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Interest Rate Modelling and Derivative Pricing

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Interest Rate Modelling and Derivative Pricing Interest Rate Modelling and Derivative Pricing Sebastian Schlenkrich HU Berlin, Department of Mathematics WS, 2019/20 Part III Vanilla Option Models p. 140 Outline Vanilla Interest Rate Options SABR Model for Vanilla Options Summary Swaption Pricing p. 141 Outline Vanilla Interest Rate Options SABR Model for Vanilla Options Summary Swaption Pricing p. 142 Outline Vanilla Interest Rate Options Call Rights, Options and Forward Starting Swaps European Swaptions Basic Swaption Pricing Models Implied Volatilities and Market Quotations p. 143 Now we have a first look at the cancellation option Interbank swap deal example Bank A may decide to early terminate deal in 10, 11, 12,.. years. p. 144 We represent cancellation as entering an opposite deal L1 Lm ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ T˜0 T˜k 1 T˜m − ✲ T0 TE Tl 1 Tn − ❄ ❄ ❄ ❄ ❄ ❄ K K [cancelled swap] = [full swap] + [opposite forward starting swap] K ✻ ✻ Tl 1 Tn − ✲ TE T˜k 1 T˜m − ❄ ❄ ❄ ❄ Lm p. 145 Option to cancel is equivalent to option to enter opposite forward starting swap K ✻ ✻ Tl 1 Tn − ✲ TE T˜k 1 T˜m − ❄ ❄ ❄ ❄ Lm ◮ At option exercise time TE present value of remaining (opposite) swap is n m Swap δ V (TE ) = K · τi · P(TE , Ti ) − L (TE , T˜j 1, T˜j 1 + δ) · τ˜j · P(TE , T˜j ) . − − i=l j=k X X future fixed leg future float leg | {z } | {z } ◮ Option to enter represents the right but not the obligation to enter swap. ◮ Rational market participant will exercise if swap present value is positive, i.e. Option Swap V (TE ) = max V (TE ), 0 . p. 146 Option can be priced via derivative pricing formula K ✻ ✻ Tl 1 Tn − ✲ t TE T˜k 1 T˜m − ❄ ❄ ❄ ❄ Lm ◮ Using risk-neutral measure, today’s present value of option is Swap V Option(T ) max V (TE ), 0 V Option(t) = B(t) · EQ E = B(t) · EQ . B(TE ) B(TE ) " # ◮ Calculation requires dynamics of future zero bonds P(TE , T ) and numeraire B(TE ). Option pricing requires specific model for interest rate dynamics. p. 147 Outline Vanilla Interest Rate Options Call Rights, Options and Forward Starting Swaps European Swaptions Basic Swaption Pricing Models Implied Volatilities and Market Quotations p. 148 A European Swaption is an option to enter into a swap Physically Settled European Swaption A physically settled European Swaption is an option with exercise time TE . It gives the option holder the right (but not the obligation) to enter into a ◮ fixed rate payer (or receiver) Vanilla swap with specified ◮ start time T0 and end time Tn (TE ≤ T0 < Tn), ◮ δ F ˜ ˜ ˜ floating rate Libor index payments L (Tj 1, Tj 1, Tj 1 + δ) paid at Tj , and ◮ − − − fixed rate K paid at Ti . All properties are specified at inception of the deal At exercise time TE swaption value is m n + Swpt δ V (TE ) = φ L (TE , T˜j 1, T˜j 1 + δ)˜τj P(TE , T˜j ) − K τi P(TE , Ti ) . − − " j=0 i=0 !# X X Hereφ = ±1 is payer/receiver swaption, [·]+ = max {·, 0}. p. 149 A European Swaption is also an option on a swap rate m n + Swpt δ V (TE ) = φ L (TE , T˜j 1, T˜j 1 + δ)˜τj P(TE , T˜j ) − K τi P(TE , Ti ) − − " j=0 i=0 !# X X n m δ + L (TE , T˜j 1, T˜j 1 + δ)˜τj P(TE , T˜j ) j=0 − − = τi P(TE , Ti ) φ n − K . τi P(TE , Ti ) i=0 P i=0 X P Float leg, annuity and swap rate m δ float leg Fl(TE ) = L (TE , T˜j 1, T˜j 1 + δ)˜τj P(TE , T˜j ) − − j=0 Xn annuity An(TE ) = τi P(TE , Ti ) i=0 Xm δ L (TE , T˜j 1, T˜j 1 + δ)˜τj P(TE , T˜j ) j=0 − − Fl(TE ) swap rate S(TE ) = n = τ P(T , T ) An(T ) P i=0 i E i E Swpt P + V (TE ) = An(TE ) · [φ (S(TE ) − K)] p. 150 Swap rate is the key quantity for Vanilla option pricing ◮ Swap rate S(TE ) always needs to be interpreted in the context of its underlying δ swap with float schedule T˜j , Libor index rates L (·) and fixed schedule j {T } . i i ◮ We omit swap details if underlying swap context is clear. ◮ Fixed rate K is the strike rate of the option. ◮ At-the-money strike K = S(TE ) is the fair fixed rate as seen at TE which prices underlying swap at par (i.e. zero present value). ◮ Float leg can be considered an asset with time-t value (t ≤ TE ) m δ Fl(t) = L (t, T˜j 