WTW792 Pricing Convertibles Project Latex Presentation.
19
The T-F Model The Binomial Model Binomial Tree Implementation Price Sensitivities Analysis and Conclusion Valuing Convertibles Within The TF Model: The Binomial Approach Sandile P. Masilela University of Pretoria [email protected] November 26, 2014 Sandile P. Masilela
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Transcript of WTW792 Pricing Convertibles Project Latex Presentation.
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
Valuing Convertibles Within The TF Model:The Binomial Approach
Sandile P. Masilela
University of Pretoria
November 26, 2014
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
Overview
1 The T-F Model
2 The Binomial Model
3 Binomial Tree Implementation
4 Price Sensitivities
5 Analysis and Conclusion
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
Falls under the equity value approach of valuing convertiblebonds proposed by McConnell and Schwartz.
convertible is split into two components; cash(debt) andequity component which are discounted at different rates.
the value of the convertible bond v is the sum of the twocomponents.
leads to coupled Black-Scholes equations which can be solvedgiven final and boundary conditions.
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
∂v
∂t+ rS
∂v
∂S+
1
2σ2S2 ∂
2v
∂S2− r(v − B) − (r + rc)B + f (t) = 0 (1)
∂B
∂t+ rS
∂B
∂S+
1
2σ2S2∂
2B
∂S2− (r + rc)B + f (t) = 0 (2)
where;v is price of the convertible bondB is the price of the cash component of the convertibler is the risk free rate, rc credit spreadf (t) is the fixed coupon rate
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
Assumptions
The stock is assumed to follow a geometric Brownian motion
dS = µSdt + σSdZ
The mean and variance of the stock price from one time stepto the next
0 < d < erδt < u and 0 < p < 1
The tree recombines i.e. ud = 1
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
one step binomial tree
Sf
dSfd
uSfu
(1 − p)
p
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
An option is replicated with a portfolio of stock and a bond
f = ∆S + B (3)
The price of this portfolio is equivalent to the price of the optionto avoid arbitrage.
∆uS + Berδt = fu
∆dS + Berδt = fd
solving the above simultaneously for ∆ and B and substituting to3 gives;
f0 = [pfu + (1 − p)fd ]e−rδt (4)
the present value of a derivative on stock (the underlying) in therisk neutral world with a payoff f at maturity
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
An option is replicated with a portfolio of stock and a bond
f = ∆S + B (3)
The price of this portfolio is equivalent to the price of the optionto avoid arbitrage.
∆uS + Berδt = fu
∆dS + Berδt = fd
solving the above simultaneously for ∆ and B and substituting to3 gives;
f0 = [pfu + (1 − p)fd ]e−rδt (4)
the present value of a derivative on stock (the underlying) in therisk neutral world with a payoff f at maturity
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
matching expected value and variance to the model takingassumptions into account give:
p =erδt−d
u − d
u = e+σ√δt
d = e−σ√δt
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
the price of the convertible bond at the final node
vT = max[Conversion value(κST ),Nominal] + Last Coupon
the convertible bond value at each node
vt = max[min(Qt ,Ct),Pt , κSt ]
where;Qt is the value calculated through the tree.Ct is call pricePt is the put priceκSt is conversion price
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
Impala shopriteStock
volatility 38% 26%Risk free rate 5.77 % 6.8%Dividend rate 1.8% 3%
BondFace value 10 000 10 000Bond yield 7.2% 8.4%conversion ratio 46.5 60coupon rate 5% 6.5%coupon Frequency 2 2call 130% 0Put 0 0No of time steps 100 100
Table: bond input parameters
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
Figure 1
0,00%
50,00%
100,00%
150,00%
200,00%
250,00%
300,00%
0 100 200 300 400 500 600
bond p
rices (
% o
f par
valu
e)
stock price
Convertible bond price phases
Parity
bond price
bond floor
Figure: Impala platinum 5 %, 2018 ZAR bond prices sensitivitySandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
Figure 2
Figure: convertible price sensitivity to credit spread
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
Figure 3
-0,01000000
0,00000000
0,01000000
0,02000000
0,03000000
0,04000000
0,05000000
0,06000000
0,07000000
0% 10% 20% 30% 40% 50% 60%
ve
ga
(accu
mu
late
d)
share price volatility
vega against volatility
Figure: cumulative vega against volatility
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
Figure 4
8800
9000
9200
9400
9600
9800
10000
10200
0 50 100 150 200 250 300
bo
nd
pri
ce Z
AR
observations
Impala 5% ZAR bond
model bond prices ZAR
bond market price ZAR
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
Figure 5
8000,00
9000,00
10000,00
11000,00
12000,00
13000,00
14000,00
0 20 40 60 80 100 120 140 160
bo
nd
pri
ces
ZA
R
observations
Shoprite 6.5% ZAR
model prices
market prices
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
References
Ayache, E., Forsyth,P. and Vetzel K.R.(2003)
Next Generation for Convertible Bonds with Credit Risk
Journal of Derivatives, vol. 11, 2003, pp9–29.
Gushchin, V. (2008)
The pricing of convertible bonds within the Tsiveriotis and Fernandesframework with exogenous credit spread: Empirical analysis
Journal of Derivatives & Hedge Funds , vol. 14, 2008, 50– 64.
Hariparsadi, S. (2009)
Valuation and Calibration of Convertible Bonds
Masters Thesis, University of Pretoria
Hull, C. (2010)
Options, futures and Other Derivatives, 8th edition,
Prentice Hall,Upeersaddle River NJ,2010
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
References
Milanov, K.and Kounchev, O. (2011)
Critical analysis of the Binomial Tree Approach in Tsiveriotis-Fernandesmodel .
Zardikov, A. (2010)
Methods of Pricing Convertible Bonds
Masters Thesis, University of Cape town
Sandile P. Masilela
The T-F ModelThe Binomial Model
Binomial Tree ImplementationPrice Sensitivities
Analysis and Conclusion
The End
Sandile P. Masilela
