume |σS σr ρr−S| 1 so that F T ' f T (24)
Consider n futures contracts with maturities T1,T2,...Tn, having, re- spectively, quotes F1 F1,F2,...Fn at some reference date.
From equation (22) and (24) we have
S −y T Fi = e i i , for i = 1, . . . , n (25) D(Ti) compute the convenience yields yi’s as 1 S ! yi = log (26) Ti D(Ti)Fi Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 33
Yield on a commodity (1/2)
What is exactly the meaning of the convenience yield? Recall that 1 + r T f = S (27) T 1 + y T and suppose we owe a commodity that is worth S today. Assume no credit risk and follow the strategy:
• Today sell the commodity; invest the money in risk-free deposit; enter in a forward contract to buy W = 1 + y T lots at T
• At T recover the money and interests from the depo, and use the money to purchase the commodity at the forward-contract price
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 34
Yield on a commodity (2/2)
In summary, starting with 1 lot of the asset Date Description Cash flows Asset exch. sell the asset at spot price S +S -1 · asset t = 0 invest the money in a deposit -S enter into W long fwd contracts 0 0 < t < T do nothing redeem the deposit +S[1 + r T ] t = T use money to service the contract −S[1 + r T ] +W · asset with W = 1 + y T . The contract payoff at redemption is given by 1 + r T ! W · f = (1 + y T ) · S = S (1 + r T ) T 1 + y T The commodity provided an interest y T !
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 35
Few convenience-yield curves for Copper 3
2.5
2
1.5
1 Copper convenience yield (%)
0.5
0 0 1 2 3 4 5 6 7 8 9 10 11 Maturity T (years)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 36
Recall the zero interest-rate curves 1.6
1.4
1.2
1
0.8
0.6 Zero interest rates (%)
0.4
0.2
0 0 1 2 3 4 5 6 Maturity T (years)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 37
Few convenience-yield curves for Corn 10
8
6
4
2
0 Corn convenience yield (%)
-2
-4
-6 0.5 1 1.5 2 2.5 3 3.5 Maturity T (years)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 38
Few convenience-yield curves for WTI Oil 5
4
3
2
1
WTI Oil convenience yield (%) 0
-1
-2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Maturity T (years)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 39
NPV of a commodity forward contract
Given the convenience yield curve y(T ) define the (continuously- compounded) convenience discount factor Dc(T ) as
Dc(T ) = e−y(T ) T (28) so that the commodity forward price can be written as Dc(T ) f(T ) = S . (29) Dr(T ) The commodity forward contract NPV is then given by h i NPV = W · Ds(T ) · f(T ) − K (30) At inception NPV=0 implies K = f(T ).
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 40
Commodities as currencies
Recall that given two currencies e and $ and their exchange rate Xe$ 1e 1 $ = Xe$ we have the arbitrage-free forward exchange rate
D$(T ) X (T ) = X e$ e$ De(T )
To be compared with the commodity relationship Dc(T ) f(T ) = S Dr(T )
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 41
Questions?
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 42
Generic pricing formula (1/2)
Remember the discount factor relationship with the money account? R T ( )d h i D(T ) = E e− 0 r t t = E e−r1 τ1−...−rn τn (31)
If one payment C is made at a future date T we have, R T ( )d PV = D(T ) C = E e− 0 r t t C (32)
Assume now n deterministic cash flows C1,C2,...,Cn, one each day, i.e. at dates T1,T2,...,Tn
PV = D(T1) C1 + D(T2) C2 + D(T3) C3 + ... (33)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 43
Generic pricing formula (2/2)
For example, in the simple case with only three dates we have
h −r τ −r τ −r τ −r τ −r τ −r τ i PV = E e 1 1 C1 + e 1 1 2 2 C2 + e 1 1 2 2 3 3 C3 −r τ n h −r τ −r τ io = e 1 1 C1 + E e 2 2 C2 + e 3 3 C3 −r τ X h −r τ −r τ i = e 1 1 C1 + P (r2) e 2 2 E C2 + e 3 3 C3|r2 r2 −r τ X −r τ X −r τ = e 1 1 C1 + P (r2) e 2 2 C2 + P (r3|r2)e 3 3 C3 r2 r3 Wouldn’t it be great if we could do the same for random cash flows?