1, T˜j 1 + δ)˜τj P(t, T˜j ). − − j=0 X ◮ Annuity can be considered a positive asset with time-t value (t ≤ TE ) n An(t) = τi P(t, Ti ). i=0 X p. 151 Libor rates can be seen as one-period swap rates ◮ Consider single period swap rate S(TE ) with m = n = 1 and τ =τ ˜1 = τ1, then δ ˜ ˜ ˜ L (TE , T0, T0 + δ)˜τ1P(t, T1) δ S(TE ) = = L (TE , T˜0, T˜0 + δ). τ1P(t, T1) ◮ δ Option on Libor rate L (TE ) is called Caplet (φ = +1) or Floorlet (φ = −1) with strike K, pay time T1 and payoff δ + τ · φ L (TE , T˜0, T˜0 + δ) − K . ◮ Time-TE price of caplet/floorlet (i.e. optionlet) is Opl δ + V (TE ) = τ · P(TE , T1) · φ L (TE , T˜0, T˜0 + δ) − K . ◮ Optionlet payoff is analogous to swaption payoff. Pricing caplets and floorlets is analogous to pricing swaptions. We focus on swaption pricing. p. 152 Swap rate is a martingale in the annuity measure Definition (Annuity measure) Consider a swap rate S(·) with corresponding underlying swap details. The annuity An(t) (t ≤ TE ) is a numeraire. The annuity measure is the equivalent martingale measure corresponding to An(t). Expectation under the annuity measure is denoted as EA [·]. Theorem (Swap rate martingale property) The swap rate S(t) is a martingale in the annuity measure and for t ≤ T ≤ TE m δ ˜ ˜ ˜ L (t, Tj 1, Tj 1 + δ)˜τj P(t, Tj ) Fl t EA j=0 − − ( ) S(t) = [S(T ) | Ft ] = n = . τ P(t, T ) An(t) P i=0 i i Swap rate S(t) is denoted forward swap rate.P Proof. Annuity measure is well defined via FTAP. The swap rate S(T ) = Fl(T )/An(T ) is a discounted asset. Thus martingale property follows directly from definition of equivalent martingale measure. p. 153 Swaption becomes call/put option in annuity measure Swpt + V (TE )= An(TE ) [φ (S(TE ) K)] . · − Derivative pricing formula yields Swpt Swpt V (t) A V (TE ) A + = E t = E [φ (S(TE ) K)] t . An(t) An(TE ) |F − |F h i ◮ + [φ (S(TE ) K)] is call (φ = +1) or put (φ = 1) option payoff. − − ◮ Requires modelling of terminal distribution of S(TE ). ◮ A Must comply with martingale property, i.e. S(t)= E [S(TE ) t ]. |F p. 154 Put-call-parity for options is an important property We can decompose a forward payoff into a long call and a short put option + + S(TE ) K =[S(TE ) K] [K S(TE )] , − − − − A A + A + E [S(TE ) K t ]= E [S(TE ) K] t E [K S(TE )] t , − |F − |F − − |F h i h i A + A + S(t) K = E [S(TE ) K] t E [K S(TE )] t . − − |F − − |F forward minus strike h undiscounted call i h undiscounted put i | {z } | {z } | {z } Put-call-parity is a general property and not restricted to Swaptions. p. 155 General swap rate dynamics are specified by martingale representation theorem Theorem (Swap rate dynamics) Consider the swap rate S(t) and a Brownian motion W (t) in the annuity measure. There exists a volatility process σ(t, ω) adapted to the filtration t F generated by W (t) such that dS(t) = σ(t, ω)dW (t). Proof. S(t) is a martingale in annuity measure. Thus, statement follows from martingale representation theorem. ◮ Theorem provides a general framework for all swap rate models. ◮ Swap rate models (in annuity measure) only differ in specification of volatility function σ(t, ω). We will discuss several models and their volatility specification. p. 156 Outline Vanilla Interest Rate Options Call Rights, Options and Forward Starting Swaps European Swaptions Basic Swaption Pricing Models Implied Volatilities and Market Quotations p. 157 Normal model is the most basic swap rate model Assume a fixed absolute volatility parameter σ and W (t) a scalar Brownian motion in annuity measure, then dS(t)= σ dW (t). · Swap rate S(T ) for t T becomes ≤ S(T )= S(t)+ σ [W (T ) W (t)] . · − Swap rate is normally distributed with S(T ) N S(t), σ2 (T t) . ∼ − p. 158 Normal model terminal distribution of S(T ) for S(0) = 0.50%, T = 1, σ =0.31% p. 159 Option price in normal model is calculated via Bachelier formula Theorem (Bachelier formula) Suppose S(t) follows the normal model dynamice dS(t) = σ dW (t). · Then the forward Vanilla option price becomes A + E [φ (S(TE ) K)] t = Bachelier S(t), K,σ√T t,φ − | F − with φ [F K] φ [F K] φ [F K] Bachelier (F , K,ν,φ) = ν Φ − − + Φ′ − · ν · ν ν and Φ( ) being the cumulated standard normal distribution function.
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