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 44
Pricing in absence of arbitrage
There is a fundamental result from Harrison and Kreps (1979) that holds for stochastic cash flows
Theorem: in absence of arbitrage, the generic pricing formula for an option with a payoff H(T, rt), depending on the interest rates rt, is − R T r dt PV = E e 0 t H(T, rt) (34) where E is the expected-value operator on the risk-neutral measure.
In terms of the money-market account M(t), recall M(0), we have "H(T, r )# PV = M(0) E t (35) M(T )
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 45
Numeraires
It turns out that formula (35) is a special case of a general theorem that holds for generic numeraire assets.
A numeraire is any tradable asset that
• is always positive • does not pay any dividends (nor coupons)
Examples: risk-less money-market account, risk-less zero-coupon bond, the price of a commodity (e.g. the gold price). See Hull book for more examples. Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 46
Equivalent Martingale Measure
Theorem: A continuous economy is arbitrage-free and every security is attain- able if for every choice of numeraire there exists a unique equivalent (martingale) measure. The security PV is then given by "H(T, r )# PV = N(0) EN t (36) N(T )
• A security is attainable if it can be replicated (recall stock options replication with delta-hedging)
• The expectation EN is taken according to the equivalent measure
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 47
Change of Numeraire
How do we compute the expectation in (36)? E.g. using the following theorem.
Given two numeraires M and N and a financial quantity G, " M(T ) N(0)# EM [G] = EN G (37) M(0) N(T )
• This result was first proved by Geman et al. in 1995
• It is incredibly useful in practice
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 48
Numeraire applications: payoff at maturity (1/2)
Consider the numeraire associated to a payoff of 1$ at time T (N(0) = D(T ), N(T ) = 1) and ET the corresponding expectation, defining G as M(0) G(T ) = H(T ) (38) M(T ) the forward payoff corresponding to H, then " M(T ) N(0)# PV = EM [G(T )] = ET G(T ) (39) M(0) N(T ) so that ...
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 49
Numeraire applications: payoff at maturity (2/2)
.. we obtain PV = D(T ) ET [H(T )] (40) Notice how the stochastic rates disappeared.
Other examples (see Hull’s book)
• Choosing N as the period-compounded deposit yields the correct measure for caplets and floorlets
• Choosing the fixed-rate annuity Afixed as numeraire provides the measure used to price swaptions
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 50
Questions?
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 51
Recall the Brownian motion
A continuous stochastic process Wt satisfying,
• W0=0
• Wt − Ws is independent form Ws (for 0 < s < t)
• Wt − Ws is normally distributed, precisely as N(0, t − s) is a standard Brownian motion.
Complex stochastic processes can be built upon the Brownian motion using stochastic differential equations. Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 52
Stochastic differential equations (1/3)
A stochastic differential equation for rt is defined as
drt = u(rt, t)dt + σ(rt, t)dWt (41)
• u is the generic drift (deterministic) • σ is the generic volatility (deterministic)
• Wt is a Brownian motion (stochastic)
Every stochastic differential equation is really a shorthand for the following stochastic (Ito’s) integral equation
Z t Z Wt rt − r0 = u(rs, s)ds + σ(rs, s)dWs (42) 0 0 Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 53
Stochastic differential equations (2/3)
We can integrate the stochastic integral equation over a short time interval [t1, t2]. The drift term becomes
u(r1, t1)(t2 − t1) (43)
Since Wt is a Brownian motion the volatility term becomes √ σ(r1, t1) (W1 − W2) = σ(r1, t1) t2 − t1 ε (44) where ε is a Gaussian random number
ε ∼ N(0, 1) (45)
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 54
Stochastic differential equations (3/3)
In summary the stochastic differential equation for rt
drt = u(rt, t)dt + σ(rt, t)dWt (46) can be interpreted, for a small interval [t, t + ∆t], as a simulation for the future values of rt+∆t given the value rt
√ rt+∆t − rt = u(rt, t)∆t + σ(rt, t) ∆t εj (47) with many samples εj’s taken “Normally” randomly
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 55
Example: Martingales
In the particular case of processes where u = 0,
dXt = σ(Xt, t)dWt (48) we have a Martingale.
Taking an expectation of any martingale yields the current value
E [Xt] = X0 (49)
Choosing Xt = Yt/Nt leads to the numeraire pricing equation " # Yt Y0 = N0 E (50) Nt
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 56
Example: Hull–White SDE
Consider a Stochastic differential equation describing a financial model. For example the Hull-White short-rate process defined by
drt = [θ(t) − a rt] dt + σ dWt (51) Consider an option V so that the underlying short rate satisfies the Hull-White model. Proceeding with no-arbitrage arguments and using Ito’s lemma it can be shown that V satisfies ∂V ∂V σ2 ∂2V + [θ(t) − a r] + − r V = 0 (52) ∂t ∂r 2 ∂r2 which is the analogous of the Black-Scholes equation for the Hull- White model
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 57
Partial differential equations
For every model described by a stochastic differential equation, the corresponding option price V satisfies a partial differential equation ∂V ∂V σ(r, t)2 ∂2V − r V + u(r, t) + = 0 (53) ∂t ∂r 2 ∂r2
• r is the source term • u is the drift (a velocity field) • σ is the volatility (the diffusion coefficients)
Equation (53) is a parabolic partial differential equation, with the ap- propriate boundary and final conditions can be solved to given V (r, T ) Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 58
Feynman-Kac formula
Richard Feynman (physicist) and Mark Kac (mathematician) demon- strated a result that applied to the equation ∂V ∂V σ(r, t)2 ∂2V − r V + u(r, t) + = 0 (54) ∂t ∂r 2 ∂r2 with the final condition, defined by the payoff H
V (T, r) = H(r) (55) Has a solution − R T r(s)ds V (t, r) = E e t H(r) (56)
The Feynman-Kac formula is another derivation of the generic price formula. Most numerical methods use this equation. Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 59
Questions?
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 60
Numerical methods for interest-rate models
Even the most sophisticated interest-rate model is useless if it cannot produce at least a single number
Definition 1: An interest-rate model is a mathematical tool that describes interest rates in the financial markets.
Definition 2: A numerical method is an algorithm that is applied to a model in order to compute numerical values for the financial variables
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 61
Analytical formulas
Analytical formulas are historically the first numerical method used to compute actual numbers from financial models
In few rare and exceptional cases it is possible to find analytical solu- tions to derivatives models. Important examples are
• Merton formula for Black-Scholes equations • Formulas for barrier and digital options priced in Black-Scholes model
• Discount-factor and bond-option formulas for Hull-White model
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 62
Analytical approximations (1/2)
Even the most basic analytical solutions needs numerical methods to be evaluated. For example the cumulative normal distribution
2 1 Z x t d N(x) = √ e− 2 t (57) 2 π −∞ needs to be approximated. A simple approximation is given by
1 −x/2 2 3 N(x) ∼ 1 − √ e a1 k + a2 k + a3 k (58) 2 π with 1 k = (59) 1 + 0.33267 x and a1 = 0.4361836, a2 = −0.1201676, and a3 = 0.937298
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 63
Analytical approximations (2/2)
Most analytical formulas are obtained from the mathematical theory of analytical functions, i.e. functions that can be obtained from a locally-convergent power series
A fairly exhaustive collection of analytical formulas for option pricing has been compiled by the collector ∗ and can be found in
• The Complete Guide to Option Pricing Formulas, Espen Gaarder Haug, Mc Graw Hill (from first edition)
∗See also the interesting picture collection
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 64
Questions?
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro Exotic interest-rate options 65
References
• Options, future, & other derivatives, John C. Hull, Prentice Hall (from fourth edition)
• Efficient methods for valuing interest rate derivatives, Antoon Pelsser, Springer Finance
• Interest rate models: theory and practice, D. Brigo and F. Mer- curio, Springer Finance (from first edition)
• The collector web site: http://www.espenhaug.com
Advanced Derivatives, Interest Rate Models 2010–2012 c Marco Marchioro